Grade 7 Mathematics. Support Document for Teachers

Transcription

1 Grade 7 Mathematics Support Documet for Teachers

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3 G r a d e 7 M a t h e m a t i c s Support Documet for Teachers 2012 Maitoba Educatio

4 Maitoba Educatio Cataloguig i Publicatio Data Grade 7 mathematics [electroic resource] : support documet for teachers Icludes bibliographical refereces ISBN: Mathematics Study ad teachig (Secodary). 2. Mathematics Study ad teachig (Secodary) Maitoba. I. Maitoba. Maitoba Educatio Copyright 2012, the Govermet of Maitoba, represeted by the Miister of Educatio. Maitoba Educatio School Programs Divisio Wiipeg, Maitoba, Caada Every effort has bee made to ackowledge origial sources ad to comply with copyright law. If cases are idetified where this has ot bee doe, please otify Maitoba Educatio. Errors or omissios will be corrected i a future editio. Sicere thaks to the authors, artists, ad publishers who allowed their origial material to be used. All images foud i this documet are copyright protected ad should ot be extracted, accessed, or reproduced for ay purpose other tha for their iteded educatioal use i this documet. Ay websites refereced i this documet are subject to chage. Educators are advised to preview ad evaluate websites ad olie resources before recommedig them for studet use. Prit copies of this resource ca be purchased from the Maitoba Text Book Bureau (stock umber 80636). Order olie at < This resource is also available o the Maitoba Educatio website at < Available i alterate formats upo request.

5 C o t e t s List of Blacklie Masters (BLMs) Grade 7 Mathematics Blacklie Masters Grades 5 to 8 Mathematics Blacklie Masters v v viii Ackowledgemets ix Itroductio 1 Overview 2 Coceptual Framework for Kidergarte to Grade 9 Mathematics 5 Assessmet 9 Istructioal Focus 11 Documet Orgaizatio ad Format 11 Number 1 Number 7.N.1 3 Number 7.N.2 19 Number 7.N.3 45 Number 7.N.4 63 Number 7.N.5 85 Number 7N Number 7.N Patters ad Relatios 1 Patters ad Relatios (Patters, ad Variables ad Equatios) 7.PR.1, 7.PR.2, 7.PR.3, 7.PR.4, 7.PR.5, 7.PR.6, 7.PR.7 3 Shape ad Space 1 Shape ad Space (Measuremet) 7.SS.1 3 Shape ad Space (Measuremet) 7.SS.2 23 Shape ad Space (3-D Objects ad 2-D Shapes) 7.SS.3 43 Shape ad Space (Trasformatios) 7.SS.4 69 Shape ad Space (Trasformatios) 7.SS.5 85 Cotets iii

6 Statistics ad Probability 1 Statistics ad Probability (Data Aalysis) 7.SP.1, 7.SP.2 3 Statistics ad Probability (Data Aalysis) 7.SP.3 25 Statistics ad Probability (Chace ad Ucertaity) 7.SP.4, 7.SP.5, 7.SP.6 45 Appedix 1 Models for Computig Decimal Numbers 1 Bibliography 1 iv Grade 7 Mathematics: Support Documet for Teachers

7 List of Blacklie Masters (BLMs) Grade 7 Mathematics Blacklie Masters Number (N) BLM 7.N.1.1: Math Laguage Crossword Puzzle (with Aswer Key) BLM 7.N.1.2: Divisibility Questios BLM 7.N.1.3: Applyig Divisibility Rules BLM 7.N.2.1: Whole ad Decimal Number Cards BLM 7.N.2.2: Operatio Cards BLM 7.N.2.3: Equivalet Percet, Fractio, ad Decimal Cards (with Aswer Key) BLM 7.N.2.4: Order of Operatios ad Skill-Testig Questios BLM 7.N.2.5: Moey Problems BLM 7.N.2.6: Restaurat Bills ad Bikig BLM 7.N.2.7: Sample Scearios 1 BLM 7.N.2.8: Sample Scearios 2 BLM 7.N.2.9: Sample Scearios 3 BLM 7.N.2.10: Decimal Problems BLM 7.N.3.1A: Tic-Tac-Toe Frames BLM 7.N.3.1B: Tic-Tac-Toe Frames (Medium Challege) BLM 7.N.3.1C: Tic-Tac-Toe Frame (Ultimate Challege) BLM 7.N.3.2: Equivalet Fractio Challege BLM 7.N.3.3: It s Betwee: Roudig Decimal Numbers BLM 7.N.3.4: Choose Your Questio BLM 7.N.3.5: Desigig to Percet Specificatios BLM 7.N.3.6: Determiig the Whole, the Part, the Percet, ad What to Fid BLM 7.N.3.7: Fidig the Missig Numbers i the Percet (Scearios) BLM 7.N.3.8: Percet Problems BLM 7.N.4.1: Table for Recordig Fractios ad Their Decimal Equivalets BLM 7.N.5.1: Iterpretig ad Recordig Differet Meaigs of Fractios BLM 7.N.5.2: Improper Fractio ad Mixed Number Cards BLM 7.N.5.3A: Ace Aviatio: Addig Fractios BLM 7.N.5.3B: Ace Aviatio: Subtractig Fractios BLM 7.N.5.4A: Represetig Recogizable Fractios ad Writig Additio Statemets BLM 7.N.5.4B: Represetig Recogizable Fractios ad Writig Subtractio Statemets BLM 7.N.5.5: Addig ad Subtractig Fractios (Scearios) BLM 7.N.5.6: Problems Ivolvig Fractios BLM 7.N.6.1: Cetimetre Number Lie BLM 7.N.6.2: Iteger Football BLM 7.N.7.1: Equivalet Fractios ad Decimals Cotets v

8 BLM 7.N.7.2: Equivalet Fractios, Decimals, ad Percets BLM 7.N.7.3: Comparig Fractio ad Decimal Equivalets BLM 7.N.7.4: Orderig Decimal Numbers BLM 7.N.7.5: Sequetial Fractios ad Their Decimal Equivalets BLM 7.N.7.6: Relatig Numbers to Bechmarks BLM 7.N.7.7: Orderig Numbers ad Verifyig the Order Patters ad Relatios (PR) BLM 7.PR.1: Patters: A Process BLM 7.PR.2: Sample Patters BLM 7.PR.3: Directios for Playig a Relatios Game BLM 7.PR.4: Uderstadig Cocepts i Patters ad Relatios BLM 7.PR.5: Possible Word Patter Cotexts to Match a Relatio BLM 7.PR.6: Formulatig Relatios to Match Word Descriptios of Patters BLM 7.PR.7: Creatig Word Descriptios of Patters ad Matchig Relatios BLM 7.PR.8: Template for Creatig ad Solvig Problems Usig Iformatio from a Graph BLM 7.PR.9: Associatig Clue Words with Operatios ad Expressios (with Aswer Key) BLM 7.PR.10: Solvig Sigle-Variable Oe-Step Equatios BLM 7.PR.11: Writig Expressios ad Solvig Equatios That Match Word Descriptios BLM 7.PR.12A: Represetig Equivalet Expressios o a Balace Scale (Sample) BLM 7.PR.12B: Represetig Equivalet Expressios o a Balace Scale (Template) BLM 7.PR.12C: Represetig Equivalet Expressios o a Balace Scale Usig Variables for Ukows (Sample) BLM 7.PR.12D: Represetig Equivalet Expressios o a Balace Scale Usig Variables for Ukows (Template) BLM 7.PR.12E: Represetig Equivalet Expressios (Template) BLM 7.PR.13: Evaluatig Expressios, Give a Value for the Variable BLM 7.PR.14A: Solvig Liear Equatios: Pictorial ad Symbolic Represetatios BLM 7.PR.14B: Solvig Liear Equatios with Costats: Applyig the Preservatio of Equality BLM 7.PR.14C: Solvig Liear Equatios with Numerical Coefficiets: Applyig the Preservatio of Equality BLM 7.PR.14D: Solvig Liear Equatios with Costats ad Numerical Coefficiets: Applyig the Preservatio of Equality BLM 7.PR.15: Problems to Represet with Liear Equatios ad with Cocrete Materials (with Aswer Key) vi Grade 7 Mathematics: Support Documet for Teachers

9 Shape ad Space (SS) BLM 7.SS.1.1: Assorted Agle Cards BLM 7.SS.1.2: Agle Classificatios, Agle Estimatios ad Measures, ad Perimeter BLM 7.SS.1.3: Cut-outs for Agles of Differet Measures BLM 7.SS.1.4: Hige Templates for Makig Agles BLM 7.SS.1.5: A Table to Compare Measures of Circles BLM 7.SS.2.1: The Area of Rectagles (Assessig Prior Kowledge) BLM 7.SS.2.2: Circles for Estimatig Area BLM 7.SS.3.1: Parallel ad Perpedicular Lies (Assessig Prior Kowledge) BLM 7.SS.3.2: Creatig Perpedicular Lies BLM 7.SS.3.3: Creatig Perpedicular Bisectors BLM 7.SS.4.1: Plottig Poits o a Cartesia Plae (with Aswer Key) BLM 7.SS.4.2: Cartesia Plae Quadrat Cards BLM 7.SS.4.3: Plot This Picture (with Aswer Key) BLM 7.SS.5.1: Comparig Poits BLM 7.SS.5.2: A Coordiate Map BLM 7.SS.5.3: Cartesia Plae Map ad UFO Templates BLM 7.SS.5.4: Explorig Trasformatios: UFO Pilot Traiig BLM 7.SS.5.5: Recordig Trasformatios: Travel Logbook BLM 7.SS.5.6: Creatig a Desig Usig Reflectios BLM 7.SS.5.7: Which Plot Is Correct? Statistics ad Probability (SP) BLM 7.SP.1.1: Fidig the Cetre of a Graph ad Comparig the Values BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy BLM 7.SP.1.3A: Simoe s Spellig Scores (Questios) BLM 7.SP.1.3B: Simoe s Spellig Performace Record BLM 7.SP.1.4: Usig Cetral Tedecy to Choose a Quarterback BLM 7.SP.3.1: Calculatig the Percet of the Total BLM 7.SP.3.2: Percet of a Circle BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs BLM 7.SP.3.4: Comparig Examples of Circle Graphs BLM 7.SP.3.5: Traslatig Percetages i a Circle Graph ito Quatities BLM 7.SP.4.1: Recordig Sheet for Fractio Decimal Percet Equivalets BLM 7.SP.4.2: What Is the Probability? BLM 7.SP.4.3: Experimetal Probability Tally Sheet ad Probability of Outcomes BLM 7.SP.5.1: Which Coditios Affect Probability? BLM 7.SP.5.2: Examples of Two Idepedet Evets BLM 7.SP.6.1: Frequecy Chart for Orgaizig Outcomes for Two Idepedet Evets BLM.7.SP.6.2: Probability Problems Ivolvig Two Idepedet Evets Cotets vii

10 Grades 5 to 8 Mathematics Blacklie Masters BLM 5 8.1: Observatio Form BLM 5 8.2: Cocept Descriptio Sheet #1 BLM 5 8.3: Cocept Descriptio Sheet #2 BLM 5 8.4: How I Worked i My Group BLM 5 8.5: Number Cards BLM 5 8.6: Blak Hudred Squares BLM 5 8.7: Place-Value Chart Whole Numbers BLM 5 8.8: Metal Math Strategies BLM 5 8.9: Cetimetre Grid Paper BLM : Base-Te Grid Paper BLM : Multiplicatio Table BLM : Fractio Bars BLM : Clock Face BLM : Spier BLM : Thousad Grid BLM : Place-Value Mat Decimal Numbers BLM : Number Fa BLM : KWL Chart BLM : Double Number Lie BLM : Algebra Tiles BLM : Isometric Dot Paper BLM : Dot Paper BLM : Uderstadig Words Chart BLM : Number Lie BLM : My Success with Mathematical Processes BLM : Percet Circle viii Grade 7 Mathematics: Support Documet for Teachers

11 A c k o w l e d g e m e t s Maitoba Educatio wishes to thak the members of the Grades 5 to 8 Mathematics Support Documet Developmet Team for their cotributio to this documet. Their dedicatio ad hard work have made this documet possible. Writer Laa Ladry Red River Valley Juior Academy Grades 5 to 8 Mathematics Support Documet Developmet Team Maitoba Educatio School Programs Divisio Staff Holly Forsyth Lida Girlig Chris Harbeck Heidi Holst Steve Hut Ja Jebse Betty Johs Diaa Kiceko Kelly Kuzyk Judy Maryiuk Greg Sawatzky Darlee Willetts Heather Aderso Cosultat (util Jue 2007) Carole Bilyk Project Maager Coordiator Louise Boissoeault Coordiator Lida Girlig Cosultat Ly Harriso Desktop Publisher Heather Kight Wells Project Leader Fort La Bosse School Divisio Louis Riel School Divisio Wiipeg School Divisio Lord Selkirk School Divisio Idepedet School Kelsey School Divisio Uiversity of Maitoba Evergree School Divisio Moutai View School Divisio Lord Selkirk School Divisio Haover School Divisio Evergree School Divisio Developmet Uit Istructio, Curriculum ad Assessmet Brach Developmet Uit Istructio, Curriculum ad Assessmet Brach Documet Productio Services Uit Educatioal Resources Brach Developmet Uit Istructio, Curriculum ad Assessmet Brach Documet Productio Services Uit Educatioal Resources Brach Developmet Uit Istructio, Curriculum ad Assessmet Brach Ackowledgemets ix

12 Maitoba Educatio School Programs Divisio Staff Susa Letkema Publicatios Editor Tim Pohl Desktop Publisher Documet Productio Services Uit Educatioal Resources Brach Documet Productio Services Uit Educatioal Resources Brach x Grade 7 Mathematics: Support Documet for Teachers

13 I t r o d u c t i o Purpose of This Documet Grade 7 Mathematics: Support Documet for Teachers provides various suggestios for istructio, assessmet strategies, ad learig resources that promote the meaigful egagemet of mathematics learers i Grade 7. The documet is iteded to be used by teachers as they work with studets i achievig the learig outcomes ad achievemet idicators idetified i Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes (Maitoba Educatio, Citizeship ad Youth). Backgroud Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes is based o The Commo Curriculum Framework for K 9 Mathematics, which resulted from ogoig collaboratio with the Wester ad Norther Caadia Protocol (WNCP). I its work, WNCP emphasizes commo educatioal goals the ability to collaborate ad achieve commo goals high stadards i educatio plaig a array of educatioal activities removig obstacles to accessibility for idividual learers optimum use of limited educatioal resources The growig effects of techology ad the eed for techology-related skills have become more apparet i the last half cetury. Mathematics ad problem-solvig skills are becomig more valued as we move from a idustrial to a iformatioal society. As a result of this tred, mathematics literacy has become icreasigly importat. Makig coectios betwee mathematical study ad daily life, busiess, idustry, govermet, ad evirometal thikig is imperative. The Kidergarte to Grade 12 mathematics curriculum is desiged to support ad promote the uderstadig that mathematics is a way of learig about our world part of our daily lives both quatitative ad geometric i ature Itroductio 1

14 Overview Beliefs about Studets ad Mathematics Learig The Kidergarte to Grade 7 mathematics curriculum is desiged with the uderstadig that studets have uique iterests, abilities, ad eeds. As a result, it is imperative to make coectios to all studets prior kowledge, experieces, ad backgrouds. Studets are curious, active learers with idividual iterests, abilities, ad eeds. They come to classrooms with uique kowledge, life experieces, ad backgrouds. A key compoet i successfully developig umeracy is makig coectios to these backgrouds ad experieces. Studets lear by attachig meaig to what they do, ad they eed to costruct their ow meaig of mathematics. This meaig is best developed whe learers ecouter mathematical experieces that proceed from the simple to the complex ad from the cocrete to the abstract. The use of maipulatives ad a variety of pedagogical approaches ca address the diversity of learig styles ad developmetal stages of studets, ad ehace the formatio of soud, trasferable mathematical cocepts. At all levels, studets beefit from workig with a variety of materials, tools, ad cotexts whe costructig meaig about ew mathematical ideas. Meaigful studet discussios ca provide essetial liks amog cocrete, pictorial, ad symbolic represetatios of mathematics. The learig eviromet should value ad respect all studets experieces ad ways of thikig, so that learers are comfortable takig itellectual risks, askig questios, ad posig cojectures. Studets eed to explore problem-solvig situatios i order to develop persoal strategies ad become mathematically literate. Learers must realize that it is acceptable to solve problems i differet ways ad that solutios may vary. First Natios, Métis, ad Iuit Perspectives First Natios, Métis, ad Iuit studets i Maitoba come from diverse geographic areas with varied cultural ad liguistic backgrouds. Studets atted schools i a variety of settigs, icludig urba, rural, ad isolated commuities. Teachers eed to recogize ad uderstad the diversity of cultures withi schools ad the diverse experieces of studets. First Natios, Métis, ad Iuit studets ofte have a whole-world view of the eviromet; as a result, may of these studets live ad lear best i a holistic way. This meas that studets look for coectios i learig, ad lear mathematics best whe it is cotextualized ad ot taught as discrete cotet. May First Natios, Métis, ad Iuit studets come from cultural eviromets where learig takes place through active participatio. Traditioally, little emphasis was 2 Grade 7 Mathematics: Support Documet for Teachers

15 placed upo the writte word. Oral commuicatio alog with practical applicatios ad experieces are importat to studet learig ad uderstadig. A variety of teachig ad assessmet strategies are required to build upo the diverse kowledge, cultures, commuicatio styles, skills, attitudes, experieces, ad learig styles of studets. The strategies used must go beyod the icidetal iclusio of topics ad objects uique to a culture or regio, ad strive to achieve higher levels of multicultural educatio (Baks ad Baks). Affective Domai A positive attitude is a importat aspect of the affective domai that has a profoud effect o learig. Eviromets that create a sese of belogig, ecourage risk takig, ad provide opportuities for success help studets develop ad maitai positive attitudes ad self-cofidece. Studets with positive attitudes toward learig mathematics are likely to be motivated ad prepared to lear, participate willigly i classroom learig activities, persist i challegig situatios, ad egage i reflective practices. Teachers, studets, ad parets* eed to recogize the relatioship betwee the affective ad cogitive domais, ad attempt to urture those aspects of the affective domai that cotribute to positive attitudes. To experiece success, studets must be taught to set achievable goals ad assess themselves as they work toward reachig these goals. Strivig toward success ad becomig autoomous ad resposible learers are ogoig, reflective processes that ivolve revisitig the settig ad assessmet of persoal goals. Middle Years Educatio Middle Years educatio is defied as the educatio provided for youg adolescets i Grades 5, 6, 7, ad 8. Middle Years learers are i a period of rapid physical, emotioal, social, moral, ad cogitive developmet. Socializatio is very importat to Middle Years studets, ad collaborative learig, positive role models, approval of sigificat adults i their lives, ad a sese of commuity ad belogig greatly ehace adolescets egagemet i learig ad commitmet to school. It is importat to provide studets with a egagig ad social eviromet withi which to explore mathematics ad to costruct meaig. Adolescece is a time of rapid brai developmet whe cocrete thikig progresses to abstract thikig. Although higher-order thikig ad problem-solvig abilities develop durig the Middle Years, cocrete, exploratory, ad experietial learig is most egagig to adolescets. * I this documet, the term parets refers to both parets ad guardias ad is used with the recogitio that i some cases oly oe paret may be ivolved i a child s educatio. Itroductio 3

16 Middle Years studets seek to establish their idepedece ad are most egaged whe their learig experieces provide them with a voice ad choice. Persoal goal settig, co-costructio of assessmet criteria, ad participatio i assessmet, evaluatio, ad reportig help adolescets take owership of their learig. Clear, descriptive, ad timely feedback ca provide importat iformatio to the mathematics studet. Askig ope-eded questios, acceptig multiple solutios, ad havig studets develop persoal strategies will help studets to develop their mathematical idepedece. Adolescets who see the coectios betwee themselves ad their learig, ad betwee the learig iside the classroom ad life outside the classroom, are more motivated ad egaged i their learig tha those who do ot observe these coectios. Adolescets thrive o challeges i their learig, but their sesitivity at this age makes them proe to discouragemet if the challeges seem uattaiable. Differetiated istructio allows teachers to tailor learig challeges to adolescets idividual eeds, stregths, ad iterests. It is importat to focus istructio o where studets are ad to see every cotributio as valuable. The eergy, ethusiasm, ad ufoldig potetial of youg adolescets provide both challeges ad rewards to educators. Those educators who have a sese of humour ad who see the woderful potetial ad possibilities of each youg adolescet will fid teachig i the Middle Years excitig ad fulfillig. Mathematics Educatio Goals for Studets The mai goals of mathematics educatio are to prepare studets to use mathematics cofidetly to solve problems commuicate ad reaso mathematically appreciate ad value mathematics make coectios betwee mathematics ad its applicatios commit themselves to lifelog learig become mathematically literate adults, usig mathematics to cotribute to society Studets who have met these goals will gai uderstadig ad appreciatio of the cotributios of mathematics as a sciece, a philosophy, ad a art exhibit a positive attitude toward mathematics egage ad persevere i mathematical tasks ad projects cotribute to mathematical discussios take risks i performig mathematical tasks exhibit curiosity 4 Grade 7 Mathematics: Support Documet for Teachers

17 Coceptual Framework for Kidergarte to Grade 9 Mathematics C ONCEPTUAL F RAME WORK F O R K 9 M A THEM A TIC S The chart below provides a overview of how mathematical processes ad the ature of mathematics The chart below provides ifluece a overview learig of how outcomes. mathematical processes ad the ature of mathematics ifluece learig outcomes. STRAND GRADE K NATURE OF MATHEMATICS CHANGE, CONSTANCY, NUMBER SENSE, PATTERNS, RELATIONSHIPS, SPATIAL SENSE, UNCERTAINTY Number Patters ad Relatios Patters Variables ad Equatios Shape ad Space Measuremet 3-D Objects ad 2-D Shapes Trasformatios Statistics ad Probability Data Aalysis Chace ad Ucertaity MATHEMATICAL PROCESSES: COMMUNICATION, CONNECTIONS, MENTAL MATHEMATICS AND ESTIMATION, PROBLEM SOLVING, REASONING, TECHNOLOGY, VISUALIZATION Mathematical Processes C o cep t u a l Fr a m ework f or K 9 M ath e m ati c s 7 There are critical compoets that studets must ecouter i mathematics to achieve the goals of mathematics educatio ad ecourage lifelog learig i mathematics. Studets are expected to commuicate i order to lear ad express their uderstadig coect mathematical ideas to other cocepts i mathematics, to everyday experieces, ad to other disciplies demostrate fluecy with metal mathematics ad estimatio develop ad apply ew mathematical kowledge through problem solvig develop mathematical reasoig select ad use techologies as tools for learig ad solvig problems develop visualizatio skills to assist i processig iformatio, makig coectios, ad solvig problems Itroductio 5

18 The commo curriculum framework icorporates these seve iterrelated mathematical processes, which are iteded to permeate teachig ad learig: Commuicatio [C]: Studets commuicate daily (orally, through diagrams ad pictures, ad by writig) about their mathematics learig. This eables them to reflect, to validate, ad to clarify their thikig. Jourals ad learig logs ca be used as a record of studet iterpretatios of mathematical meaigs ad ideas. Coectios [CN]: Mathematics should be viewed as a itegrated whole, rather tha as the study of separate strads or uits. Coectios must also be made betwee ad amog the differet represetatioal modes cocrete, pictorial, ad symbolic (the symbolic mode cosists of oral ad writte word symbols as well as mathematical symbols). The process of makig coectios, i tur, facilitates learig. Cocepts ad skills should also be coected to everyday situatios ad other curricular areas. Metal Mathematics ad Estimatio [ME]: The skill of estimatio requires a soud kowledge of metal mathematics. Both are ecessary to may everyday experieces, ad studets should be provided with frequet opportuities to practise these skills. Problem Solvig [PS]: Studets are exposed to a wide variety of problems i all areas of mathematics. They explore a variety of methods for solvig ad verifyig problems. I additio, they are challeged to fid multiple solutios for problems ad to create their ow problems. Reasoig [R]: Mathematics reasoig ivolves iformal thikig, cojecturig, ad validatig these help studets uderstad that mathematics makes sese. Studets are ecouraged to justify, i a variety of ways, their solutios, thikig processes, ad hypotheses. I fact, good reasoig is as importat as fidig correct aswers. Techology [T]: The use of calculators is recommeded to ehace problem solvig, to ecourage discovery of umber patters, ad to reiforce coceptual developmet ad umerical relatioships. They do ot, however, replace the developmet of umber cocepts ad skills. Carefully chose computer software ca provide iterestig problem-solvig situatios ad applicatios. Visualizatio [V]: Metal images help studets to develop cocepts ad to uderstad procedures. Studets clarify their uderstadig of mathematical ideas through images ad explaatios. These processes are outlied i detail i Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. 6 Grade 7 Mathematics: Support Documet for Teachers

19 Strads The learig outcomes i the Maitoba curriculum framework are orgaized ito four strads across Kidergarte to Grade 9. Some strads are further subdivided ito substrads. There is oe geeral learig outcome per substrad across Kidergarte to Grade 9. The strads ad substrads, icludig the geeral learig outcome for each, follow. Number Develop umber sese. Patters ad Relatios Patters Use patters to describe the world ad solve problems. Variables ad Equatios Represet algebraic expressios i multiple ways. Shape ad Space Measuremet Use direct ad idirect measure to solve problems. 3-D Objects ad 2-D Shapes Describe the characteristics of 3-D objects ad 2-D shapes, ad aalyze the relatioships amog them. Trasformatios Describe ad aalyze positio ad motio of objects ad shapes. Statistics ad Probability Data Aalysis Collect, display, ad aalyze data to solve problems. Chace ad Ucertaity Use experimetal or theoretical probabilities to represet ad solve problems ivolvig ucertaity. Itroductio 7

20 Learig Outcomes ad Achievemet Idicators The Maitoba curriculum framework is stated i terms of geeral learig outcomes, specific learig outcomes, ad achievemet idicators: Geeral learig outcomes are overarchig statemets about what studets are expected to lear i each strad/substrad. The geeral learig outcome for each strad/substrad is the same throughout the grades from Kidergarte to Grade 9. Specific learig outcomes are statemets that idetify the specific skills, uderstadig, ad kowledge studets are required to attai by the ed of a give grade. Achievemet idicators are oe example of a represetative list of the depth, breadth, ad expectatios for the outcome. Achievemet idicators are pedagogyad cotext-free. I this documet, the word icludig idicates that ay esuig items must be addressed to meet the learig outcome fully. The phrase such as idicates that the esuig items are provided for illustrative purposes or clarificatio, ad are ot requiremets that must be addressed to meet the learig outcome fully. Summary The coceptual framework for Kidergarte to Grade 9 mathematics describes the ature of mathematics, the mathematical processes, ad the mathematical cocepts to be addressed i Kidergarte to Grade 9 mathematics. The compoets are ot meat to stad aloe. Learig activities that take place i the mathematics classroom should stem from a problem-solvig approach, be based o mathematical processes, ad lead studets to a uderstadig of the ature of mathematics through specific kowledge, skills, ad attitudes amog ad betwee strads. Grade 7 Mathematics: Support Documet for Teachers is meat to support teachers to create meaigful learig activities that focus o formative assessmet ad studet egagemet. 8 Grade 7 Mathematics: Support Documet for Teachers

21 Assessmet Authetic assessmet ad feedback are a drivig force for the suggestios for assessmet i this documet. The purposes of the suggested assessmet activities ad strategies are to parallel those foud i Rethikig Classroom Assessmet with Purpose i Mid: Assessmet for Learig, Assessmet as Learig, Assessmet of Learig (Maitoba Educatio, Citizeship ad Youth). These iclude the followig: assessig for, as, ad of learig ehacig studet learig assessig studets effectively, efficietly, ad fairly providig educators with a startig poit for reflectio, deliberatio, discussio, ad learig Assessmet for learig is desiged to give teachers iformatio to modify ad differetiate teachig ad learig activities. It ackowledges that idividual studets lear i idiosycratic ways, but it also recogizes that there are predictable patters ad pathways that may studets follow. It requires careful desig o the part of teachers so that they use the resultig iformatio to determie ot oly what studets kow, but also to gai isights ito how, whe, ad whether studets apply what they kow. Teachers ca also use this iformatio to streamlie ad target istructio ad resources, ad to provide feedback to studets to help them advace their learig. Assessmet as learig is a process of developig ad supportig metacogitio for studets. It focuses o the role of the studet as the critical coector betwee assessmet ad learig. Whe studets are active, egaged, ad critical assessors, they make sese of iformatio, relate it to prior kowledge, ad use it for ew learig. This is the regulatory process i metacogitio. It occurs whe studets moitor their ow learig ad use the feedback from this moitorig to make adjustmets, adaptatios, ad eve major chages i what they uderstad. It requires that teachers help studets develop, practise, ad become comfortable with reflectio, ad with a critical aalysis of their ow learig. Assessmet of learig is summative i ature ad is used to cofirm what studets kow ad ca do, to demostrate whether they have achieved the curriculum learig outcomes, ad, occasioally, to show how they are placed i relatio to others. Teachers cocetrate o esurig that they have used assessmet to provide accurate ad soud statemets of studets proficiecy so that the recipiets of the iformatio ca use the iformatio to make reasoable ad defesible decisios. Itroductio 9

22 Overview of Plaig Assessmet Assessmet for Learig Assessmet as Learig Assessmet of Learig Why Assess? to eable teachers to determie ext steps i advacig studet learig Assess What? each studet s progress ad learig eeds i relatio to the curriculum outcomes to guide ad provide opportuities for each studet to moitor ad critically reflect o his or her learig ad idetify ext steps each studet s thikig about his or her learig, what strategies he or she uses to support or challege that learig, ad the mechaisms he or she uses to adjust ad advace his or her learig to certify or iform parets or others of the studet s proficiecy i relatio to curriculum learig outcomes the extet to which each studet ca apply the key cocepts, kowledge, skills, ad attitudes related to the curriculum outcomes What Methods? a rage of methods i differet modes that make a studet s skills ad uderstadig visible a rage of methods i differet modes that elicit the studet s learig ad metacogitive processes a rage of methods i differet modes that assess both product ad process Esurig Quality accuracy ad cosistecy of observatios ad iterpretatios of studet learig accuracy ad cosistecy of a studet s self-reflectio, self-moitorig, ad self-adjustmet accuracy, cosistecy, ad fairess of judgmets based o high-quality iformatio clear, detailed learig expectatios accurate, detailed otes for descriptive feedback to each studet egagemet of the studet i cosiderig ad challegig his or her thikig the studet records his or her ow learig clear, detailed learig expectatios fair ad accurate summative reportig Usig the Iformatio provide each studet with accurate descriptive feedback to further his or her learig differetiate istructio by cotiually checkig where each studet is i relatio to the curriculum outcomes provide parets or guardias with descriptive feedback about studet learig ad ideas for support provide each studet with accurate, descriptive feedback that will help him or her develop idepedet learig habits have each studet focus o the task ad his or her learig (ot o gettig the right aswer) provide each studet with ideas for adjustig, rethikig, ad articulatig his or her learig provide the coditios for the teacher ad studet to discuss alteratives the studet reports about his or her ow learig idicate each studet s level of learig provide the foudatio for discussios o placemet or promotio report fair, accurate, ad detailed iformatio that ca be used to decide the ext steps i a studet s learig Source: Maitoba Educatio, Citizeship ad Youth. Rethikig Classroom Assessmet with Purpose i Mid: Assessmet for Learig, Assessmet as Learig, Assessmet of Learig. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, 2006, Grade 7 Mathematics: Support Documet for Teachers

23 Istructioal Focus The Maitoba curriculum framework is arraged ito four strads. These strads are ot iteded to be discrete uits of istructio. The itegratio of learig outcomes across strads makes mathematical experieces meaigful. Studets should make the coectio betwee cocepts both withi ad across strads. Cosider the followig whe plaig for istructio: Itegratio of the mathematical processes withi each strad is expected. By decreasig emphasis o rote calculatio, drill, ad practice, ad the size of umbers used i paper-ad-pecil calculatios, teachers make more time available for cocept developmet. Problem solvig, reasoig, ad coectios are vital to icreasig mathematical fluecy, ad must be itegrated throughout the mathematics programmig. There is to be a balace amog metal mathematics ad estimatio, paper-ad-pecil exercises, ad the use of techology, icludig calculators ad computers. Cocepts should be itroduced usig maipulatives ad gradually developed from the cocrete to the pictorial to the symbolic. Documet Orgaizatio ad Format This documet cosists of the followig sectios: Itroductio: The Itroductio provides iformatio o the purpose ad developmet of this documet, discusses characteristics of ad goals for Middle Years learers, ad addresses Aborigial perspectives. It also gives a overview of the followig: Coceptual Framework for Kidergarte to Grade 9 Mathematics: This framework provides a overview of how mathematical processes ad the ature of mathematics ifluece learig outcomes. Assessmet: This sectio provides a overview of plaig for assessmet i mathematics, icludig assessmet for, as, ad of learig. Istructioal Focus: This discussio focuses o the eed to itegrate mathematics learig outcomes ad processes across the four strads to make learig experieces meaigful for studets. Documet Orgaizatio ad Format: This overview outlies the mai sectios of the documet ad explais the various compoets that comprise the various sectios. Q Q Number: This sectio correspods to ad supports the Number strad for Grade 7 from Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. Itroductio 11

24 Patters ad Relatios: This sectio correspods to ad supports the Patters ad Variables ad Equatios substrads of the Patters ad Relatios strad for Grade 7 from Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. Shape ad Space: This sectio correspods to ad supports the Measuremet, 3-D Objects ad 2-D Shapes, ad Trasformatios substrads of the Shape ad Space strad for Grade 7 from Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. Statistics ad Probability: This sectio correspods to ad supports the Data Aalysis ad Chace ad Ucertaity substrads of the Statistics ad Probability strad for Grade 7 from Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. Blacklie Masters (BLMs): Blacklie masters are provided to support studet learig. They are available i Microsoft Word format so that teachers ca alter them to meet studets eeds, as well as i Adobe PDF format. Bibliography: The bibliography lists the sources cosulted ad cited i the developmet of this documet. Guide to Compoets ad Icos Each of the sectios supportig the strads of the Grade 7 Mathematics curriculum icludes the compoets ad icos described below. Edurig Uderstadig(s): These statemets summarize the core idea of the particular learig outcome(s). Each statemet provides a coceptual foudatio for the learig outcome. It ca be used as a pivotal startig poit i itegratig other mathematics learig outcomes or other subject cocepts. The itegratio of cocepts, skills, ad strads remais of utmost importace. Geeral Learig Outcome(s): Geeral learig outcomes (GLOs) are overarchig statemets about what studets are expected to lear i each strad/substrad. The GLO for each strad/substrad is the same throughout Kidergarte to Grade Grade 7 Mathematics: Support Documet for Teachers

25 Specific Learig Outcome(s): Achievemet Idicators: Specific learig outcome (SLO) statemets defie what studets are expected to achieve by the ed of the grade. A code is used to idetify each SLO by grade ad strad, as show i the followig example: 7.N.1 The first umber refers to the grade (Grade 7). The letter(s) refer to the strad (Number). The last umber idicates the SLO umber. [C, CN, ME, PS, R, T, V] Each SLO is followed by a list idicatig the applicable mathematical processes. Achievemet idicators are examples of a represetative list of the depth, breadth, ad expectatios for the learig outcome. The idicators may be used to determie whether studets uderstad the particular learig outcome. These achievemet idicators will be addressed through the learig activities that follow. Prior Kowledge Prior kowledge is idetified to give teachers a referece to what studets may have experieced previously. Related Kowledge Related kowledge is idetified to idicate the coectios amog the Grade 7 Mathematics learig outcomes. Backgroud Iformatio Backgroud iformatio is provided to give teachers kowledge about specific cocepts ad skills related to the particular learig outcome(s). Mathematical Laguage Lists of terms studets will ecouter while achievig particular learig outcomes are provided. These terms ca be placed o mathematics word walls or used i a classroom mathematics dictioary. Kidergarte to Grade 8 Mathematics Glossary: Support Documet for Teachers (Maitoba Educatio, Citizeship ad Youth) provides teachers with a uderstadig of key terms foud i Kidergarte to Grade 7 mathematics. The glossary is available o the Maitoba Educatio website at < Itroductio 13

26 Learig Experieces Suggested istructioal strategies ad assessmet ideas are provided for the specific learig outcomes ad achievemet idicators. I geeral, learig activities ad teachig strategies related to specific learig outcomes are developed idividually, except i cases where it seems more logical to develop two or more learig outcomes together. Suggestios for assessmet iclude iformatio that ca be used to assess studets progress i their uderstadig of a particular learig outcome or learig experiece. Assessig Prior Kowledge Suggestios are provided to assess studets prior kowledge ad to help direct istructio. Observatio Checklist Checklists are provided for observig studets resposes durig lessos. Suggestios for Istructio Achievemet idicators appropriate to particular learig experieces are listed. The istructioal suggestios iclude the followig: Materials/Resources: Outlies the resources required for a learig activity. Orgaizatio: Suggests groupigs (idividual, pairs, small group, ad/or whole class). Procedure: Outlies detailed steps for implemetig suggestios for istructio. Some learig activities make use of BLMs, which are foud i the Blacklie Masters sectio i Microsoft Word ad Adobe PDF formats. Puttig the Pieces Together Puttig the Pieces Together tasks, foud at the ed of the learig outcomes, cosist of a variety of assessmet strategies. They may assess oe or more learig outcomes across oe or more strads ad may make cross-curricular coectios. 14 Grade 7 Mathematics: Support Documet for Teachers

27 G r a d e 7 M a t h e m a t i c s Number

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29 Number (7.N.1) Edurig Uderstadig(s): Number sese ad metal mathematics strategies are used to estimate aswers ad lead to flexible thikig. Geeral Learig Outcome(s): Develop umber sese. Specific Learig Outcome(s): 7.N.1 Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] Achievemet Idicators: Determie if a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad explai why. Sort a set of umbers based upo their divisibility usig orgaizers, such as Ve or Carroll diagrams. Determie the factors of a umber usig the divisibility rules. Explai, usig a example, why umbers caot be divided by 0. Prior Kowledge Studets should be able to select from a repertoire of metal mathematics strategies for additio, subtractio, multiplicatio, ad divisio. Q Q (3.N.10) Determie additio facts ad related subtractio facts (to 18). Q Q Q Q (4.N.4) Explai the properties of 0 ad 1 for multiplicatio, ad the property of 1 for divisio. (4.N.5) Describe ad apply metal mathematics strategies, such as skip-coutig from a kow fact usig doublig or halvig usig doublig ad addig oe more group usig patters i the 9s facts usig repeated doublig to develop recall of basic multiplicatio facts to 9 9 ad related divisio facts. Number 3

30 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q (4.N.6) Demostrate a uderstadig of multiplicatio (2- or 3-digit umerals by 1-digit umerals) to solve problems by usig persoal strategies for multiplicatio with ad without cocrete materials usig arrays to represet multiplicatio coectig cocrete represetatios to symbolic represetatios estimatig products (4.N.7) Demostrate a uderstadig of divisio (1-digit divisor ad up to 2-digit divided) to solve problems by usig persoal strategies for dividig with ad without cocrete materials estimatig quotiets relatig divisio to multiplicatio (4.PR.4) Idetify ad explai mathematical relatioships usig charts ad diagrams to solve problems. (5.N.3) Determie multiplicatio facts (to 81) ad related divisio facts. (5.N.4) Apply metal mathematics strategies for multiplicatio, such as aexig, the addig zeros halvig ad doublig usig the distributive property (5.N.5) Demostrate a uderstadig of multiplicatio (2-digit umerals by 2-digit umerals) to solve problems. (5.N.6) Demostrate a uderstadig of divisio (3-digit umerals by 1-digit umerals) with ad without cocrete materials, ad iterpret remaiders to solve problems. (6.N.1) Demostrate a uderstadig of place value for umbers greater tha oe millio less tha oe-thousadth (6.N.3) Demostrate a uderstadig of factors ad multiples by determiig multiples ad factors of umbers less tha 100 idetifyig prime ad composite umbers solvig problems ivolvig factors or multiples 4 Grade 7 Mathematics: Support Documet for Teachers

31 Backgroud Iformatio Divisibility A divided is cosidered to be divisible by a divisor if it ca be divided by that divisor to make a quotiet that is a whole umber (with o remaiders). Example: 36 is divisible by 4 because it gives 9 sets, with o remaiders. If a divided is divisible by a divisor, that divisor is a factor of the divided. Example: Sice 36 is divisible by 4, 4 is a factor of 36. Sice the divisor is a factor of the divided, the divided is a multiple of the divisor. Example: Sice 4 is a factor of 36, 36 is a multiple of 4. If a umber is divisible by more tha two factors, it is also divisible by the product of ay combiatio of its prime factors.* Example: 36 is divisible by both 2 ad 3. 2 ad 3 are both prime factors, so 36 is also divisible by 6, the product of those two prime factors (2 3 = 6). A clear grasp of divisibility is fudametal to achievig may other learig outcomes. It helps studets to idetify factors ad * Note: I Grade 7, studets are ot formally exposed to prime factorizatio. It is a achievemet idicator i Grade 8 Mathematics i the study of squares ad square roots, ad a learig outcome i Grade 10 Itroductio to Applied ad Pre-Calculus Mathematics. uderstad relatioships betwee umbers. It makes it easier for them to solve problems, sort umbers, work with fractios, uderstad percets ad ratios, ad work with algebraic equatios. Whe studets ca idetify factors with ease, they ca readily idetify prime ad composite umbers, idetify commo factors ad multiples, ad fid both the greatest commo factors ad the least commo multiples. Uderstadig divisibility ehaces studets ability to reame fractios with commo deomiators ad to represet fractios i lowest terms, thereby makig it easier for them to compare fractios ad to perform operatios with fractios. If studets uderstad place value ad have facility i usig metal mathematics strategies ad facts, it will be easier for them to fid patters i multiples of factors, to add the digits of multiples, ad to recogize umbers that are divisible by a particular factor. Proficiecy with these skills will help studets to discover divisibility rules, uderstad ad explai why divisibility rules work, ad use divisibility rules effectively to determie divisibility. Number 5

32 Uderstadig divisibility rules ad the reasos why the rules work icreases studets umber sese ad their uderstadig of our umber system ad patters withi the system. Explorig these relatioships ad developig divisibility rules or explaatios for the rules ca be challegig ad time-cosumig, but will provide studets with rich opportuities to practise the mathematical processes of problem solvig, reasoig, makig coectios, ad commuicatig. Whe selectig learig experieces, verify that studets have the required backgroud kowledge ad skills, clearly outlie the tasks ad expectatios, provide a warm-up learig activity with the simpler factors (e.g., 2, 5, 10), ad provide appropriate hits to guide studet discovery without beig prescriptive. Divisibility Rules Below are some possible divisibility rules for commo factors, alog with explaatios ad examples. Provide studets with opportuities to make their ow discoveries ad to develop their uderstadig through learig experieces, rather tha askig them to memorize the divisibility rules. Divisible by 2 The umber is eve. OR The fial digits are 2, 4, 6, 8, or 0. 3 The sum of the digits is divisible by 3. Cotiually addig the digits util you ed up with a sigle digit will ultimately result i a total of 3, 6, or 9. Divisibility Rules for Commo Factors Rule Explaatio Examples ad No-examples Eve umbers are composed of groups of 2. Therefore, it is ecessary oly to examie the uits (or oes) place whe determiig divisibility by 2. Use place value ad the logic of remaiders. Each hudred ca be divided ito 33 groups of 3 ad leaves 1 uit remaiig. Each te divides ito three groups of 3 ad leaves 1 uit remaiig. The oes are already idividual uits. Add all the remaiig uits (or remaiders). If this sum divides evely by 3, the origial umber is divisible by is divisible by 2 because the digit i the uits place (8) is eve. 89 is ot divisible by 2 because the digit i the uits place (9) is odd. 351 is divisible by 3 because dividig 3 hudreds by 3 leaves 3 uits remaiig, 5 tes leaves 5 uits remaiig, ad 1 uit is the remaider i the uits place. Add up the remaiders: = 9. Sice the remaiders are divisible by 3, the etire umber is divisible by is ot divisible by 3 because dividig 2 hudreds by 3 leaves 2 uits remaiig, 3 tes leaves 3 uits remaiig, ad 8 uits are the uits remaider. Add up the remaiders: = 13; = 4. Sice 4 is ot divisible by 3, the etire umber is ot. (cotiued) 6 Grade 7 Mathematics: Support Documet for Teachers

33 Divisible by 4 The umber formed by the fial two digits is divisible by 4. OR The umber formed by the fial two digits is divisible by 2 twice. 5 The fial digit is 0 or 5. 6 The umber is eve ad divisible by 3. OR The umber has both 2 ad 3 as factors (is divisible by both 2 ad 3). 8 The umber formed by the fial three digits is divisible by 8. OR The umber formed by the fial three digits is divisible by 2 three times. Divisibility Rules for Commo Factors (cotiued) Rule Explaatio Examples ad No-examples Use place value logic. 100 is the smallest place value positio divisible by 4 (100 4 = 25). Ay umber greater tha 100 ca be expressed as x umber of hudreds. Therefore, oly the umber formed by the digits i the tes ad uits places must be examied. Use place value logic. Every group of 10 forms two groups of 5. Therefore, it is ecessary oly to examie the uits (or oes) place whe determiig divisibility by 5. If the umber is divisible by both the prime factors 2 ad 3, it must also be divisible by 6 because two groups of 3 make a group of 6. Use place value logic is the smallest place value positio divisible by 8 ( = 125). Ay umber larger tha 1000 ca be expressed as x umber of thousads. Therefore, oly the umber formed by the digits i the hudreds, tes, ad uits places must be examied. 524 is divisible by 4 because 100 is divisible by 4, ad so is divisible by 4, ad 24 is divisible by is ot divisible by 4. Although is divisible by 4, 90 is ot divisible by 2 twice (ot divisible by 4). 130 is divisible by 5 because the digit i the oes place (0) is 0 or is ot divisible by 5 because the digit i the oes place (9) is ot 0 or is divisible by 6 because it is divisible by both 2 (it is a eve umber) ad 3 (the sum of its digits is divisible by 3). 153 is ot divisible by 6 because it is ot divisible by 2 (it is a odd umber), but it is divisible by 3 (the sum of its digits is divisible by 3) is divisible by 8 because 480 is divisible by 2 three times (480 2 = 240; = 120; = 60) is ot divisible by 8 because 220 is ot divisible by 8. (cotiued) Number 7

34 Divisible by 9 The sum of the digits is divisible by 9. OR The umber is divisible by 3 twice. Divisibility Rules for Commo Factors (cotiued) Rule Explaatio Examples ad No-examples Use place value ad the logic of remaiders. Each hudred ca be divided ito 11 groups of 9 ad leaves 1 uit remaiig. Each te divides ito oe group of 9 ad leaves 1 uit remaiig. The oes are already idividual uits. Add all the remaiig uits (or remaiders). If this sum divides evely by 9, the origial umber is divisible by The fial digit is 0. All writte multiples of 10 ed i 0. The followig are ways to show a umber caot be divided by zero: 351 is divisible by 9 because dividig 3 hudreds by 9 leaves 3 uits remaiig, 5 tes leaves 5 uits remaiig, ad 1 uit is the remaider i the uits place. Add up the remaiders: = 9. Sice the remaiders are divisible by 9, the etire umber is. 418 is ot divisible by 9 because dividig 4 hudreds by 9 leaves 4 uits remaiig, 1 te leaves 1 uit remaiig, ad 8 uits are the remaider i the uits place. Add up the remaiders: = 13. Sice 13 is ot divisible by 9, the etire umber is ot. 130 is divisible by 10 because the digit i the oes place (0) is is ot divisible by 10 because the digit i the oes place (9) is ot 0. Usig a calculator to divide a umber by zero results i a error message. Applyig the actio of divisio results i a impossible situatio. Example: If you have a quatity x, how may groups of zero ca you make? You would be tryig to make groups of zero forever. If you had to share a quatity ito zero groups, you would have o groups to share the quatity with. Both scearios are impossible. Usig the patter ad logic of related facts provides o solutio to dividig by zero. Multiplicatio ad divisio are iverse operatios. Thik of related statemets such as the followig: 4 2 = 8 ad 8 2 = 4 0? = 8 ad 8 0 =? (There is o aswer.) 8 Grade 7 Mathematics: Support Documet for Teachers

35 Mathematical Laguage added Carroll diagram differece divided divisibility divisible divisor factor multiple prime product quotiet sum Ve diagram Learig Experieces Assessig Prior Kowledge Materials: BLM 7.N.1.1: Math Laguage Crossword Puzzle grid paper (optioal) Orgaizatio: Idividual, pairs, whole class Procedure: 1. Distribute copies of BLM 7.N.1.1: Math Laguage Crossword Puzzle. Have studets read the clues ad complete the puzzle. Studets may cosult refereces for assistace i usig mathematical terms. 2. After givig studets sufficiet time to complete the puzzle, have them share their aswers. Discuss resposes as a class. Ecourage studets to ask questios, explai their resposes, or exted the cocepts. Number 9

36 Variatios: Iclude or omit the word bak foud o BLM 7.N.1.1: Math Laguage Crossword Puzzle. Supply studets with mathematical terms ad have them create clues ad puzzles usig olie puzzle geerators or grid paper. Sample Website: Crossword Puzzle Games. Create a Crossword Puzzle < Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Respod correctly to the mathematical terms whe hearig or readig them. r Use the terms appropriately i commets. Suggestios for Istructio Determie if a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad explai why. Sort a set of umbers based upo their divisibility usig orgaizers, such as Ve or Carroll diagrams. Materials: BLM 7.N.1.2: Divisibility Questios math jourals or otebooks BLM : Multiplicatio Table (optioal) BLM : Isometric Dot Paper (optioal) calculators (optioal) For additioal problems, refer to the followig books: Sachar, Louis. More Sideways Arithmetic from Wayside School. New York, NY: Scholastic Ic., Sideways Arithmetic from Wayside School. New York, NY: Scholasatic Ic., Orgaizatio: Idividual, pairs or small groups, whole class 10 Grade 7 Mathematics: Support Documet for Teachers

37 Procedure: 1. Distribute copies of BLM 7.N.1.2: Divisibility Questios for studets to complete. Studets will eed to use metal mathematics strategies, as well as Ve ad Carroll diagrams. 2. Have studets work aloe at first, aswerig as may questios as they ca. 3. The have studets work i pairs or i small groups to see whether they ca aswer more questios together, makig sure each studet is able to explai the solutios. 4. After studets have had sufficiet time to work o the questios, have them gather as a class to share resposes ad reasoig. 5. Have each studet create a questio similar to the questios o the BLM. 6. Have studets select oe problem to solve, ad ask them to explai their solutio i their math jourals or otebooks. Variatio: If ecessary, allow some studets to use multiplicatio charts (e.g., BLM : Multiplicatio Table), array charts (e.g., BLM : Isometric Dot Paper), or calculators. Allow studets to use the costat key o a calculator to geerate multiples. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Use a repertoire of metal mathematics strategies. r Multiply ad divide umbers. r Uderstad place value ad reame umbers (e.g., the umber has 654 hudreds ad 32 uits). r Idetify multiples ad factors of a umber. r Use Ve ad Carroll diagrams to represet relatios betwee umber sets. r Commuicate mathematically. Number 11

38 Suggestios for Istructio Determie if a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad explai why. Sort a set of umbers based upo their divisibility usig orgaizers, such as Ve or Carroll diagrams. Materials: display board calculators Orgaizatio: Whole class Procedure: 1. Challege the class to a teacher-versus-studets cotest to see who ca determie whether a particular umber is divisible by a desigated factor (1 to 10). 2. Name a factor. 3. Have a studet secretly write a umber (two to six digits) o a display board. 4. Desigate two studets to use calculators to verify the correct respose. (Yes, it is divisible, or o, it is ot divisible.) 5. Reveal the umber ad begi the cotest. 6. Whoever replies correctly first, wis. 7. Keep score if you wish (teacher versus studets). Before log, you will dazzle them with your speed, ad pique their curiosity. Stop the game whe it becomes evidet to studets that there is a trick to this. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Sort umbers accordig to their divisibility. r Apply reasoig skills ad kowledge of umber properties ad operatios to develop a persoal meaig for the divisibility of a umber by 2, 3, 4, 5, 6, 8, 9, or 10. r Commuicate mathematically. 12 Grade 7 Mathematics: Support Documet for Teachers

39 Suggestios for Istructio Determie if a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad explai why. Sort a set of umbers based upo their divisibility usig orgaizers, such as Ve or Carroll diagrams. Materials: Maitoba Educatio. Middle Years Activities ad Games. Mathematics. < BLM 5 8.6: Blak Hudred Squares couters pecil crayos or markers math jourals or otebooks calculators Ve diagram or Carroll diagram templates (optioal) Orgaizatio: Whole class, idividual, pairs, small groups Guided Discovery: As some of the exploratios i this three-part learig experiece may ot be ituitive for studets, the followig suggestios are meat to help you prepare for this guided discovery: Studets ca use patters ad relatios to create geeralizatios about umbers that are divisible by the factor, ad the test the uiversality of their predictios. Challege studets to explai why their divisibility rules work. Lead studets to discover simple practical tests that ca be applied to ay umber to determie easily whether the umber is divisible by the factors 2, 3, 4, 5, 6, 8, 9, or 10. List multiples of a factor. Begi with the first 10 multiples of the factor, ad look for patters. If o patters appear, exted the list of multiples. Circle multiples o BLM 5 8.6: Blak Hudred Squares to reveal patters. Look for further patters or relatios that may iclude place values, eve or odd fial digits, similarities i the fial two or fial three digits, ad similarities i the sums of digits ad their commo factors. Number 13

40 Hits: Choose oe colour marker, work with BLM 5 8.6: Blak Hudred Squares, ad circle all the multiples of 10. Examie the multiples. What do you otice? Test your idea o larger umbers. Divide umbers edig i zero by 10 to esure there are o remaiders. The try dividig umbers that do ot ed i zero by 10 to prove there are remaiders. Procedure: The followig procedure will take place over the course of several classes. Part A: Divisibility by 10, 5, ad 2 1. As a class, begi with the followig questio: How ca we kow which umbers are divisible by 10? 2. Discuss studets resposes to the questio. (Usig the Thik-Pair-Share strategy with a parter may help studets prepare for whole-class discussio. Studets thik about the questio idividually, discuss their ideas with a parter, ad the share their resposes with the class.) If studets reply that all the oes digits are 0, ask them how they kow this or how they could prove it to someoe who did ot agree. The test results could be show i a Carroll diagram to illustrate the test s reliability. Example: Divisible by 10 Not Divisible by 10 Numbers with 0 uits All examples will be here Numbers with other tha 0 uits All examples will be here 3. Ask studets why the test works. 4. Followig the discussio, have studets record their fidigs i their math jourals, usig words, pictures, diagrams, or symbols, or create a Divisibility Study Booklet. Divisibility rule for the factor This rule works because Repeat steps 1 to 4 for the followig questio: How ca we kow a umber is divisible by 5? Aim for less teacher guidace ad more studet cotrol of the discussio. Remid studets to write a summative joural etry. 6. Repeat steps 1 to 4 for the followig questio: How ca we kow a umber is divisible by 2? 14 Grade 7 Mathematics: Support Documet for Teachers

41 Agai, aim for less teacher guidace ad more studet cotrol of the discussio. Remid studets to write a summative joural etry. Part B: Divisibility by 4, 8, 3, 9, ad 6 After workig through the examples i Part A, studets will have a geeral idea of how to determie whether a umber is divisible by a specific factor. Further ideas o how to proceed are provided below, i a suggested order. 1. How ca we kow which umbers are divisible by 4? This is a subset of the umbers divisible by 2, so studets ca cotiue their fidigs about factors of 2. Every secod multiple of 2 is a factor of 4. It may be helpful to cosider that 100 is divisible by 4 (as 10 is divisible by 5). 2. Determie whether a umber is divisible by 8. Simply apply ad exted the same strategies is divisible by 8. Show multiples of 2 ad 4 ad 8 i a Ve diagram. 3. Determie whether a umber is divisible by 3. Cosider makig groups of 3 for each place value i each multiple, ad the iclude multiples with three or more digits. Repeat the process with umbers that are ot multiples of 3. Cosider what happes to the remaiig uits as you group each place value. Next, examie the sums of the digits i the multiples. Arrage the multiples accordig to the sums. Look for patters. Put patters ito charts, or sort them with Ve or Carroll diagrams. 4. Determie whether a umber is divisible by 9. Apply the strategies used for the factor Determie whether a umber is divisible by 6. This set cotais umbers divisible by both 2 ad 3. Cosider usig a Ve diagram to show the itersectio. Esure studets record their rules ad explaatios i their math jourals or otebooks, or i their Divisibility Study Booklets, ad that they make ay desired additios or chages durig whole-class discussio. Part C: Commuicatig Ideas 1. Iform studets that to celebrate ad cosolidate the learig resultig from this iquiry, they will create a display of their divisibility rules, with brief explaatios of why their rules work. 2. Allow displays to be small ad persoal. They may be created idividually or i small groups, with tasks divided amog group members. Create a larger wall display. 3. The class could orgaize a divisibility challege evet similar to the learig experiece suggested for learig outcome 7.N.1 followig Assessig Prior Kowledge. Number 15

42 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Sort umbers accordig to their divisibility. r Apply reasoig skills ad kowledge of umber properties ad operatios to develop a persoal meaig for the divisibility of a umber by 2, 3, 4, 5, 6, 8, 9, or 10. r Commuicate mathematically. Suggestios for Istructio Explai, usig a example, why umbers caot be divided by 0. Materials: calculators math jourals or otebooks couters Orgaizatio: Idividual Procedure: 1. Let studets kow that they will explore divisibility by zero. 2. Ask studets to divide umbers by zero o their calculators ad record the results i their math jourals. Have them explai why they thik there is a error message. 3. Ask studets to review the meaig of divisio ad to model divisio by some factor usig cocrete materials (e.g., couters) or a diagram. Next, ask studets to model divisio usig zero as a factor (divisor) ad to record their discovery i their math jourals. 4. Ask studets to create a table of multiplicatio facts for a give product, ad the to write the related divisio statemets. Next, ask studets to fid 0 = that product ad the related divisio fact. (There is o aswer.) 16 Grade 7 Mathematics: Support Documet for Teachers

43 Example: Multiplicatio Facts Related Divisio Facts 4 3 = = 3 4 = = 2 6 = = 1 12 = = ot possible 0 = = ot possible 5. Have studets record i their math jourals what they have discovered about dividig by zero, ad the ask them to aswer the followig questio: Based o what you have discovered, do you thik there should be a divisibility rule for zero? Explai your respose. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie that a umber caot be divided by zero. r Explai, usig examples, that a umber caot be divided by zero. Suggestios for Istructio Determie if a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad explai why. Determie the factors of a umber usig the divisibility rules. Materials: BLM 7.N.1.3: Applyig Divisibility Rules display board pes or markers of differet colours math jourals or otebooks Orgaizatio: Idividual, pairs or small groups, whole class Number 17

44 Procedure: 1. Distribute copies of BLM 7.N.1.3: Applyig Divisibility Rules. 2. Write 10 umbers o the display board. 3. Allow studets to choose ay five of the umbers ad complete the table provided o BLM 7.N.1.3: Applyig Divisibility Rules. 4. After givig studets time for idividual work, assig pairs or small groups of studets oe of the questios to preset to the class. Have them meet to discuss their respose ad to determie how they arrived at their aswer, how they kow their aswer is correct, ad how they will preset their questio ad respose. 5. Durig the class presetatios, the audiece members may ask questios of the preseters, or share their opiios. They may make desired adjustmets to their ow papers as they participate i the discussio. (Usig writig istrumets of differet colours reveals what ew kowledge or coectios were acquired durig that time.) 6. Have studets respod to the followig i their math jourals: Determie which digit(s) could be placed i the followig umber to make it divisible by 2, 3, ad 9: Explai your thikig. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Demostrate proficiecy with a variety of divisibility rules. r Use divisibility rules to determie factors of umbers. 18 Grade 7 Mathematics: Support Documet for Teachers

45 Number (7.N.2) Edurig Uderstadig(s): The priciples of operatios ad algorithms used with whole umbers also apply to operatios with decimals, fractios, ad itegers. Number sese ad metal mathematics strategies are used to estimate aswers ad lead studets to develop persoal algorithms. Geeral Learig Outcome(s): Develop umber sese. Specific Learig Outcome(s): 7.N.2 Demostrate a uderstadig of the additio, subtractio, multiplicatio, ad divisio of decimals to solve problems (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). [ME, PS, T] Achievemet Idicators: Solve a problem ivolvig the additio of two or more decimal umbers. Solve a problem ivolvig the subtractio of decimal umbers. Solve a problem ivolvig the multiplicatio or divisio of decimal umbers (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Place the decimal i a sum or differece usig frot-ed estimatio (e.g., for , thik , so the sum is greater tha 260). Place the decimal i a product usig froted estimatio (e.g., for $ , thik $12 2, so the product is greater tha $24). Place the decimal i a quotiet usig froted estimatio (e.g., for m 2.1, thik 50 m 2, so the quotiet is approximately 25 m). Check the reasoableess of aswers usig estimatio. Solve a problem that ivolves operatios o decimals (limited to thousadths), takig ito cosideratio the order of operatios. Explai, usig a example, how to use metal mathematics for products or quotiets whe the multiplier or the divisor is 0.1 or 0.5 or Number 19

46 Prior Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q (3.N.6) Describe ad apply metal mathematics strategies for addig two 2-digit umerals, such as addig from left to right takig oe added to the earest multiple of 10 ad the compesatig usig doubles (3.N.7) Describe ad apply metal mathematics strategies for subtractig two 2-digit umerals, such as takig the subtrahed to the earest multiple of 10 ad the compesatig thikig of additio usig doubles (4.N.5) Describe ad apply metal mathematics strategies, such as skip-coutig from a kow fact usig doublig or halvig usig doublig ad addig oe more group usig patters i the 9s facts usig repeated doublig to develop recall of basic multiplicatio facts to 9 9 ad related divisio facts. (5.N.2) Apply estimatio strategies, icludig frot-ed roudig compesatio compatible umbers i problem-solvig cotexts. (5.N.3) Determie multiplicatio facts (to 81) ad related divisio facts. (5.N.4) Apply metal mathematics strategies for multiplicatio, such as aexig, the addig zeros halvig ad doublig usig the distributive property (5.N.5) Demostrate a uderstadig of multiplicatio (2-digit umerals by 2-digit umerals) to solve problems. (5.N.6) Demostrate a uderstadig of divisio (3-digit umerals by 1-digit umerals) with ad without cocrete materials, ad iterpret remaiders to solve problems. 20 Grade 7 Mathematics: Support Documet for Teachers

47 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q (5.N.8) Describe ad represet decimals (teths, hudredths, thousadths) cocretely, pictorially, ad symbolically. (5.N.11) Demostrate a uderstadig of additio ad subtractio of decimals (limited to thousadths). (6.N.1) Demostrate a uderstadig of place value for umbers greater tha oe millio less tha oe-thousadth (6.N.2) Solve problems ivolvig large umbers, usig techology. (6.N.6) Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. (6.N.8) Demostrate a uderstadig of multiplicatio ad divisio of decimals ivolvig 1-digit whole-umber multipliers 1-digit atural umber divisors multipliers ad divisors that are multiples of 10 (6.N.9) Explai ad apply the order of operatios, excludig expoets (limited to whole umbers). (6.SS.3) Develop ad apply a formula for determiig the perimeter of polygos area of rectagles volume of right rectagular prisms Backgroud Iformatio Grade 7 provides a opportuity for studets to review decimal cocepts, to estimate, ad to solve problems usig operatios with decimal umbers. I Grade 5, studets added ad subtracted decimals, ad i Grade 6, they multiplied decimals by whole umbers ad powers of 10 ad divided decimals by whole umbers ad powers of 10. I Grade 7, studets exted multiplicatio ad divisio to iclude decimal umbers with 2-digit multipliers ad 1-digit divisors. They also explai metal mathematics strategies for dividig by 0.1, 0.5, ad For may practical situatios ivolvig decimals (e.g., calculatig tips o restaurat bills, averagig umbers of poits or people, purchasig goods sold by area or volume), estimated aswers are ofte preferred over precise calculatios. I other istaces, such as whe umbers are very large or very small, scietific otatio is used.* *Note: Scietific otatio is o loger formally taught i the mathematics classroom. Number 21

48 Situatios that ivolve may decimal poits ad require precise aswers are ofte techical i ature, ad techology is used to calculate these aswers. Because estimates are commo i everyday use, they provide a effective approach to teach operatios with decimals. The same priciples that studets have previously used with operatios, estimatios, models, ad algorithms for whole umbers also apply to decimal umbers. Whe you choose learig activities, focus o umber sese before addressig procedural operatios. If studets have difficulty with decimal operatios, esure that they have a good uderstadig of place value ad that they uderstad the role of the decimal poit. For example, certai statemets (e.g., dividig by 100 meas you move the decimal poit two places to the left) ca actually lead to miscoceptios ad emphasizes memorizatio of rules rather tha uderstadig of cocepts. I learig activities that require estimatig, provide studets with opportuities to develop methods of determiig where to positio the decimal poit i their estimates, ad the have them attempt to calculate precise aswers. As they work through this process, usig their uderstadig of place value, expaded otatio, ad operatios, they will develop a persoal method or algorithm based o meaigs. To be reliable, their algorithms must apply to all situatios. Ivite studets to share, explai, ad evaluate each other s methods. The itroduce traditioal algorithms ad explai their methodology ad how these lik to the studets persoal methods. Avoid favourig oe method over aother; studets should be free to select a method, providig it is mathematically soud ad allows studets to be efficiet ad accurate. If studets uderstad operatios as actios (e.g., additio as a combiig operatio, subtractio as a takig away operatio, multiplicatio as repeated additio or x groups of, ad divisio as repeated subtractio, or partitioig a umber ito groups, or how may groups of x are i this umber), they ca represet the operatios cocretely or pictorially. Maipulatives ca be used to represet decimal umbers. Operatios ca be visually represeted usig umber lies, place value mats, ad/or grid paper (e.g., BLM : Base-Te Grid Paper). Whe represetig decimals with base-10 blocks, it is ecessary to establish which maipulatives represet the value of 1. Example 1: If a flat represets 1 uit, the a rod ca represet 1 10 or 0.1 uit, ad a cube ca represet 1 or 0.01 uit Grade 7 Mathematics: Support Documet for Teachers

49 Example 2: If a large block represets 1 uit, the a flat ca represet 1 10 or 0.1 uit, ad a rod ca represet ad a cube ca represet or 0.01 uit, 1 or uit For illustratios, refer to the Appedix at the ed of this documet. Mathematical Laguage added additio array brackets decimals differece divided divisio divisor estimatio expoets* frot-ed estimatio hudredths * Note: The term expoet may arise durig discussio of the order of operatios, but studets are ot required to kow the term. Studets will study expoets formally i Grade 9. Number 23

50 metal mathematics multiplicad multiplicatio multiplier order of operatios paretheses product quotiet subtractio sum teths thousadths Learig Experieces Assessig Prior Kowledge Materials: list of mathematical terms paper for persoal or class posters referece books Orgaizatio: Idividual or pairs Procedure: 1. Have studet create posters that illustrate the terms i the mathematical laguage list provided. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Illustrate ad use mathematical terms correctly. 24 Grade 7 Mathematics: Support Documet for Teachers

51 Assessig Prior Kowledge Materials: BLM : Number Fas BLM 7.N.2.1: Whole ad Decimal Number Cards BLM 7.N.2.2: Operatio Cards brass tacks card stock (paper) display board (optioal) Orgaizatio: Whole class, a caller ad a verifier (teacher or studet) Procedure: 1. Copy BLM : Number Fas oto card stock ad attach two copies of each digit with a brass tack at the base of the fa. 2. Give each studet a umber fa. 3. Iform studets that for this learig activity, either the teacher or studets will take turs callig out whole ad decimal umbers. 4. Esure that the caller has access to BLM 7.N.2.1: Whole ad Decimal Number Cards. 5. Iform studets that they will proceed as follows: a) The caller chooses a umber from the list of cards, ad says it slowly three times. b) Players arrage the umber usig their respective umber fas ad hold it i frot of their chests. c) The verifier checks whether or ot studets resposes are correct. d) Studets perform some operatio o their umber usig BLM 7.N.2.2: Operatio Cards. e) The verifier checks whether the resposes are correct. Variatio: Form teams. Team members take turs writig umbers o the display board, or each team member records the umbers idividually, ad oe team member is amed to hold up the group s aswer. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Represet decimal umbers to thousadths correctly. r Uderstad place value. r Use metal mathematics strategies. r Perform mathematical operatios o decimal umbers. Number 25

52 Assessig Prior Kowledge Materials: BLM 7.N.2.3: Equivalet Percet, Fractio, ad Decimal Cards (sets of cards) Orgaizatio: Small groups (two to five studets) Procedure: 1. The objective of this game is to create a set of four matchig cards. A set cosists of a percet card the decimal umber expressed as teths or hudredths the decimal umber expressed as thousadths the umber expressed as a fractio i lowest terms 2. Have studets form small groups to play the game accordig to the geeral rules of Go Fish. 3. Studets shuffle the cards, ad deal five cards to each player i a group. They place the remaiig cards face dow i a pile. 4. Oe player asks aother for a specific card. The asker must have at least oe card of the set requested. 5. If the asked player has the card, he or she must give it to the asker. 6. If the asked player does ot have the card, he or she says, Go fish. 7. The asker the draws a card from the pile. If the asker is successful (picks up the card requested), he or she takes aother tur. If ot, play passes to the ext player. Variatios: Use the cards to play Cocetratio. Arrage a umber of matchig cards face dow. Players take turs turig over two cards to create sets. If the cards do t match, they are tured face dow agai. Observatio Checklist Note: This learig activity could be used to assess the followig competecy o the Grade 7 Numeracy Assessmet: Studet uderstads that a give umber may be represeted i a variety of ways. Referece: Maitoba Educatio, Citizeship ad Youth. Middle Years Assessmet: Grade 7 Mathematics: Support Documet for Teachers: Eglish Program. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at <www. edu.gov.mb.ca/k12/assess/ support/math7/>. Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create sets of equivalet fractios, decimals, ad percets. 26 Grade 7 Mathematics: Support Documet for Teachers

53 Assessig Prior Kowledge Materials: BLM 7.N.2.4: Order of Operatios ad Skill-Testig Questios (or similar questios) calculators Orgaizatio: Idividual or pairs, whole class Procedure: 1. Review the order of operatios with the class. Commet o how the order of operatios is required i may skill-testig questios for cotests (e.g., a draw at a store to wi a bicycle or a electroic device). 2. Distribute copies of BLM 7.N.2.4: Order of Operatios ad Skill- Testig Questios, ad have studets aswer the questios idividually or i pairs. 3. After a set amout of time has passed, review resposes as a class by havig studets share ad justify their aswers. 4. Have studets respod to the questios usig calculators to check whether or ot their calculators are programmed to follow the order of operatios. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Follow the order of operatios. r Fid solutios usig metal mathematics strategies. Number 27

54 Suggestios for Istructio Solve a problem ivolvig the additio of two or more decimal umbers. Solve a problem ivolvig the subtractio of decimal umbers. Place the decimal i a sum or differece usig frot-ed estimatio (e.g., for , thik , so the sum is greater tha 260). Check the reasoableess of aswers usig estimatio. Materials: BLM 7.N.2.5: Moey Problems BLM 7.N.2.6: Restaurat Bills ad Bikig BLM : Place-Value Mat Decimal Numbers (optioal) base-10 blocks (optioal) Orgaizatio: Idividual or pairs, whole class Procedure: 1. Distribute copies of BLM 7.N.2.5: Moey Problems ad/or BLM 7.N.2.6: Restaurat Bills ad Bikig. 2. Have studets estimate aswers before they do the calculatios. 3. Studets may model the additio ad subtractio usig BLM : Place-Value Mat Decimal Numbers. 4. Circulate amog studets while they are workig to assess their progress, ad supply guidace if ecessary. After studets have had sufficiet time to work o the problems, hold a debriefig discussio with the class, or collect ad assess the completed papers. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Add decimal umbers correctly. r Subtract decimal umbers correctly. r Apply appropriate strategies to solve problems. 28 Grade 7 Mathematics: Support Documet for Teachers

55 Suggestios for Istructio Solve a problem ivolvig the additio of two or more decimal umbers. Solve a problem ivolvig the subtractio of decimal umbers. Solve a problem ivolvig the multiplicatio or divisio of decimal umbers (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Place the decimal i a sum or differece usig frot-ed estimatio (e.g., for , thik , so the sum is greater tha 260). Place the decimal i a product usig frot-ed estimatio (e.g., for $ , thik $12 2, so the product is greater tha $24). Place the decimal i a quotiet usig frot-ed estimatio (e.g., for m 2.1, thik 50 m 2, so the quotiet is approximately 25 m). Check the reasoableess of aswers usig estimatio. Materials: BLM : Base-Te Grid Paper base-10 blocks coloured pecils math jourals BLM 7.N.2.7: Sample Scearios 1 (optioal) Orgaizatio: Idividual, pairs, whole class Procedure: 1. Preset the class with scearios that require multiplyig by decimals, ad ask studets to estimate reasoable solutios. Samples are provided o BLM 7.N.2.7: Sample Scearios 1. Exact solutios are ot required at this poit. These scearios ivolve percet values that eed to be coverted to decimal umbers. Preset oe sceario at a time, or preset several scearios together as a set, depedig o the eeds of the class. 2. Ask studets to record their estimates ad the strategies they used to arrive at them. 3. Multiplicatio ad divisio are iverse operatios. Have studets ivestigate the related divisio statemets that match the scearios. Compare the relatios, ad explore divisio strategies. Number 29

56 4. Begi with a brief Thik-Pair-Share strategy to help studets prepare for a wholeclass discussio. Ask studets to share their estimates, explai the strategies they used, justify where they placed the decimal, ad explai why they thik their solutios are reasoable. Possible resposes may iclude frot-ed estimatio roudig to the earest whole expaded otatio Ecourage studets to agree with, questio, or challege resposes respectfully. 5. Followig the discussio, have studets retur to the problems ad calculate exact solutios, record the solutios i their math jourals, ad prepare to articulate the strategies they used ad to explai why they thik their solutios are correct. I a later learig experiece, studets will devise a method for multiplyig by decimals, ad apply it to other situatios. 6. Multiplicatio ad divisio are iverse operatios. Ask studets to write related divisio statemets for each of their multiplicatio statemets, compare the resposes, ad ivestigate strategies for dividig decimal umbers. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create multiplicatio estimates with the decimal i the correct positio. r Calculate exact solutios. r Solve a problem ivolvig the multiplicatio of decimal umbers. r Place the decimal i a product usig frot-ed estimatio. r Check the reasoableess of aswers usig estimatio. r Commuicate mathematically. 30 Grade 7 Mathematics: Support Documet for Teachers

57 Suggestios for Istructio Solve a problem ivolvig the multiplicatio or divisio of decimal umbers (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Place the decimal i a product usig frot-ed estimatio (e.g., for $ , thik $12 2, so the product is greater tha $24). Place the decimal i a quotiet usig frot-ed estimatio (e.g., for m 2.1, thik 50 m 2, so the quotiet is approximately 25 m). Check the reasoableess of aswers usig estimatio. Materials: BLM : Base-Te Grid Paper BLM : Number Lie BLM 7.N.2.8: Sample Scearios 2 base-10 blocks coloured pecils math jourals Orgaizatio: Idividual, pairs, whole class Procedure: 1. Ask studets to defie divisio ad to provide a illustratio of the operatio. Divisio may be uderstood as repeated subtractio, or partitioig ito groups of a particular size or ito a particular umber of groups. Repeated subtractio ca be show o a umber lie, ad partitioig ca be show with base-10 blocks. Divisio may be demostrated by formig a array with base-10 blocks or drawig o base-10 grid paper (e.g., BLM : Base-Te Grid Paper). For some examples, refer to the Appedix at the ed of this documet. 2. Preset the class with scearios requirig dividig by decimals, ad ask studets to estimate reasoable solutios. Samples are provided o BLM 7.N.2.8: Sample Scearios 2. Preset oe sceario at a time, or preset several scearios together as a set, depedig o the eeds of the class. 3. Ask studets to record their estimates ad the strategies they used to arrive at them. Have studets record their thikig. 4. Divisio ad multiplicatio are iverse operatios. Have studets ivestigate the related multiplicatio statemets that match the scearios. Compare the relatios ad multiplicatio strategies. 5. Allow a few miutes for the use of a Thik-Pair-Share strategy to help studets prepare for a whole-class discussio i which they share their estimates ad explai the strategies they used, how they decided where to place the decimal, ad why they thik the solutios are reasoable. Ecourage studets to agree with, questio, or challege resposes respectfully. Record the reasoable estimates i a chart. Number 31

58 6. Followig the discussio, have studets retur to the problems ad calculate exact solutios, agai recordig their solutios i their math jourals, ad prepare to articulate the strategies they used ad explai why they thik their solutios are correct. 7. Divisio ad multiplicatio are iverse operatios. Ask studets to write related multiplicatio statemets for each of their divisio statemets, compare the resposes, ad ivestigate strategies for multiplyig decimal umbers. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create estimates for divisio with the decimal i the correct positio. r Calculate exact solutios. r Solve a problem ivolvig the divisio of decimal umbers. Suggestios for Istructio Solve a problem ivolvig the multiplicatio or divisio of decimal umbers (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Place the decimal i a product usig frot-ed estimatio (e.g., for $ , thik $12 2, so the product is greater tha $24). Place the decimal i a quotiet usig frot-ed estimatio (e.g., for m 2.1, thik 50 m 2, so the quotiet is approximately 25 m). Check the reasoableess of aswers usig estimatio. Materials: BLM : Base-Te Grid Paper BLM : My Success with Mathematical Processes base-10 blocks coloured pecils or highlighters math jourals calculators Orgaizatio: Idividual, pairs, whole class 32 Grade 7 Mathematics: Support Documet for Teachers

59 Procedure: 1. Remid studets that by applyig their preset kowledge of mathematics, they have successfully solved multiplicatio questios with 1- ad 2-digit decimal multipliers, ad have explored strategies for divisio ivolvig decimal umbers. 2. Challege studets to articulate a method that will allow them to solve ay such multiplicatio ad divisio questios, ad to share those methods with others. 3. If ecessary, prompt studets to orgaize their ivestigatio. 4. First, they will eed to uderstad the problem. 5. The, they will collect data by selectig umbers with which to experimet. Numbers that are easy to work with usig metal mathematics, ad sequetial series, will geerate data with the most obvious patters. 6. Suggest that studets collect data by varyig both the multiplicad ad the multiplier, usig decimal umbers i both. 7. Later, compare the related divisio statemets for the same data. 8. Studets may wish to use base-10 blocks, drawigs, or base-10 grid paper (e.g., BLM : Base-Te Grid Paper) to help them solve equatios. 9. Illustratios usig base-10 blocks are outlied i the Appedix at the ed of this documet. Studets may geerate data such as the followig: = = = = 8 Notice that these umbers ca also be writte as 5.0, 6.0, 7.0, ad = = = = = = = = = = = = = = = = = = As studets examie data, they may make the followig observatios or coclusios: The resultig digits are the same as i whole-umber multiplicatio or divisio, but the place value positios are differet. 11. Studets may use calculators to compare the products or quotiets of decimal umbers to products or quotiets of whole umbers with the same digits, but o decimals. They ca temporarily disregard the decimal, ad use ay previous algorithms for multiplicatio or for determiig the digits, ad the decide where to place the decimal i the product based o estimatio strategies. They ca remove the decimal by multiplyig by a power of 10, ad, after fidig the product, compesate by dividig by that power of 10. Number 33

60 12. Studets may use their kowledge of place value ad aexig zeros whe multiplyig ad dividig by powers of 10, ad may otice geeralizatios (e.g., the umber of digits to the right of the decimal poit i the product is equal to the umber of digits to the right of the decimal poit i the factors).* 13. Ask studets to use their math jourals to describe a procedure for multiplyig decimal umbers. The ask studets to write related divisio statemets for each of their multiplicatio statemets, compare the resposes, ad describe a procedure for dividig decimal umbers. * Note: It is imperative that studets have ample exploratio time to discover these geeralizatios without merely beig told what they are. Remai ope to hearig all discoveries studets make ad ecourage studets to discuss them. 14. Have studets reflect o ad record their success with mathematical processes, usig BLM : My Success with Mathematical Processes. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Uderstad problem-solvig strategies. r Apply appropriate problem-solvig strategies. r Commuicate ideas clearly. r Questio ideas. r Describe a effective method for solvig multiplicatio with decimal multipliers. r Describe a effective method for solvig divisio questios with decimal multipliers. 34 Grade 7 Mathematics: Support Documet for Teachers

61 Suggestios for Istructio Solve a problem ivolvig the multiplicatio or divisio of decimal umbers (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Place the decimal i a product usig frot-ed estimatio (e.g., for $ , thik $12 2, so the product is greater tha $24). Place the decimal i a quotiet usig frot-ed estimatio (e.g., for m 2.1, thik 50 m 2, so the quotiet is approximately 25 m). Check the reasoableess of aswers usig estimatio. Materials: BLM : Base-Te Grid Paper BLM : Number Lie base-10 blocks coloured pecils calculators math jourals Orgaizatio: Idividual, pairs, whole class Procedure: 1. Ask studets to defie divisio ad to provide a illustratio of the operatio. Divisio may be uderstood as repeated subtractio, or partitioig ito groups of a particular size or ito a particular umber of groups. Repeated subtractio ca be show o a umber lie (see BLM : Number Lie), ad partitioig ca be show with base-10 blocks. Divisio may be demostrated by formig a array with base-10 blocks or drawig o base-10 grid paper (see BLM 5 8:10: Base-Te Grid Paper). Illustratios usig base-10 blocks are outlied i the Appedix at the ed of this documet. 2. Ask studets to illustrate the followig divisio questios. After givig studets sufficiet time to cosider the questios, ask them to discuss their defiitios, illustratios, ad discoveries. a) 8 4 b) c) d) Have studets create other divisio questios ad vary the place value positio of the digits. They may use techology for divisors with two or more digits. Studets record the equatios i charts, ad develop a persoal method or algorithm for dividig with decimals. They prepare to discuss fidigs with the class. Number 35

62 4. Studets may make the followig observatios or coclusios: If the digits i the divisio questios remai the same, the digits i the quotiets are the same. As with multiplicatio, the differece is i the place value positio of the digits. Studets ca use kow divisio algorithms ad either igore the decimal place or fid the digits i the solutio, ad the place the decimal accordig to their estimatio. They ca reame the decimal umbers so they have the same place value (e.g., is equivalet to 40 teths divided by 5 teths, which is 8). They ca use a balace scale priciple ad will get the same quotiet if they multiply both umbers i the questio by the same power of 10 (e.g., is equivalet to multiplyig both umbers by 100, which becomes = 50). 5. Have studets record i their math jourals how to divide decimal umbers. 6. Ask studets to write related multiplicatio statemets for each of their divisio statemets, ad to compare the resposes ad procedures for dividig ad multiplyig decimal umbers. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Uderstad problem-solvig strategies. r Apply appropriate problem-solvig strategies. r Commuicate ideas clearly. r Questio ideas. r Describe a effective method for solvig divisio with decimals. r Solve a problem ivolvig the divisio of decimal umbers. r Place the decimal i a quotiet usig frot-ed estimatio. r Check the reasoableess of aswers usig estimatio. 36 Grade 7 Mathematics: Support Documet for Teachers

63 Suggestios for Istructio Solve a problem ivolvig the multiplicatio or divisio of decimal umbers (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Place the decimal i a product usig frot-ed estimatio (e.g., for $ , thik $12 2, so the product is greater tha $24). Place the decimal i a quotiet usig frot-ed estimatio (e.g., for m 2.1, thik 50 m 2, so the quotiet is approximately 25 m). Check the reasoableess of aswers usig estimatio. Materials: BLM 7.N.2.9: Sample Scearios 3 Orgaizatio: Idividual Procedure: 1. Whe studets have a method for solvig multiplicatio problems with decimal umbers, give them opportuities to solve a variety of problems, such as those provided o BLM 7.N.2.9: Sample Scearios Remid studets that estimatig aswers before calculatig solutios is importat to verify that aswers are reasoable. This strategy is importat whe usig techology. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Place decimal poits i the correct positio. r Solve problems correctly. r Solve a problem ivolvig the multiplicatio or divisio of decimal umbers. r Place the decimal i a product usig frot-ed estimatio. r Place the decimal i a quotiet usig frot-ed estimatio. r Check the reasoableess of aswers usig estimatio. Number 37

64 Suggestios for Istructio Solve a problem ivolvig the multiplicatio or divisio of decimal umbers (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Place the decimal i a quotiet usig frot-ed estimatio (e.g., for m 2.1, thik 50 m 2, so the quotiet is approximately 25 m). Check the reasoableess of aswers usig estimatio. Solve a problem that ivolves operatios o decimals (limited to thousadths), takig ito cosideratio the order of operatios. Materials: BLM 7.N.2.10: Decimal Problems idex cards (optioal) Orgaizatio: Idividual Procedure: 1. Studets solve divisio ad order of operatio questios such as those foud o BLM 7.N.2.10: Decimal Problems. Variatio: Have studets create their ow scearios ad solutios o idex cards. Studets ca exchage their scearios ad solutios with a classmate, or add them to a questio bak to use as Exit Slips or for i-class challeges. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Place the decimal correctly while estimatig. r Use the order of operatios to aswer multi-step problems correctly. r Solve a problem ivolvig the multiplicatio or divisio of decimal umbers. r Place the decimal i a quotiet usig frot-ed estimatio. r Check the reasoableess of aswers usig estimatio. r Solve a problem that ivolves operatios o decimals, takig ito cosideratio the order of operatios. 38 Grade 7 Mathematics: Support Documet for Teachers

65 Suggestios for Istructio Explai, usig a example, how to use metal mathematics for products or quotiets whe the multiplier or the divisor is 0.1 or 0.5 or Materials: BLM : Base-Te Grid Paper base-10 blocks math jourals Orgaizatio: Pairs or small groups, whole class, idividual Procedure: 1. Preset studets with a missio: Discover a quick metal mathematics strategy to use whe multiplyig ay umber by 0.1. Later, the problem will be exteded to fidig the quotiet for dividig by 0.1, ad the to fidig products ad quotiets for multiplyig ad dividig by 0.5 ad If studets eed guidace to orgaize their ivestigatio, assist them with the followig hits. Ecourage them to use a problem-solvig strategy, rather tha follow a set of steps. Example: What must you do? What must you kow i order to do what you have determied you eed to do? How ca you get that iformatio? What does it mea to multiply? What does it mea to multiply by 0.1? What is 0.1? What do you eed i order to fid a patter? a) First, esure that studets uderstad the problem ad ca articulate the meaig of multiplicatio by 0.1. b) Next, have them explore the patters created by multiplyig a set of umbers by 0.1. c) Fially, have them examie the patters they see, ad create a metal mathematics strategy to fid the product whe multiplyig by 0.1. Test the strategy to esure it works for all umbers. Try umbers such as 120 or 122, or smaller umbers such as or After studets (workig i pairs or i small groups) have had sufficiet time to explore ad develop a strategy, have them reassemble as a class to debrief, givig them a opportuity to commuicate their ideas ad questios ad revise their strategies as ecessary. The have them use their math jourals to record their strategies ad explai why the strategies work. 4. Exted the problem to fidig the quotiet for dividig by 0.1, ad the products ad quotiets for multiplyig ad dividig by 0.5 ad Number 39

66 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Use a mathematically correct metal mathematics strategy for each operatio. r Provide a explaatio that supports the strategy. r Use effective problem-solvig strategies. r Explai, usig a example, how to use metal mathematics for products or quotiets whe the multiplier or the divisor is 0.1 or 0.5 or Grade 7 Mathematics: Support Documet for Teachers

67 Puttig the Pieces Together School-Supply Kits Itroductio: Studets receive a budget for creatig 20 school-supply kits to be give as welcome gifts to ew studets i eed. Purpose: I this ivestigatio, studets will demostrate the ability to do the followig (coectios to learig outcomes are idetified i paretheses): Perform operatios with decimals to solve problems. (7.N.2) Solve problems ivolvig percets from 1% to 100%. (7.N.3) Compare ad order decimals ad itegers. (7.N.7) Studets will also demostrate the followig mathematical processes: Commuicatio Coectios Metal Mathematics ad Estimatio Problem Solvig Reasoig Techology Materials/Resources: office- ad school-supply flyers ad prit ad olie catalogues, ad/or access to office ad school-supply stores calculators shoppig list purchase order Orgaizatio: Idividual or small group Procedure: 1. Several ew studets have bee arrivig at your school without ay school supplies or ay meas to obtai them. This has added to the studets difficulties i tryig to adjust to their ew school experiece. The studet coucil has held a fudraisig drive to help future ew studets adjust to their ew school experiece. They have raised $500 to create 20 school-supply kits. I the future, a kit will be give as a welcome gift to a ew studet i eed. The studet coucil has chose your group to prepare the kits. 2. Create a list of school-supply items ad quatities required for each kit. Number 41

68 3. Calculate the total umber of each item required for all the kits. 4. Shop to fid prices for the items. Use estimatios to help guide your choices. Remember to add PST ad GST ad ay applicable shippig or hadlig costs. Stay as close to budget as possible. 5. Prepare a clearly writte letter to iform the store of your project ad request their assistace. You receive a respose to your letter from the store offerig a 25% discout to help with your project. 6. Make ay desired adjustmets to your shoppig list. 7. Fialize the items to purchase. Remember that PST ad GST will be added ad shippig or hadlig charges may be added. Stay as close to budget as possible. 8. Prepare a purchase order for the store. List the items from least expesive to most expesive, the quatities to order, uit price, exteded price, discouted or adjusted price, subtotal, taxes, shippig ad hadlig charges (if applicable), ad the grad total. 9. Prepare a clearly writte letter to the studet coucil regardig your decisio. Iclude a list of the items to be icluded i each school-supply kit, idetify ay extra items that will be left over, compare the cost to the budget, ad iclude the purchase order. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create a adequate list of school supplies to iclude i each kit. r Make reasoable estimates to guide decisio makig. r Perform operatios with decimal umbers correctly. r Calculate percets accurately (discout, GST, PST). r Make decisios to match a budget withi 5%. r Order ad calculate the purchase order correctly. r Prepare a clear letter requestig assistace. r Prepare a clear summary letter outliig actios ad comparig cost to budget. 42 Grade 7 Mathematics: Support Documet for Teachers

69 Shoppig List for School-Supply Kits Item Quatity Required for Each Kit Quatity for 20 Kits Purchase Price Cost Per Item Cost Per 20 Kits Number 43

70 Purchase Order for School-Supply Kits To: From: Item Descriptio Quatity Required Uit Price Exteded Price Adjusted Price Subtotal GST (5%) PST (7%) Shippig ad Hadlig Charges Grad Total 44 Grade 7 Mathematics: Support Documet for Teachers

71 Number (7.N.3) Edurig Uderstadig(s): Percets ca be thought of as a ratio comparig to 100 or a fractio out of 100. Circle graphs show a compariso of each part to a whole usig ratios. Percets, fractios, decimals, ad ratios are differet represetatios of the same quatity. Number sese ad metal mathematics strategies are used to estimate aswers ad lead to flexible thikig. Geeral Learig Outcome(s): Develop umber sese. Specific Learig Outcome(s): 7.N.3 Solve problems ivolvig percets from 1% to 100%. [C, CN, PS, ME, R, T] Achievemet Idicators: Express a percet as a decimal or fractio. Solve a problem that ivolves fidig a percet. Determie the aswer to a percet problem where the aswer requires roudig, ad explai why a approximate aswer is eeded (e.g., total cost icludig taxes). Prior Kowledge Studets should be able to do the followig: Q Q (5.N.2) Apply estimatio strategies, icludig frot-ed roudig compesatio compatible umbers i problem-solvig cotexts. Q Q (5.N.4) Apply metal mathematics strategies for multiplicatio, such as aexig, the addig zeros halvig ad doublig usig the distributive property Number 45

72 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q (5.N.7) Demostrate a uderstadig of fractios by usig cocrete ad pictorial represetatios to create sets of equivalet fractios compare fractios with like ad ulike deomiators (5.N.8) Describe ad represet decimals (teths, hudredths, thousadths) cocretely, pictorially, ad symbolically. (5.N.9) Relate decimals to fractios (teths, hudredths, thousadths). (5.N.10) Compare ad order decimals (teths, hudredths, thousadths) by usig bechmarks place value equivalet decimals (6.N.1) Demostrate a uderstadig of place value for umbers greater tha oe millio less tha oe-thousadth (6.N.3) Demostrate a uderstadig of factors ad multiples by determiig multiples ad factors of umbers less tha 100 idetifyig prime ad composite umbers solvig problems ivolvig factors or multiples (6.N.5) Demostrate a uderstadig of ratio, cocretely, pictorially, ad symbolically. (6.N.6) Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. Related Kowledge Studets should be able to do the followig: Q Q Q Q (7.N.2) Demostrate a uderstadig of the additio, subtractio, multiplicatio, ad divisio of decimals to solve problems (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). (7.N.4) Demostrate a uderstadig of the relatioship betwee repeatig decimals ad fractios, ad termiatig decimals ad fractios. 46 Grade 7 Mathematics: Support Documet for Teachers

73 Backgroud Iformatio People regularly ecouter practical situatios requirig them to uderstad ad solve problems related to percet. These situatios iclude problems related to sports statistics, discouts, price icreases, taxes, polls, social chages ad treds, the likelihood of precipitatio, ad so o. The media provide sources of cotextual data for creatig problems ivolvig percet. Learig outcome 7.N.3 builds o uderstadigs related to fractios, ratios, decimals, percets, ad problem solvig that studets have developed i previous grades. Fractios ad Decimals Before studets become skilful at solvig problems ivolvig percet, they must have a strog coceptual uderstadig of fractios, decimals, ad percets, ad they must be able to iterchage equivalet ames to represet the cocepts. The term fractio has several meaigs. A expert bleds ad separates these meaigs for coveiece, but this bledig ca cofuse studets who lack fluecy i applyig the differet meaigs of fractio. Fractio otatio is used to represet a cut or a part of a uit, a part of a group or set, a measuremet, or a poit o a umber lie. It is also used to represet a ratio or a portio of a tur, ad to idicate the divisio operatio. Decimals are a coveiet meas of represetig fractioal quatities usig a place value system. Fractios may be coverted to decimals by usig the divisio operatio meaig of fractio ad dividig the umerator by the deomiator (e.g., 3 4 may be viewed as 3 4 = 0.75). Fractios may also be coverted to decimals by fidig a equivalet fractio with a deomiator of ay power of 10 (e.g., 100), ad the writig 7 14 the fractio i stadard otatio e.g., = =014.. It is useful to commit to memory some commo fractio decimal equivalets, such as halves, quarters, ad teths. A decimal poit is used to separate whole uits from parts of uits. Each positio to the right of the decimal represets a teth part of oe of the previous uits. I stadard otatio, the first positio followig the decimal represets teth parts of oe whole uit, ad the secod place represets teth parts of a teth, or hudredth parts of oe uit. thousads hudreds tes oes/uits teths hudredths thousadths Number 47

74 Problems Ivolvig Percet Whe traslatig stadard otatio to percet, the decimal poit idicates where to read the hudredths i a umber. The word percet meas per hudred ad may be 7 substituted for the word hudredths whe readig a umber. Therefore, or 0.07 may 100 be read as 7 hudredths ad also as 7 percet. Percet may also be used to represet fractioal quatities that are a little larger tha a hudredth. Place value positios to the right of the decimal represet the cut meaig of a fractio, ad each successive positio represets oe of the previous uits cut ito 10. I stadard otatio, the third positio represets thousadth parts of oe uit, but it may also be viewed as teths of a hudredth. A uderstadig of place value allows us to express ay umber as a umber of selected uits. Just as 141 ca represet 14 tes ad 1 oe, 0.141, which represets a umber that is a little larger tha oe teth of oe whole, may be expressed as 1.41 teths, 14.1 hudredths, or 141 thousadths. Substitutig the word percet as aother word for hudredth, the decimal umber may be cosidered as 14.1 hudredths, or 14.1 percet (%). I Grade 7, studets eed to work oly with umbers from 1% to 100%. The percets may represet a part of oe whole item or a part of oe whole group. Percet represets a special type of fractio with a deomiator of 100. The quatity represeted by the percet depeds o the amout i the whole. For example, 1% may be a large or small quatity, depedig o the whole. Cosider 1% of the moey i a idividual s piggy bak, versus 1% of the moey i the bak s books. The same quatity may also represet differet percet values. For example, 20 is 20% of 100, but it is also 100% of 20. Idetifyig which umber i a situatio represets the whole ad which umber represets the part is importat whe solvig problems ivolvig percet. Whe studets have developed multiple views of fractios ad percets, they will beefit from havig multiple meaigful approaches to fid percet values. To fid 25% of 80, studets may approach the problem as follows: Thik of 25% as , ad the equivalet fractio 1 4, ad the fid 1 of 80 by 4 dividig 80 ito 4 groups: 80 4 = 20, so 25% of 80 is 20. Thik of a equivalet fractio This view is less coveiet i this situatio, but more coveiet i other situatios. Thik of a part-to-whole ratio ad proportio: Thik of a circle with begiig ad ed poits 0 ad 80, ad thik of 25% as 1 4 of a tur. 48 Grade 7 Mathematics: Support Documet for Teachers

75 Thik of a circle graph that shows 25% of the studets i a class like Caesar salad ad 75% like taco salad. If there are 80 studets i the class, how may like Caesar salad? Thik of a umber lie with ed poits of 0 ad 80, ad correspodig poits 0% ad 100%. Studets may thik 25% is half of 50%. Half of 80 is 40, ad half of 40 is 20. Thik of the decimal equivalet, ad chage the word expressio ito a umber expressio. 25% of 80 is = Number 49

76 Studets may also use metal mathematics ad the distributive property to solve problems ivolvig percet. To fid 35% of 80, thik of 35% as 25% + 10%. I the above problem, 25% of 80 is 20, ad 10% of 80 is = 28, so 35% of 80 is 28. Studets may also use related fractios to solve percet problems. If 1 4 of 80 is 20, the 3 of 80 must be 3 20 or With multiple approaches to fidig percets, studets ca choose the most coveiet approach for each problem. Whe settig up examples ad creatig problems for studets, frequetly choose umbers that are coveiet to work with, so that studets will be able to cocetrate o the processes ivolved, rather tha o the arithmetic. Also ecourage studets to use a variety of approaches ad ot to over-rely o oe specific method. For example, they may develop a habit of usig factors of 10, ad forget to use equivalet fractios or the commoly used, very effective part-to-whole ratio approach. To fid 25% of a umber, you may thik 25% is 10% + 10% + half of 10%, but it may be much more coveiet to thik of 25% as 1, ad divide the whole by 4. 4 Mathematical Laguage decimal equivalet factor fractio multiple percet proportio ratio simplify 50 Grade 7 Mathematics: Support Documet for Teachers

77 Learig Experieces Assessig Prior Kowledge Materials: markers or pes of two differet colours (for each pair of studets) two regular umber cubes (providig factors 1 to 12) or a multi-sided cube grid paper or tic-tac-toe grids or frames (of various sizes), such as the followig: BLM 7.N.3.1A: Tic-Tac-Toe Frames BLM 7.N.3.1B: Tic-Tac-Toe Frames (Medium Challege) BLM 7.N.3.1C: Tic-Tac-Toe Frame (Ultimate Challege) Orgaizatio: Pairs Procedure: 1. Have pairs of studets take turs fillig i the squares of their tic-tactoe grids with umbers 1 to 99, or multiples that correspod to the umbers o their umber cubes. 2. Studets choose a colour ad a X or a O mark, ad determie who will play first. 3. Studets take turs rollig the umber cube(s), ad use their colour markers to mark a X or a O o a multiple of the umber they rolled. Ecourage them to practise usig mathematical laguage with statemets such as the followig: 27 is a multiple of 9 because 3 9 = ad 3 are factors of 27 because 3 9 = is a prime umber. Its oly factors are 1 ad Studets will eed to agree about what to do if someoe makes a error. 5. The first studet who creates a horizotal, vertical, or diagoal lie with his or her marks wis. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Use vocabulary for multiples, factors, ad primes correctly. r Idetify multiples of various umbers correctly. r Idetify prime umbers correctly. Number 51

78 Assessig Prior Kowledge Materials: BLM 7.N.3.2: Equivalet Fractio Challege a pair of six-sided umber cubes, or a multi-sided umber cube, or a spier (for each pair of studets) Orgaizatio: Whole class, pairs Procedure: 1. As a class, review procedures for creatig equivalet fractios by multiplyig or dividig by a fractio ame for 1, or by multiplyig or dividig each term i the fractio by the same factor. 2. Demostrate oe roud of the game, followig the procedures outlied o BLM 7.N.3.2: Equivalet Fractio Challege, ad usig the game cards provided o the BLM. I summary, studets create a target fractio, take turs rollig the umber cube(s) to determie a chage factor, ad the create a equivalet fractio. The player who returs the fractio to its origial target ame wis. 3. Distribute game cards. 4. Have studets play the game i pairs. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create equivalet fractios correctly ad simplify fractios with ease. 52 Grade 7 Mathematics: Support Documet for Teachers

79 Assessig Prior Kowledge Materials: BLM 7.N.3.3: It s Betwee: Roudig Decimal Numbers demostratio board Orgaizatio: Idividual, or pairs, or whole class Procedure: 1. Review the place value positios for decimal umbers, ad review how to read ad write decimal umbers. 2. Review strategies for roudig decimal umbers to a give place value positio by idetifyig which umbers the give umber is betwee ad the determiig which umber it is closest to. Example: Roud to the earest teth. 3. Idetify the value of the required place value. I the above example, it is teths (0.652 has 6 teths ad a little more). 4. Idetify oe uit higher for the same place value. Oe uit higher tha 6 teths is 7 teths (0.652 is betwee 6 teths ad 7 teths). 5. Determie whether is closer to 6 teths or to 7 teths by thikig of the umbers o a lie, with the midway poit betwee the umbers beig Sice is a little more tha 0.65, it is closer to 0.7 tha it is to 0.6. So, rouded to the earest teth is Repeat the same process for roudig to the earest hudredth, ad the to the earest thousadth. 7. Have studets verify their skill by roudig a set of umbers such as those provided o BLM 7.N.3.3: It s Betwee: Roudig Decimal Numbers. Variatio: Usig the demostratio board, preset oe questio at a time ad have studets solve it. Call upo oe studet to idetify the umbers betwee which a give umber lies, ad aother studet to idetify which umber it is closest to, ad therefore rouds to. Number 53

80 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify place values correctly. r Determie the umbers betwee which the umber beig rouded lies. r Roud a umber to the required decimal place correctly. Suggestios for Istructio Express a percet as a decimal or fractio. Determie the aswer to a percet problem where the aswer requires roudig, ad explai why a approximate aswer is eeded (e.g., total cost icludig taxes). Materials: BLM 7.N.3.4: Choose Your Questio (Poit Sheet, Game Sheet 1 ad Game Sheet 1 Aswer Key, Blak Game Sheet) scissors low-tack glue corrugated board pis, tacks, or staples Orgaizatio: Whole class, small groups (three to five studets) Procedure: 1. Review the place value positios for decimal umbers, ad review readig ad writig decimal umbers. 2. Review strategies for roudig decimal umbers to a give place value positio, as described i the precedig learig activity. 3. Review readig decimal umbers as percets, as outlied i the Backgroud Iformatio for learig outcome 7.N.3 (e.g., may be read as 45.6%). 54 Grade 7 Mathematics: Support Documet for Teachers

81 4. Play the game outlied i BLM 7.N.3.4: Choose Your Questio, usig the followig procedure: a) Form studet groups, each cotaiig oe quizmaster ad three to five cotestats. b) Distribute copies of BLM 7.N.3.4: Choose Your Questio (Poit Sheet, Game Sheet 1 ad Aswer Key, Blak Game Sheet), alog with other required materials, ad have studet groups make game boards. The cotestats receive the category Poit Sheet ad cut up the sectios, which they give to the quizmaster. The quizmaster receives the Game Sheet ad Aswer Key. He or she hides the Aswer Key, ad uses the Game Sheet to create a game board by tackig the Poit Sheet sectios over the questios with temporary low-tack glue, or by placig the Game Sheet o the corrugated board ad tackig the poit cards i place with pis, tacks, or staples. c) Studets decide which cotestat will go first. d) That cotestat chooses a category ad poit value. e) The quizmaster ucovers the questio. If the cotestat respods correctly, he or she receives the poit card. f) Play the moves to the ext cotestat. The player with the most poits at the ed wis. The wier becomes the quizmaster for additioal rouds. A Blak Game Sheet is icluded for additioal rouds. Variatio: Alter the categories to iclude ay skills you wish to review (Game Sheet 2 icludes part-to-whole ratios ad percets as decimals ad fractios). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Roud decimal umbers to the desigated place value correctly. r Express a decimal umber as a percet. Number 55

82 Suggestios for Istructio Express a percet as a decimal or fractio. Materials: BLM 7.N.3.4: Choose Your Questio (Poit Sheet, Game Sheet 2, Blak Game Sheet) (optioal) scissors Orgaizatio: Small groups (three to five studets) Procedure: 1. Review expressig a situatio as a part-towhole ratio, ad vice versa. 2. Review expressig a percet as a decimal umber, ad vice versa (iclude percets such as % or %, which are commo 4 2 i iterest rates or pay-icrease rates). 3. Review expressig a percet as a decimal ad as a fractio, ad vice versa. 4. Follow the directios for Game Sheet 1 i the previous learig activity. Note: This learig activity could be used to assess the followig competecy o the Grade 7 Numeracy Assessmet: Studet uderstads that a give umber may be represeted i a variety of ways. Referece: Maitoba Educatio, Citizeship ad Youth. Middle Years Assessmet: Grade 7 Mathematics: Support Documet for Teachers: Eglish Program. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at <www. edu.gov.mb.ca/k12/assess/ support/math7/>. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express situatios as ratios ad percets correctly. r Express a percet as a decimal correctly. r Express a percet as a fractio correctly. r Express a fractio as a percet correctly 56 Grade 7 Mathematics: Support Documet for Teachers

83 Suggestios for Istructio Express a percet as a decimal or fractio. Materials: various card sets for each group from the followig website: Maitoba Educatio. Middle Years Activities ad Games. Mathematics. < BLM 7.N.2.3: Equivalet Percet, Fractio, ad Decimal Cards Orgaizatio: Small groups (three or four studets) Procedure: 1. Tell studets they will play a game accordig to the rules of Rummy. The objective of the game is to create sets of four matchig cards (a percet, a fractio, a decimal, ad a illustratio) ad to be the first perso to dispose of all his or her cards. 2. Have studets form small groups ad choose a dealer. The dealer shuffles the cards ad deals seve cards to each player, who the secretly sorts them ito sets. The dealer places the remaiig cards face dow o the table to form a draw pile, ad turs the top card of the draw pile face up to form a discard pile. 3. The first player begis by drawig a card from either the draw pile or the discard pile. The player may lay ay sets of three or four equivalet cards face up o the table i frot of him or her. I the same tur, the player may also play equivalet cards o top of other players equivalet sets. The player completes a tur by playig a card o the discard pile. This card may ot be the same card the player selected from the discard pile at the begiig of the tur. 4. Play passes to the ext player. 5. The roud is over whe the first player discards his or last card o the discard pile. Variatios: Note: This learig activity could be used to assess the followig competecy o the Grade 7 Numeracy Assessmet: Studet uderstads that a give umber may be represeted i a variety of ways. Use the cards from BLM 7.N.2.3: Equivalet Percet, Fractio, ad Decimal Cards (which iclude equivalet fractios i lowest terms ad decimals expressed i teths, hudredths, thousadths, ad percets). Use the cards to play Go Fish, usig rules outlied i the suggested learig experieces for learig outcome 7.N.2 for fractio, decimal, ad percet equivalets. Number 57

84 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express a percet as a decimal or fractio correctly. Suggestios for Istructio Express a percet as a decimal or fractio. Materials: BLM 7.N.3.5: Desigig to Percet Specificatios coloured pecils or markers Orgaizatio: Idividual Procedure: 1. Create a desig o 100-grid paper that meets set percet specificatios. Grid paper is provided o BLM 7.N.3.5: Desigig to Percet Specificatios. 2. Express each percet as a decimal ad as a fractio. 3. Simplify the fractios. Example: 10% of the desig is red. 40% of the desig is blue. 15% of the desig is gree. The remaider of the desig is yellow. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express a percet as a decimal or fractio correctly. 58 Grade 7 Mathematics: Support Documet for Teachers

85 Suggestios for Istructio Express a percet as a decimal or fractio. Solve a problem that ivolves fidig a percet. Materials: BLM 7.N.3.6: Determiig the Whole, the Part, ad the Percet math jourals demostratio board markig pe Orgaizatio: Whole class Procedure: Part A 1. Iform studets that percets are a special type of fractios out of 100. Therefore, each percet problem is a part-to-whole relatio ad may be represeted as a fractio problem. 2. Preset situatios that ivolve fidig the whole, the part, ad the percet. Preset oe situatio at a time, recordig it o the board. 3. For each situatio preseted, ask studets to idetify the whole, the part, ad the percet, either o BLM 7.N.3.6: Determiig the Whole, the Part, ad the Percet or i their math jourals, ad the highlight what they are to fid. Next, they write a word phrase or a umber expressio to represet the situatio. Studets do ot fid the solutios at this time. The problem will be solved i the ext step. 4. Ask a studet to share his or her respose ad record it o the board. Ecourage studets to cofirm or questio the respose. Durig this discussio time, address ay questios studets have, ad correct ay errors with a markig pe. 5. Cotiue with the ext situatio, util sufficiet examples have bee explored. Iclude several examples of each of the followig three types of situatios studets will ecouter i solvig problems with percet: A desigated percet of a desigated umber is what umber? Example: There are 80 cars i a shipmet, ad 40% of them are silver. How may are silver? 40% of 80 is. Number 59

86 Part B A desigated umber is what percet of aother desigated umber? Example: There are 60 cars i a shipmet, ad 15 of them are red. What percet are red? 15 is % of 60. A desigated umber is a desigated percet of what umber? Example: 25% of the cars i a shipmet are blue. There are 50 blue cars. How may cars are i the shipmet? 50 is 25% of. 1. Retur to the first situatio preseted, ad ask studets to fid the solutio. 2. After studets have had sufficiet time to fid the solutio, ask idividual studets to share their respose ad describe the strategy they used to fid the solutio. I the class discussio, ecourage studets to cofirm or questio the shared resposes, ad to suggest alterative strategies. Address ay questios studets have, ad correct ay errors with a markig pe. 3. Solve all the situatios i this maer. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express a percet as a decimal or fractio. r Solve a problem that ivolves fidig a percet. Suggestios for Istructio Express a percet as a decimal or fractio. Solve a problem that ivolves fidig a percet. Materials: BLM 7.N.3.7: Fidig the Missig Numbers i the Percet (Scearios) Orgaizatio: Idividuals, small groups or whole class (for sharig resposes) 60 Grade 7 Mathematics: Support Documet for Teachers

87 Procedure: 1. Distribute copies of BLM 7.N.3.7: Fidig the Missig Numbers i the Percet (Scearios). 2. Ask studets to complete three of the percet problems preseted o the BLM, oe problem requirig them to fid the whole, oe to fid the part, ad oe to fid the percet. Remid studets to idetify the whole, the part, ad the percet of each problem, before attemptig a solutio (as they did i the previous learig activity). They may fid it helpful to create a expressio to summarize the problem. Havig them show two strategies to solve each problem will help them stay flexible i their problem-solvig strategies. 3. Whe studets have had sufficiet time for their idividual work, have them meet i small groups or as a whole class to share their strategies ad aswers. Have studets discuss their preferred strategy for each problem. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express a percet as a decimal or fractio. r Solve a problem that ivolves fidig a percet. Suggestios for Istructio Determie the aswer to a percet problem where the aswer requires roudig, ad explai why a approximate aswer is eeded (e.g., total cost icludig taxes). Materials: BLM 7.N.3.8: Percet Problems demostratio board math jourals or otebooks markig pes Orgaizatio: Whole class (for demostratios), idividual or small groups (for practice) Procedure: 1. Explai to studets that some problems cotai situatios that require the solutio to be rouded for practical reasos. Number 61

88 2. Review decimal, fractio, ad percet equivalets with a quick oral quiz or a cotest (e.g., a spellig bee), with teams lied up perpedicular to the board. Each team seds a represetative to the board. The represetative writes the decimal, fractio, or percet equivalets o the board. If the respose is correct, the player returs to the ed of the lie; if the respose is icorrect, the player steps out of the lie ad watches. The last team with a player at the board wis. 3. Preset the class with a problem, which may or may ot require roudig. Example: Your gradmother is goig to buy a hooded sweatshirt for your birthday. She heads dow to the Pretty Tredy Clothig Store, ad fids that the store has a aiversary special. All regular prices are reduced by 20%. She selects a sweatshirt that is regularly priced at $59.99 ad pays GST of 5%. How much does she pay for the sweatshirt? 4. Ask studets to solve the problem i their math jourals or otebooks. 5. After studets have had sufficiet time to work o the problem, ask idividuals to share the strategies they used to solve the problem. Compare studets solutios, otig whether or ot studets used roudig, ad discuss reasos for their decisios. Discuss which strategies studets prefer for this problem. Esure that studets cosider the optio of calculatig the remaiig cost versus calculatig the value of the sale ad subtractig it from the origial price (80% of $59.99 versus $ % of $59.99). Durig the discussio, have studets make chages or add commets to their work with a markig pe. They may make a math joural etry suggestig hits for solvig problems with percets. 6. Follow the same procedure for a few more problems such as the oes listed o BLM 7.N.3.8: Percet Problems. Variatio: After presetig situatios requirig roudig ad providig a few examples, have studets create their ow problems ad solutio keys. The have studets exchage problems with group members, solve the problems, ad later reassemble as a group to discuss solutios. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express a percet as a decimal or fractio. r Solve a problem that ivolves fidig a percet. r Determie the aswer to a percet problem where the aswer requires roudig, ad explai why a approximate aswer is eeded. 62 Grade 7 Mathematics: Support Documet for Teachers

89 Number (7.N.4) Edurig Uderstadig(s): Percets, fractios, decimals, ad ratios are differet represetatios of the same quatity. Number sese ad metal mathematics strategies are used to estimate aswers ad lead to flexible thikig. Circle graphs show a compariso of each part to a whole usig ratios. Geeral Learig Outcome(s): Develop umber sese. Specific Learig Outcome(s): 7.N.4 Demostrate a uderstadig of the relatioship betwee repeatig decimals ad fractios, ad termiatig decimals ad fractios. [C, CN, R, T] Achievemet Idicators: Predict the decimal represetatio of a 1 fractio usig patters e.g., = 009., =., =? Match a set of fractios to their decimal represetatios. Sort a set of fractios as repeatig or termiatig decimals. Express a fractio as a termiatig or repeatig decimal. Express a repeatig decimal as a fractio. Express a termiatig decimal as a fractio. Provide a example where the decimal represetatio of a fractio is a approximatio of its exact value. Prior Kowledge Studets should be able to do the followig: Q Q (5.N.2) Apply estimatio strategies, icludig frot-ed roudig compesatio compatible umbers i problem-solvig cotexts. Number 63

90 Q Q Q Q Q Q Q Q Q Q Q Q Q Q (5.N.4) Apply metal mathematics strategies for multiplicatio, such as aexig, the addig zeros halvig ad doublig usig the distributive property (5.N.6) Demostrate a uderstadig of divisio (3-digit umerals by 1-digit umerals) with ad without cocrete materials, ad iterpret remaiders to solve problems. (5.N.8) Describe ad represet decimals (teths, hudredths, thousadths) cocretely, pictorially, ad symbolically. (5.N.9) Relate decimals to fractios (teths, hudredths, thousadths). (5.N.10) Compare ad order decimals (teths, hudredths, thousadths) by usig bechmarks place value equivalet decimals (6.N.1) Demostrate a uderstadig of place value for umbers greater tha oe millio less tha oe-thousadth (6.N.3) Demostrate a uderstadig of factors ad multiples by determiig multiples ad factors of umbers less tha 100 idetifyig prime ad composite umbers solvig problems ivolvig factors or multiples Related Kowledge Studets should be able to do the followig: Q Q Q Q Q Q (7.N.2) Demostrate a uderstadig of the additio, subtractio, multiplicatio, ad divisio of decimals to solve problems (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). (7.N.7) Compare ad order fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals (7.PR.1) Demostrate a uderstadig of oral ad writte patters ad their correspodig relatios. 64 Grade 7 Mathematics: Support Documet for Teachers

91 Q Q Q Q (7.SS.1) Demostrate a uderstadig of circles by describig the relatioships amog radius, diameter, ad circumferece of circles relatig circumferece to pi determiig the sum of the cetral agles costructig circles with a give radius or diameter solvig problems ivolvig the radii, diameters, ad circumfereces of circles (7.SP.4) Express probabilities as ratios, fractios, ad percets. Backgroud Iformatio I previous grades, studets leared that fractios ad decimals are iterchageable ames for the same quatity. I Grade 7 (learig outcome 7.N.3), studets review multiple meaigs for the term fractio ad lear strategies for fidig decimal ad fractio equivalets. For learig outcome 7.N.4, studets will frequetly use calculators ad the uderstadig that a fractio also represets divisio to fid the decimal equivalets for fractios that are ot coveietly reamed i other ways (e.g., 1 may be read as 1 8, which equals 0.125). 8 Fractio ad Decimal Equivalets All fractios have equivalet decimal ames. The decimal ames may have a defiite umber of digits. These are termiatig decimals. A termiatig decimal ca be easily reamed as a fractio with a deomiator that is a power of 10 (e.g., 0.125, read as 125 thousadths, ad writte as a fractio , which ca be simplified to 1 8 ). Whe some fractios are reamed as decimals, the decimal umber cotais oe or more digits that repeat i a cotiuous patter idefiitely (e.g., 1 = ). These 3 are repeatig decimals. The three dots idicatig the digits cotiue without ed are called a ellipsis. I North America, the commo represetatio for repeatig decimals is to write the umber with oe set of the repeatig digits, ad the draw a bar over the digits that form the repeatig patter 03.. a period. The bar is called a viculum. Other otatios iclude placig a dot over the digits at each ed of the repeatig sequece ( 03..,) or eclosig the repeatig sequece i paretheses [0.(3)]. Repeatig decimals may also be reamed 1 as fractios. Characteristic patters may 3 be used to predict the decimal represetatio of these fractios ad to predict the fractio represetatio of repeatig decimals. ( ) The series of digits that repeat may be called Note: Studets from varyig cultural backgrouds may have differet covetios for represetig the repeatig sequece. Studets should be aware of all covetios, but should ot be required to memorize the represetatios. Number 65

92 Provide studets with learig activities that lead them to discover iterestig patters i the relatioships betwee fractio ad decimal equivalets. Ivestigatig the patters provides a opportuity to explore umber sese. The situatio with iths may lead to a questio of 09. versus 1. (Some patters are listed for teacher referece i a table o the ext page.) To determie whether a fractio will result i a termiatig or repeatig decimal, simplify the fractio ad cosider the prime factors of the deomiator. If the oly prime factors of the deomiator are 2s ad/or 5s, the fractio will have a termiatig decimal equivalet. The umber of 2s ad 5s may be used to predict the umber of place value positios i the decimal equivalet. The relatioship is described below. If the deomiator cotais prime factors other tha 2 or 5, the decimal umber will be a repeatig decimal. The maximum umber of digits that may repeat will be oe less tha the deomiator. This is sometimes the case whe the deomiator is a prime umber. The decimal equivalets for fractios with the prime deomiator 7 form a cyclic repeatig decimal patter with six repeatig digits (142857). To predict the umber of digits i a termiatig decimal umber, simplify the fractio ad express the deomiator as a product of prime factors. Cout the umber of 2s ad the umber of 5s i the product. Determie whether there are more 2s or more 5s. The umber of times the most frequetly occurrig digit occurs i the prime product equals the umber of place value positios i the decimal umber. Example: writte as a product of prime factors = There are three 2s i the product ad three decimal places i the decimal equivalet for 1, which is = writte as a product of prime factors = 2 2. There are two decimal places i the decimal equivalet Note: These are iterestig patters for studets to discover, but studets are ot required to memorize these relatioships. 66 Grade 7 Mathematics: Support Documet for Teachers

93 Some fractios ad their repeatig decimal equivalets are listed i the followig table for referece. Give studets opportuities to discover these patters. Deomiator of the Fractio 7ths Fractios ad Their Repeatig Decimal Equivalets Patter i the Repeatig Decimal six repeatig digits digits are i a cyclic patter 9ths sigle repeatig digit 99ths 999ths the umerator is the repeatig digit two repeatig digits the umerator is the repeatig sequece three repeatig digits the umerator is the repeatig sequece 11ths two repeatig digits that are a multiple of 9 the umerator is the factor 9 that equals the repeatig sequece Example Roudig a decimal equivalet results i a approximatio of the value of the fractio. Cotexts or circumstaces may dictate that a decimal umber must be rouded to a specified umber of digits. May calculators roud to the fial digit i their display (e.g., a studet s score of 12 of a metre may be measured 18 may be reported as 67%, 2 3 as 67 cm). Each of these situatios represets a approximatio of the true value of the umber. To express a exact value for a repeatig decimal, idicate the repeatig sectio with a viculum, or write the fractio equivalet. To idicate that the umber is a approximatio of the true value, use a tilde mark (~), a approximately equal sig (»), or a equal sig with a dot over it ( ). Note: Similar to the otatio for repeatig decimals, studets may have had prior exposure to oe of these represetatios. All represetatios should be cosidered correct. Number 67

94 Mathematical Laguage decimal equivalet decimal umber deomiator factor fractio fractio equivalet multiple umerator prime umber product of prime factors repeatig decimal simplify a fractio to lowest terms termiatig decimal Optioal laguage that may be used by teachers, but is ot required of studets ellipsis tilde uit fractio viculum 68 Grade 7 Mathematics: Support Documet for Teachers

95 Learig Experieces Assessig Prior Kowledge Materials: math otebooks calculators (optioal) Orgaizatio: Idividual, pairs Procedure: 1. Review the meaig of a prime umber, factors, prime factors, ad writig a umber as the product of prime factors.* 2. Have each studet use each of the digits 0 to 9 oce to write five 2-digit umbers i his or her math otebook. 3. Next, have studets write each of their umbers as a product of prime factors. 4. Whe studets are fiished with their idividual work, ask them to exchage otebooks with a parter. Have them verify that the umbers i the product are all prime umbers ad that the product is equal to the origial umber. Variatios: Supply umbers ad a template for fidig the prime factors. Have studets create 3-digit umbers, usig each digit o more tha twice. Observatio Checklist * Note: I Grade 7, studets are ot formally exposed to prime factorizatio. It is a achievemet idicator i Grade 8 Mathematics i the study of squares ad square roots, ad a learig outcome i Grade 10 Itroductio to Applied ad Pre-Calculus Mathematics. Provide studets with guided support whe prime factorizatio is required. Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify factors ad multiples of a umber correctly. r Idetify prime factors correctly, with guidace. r Use metal mathematics strategies, icludig divisibility rules, ad usig calculators whe required. Number 69

96 Assessig Prior Kowledge Materials: demostratio board, chalk or markers, erasers a list of umbered divisio questios specifyig the format of the aswers (e.g., fractio remaider, umber of decimal places) ad the aswers copies of each umbered questio o idividual papers (the umber of copies equals the umber of teams) a reward for the wiers (optioal) Orgaizatio: Teams groups of three (of mixed ability) are ideal, but the umber depeds o the size of the board space available ad the umber of studets i the class. Procedure: 1. Have teams lie up perpedicular to the board. The first players o all teams positio themselves at the board ad draw a poit box at the top of the board, while the other players remai about two metres behid (or whatever distace works). Remid players to write large eough ad high eough so you ca see their work. 2. Provide the first player o each team with a divisio questio, ad state the form for expressig the aswer (e.g., fractio remaider, umber of decimal places). Studets record their respective questios o the board, fid the solutios, ad draw a box aroud the quotiets whe their resposes are complete. Establish a sigal that studets ca use to get your attetio for assessig their resposes. 3. Respod to a studet s sigal ad verify his or her respose. If the respose is correct, the studet records a poit i the team s poit box. This player the comes to get the ext questio from you, dictates the questio to the ext team member i lie, ad the returs to the ed of the lie for his or her team. 4. The ext team member goes to the board, ad repeats the process. 5. Whe studets have had sufficiet time to work o the divisio questios, ed the game, ad declare the wiig team to be the oe with the most poits. Provide a reward (optioal). 70 Grade 7 Mathematics: Support Documet for Teachers

97 Variatios: Establish ay desired rules about obtaiig help from teammates. Have all teams work o the same questio at the same time. You decide the time available to complete the questio, call out the time, ad provide the aswer. Studets award their team with a poit if their aswer is correct. Players retur to the ed of their team lie, ad you call out the ext questio. Have studets create the questios ad aswer keys. Have studets call out the questios ad the aswers, givig you more time to observe. Have teams complete the work o large pieces of scrap paper at table groups ad show their work o request. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Apply a appropriate divisio algorithm. r Recogize that whe a remaider repeats, the quotiet has etered a repeatig patter. r Express a fractio as a termiatig or repeatig decimal. r Sort a set of fractios as repeatig or termiatig decimals. r Provide a example where the decimal represetatio of a fractio is a approximatio of its exact value. Number 71

98 Suggestios for Istructio Predict the decimal represetatio of a fractio usig patters e.g., 1 = = ,., =? Sort a set of fractios as repeatig or termiatig decimals. Express a fractio as a termiatig or repeatig decimal. Express a repeatig decimal as a fractio. Express a termiatig decimal as a fractio. Provide a example where the decimal represetatio of a fractio is a approximatio of its exact value. Materials: BLM : Base-Te Grid Paper base-10 blocks (10 flats, 10 rods, ad 10 uits for each studet) three-dimesioal or paper models small strips to represet a uit block divided ito 10 colums ad o rows three food items (e.g., cady bars, graola bars), of uiform size, that ca easily be cut ito equal portios (or eough to share with the whole class) demostratio board math otebooks calculators (at least oe of which rouds repeatig decimals) Orgaizatio: Whole class Procedure: 1. Lead a class demostratio ad discussio about represetig a fractio quatity with a base-10 place value system. Use base-10 blocks, ad iclude the cocepts of termiatig ad repeatig decimals, usig divisio to represet a fractio, otatios that represet exact quatities, ad approximatio. 2. The purpose of umbers is to provide a way to commuicate about quatity. Share a cady bar equally betwee two studets. Ask how much of the cady bar each oe received. Write 1 o the board. Oe-half of the cady bar describes exactly what 2 each studet received. It s a exact umber. Ask whether it is possible to ame that umber with the base-10 place value system. 72 Grade 7 Mathematics: Support Documet for Teachers

99 3. Have studets demostrate the actio with base-10 models. The flat represets oe cady bar. The place value system dictates that if the flat is cut, it must be cut ito 10 equal pieces. Review that there are 10 pieces (use the digits 0 to 9) i each place value positio. If studets have ot used base-10 blocks to represet decimals or fractios before, you may eed to make it very clear that they are usig the blocks differetly here tha whe they used them for whole-umber operatios. They are usig them because they are icely cut ito teths. If oe flat represets oe cady bar, ad idividuals are goig to share it, the oe flat must first be cut ito 10 pieces. Do t actually cut the block; istead, swap it for 10 rods. Each rod represets oe-teth of oe. Now share the cady bar block with two imagiary people. How much does each oe receive? Each oe receives five rods, which are amed five-teths, ad writte as 0.5. Record 0.5 o the board. It, too, is a exact umber. Five-teths describes the exact quatity here. 4. Repeat the process (steps 2 ad 3) with the secod cady bar, sharig it amog three studets. Ask how much cady bar each oe received. Record 1 o the board. Oethird of the cady bar describes exactly what each studet received. It s a exact 3 umber. Ask whether we are able to ame that umber with the base-10 place value system. 5. Have studets model sharig their flat cady bar with three imagiary people. Each will get three-teths, ad oe-teth will be left over. Explore optios for writig the umber. Ca we write 0.3 as 1? Record this questio o the board with 3 a questio mark. That would be mixig two laguages i the same word, which would be cofusig. Go back ad cut up the teth rod ad share the pieces. Remember, the oly possible cut is ito 10 equal pieces. Swap the rod for 10 small cubes. This results i 0.33 ad oe-teth of a teth left to share amog three. You eed to cut that piece ito 10 agai i order to share it. You are out of blocks, so use the grid paper to represet the actio. The grid represets oe little block cut ito 10 pieces. Share the little pieces, ad oe will be left agai. Studets should realize this is goig to go o forever. A similar situatio was ecoutered whe studets explored divisibility rules for 3. Explore possibilities for writig the umber. Itroduce the term repeatig decimal umber ad the cocept of drawig a bar over the 3 as a otatio that this 3 repeats forever. 6. Repeat the process (steps 2 ad 3), with the third cady bar shared amog four studets. Write 1 o the board, ad the Both are exact umbers. They 4 describe the exact quatity. 7. Note that 1 is amed a repeatig decimal because the sharig is ever complete, as 3 the value 3 repeats i every place value positio idefiitely. A repeatig decimal may have more tha oe repeatig digit, but the patter will ever stop. Observe that 1 1 ad were shared completely i the place value model. The umerals had 2 4 a defiite umber of digits. Decimal umbers with a defiite umber of digits are called termiatig decimals. Termiatig decimals may have may digits, but there is o repeatig patter, ad they stop, or there is zero repeatig. Number 73

100 8. I this learig activity, the actios of cuttig ad sharig the bars deote divisio. We ca read fractios as divisio operatios ad obtai ames for the fractio i our place value system. Demostrate log divisio o the board, or have studets use their math otebooks to perform log divisio for 1 2, 1 3, ad 1 4, ad compare their results to the models. Ask studets to obtai a decimal ame for 1 6. Ask whether these are exact ames for the fractios. 9. Ask studets to use calculators to covert the uit fractios from 1 1 to to decimal 2 6 umbers, ad have them record the equivalets i a table i their math otebooks. Ask studets to share the results. Some studets will likely have calculators that roud 1 6 to Note: Roudig a decimal umber, or failig to provide a idicatio that the decimal repeats, is a approximatio, ot a exact value, ad it ought to be oted as such. It also shows studets whether the umber of digits equals or exceeds the capacity of the calculator display. Studets caot rely o the calculator to verify whether a fractio is represeted as a termiatig or repeatig decimal. Variatio: Use smaller decimal examples, sharig with fifths, sixths, or eighths. With these variatios, the model is a little more tedious to use ad the cady bar pieces are small. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Predict the decimal represetatio of a fractio usig patters 1 e.g., =., =., =? r Sort a set of fractios as repeatig or termiatig decimals. r Express a fractio as a termiatig or repeatig decimal. r Express a repeatig decimal as a fractio. r Express a termiatig decimal as a fractio. r Provide a example where the decimal represetatio of a fractio is a approximatio of its exact value. 74 Grade 7 Mathematics: Support Documet for Teachers

101 Suggestios for Istructio Express a fractio as a termiatig or repeatig decimal. Provide a example where the decimal represetatio of a fractio is a approximatio of its exact value. Materials: coloured couters chart paper Orgaizatio: Pairs, whole class Procedure: 1. Have pairs of studets make up oe multiple-choice questio to ask the class, or a target group withi the class. Sample Questios: For all studets: Is your favourite seaso sprig, summer, fall, or witer? For the girls: As a luchtime activity, do you prefer itramurals, school goverace, free time, homework help, or games club? For the boys: If you could have a superpower, would it be time travel, rocket speed, or immortality? 2. Have studets collect the data from the class or target group. 3. Provide studets with access to coloured couters, ad have them represet the collected data i the form of a circle graph. Oce you have had a chace to look at their circle graphs, ask studets to represet their circle graphs o chart paper, labellig the titles of each piece of the pie o the frot of the chart paper ad recordig their resposes to the followig questios o the reverse side of the chart paper for future referece: a) What percet of this class or target group is represeted by each piece of the pie? b) What fractio would best represet each piece of the pie? c) What decimal would best represet each piece of the pie? d) Make a statemet about the largest sectio of the pie or circle graph. Note: Studets work i this learig activity is meat to address repeatig ad termiatig decimals ad their fractioal equivalets, ad should remai geeral with respect to circle graphs. Learig outcome 7.SP.3 relates specifically to circle graphs. Number 75

102 4. Have studets rotate, i pairs, to the circle graphs created by the rest of the class ad aswer the followig questios: a) What percet of this class or target group is represeted by each piece of the pie? b) What fractio would best represet each piece of the pie? c) What decimal would best represet each piece of the pie? d) Make a statemet about the largest sectio of the pie or circle graph. 5. Gather as a class ad discuss the followig questios: a) What ca be said about represetig the data o a circle graph? b) Which is the best way to view this data (as fractios, decimals, or percets)? c) Are exact or approximate values eeded o the circle graph to represet the fractio of studets? Give examples of whe a exact represetatio is eeded ad whe a approximate represetatio is eeded. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express a fractio as a repeatig decimal. r Express a fractio as a termiatig decimal. r Represet a umber i a variety of ways. r Estimate a value based o a pictorial represetatio. r Commuicate mathematically. Suggestios for Istructio Predict the decimal represetatio of a fractio usig patters e.g., 1 = = ,., =? Sort a set of fractios as repeatig or termiatig decimals. Express a fractio as a termiatig or repeatig decimal. Materials: BLM 7.N.4.1: Table for Recordig Fractios ad Their Decimal Equivalets (or aother chart for recordig fractio-decimal equivalets, such as the oe studets bega i their math otebooks i the previous learig activity) demostratio board calculators Ve diagram or T-chart (optioal) 76 Grade 7 Mathematics: Support Documet for Teachers

103 Orgaizatio: Whole class, idividual, small groups Procedure: I the previous learig activity, studets idetified the decimal equivalets for the uit fractios 1 1 to ad classified them as repeatig or termiatig decimals. They used 2 6 calculators ad log divisio to cofirm their classificatios ad oted the differece betwee exact ad approximate represetatios. I this learig activity, studets ivestigate patters i the uit fractios 1 1 to to develop geeralizatios useful for 2 20 predictig termiatig or repeatig decimals ad expressig fractio ad decimal equivalets. 1. Review the cocepts addressed i the previous learig activity. Stimulate studet curiosity by askig if they thik it is possible to predict whether a decimal represetatio will termiate or repeat. 2. Lauch a iquiry task to fid a aswer to the questio of predictability. Lookig at examples ad discoverig patters gives a basis for makig predictios that ca the be cofirmed ad geeralized ito rules. Possible steps that studets ca follow for the iquiry task are listed below. a) Specify the questio to ivestigate. (How do you predict whether a fractio umber is represeted by a termiatig or repeatig decimal umber?) b) Select a elemet that you thik may be the determiig factor (e.g., umerator or deomiator). It is ot possible to examie all elemets at oce, so select oe to ivestigate. The oly elemets i a fractio are the umerator ad the deomiator. Look at the list created so far for a hit as to which oe to choose. The fractio 1 3 repeats, as does 1 6. What about 2 2 or? Fractio ames for eed ot be cosidered because they Note: The importat thig here is to have studets ivestigate the deomiator as the determiig factor. all equal 1.0. The fractio 1 4 termiates. What about 2 3 ad? 4 4 c) Geerate a list of examples that isolate the chose factor. (Suggest begiig with uit fractios with deomiators 2 to 10 ad umerators of 1.) d) Examie the list of examples for a commo elemet or patters that describe the relatio. Use a Ve diagram or T-chart to orgaize fidigs. (These repeatig decimals have deomiators 3, 6, 7, ad 9. The termiatig decimals have deomiators 2, 4, 5, 8, ad 10. Notice 2, 4, ad 8 are all divisible by 2, ad 5 ad 10 are divisible by 5. To add more examples, exted the list to deomiators to 20. A group effort is advised for 17 ad 19, as these deomiators have a log repeatig period. Studets may be excited to ote that the deomiators of all the termiatig decimals have 2 ad/or 5 as factors. See the 2 5 = 10 base-10 coectio.) Number 77

104 e) Create a descriptive phrase o which to base predictios. (Termiatig decimals have deomiators that are multiples of 2 ad/or 5. The deomiators have o prime factors other tha 2 ad/or 5. Repeatig decimals have deomiators with prime factors other tha 2 or 5.) f) Test your predictios with several ew examples. (Ay fractios will do. Cosider icludig various umerators such as 8 11 ad, as well. Studets should ote that equivalet fractios have the same decimal represetatios.) g) If all predictios are correct, it s time to create a rule. Be ope to the fact that some examples may disprove your rule, ad you will eed to begi at step (d) agai, or ote exceptios to your rule. (Esure that, at some poit, studets realize a fractio must be simplified to apply their rule.) 3. Record ad celebrate studets iquiry fidigs by playig a termiatig or repeatig decimal game, such as the oe suggested i the ext learig activity. Variatios: Guide the whole class as a group, or have studets work idividually or i small groups based o studets ability to coduct iquiries. Provide templates to guide sample selectios, ad provide questios to prompt geeralizatios for iquiries. Supply studets with the rule, ad explai how it is based o factors. Have them use prime factor deomiators i several predetermied fractios to determie whether the fractios are represeted by termiatig or by repeatig decimals. Followig the iquiry, preset studets with a list of fractios, ad ask them to sort the fractios accordig to whether they have termiatig or repeatig decimal equivalets. Have studets supply prime factors as evidece of their decisio. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Express a fractio as a termiatig or repeatig decimal. r Sort a set of fractios as repeatig or termiatig decimals. r Express a fractio as a termiatig or repeatig decimal. r Make coectios amog repeatig decimals, termiatig decimals, ad place value. 78 Grade 7 Mathematics: Support Documet for Teachers

105 Suggestios for Istructio Sort a set of fractios as repeatig or termiatig decimals. Express a fractio as a termiatig or repeatig decimal. Materials: tic-tac-toe frames or grids (of various sizes), such as the followig: BLM 7.N.3.1A: Tic-Tac-Toe Frames BLM 7.N.3.1B: Tic-Tac-Toe Frames (Medium Challege) BLM 7.N.3.1C: Tic-Tac-Toe Frame (Ultimate Challege) calculators otebook paper pes of two differet colours Orgaizatio: Pairs Procedure: Pairs of studets play a tic-tac-toe game o a grid of ay size. The object is to create a lie of fractios whose equivalets are either termiatig or repeatig decimals ad to moitor the fractios played by the oppoet to verify whether they represet termiatig or repeatig decimal umbers. 1. The players each choose a colour ad decide who will play fractios represeted by termiatig decimals, ad who will play fractios represeted by repeatig decimals. They decide o the size of grid o which to play, ad who will go first. 2. O the first move, a player selects which square to play i, creates a fractio represeted by her or his type of decimal represetatio, ad writes the fractio clearly i the selected square. 3. The oppoet verifies the play, usig a calculator if ecessary. If the player has played a fractio represeted by a termiatig decimal istead of a repeatig decimal, or vice versa, the oppoet may capture the play by circlig or re-colourig the fractio with his or her ow colour. 4. The ext player plays, repeatig steps 2 ad Play cotiues util oe player wis the roud by coectig a horizotal, vertical, or diagoal lie of repeatig or termiatig decimals. If a player fails to otice a coectig lie o his or her tur, the oppoet may draw the coectio ad declare himself or herself the wier. If the challeger is icorrect, the other player is a double wier. Number 79

106 6. Studets have two optios to ed the game: a) The game is over after a specified amout of time has passed. b) Play stops whe oe player reaches a target umber of wis. Variatios: Vary the size of the grid the players use for the game. Have studets desig their ow game. Prepare a list of fractios. Have studets sort the fractios ito two groups, repeatig decimals or termiatig decimals. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Sort a set of fractios as repeatig or termiatig decimals. r Express a fractio as a termiatig or repeatig decimal. Suggestios for Istructio Predict the decimal represetatio of a fractio usig patters e.g., 1 = = ,., =? Sort a set of fractios as repeatig or termiatig decimals. Express a fractio as a termiatig or repeatig decimal. Express a repeatig decimal as a fractio. Express a termiatig decimal as a fractio. Materials: BLM 7.N.4.1: Table for Recordig Fractios ad Their Decimal Equivalets (or aother chart for recordig fractio-decimal equivalets, such as the oe studets bega i their math otebooks i a previous learig activity) demostratio board calculators Ve diagrams or T-charts (optioal) paper (small size), colours, glue, scissors, or computer techology for posters (optioal) 80 Grade 7 Mathematics: Support Documet for Teachers

107 Orgaizatio: Idividual, small group, or whole class (depedig o the iterest ad iquiry ability of studets i the class) Procedure: 1. Ask studets to suggest other predictios that may be iterestig or useful to ivestigate. Ca we predict the umber of digits that will appear i the decimal, or i the repeatig period? Ca the idetity of the digits i a fractio decimal equivalet be predicted? Be sure to ivestigate fractios with deomiators of 9, 99, 999, ad 11. The patters of deomiators 90 ad 7 are also iterestig ivestigatios. 2. Have studets coduct iquiries to fid patters o which to base predictios. Geeral rules for each ivestigatio are listed below. a) Specify the questio to ivestigate. b) Select a elemet that you thik may be the determiig factor. c) Geerate a list of examples that isolate the chose factor. d) Examie the list of examples for a commo elemet or patters that describe the relatio. e) Create a descriptive phrase o which to base predictios. f) Test your predictio with several ew examples. This may lead back to step (d). g) Create a rule or geeralizatio. 3. Whe studets have a set of rules or geeralizatios, ask whether they could make predictios about the fractio represeted by a decimal umber. From all their observatios, studets should ote the reverses. For repeatig decimals, if 07. is a equivalet represetatio of 7 7,the is the equivalet represetatio of Have studets share their geeralizatios ad create posters that outlie geeral rules about covertig fractio ad decimal equivalets. Variatios: Provide studets with templates to guide sample selectios, ad questios to prompt geeralizatios for iquiries. Provide studets with a list of geeralizatios. Guide them through examples that prove each geeralizatio. Number 81

108 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Predict the decimal represetatio of a fractio usig patters 1 e.g., =., =., =? r Sort a set of fractios as repeatig or termiatig decimals. r Express a fractio as a termiatig or repeatig decimal. r Express a repeatig decimal as a fractio. r Express a termiatig decimal as a fractio. r Commuicate mathematical uderstadig. r Reaso i order to make coectios to prior uderstadig. Suggestios for Istructio Predict the decimal represetatio of a fractio usig patters e.g., 1 = = ,., =? Match a set of fractios to their decimal represetatios. Sort a set of fractios as repeatig or termiatig decimals. Express a fractio as a termiatig or repeatig decimal. Express a repeatig decimal as a fractio. Express a termiatig decimal as a fractio. Materials: calculators a list of geeralizatios about covertig fractio ad decimal equivalets (created i previous learig activities) BLM 7.N.3.4: Choose Your Questio paper (small size), colours, glue, scissors, or computer techology for posters (optioal) computers or other techology (optioal) Orgaizatio: Idividual or small groups 82 Grade 7 Mathematics: Support Documet for Teachers

109 Procedure: As a culmiatig activity, have studets create ad participate i a variety of games that will help them to demostrate a uderstadig of the relatioship betwee repeatig decimals ad fractios, ad termiatig decimals ad fractios. 1. Have studets use the geeral rules about covertig fractio ad decimal equivalets that they created previously to create a Fractio ad Decimal Expressios Trivia game, with players havig to collect desigated poits from each category. Alteratively, have studets use the templates from BLM 7.N.3.4: Choose Your Questio to create a Choose the Questio ad Poits game i a effort to score the highest poits. Studets may also wish to play some other game of their choice. 2. The game categories for the selected game are as follows: fractio/decimal patters sortig fractios as repeatig or termiatig decimals expressig a fractio as a termiatig or repeatig decimal expressig a repeatig decimal as a fractio expressig a termiatig decimal as a fractio 3. The class could host a grad competitio. Variatios: Have studets create a challege quiz sheet ad a aswer key with a assortmet of termiatig ad repeatig decimals ad fractio represetatios. Create criteria for the quiz, icludig questios about fractio/decimal patters sortig fractios as repeatig or termiatig decimals expressig a fractio as a termiatig or repeatig decimal expressig a repeatig decimal as a fractio expressig a termiatig decimal as a fractio Studets may use computers or other techology to create their quiz. Photocopy the quiz sheets, ad ask studets to exchage challeges with a parter. The parters complete ad correct each other s challeges. Number 83

110 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Predict the decimal represetatio of a fractio usig patters 1 e.g., =., =., =? r Sort a set of fractios as repeatig or termiatig decimals. r Express a fractio as a termiatig or repeatig decimal. r Express a repeatig decimal as a fractio. r Express a termiatig decimal as a fractio. 84 Grade 7 Mathematics: Support Documet for Teachers

111 Number (7.N.5) Edurig Uderstadig(s): The priciples of operatios used with whole umbers also apply to operatios with decimals, fractios, ad itegers. Number sese ad metal mathematics strategies are used to estimate aswers ad lead to flexible thikig. Geeral Learig Outcome(s): Develop umber sese. Specific Learig Outcome(s): 7.N.5 Demostrate a uderstadig of addig ad subtractig positive fractios ad mixed umbers, with like ad ulike deomiators, cocretely, pictorially, ad symbolically (limited to positive sums ad differeces). [C, CN, ME, PS, R. V] Achievemet Idicators: Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. Determie the sum of two positive fractios or mixed umbers with like deomiators. Determie the differece of two positive fractios or mixed umbers with like deomiators. Determie a commo deomiator for a set of positive fractios or mixed umbers. Determie the sum of two positive fractios or mixed umbers with ulike deomiators. Determie the differece of two positive fractios or mixed umbers with ulike deomiators. Simplify a positive fractio or mixed umber by idetifyig the commo factor betwee the umerator ad deomiator. Simplify the solutio to a problem ivolvig the sum or differece of two positive fractios or mixed umbers. Solve a problem ivolvig the additio or subtractio of positive fractios or mixed umbers, ad determie if the solutio is reasoable. Number 85

112 Prior Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q Q Q (3.N.13) Demostrate a uderstadig of fractios by explaiig that a fractio represets a portio of a whole divided ito equal parts describig situatios i which fractios are used comparig fractios of the same whole with like deomiators (4.N.5) Describe ad apply metal mathematics strategies, such as skip-coutig from a kow fact usig doublig or halvig usig doublig ad addig oe more group usig patters i the 9s facts usig repeated doublig to develop recall of basic multiplicatio facts to 9 9 ad related divisio facts. (4.N.8) Demostrate a uderstadig of fractios less tha or equal to oe by usig cocrete ad pictorial represetatios to ame ad record fractios for the parts of a whole or a set compare ad order fractios model ad explai that for differet wholes, two idetical fractios may ot represet the same quatity provide examples of where fractios are used (5.N.4) Apply metal mathematics strategies for multiplicatio, such as aexig, the addig zeros halvig ad doublig usig the distributive property (5.N.7) Demostrate a uderstadig of fractios by usig cocrete ad pictorial represetatios to create sets of equivalet fractios compare fractios with like ad ulike deomiators (6.N.3) Demostrate a uderstadig of factors ad multiples by determiig multiples ad factors of umbers less tha 100 Q Q idetifyig prime ad composite umbers solvig problems ivolvig factors or multiples (6.N.4) Relate improper fractios to mixed umbers. 86 Grade 7 Mathematics: Support Documet for Teachers

113 Q Q Q Q (6.N.5) Demostrate a uderstadig of ratio, cocretely, pictorially, ad symbolically. (6.N.6) Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. Related Kowledge Studets should be able to do the followig: Q Q Q Q (7.N.4) Demostrate a uderstadig of the relatioship betwee repeatig decimals ad fractios, ad termiatig decimals ad fractios. (7.N.7) Compare ad order fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals Backgroud Iformatio Topic Overview Fractios are commoly used i a variety of cotexts i daily life. They are used to idicate quatities other tha a whole. Fractios ca be used to measure moey, time, ad distace. Fractio measuremets are used i costructio ad cabietry, i cookig, i art ad desig projects, ad i sports. They are used to coduct tests ad experimets, to gauge liquids, to keep track of periods at sportig evets, ad so o. Fractios are importat for sharig aythig from chocolate bars ad pies to splittig restaurat bills. I Grade 7, studets develop their mathematical literacy by extedig their coceptual uderstadig of fractios to combiig ad comparig fractioal quatities usig additio ad subtractio. Fractios ame quatities betwee whole umbers. The part betwee the wholes ca be divided ito ay umber of equal parts. The umber of equal parts i oe whole is the deomiator of the fractio, ad the umber of parts beig referred to forms the umerator. Fractios are a extesio of the whole umber system, ad the same priciples for addig ad subtractig whole umbers apply to addig ad subtractig fractios. Uderstadig operatios with whole umbers ad havig a good coceptual uderstadig of fractios provides a importat foudatio for both addig ad subtractig fractios. Whe teachig operatios with fractios, ecourage studets to focus o meaig ad to make sese of a variety of cotextual problems, regardless of whether they are workig with proper fractios or with mixed umbers. Number 87

114 May people, both studets ad adults, have ufriedly relatioships with fractios. This idicates the importace of emphasizig umber sese ad cocepts whe workig with fractios. Coceptual Uderstadigs Before studets perform operatios, it is importat to verify their coceptual uderstadigs. The term fractio has several meaigs. Fractio otatio is used to represet a cut or a part of a whole uit or regio, a part of a group or set, a measuremet, or a poit o a umber lie. It is also used to represet a ratio or a portio of a tur, ad to idicate the divisio operatio. Esure that studets ca record ad iterpret these differet meaigs. The quatity represeted by a fractio depeds o the size of the whole. For example, 1 4 of Price Edward Islad represets a differet area tha 1 4 of Quebec, ad 1 of represets a differet quatity tha 1 of The umerator, or the top umber of the fractio, idicates the umber of parts i the fractio, ad the deomiator, or the bottom umber, represets the type of part or uit size of the fractio. Whe addig or subtractig ay umbers, the value or size of uits must be the same. If the distace from a give poit to your house is 2 km, ad 500 m more to the park, it is icorrect to combie the umbers for a total distace of 502. It is ecessary to covert the measuremets to commo uits before combiig them. For example, 2 km plus 0.5 km totals 2.5 km, or 2000 m plus 500 m totals 2500 m. These distaces are equivalet. Fractios that have the same deomiator, such as 1 3, 5 5 ca easily be combied as 4, but a collectio of fractios might have differet 5 deomiators. The differet deomiators idicate differet types or uits of measure that must be related to commo referece poits, or coverted to commo uits, before addig or subtractig them. Esure that studets are able to create ad idetify equivalet ames for fractios. Some equivalet fractios, such as 1 2 ad, are easily recogizable. Proficiecy i 2 4 idetifyig commo factors ad multiples facilitates reamig less recogizable fractios. Studet will have a much easier time idetifyig factors ad multiples if they have ready access to multiplicatio ad divisio facts, ad if they ca apply divisibility rules. 88 Grade 7 Mathematics: Support Documet for Teachers

115 Fractio otatios may represet differet meaigs, ad ot all fractio meaigs ca be combied i the same maer. Whe cosiderig the cut meaig of a fractio, it is quite clear that 1 2 of a pizza ad 1 of a pizza ca be combied to form the equivalet 2 of 1 whole pizza, as log as all the pizzas are the same size. However, if you wrote a test, ad aswered 1 2 of the questios i Part A correctly ad 1 of the questios i Part B 2 correctly, you caot combie the two parts ad say you aswered 1 test correctly. I this case, the fractios are parts of sets. Whe you combie them, you icrease both the umber of selected parts ad the umber of parts i the set. The score totals 2 4. Likewise, if you represet 1 of the members i your family, ad your fried represets 1 6 of the members i her family, ad you are asked to combie these fractios, you may 3 be tempted to say =. Ask, of what? You ad your fried do ot represet 1 2 of the members of both families. These fractios represet part of a set. The sets i the addeds are ot the same set referred to i the aswer. Your fried ad you would represet 2 of both families. You eed to combie the members composig the set, as 9 well as the umerators i each set. If you wated to view the fractios as ratios, ad you wated to fid the average portio of each family you represet, you could add the fractios ad divide by 2 to obtai 1. This illustrates the importace of studets havig 4 both umber sese ad a uderstadig of the differet meaigs of fractios i order to add ad subtract them correctly. Addig ad subtractig whole umbers sometimes requires regroupig, or carryig ad borrowig. Coversios ad regroupig betwee mixed umbers ad improper fractios provide regroupig opportuities for operatios with fractios. Studets require the ability to make these coversios. Focus of Istructio Some mathematics teachig resources begi istructio with commo deomiators ad addeds with sums that are less tha oe. They the move to subtractio with commo deomiators, with the stipulatio that the smaller fractio is removed from the larger fractio. Next, they progress to questios i which oe deomiator is a multiple of the other, ad the to questios i which both deomiators must be chaged. The they move to sums larger tha oe, ad fially to mixed umbers. The progressio is logical ad icreases i complexity. Studets may, however, have difficulty with fractio operatios if they focus or deped o rememberig a sequece of steps they must follow to complete the operatio. Studets are more likely to iteralize cocepts if they have the opportuity to apply umber sese to solvig problems. Number 89

116 If studets have a strog uderstadig of fractio cocepts ad whole umber operatios, ad if teachig is focused o makig meaig i a problem-solvig cotext, it is ot ecessary to begi istructio with commo deomiators, ad follow a set progressio. Istead, preset problems as realistic scearios, ad ecourage studets to use maipulatives ad iformal methods to arrive at solutios. Use friedly fractios that ca easily be represeted with maipulatives or drawigs, ad fractios that ca easily be related to oe aother, such as quarters ad eighths. A example of a friedly fractio combiatio is thirds ad sixths. A example of a ufriedly fractio combiatio is fifths ad twelfths. Usig friedly fractios makes it easier for studets to fid equivalet uits ad helps them build cofidece i the strategies they are developig. Highlight the geeralizatios that studets make by coectig them to symbolic models for addig ad subtractig fractios. Ecourage studets to use roudig ad bechmarks to make estimates. Estimates help studets to focus o meaig ad to create target zoes for their solutios. The bechmarks of 0, 1 2, ad 1 (or 5, 5 1, 6, ad so o) are useful whe addig ad 2 subtractig fractios. Iclude problems relatig to the various meaigs of subtractio, such as take away or compare, ad problems relatig subtractio to additio by fidig the missig added. Provide studets with opportuities to share ad to assess oe aother s strategies. Cocrete ad Pictorial Models Cocrete materials iclude fractio circles, fractio bars, fractio strips combied with umber lies, metric rulers, ad metre sticks, a collectio of equivalet umbers lies with differet uit divisios, base-10 blocks, Cuiseaire rods, patter blocks, clocks, ad moey. Use commercially available products, or have studets costruct the materials by measurig them or by usig blacklie masters provided. Studets ca geerate simple pictures ad diagrams to represet fractios ad their combiatios. Be aware that iaccurate drawigs ca lead to iaccurate results, especially whe usig circles. Miimize iaccuracies by supplyig grid paper for drawig rectagles. Templates to represet circles, rulers, or umber lies with equal itervals are also useful. Equivalet fractios ad combiatios of fractios with ulike deomiators ca be represeted by usig grid drawigs ad by placig couters o the specified umbers of squares, or by colourig them. The deomiators of the two fractios determie the umber of rows ad colums i the grid, ad the umerators determie the umber of squares i the grid that must be coloured or covered with couters. 90 Grade 7 Mathematics: Support Documet for Teachers

117 Example: Represet 2 1 with a grid havig five colums ad three rows. Cover two colums 5 3 to represet 2 5, ad cover oe row to represet 1. Rearrage the couters so there is 3 oly oe couter per square. Of the 15 squares, 11 are covered, so the sum is Equivalet fractios with commo deomiators are also evidet i the drawig. 2 5 covers six squares or 6 15 of the grid, so covers five squares or of the grid Therefore, 1 5. Combied, plus total 15. Similar grids ca also be created by foldig a piece of paper horizotally x umber of times to represet oe deomiator, ad the vertically y umber of times to represet the other deomiator. The umerators ca be represeted by placig couters i the rectagles of the grid or by colourig them. Examples with Patter Blocks: Use patter blocks for fractios of halves, thirds, ad sixths. Studets may cover the hexago with six triagles. So, = = This is also = 1 or, or + = 1 or Number 91

118 Also iclude related subtractio possibilities. Models may also iclude combiatios of fractios, such as = Also iclude related subtractio possibilities. Examples with Rods: The use of rods allows for various equivalet fractio represetatios, depedig o which rod represets the whole (e.g., halves ad quarters of 4 ad 8, halves ad thirds of 6, thirds ad iths of 9, halves ad fifths of 10), ad allows for coectios to commo factors ad multiples. With Cuiseaire rods, studets ca choose ay rod to represet a whole. If they choose the rod that is eight white squares log, the the 8 cm brow rod is the whole. The 8 cm rod ca be covered with four red 2 cm rods or with two purple 4 cm rods, or a combiatio of 2 cm ad 4 cm rods. Note that fractios are equal parts of a particular whole. Examples with Circles: Circles ca be divided ito ay fractioal segmets. For example, a blak clock face with miute divisios ca represet multiple fractios ad may equivalets, such as halves, thirds, fourths, fifths, sixths, teths, twelfths, fifteeths, twetieths, ad thirtieths. Examples with Fractio Strips: Fractio strips are coveiet models because there are multiple fractio sizes for the same size of a whole. They represet fractios as parts of whole umbers. Also, these strips ca be joied together o the coordiatig umber lies to match sums that are greater tha 1. Demostrate how this works. 92 Grade 7 Mathematics: Support Documet for Teachers

119 Studets may wish to make their ow fractio strip models ad umber lies usig BLM : Fractio Bars. Developig Algorithms Allow studets to use models as log as they eed them. As studets model addig ad subtractig fractios, ad discuss the strategies they use, they will develop algorithms. Multiple models help studets to focus o meaig ad ecourage them to be flexible i their thikig. They provide opportuities to create equivalet fractios ad to reame improper fractios ad mixed umbers. By usig models, studets lear that reamig fractios makes statemets easier to solve ad that the equivalet statemets are merely differet ames for the same actio. As with algorithms for whole-umber ad decimal operatios, itroduce algorithms for fractios after studets have had time to develop their uderstadig. Mathematical Laguage deomiator differece equivalet fractios factor, greatest commo factor fractio improper fractio mixed umber multiple, least commo multiple umerator proper fractio simplify sum Ve diagram Number 93

120 Learig Experieces Assessig Prior Kowledge Materials: BLM 7.N.5.1: Iterpretig ad Recordig Differet Meaigs of Fractios demostratio board markig pes projector (optioal) poster paper (optioal) Orgaizatio: Idividual or pairs Procedure: 1. Distribute copies of BLM 7.N.5.1: Iterpretig ad Recordig Differet Meaigs of Fractios, ad have studets complete them. 2. Correct the resposes together with studets, ad review various meaigs of fractios. Durig the review, ecourage discussio, questios, ad additioal scearios, ad clear up ay misuderstadigs. If studets have errors o their sheets, have them make chages ad add otes with a markig pe. A fractio may represet a cut or a part of a whole, a part of a set, a ratio, a portio of a tur, a measuremet, a poit o a umber lie, or a divisio statemet. Variatios: Vary the complexity of the questios. Have studets create, model, ad ame their ow situatios. Have studets record represetatios of stated fractios o paper or at the board. Have studets create posters or collages to represet various meaigs of fractios. Use a projector to preset visuals of various fractioal represetatios, ad have studets record matchig fractio ames. Have studets create fractio spiders. Draw a circle to represet the spider body ad record the ame of the fractio o the body. Draw eight legs comig out from the body. At the ed of each leg, iclude some represetatio of the fractio. The feet may be illustratios, equivalet fractios, equivalet umber seteces, word seteces, ad so o. 94 Grade 7 Mathematics: Support Documet for Teachers

121 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Iterpret ad record various meaigs of fractios correctly. Assessig Prior Kowledge Materials: umber cubes (10-sided or regular) or spiers recordig paper idex cards (optioal) grid paper ad circle templates (optioal) Orgaizatio: Whole class, idividual, pairs Procedure: 1. As a class, review defiitios of proper fractios, improper fractios, ad mixed umbers. Also review procedures for covertig improper fractios to mixed umbers, ad vice versa. 2. Have studets, workig idividually, roll a umber cube or spi a spier to geerate a list of 15 fractios. The first roll or spi determies the umerator ad the secod roll or spi determies the deomiator. Usig recordig paper, studets umber ad record each fractio eatly i a colum, double spacig betwee fractios. Some fractios will be proper fractios ad some will be improper fractios. 3. Have studets covert the improper fractios to mixed umbers ad record the ew ames i a adjacet colum (about 3 cm away). They may also choose to simplify ay fractios ot i lowest terms. 4. Studets the fold their papers vertically, so that the origial fractios are ot visible, but the mixed umbers are. They exchage papers with a parter. The parter eatly records the improper fractio equivalets for each mixed umber i aother colum. 5. Studets retur the papers to the origial owers, who compare the resposes i all three colums. The parters discuss ay discrepacies i the coversios. For example, someoe may have simplified or failed to simplify a proper fractio, resultig i a differet aswer. Discuss the reaso for the differeces, ad whether these differeces are really differeces i umbers, or i ame oly. Number 95

122 Variatios: Have studets add oe whole to each of the fractios. Record the sum, ad repeat the precedig process (e.g., 6 4 becomes 10 4, becomes Have studets record the fractios o oe face of a idex card ad use the reverse side to illustrate the fractio ad reame it as a mixed umber, ad perhaps simplify it. Provide studets with grid paper ad circle templates to esure their drawigs are accurate. Cards ca be saved ad used for future additio ad subtractio learig activities. Create a huma umber lie. Ask studets to record oe of their fractios o paper with large writig. Call small groups to the frot of the class. Studets i the group hold their fractios i frot of their chests, ad stad i order from smallest to greatest fractio. Call the ext group to fit ito the lie. Discuss studets strategies for orderig fractios ad for what to do with equivalet fractios. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Correctly covert mixed umbers to improper fractios, ad vice versa. r Create ad idetify equivalet fractios. 96 Grade 7 Mathematics: Support Documet for Teachers

123 Assessig Prior Kowledge Materials: BLM 7.N.5.2: Improper Fractio ad Mixed Number Cards various card sets (for each group) from the followig website: Maitoba Educatio. Middle Years Activities ad Games. Mathematics. < idex.html>). maipulatives (optioal) grid paper or circle templates (optioal) Orgaizatio: Pairs or groups of three Procedure: 1. Have studets form pairs or groups of three to play Cocetratio. 2. Choose four sets of matchig cards (e.g., from BLM : Improper Fractio ad Mixed Number Cards). 3. Shuffle the cards ad arrage them face dow i a square. 4. Have players take turs turig over two cards. If the cards represet the same quatity, the players keep the cards. Decide whether a match set warrats aother tur. 5. The game eds whe all the cards have bee matched. The player with the most cards wis. Variatios: Studets vary the umber of card sets, or place the cards face up ad match the sets. Shuffle a radom umber of fractio cards ad place the pile face dow. Players draw a card from the pile ad build the fractio usig maipulatives, or illustrate that fractio usig grid paper or circle templates. They write both the improper fractio ad mixed umber ames for the quatity. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Correctly covert mixed umbers to improper fractios, ad vice versa. Number 97

124 Assessig Prior Kowledge Materials: markers or pes of two differet colours (for each pair of studets) two regular umber cubes (providig factors 1 to 12) or a multi-sided umber cube (for each pair of studets) grid paper or tic-tac-toe frames (of various sizes), such as the followig: BLM 7.N.3.1A: Tic-Tac-Toe Frames BLM 7.N.3.1B: Tic-Tac-Toe Frames (Medium Challege) BLM 7.N.3.1C: Tic-Tac-Toe Frame (Ultimate Challege) spiers (optioal) Orgaizatio: Pairs Procedure: 1. Studets draw a tic-tac-toe grid ad take turs fillig i the squares with umbers 1 to 99, or with multiples that correspod to the umbers o their umber cube(s). 2. Explai the procedure for this learig activity to studets: Studets choose a colour ad a X or a O mark, ad determie who will play first. Studets take turs rollig the umber cube(s), ad use their colour marker to mark a X or a O o a multiple of the umber they rolled. Ecourage them to practise usig mathematical laguage with statemets such as the followig: 27 is a multiple of 9 because 3 9 = ad 3 are factors of 27 because 3 9 = is a prime umber. Its oly factors are 1 ad 17. Studets will eed to agree about what to do if someoe makes a error. They may lose a tur, forfeit their play to their oppoet, or just accept the correctio. The first studet who creates a horizotal, vertical, or diagoal lie with his or her marks wis. 98 Grade 7 Mathematics: Support Documet for Teachers

125 Variatios: Exted the grid to 4 4 or 5 5, or whatever dimesios studets are able to hadle. Vary the shape or size of the wiig lie. Iclude multiples of 12, 15, ad 25, or target specific factors with custom-labelled umber cubes or spiers. Prepare boards with selected umbers for studets to practise. Ecourage studets to practise strategies whe selectig their multiples. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Use vocabulary for multiples, factors, ad prime umbers correctly. r Idetify multiples of various umbers correctly. r Idetify prime umbers correctly. Suggestios for Istructio Simplify a positive fractio or mixed umber by idetifyig the commo factor betwee the umerator ad deomiator. Materials: BLM 7.N.3.2: Equivalet Fractio Challege a pair of six-sided umber cubes, or a multi-sided cube, or a spier (for each pair of studets) calculators or multiplicatio charts (optioal) Orgaizatio: Whole class, pairs Procedure: 1. As a class, review procedures for creatig equivalet fractios by multiplyig or dividig by a fractio ame for 1, or by multiplyig or dividig each term i the fractio by the same factor. Number 99

126 2. Demostrate oe roud of the game, followig the procedures outlied o BLM 7.N.3.2: Equivalet Fractio Challege, ad usig the game cards provided o the BLM. I summary, studets create a target fractio, take turs rollig the umber cube(s) to determie a chage factor, ad the create a equivalet fractio. The player who returs the fractio to its origial target ame wis. 3. Distribute game cards. 4. Have studets play the game i pairs. Variatios: Vary the complexity of the arithmetic by cotrollig the optios o the type of umber cubes. Use basic six-sided umber cubes for umbers 1 to 6, a pair of umber cubes for umbers 1 to 12, custom-labelled umber cubes, or multi-sided umber cubes. If you use a umber cube with a zero, make a rule pertaiig to zero (e.g., the player who rolls zero forfeits his or her tur, or forfeits the game). A spier may also be used. Allow studets who have difficulty with multiplicatio ad divisio facts to use a calculator, a multiplicatio chart, or some other aid. Cotiue to work o developig studets uderstadig of multiplicatio ad divisio facts so that they may develop recall. The game could be played with larger groups, or as a class. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create equivalet fractios correctly ad simplify fractios with ease. r Use mathematical laguage to commuicate about fractios. Suggestios for Istructio Simplify a positive fractio or mixed umber by idetifyig the commo factor betwee the umerator ad deomiator. Materials: presetatio board math jourals or otebooks Orgaizatio: Whole class, idividual 100 Grade 7 Mathematics: Support Documet for Teachers

127 Procedure: 1. Preset the class with a fractio that ca be easily simplified e.g., 2 2 6,,. Record the fractio o the board, ask for its simplified form, ad record that fractio o the board. Cotiue recordig ad simplifyig fractios, icreasig the demad of the task. Ask studets to explai the reasos behid their simplificatios, ad ask whether the ew fractios ca be simplified further. Evetually, it should become evidet that it would be desirable to have a reliable, simple procedure to simplify less obvious fractios e.g., 24 14, Solicit fractio suggestios from studets. Someoe may suggest that if you kew the largest factor of both umbers, you would eed to divide each umber oly oce. Tell studets this strategy is called fidig the greatest commo factor. 3. Write a fractio e.g., 24 o the board. List the umerator ad the deomiator, ad 36 ask studets to idetify the factors for each umber i a systematic progressio. Example: Begi with the smallest factor of 24 (which is 2) ad record it toward the left ed of the row. Record its correspodig factor (12) toward the right ed of the row. Cotiue workig toward the cetre util both factors begi to repeat. Look for the largest factor that is commo to both umbers. I this case, the largest factor is 12. Use this factor to simplify the fractio by dividig by as a ame for 1. Or divide both the umerator ad the deomiator by = Have studets geerate a list of obvious ad less obvious fractios for which they will fid the greatest commo factor, ad which they will simplify. Iclude proper fractios, improper fractios, ad mixed umbers. 5. Record the list of fractios o the board, ad have studets record it i their math jourals or otebooks. Complete oe more sample together with the class, ad the have studets fid the factors ad simplify the fractios o their ow. Whe they have completed the task, correct resposes as a class. Solicit questios ad commets, ad have studets record correctios ad otes i their math jourals or otebooks. Number 101

128 Variatio: Supply studets with a sheet cotaiig a collectio of fractios ad proper, improper, ad mixed umbers. Studets list the factors for the umerator ad the deomiator, idicate the greatest commo factor, ad simplify the fractio. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Simplify a positive fractio or mixed umber by idetifyig the commo factor betwee the umerator ad deomiator. Suggestios for Istructio Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. Determie a commo deomiator for a set of positive fractios or mixed umbers. Materials: presetatio board math jourals or otebooks rulers pes or markers of differet colours maipulatives to represet fractios (e.g., patter blocks, fractio circles, Cuiseaire rods, grids, couters) templates copied o card stock (e.g., fractio bars, circles, patter blocks, rods, grid paper) for studets to copy or cut ad glue ito their math otebooks scissors, glue (optioal) Orgaizatio: Small groups, whole class, idividual 102 Grade 7 Mathematics: Support Documet for Teachers

129 Procedure: Part A: Modellig Fractios Equallig a Whole 1. Have studets, workig i small groups, use differet types of maipulatives to explore differet ways to represet a whole. Cosider givig each group differet types of maipulatives, depedig o the umber of resources available. Decide whether to iclude oly maipulatives that represet the cut (part of a whole) meaig of a fractio, or whether to iclude a poit o a lie. Have studets talk about their models withi their groups, ad draw illustratios (or cut ad paste templates) of two or more models i their math jourals or otebooks. 2. Have studets reassemble as a class. Ask a few studets to share their models with the class. Show their illustratios o the presetatio board. 3. Solicit ideas from studets about how to write a additio statemet to match each model of fractios equallig a whole. Example: Here is oe example usig patter blocks: A hexago ca represet oe whole. Three rhombuses cover the hexago ad occupy the same space, so each rhombus represets 1 of the whole = 1 or Combie fractioal pieces if you wish. (See Cocepts to Review durig Discussio, followig Parts A to C of the procedures.) = ad = ad 3 3 =1 Subtractio ca be represeted with the same model. Demostrate a take-away actio. Also, remove a fractioal part ad fid the differece by comparig what is left of the whole. Ask what part is missig. Number 103

130 Write the matchig subtractio statemets: 3 1 or = 3 Take away some more if you wish: = 3 Fid the differece betwee two of the fractioal parts: 1 1 =0 3 3 Whe the appropriate model arises, exted the example to the followig: = 6 4. Have studets retur to their math jourals or otebooks ad write additio ad subtractio statemets to match the models they illustrated. Part B: Modellig Fractios, Icludig Proper Fractios 5. Have the groups chage maipulatives ad repeat the process outlied i Part A as may times as seems useful. If studets are ready, have them cosider equivalet represetatios, simplified fractios, ad statemets ivolvig proper fractios less tha a whole. Remid studets to talk with their groups about their models ad the matchig of additio ad subtractio statemets. Have studets record two or more ew models, ad write matchig additio ad subtractio statemets. 6. Have studets reassemble as a class to share iterestig discoveries ad to verify resposes. Part C: Modellig Proper Fractios, Improper Fractios, ad Mixed Numbers 7. Whe appropriate, ivite studets to iclude combiatios that represet more or less tha a whole (e.g., mixed umbers such as pizzas, 2 1 cas of juice, of a chocolate bar, 5 of a ich). Remid studets to verify that the fractioal 8 pieces are equal parts of the particular whole. Idetifyig the whole is importat to uderstadig the fractioal relatios. Have studets talk about their models ad statemets withi their groups. Remid them to draw illustratios (or cut ad paste templates) of two or more of the models i their math jourals or otebooks. 8. Reassemble as a class, ad have a few studets share their ew models ad additio or subtractio statemets. This sharig process provides both you ad the studets with a opportuity to verify resposes. Whe modellig subtractio, compare two fractios ad fid the differece or missig part betwee them. Also model subtractio as takig away a fractioal part from a mixed umber. Have studets write or verify their additio ad subtractio statemets for the models. 104 Grade 7 Mathematics: Support Documet for Teachers

131 9. Have the groups chage maipulatives ad repeat the process as may times as seems useful. Have studets discuss their models ad statemets withi their groups, ad record two or more models ad matchig additio ad subtractio statemets. Cocepts to Review durig Discussio A. The fractio models provide opportuities for rewritig additio ad subtractio statemets by combiig fractios with like deomiators, ad reamig ulike deomiators with equivalet fractios to obtai commo deomiators. B. As you write equatios for the models, discuss with studets why these fractios with differet ames ca be combied. The 1 sectios represet three equal parts of 3 a whole, ad the 1 sectios represet six equal parts of the same whole. They are all 6 parts of the same whole. C. Not all fractio sizes of the same whole ca be combied to equal oe whole. For example, combiig halves ad fifths will always result i either more or less tha a whole. D. The same fractio ame ca be used to represet differet quatities. For example, 1 2 of the water i my glass is differet tha 1 of the water i my bathtub. 2 Furthermore, 1 3 of my whole patter block is ot the same as 1 of your 3 Cuiseaire rod, or the same as 1 of someoe s fractio block. Whe usig a cut 3 (part of a whole) meaig of a fractio, the parts must be a fractio of the same whole i order to combie them. Revisit this topic frequetly whe discussig combiig differet types of fractios. Whe usig Cuiseaire rods, studets ca explore to fid that the red 2 cm blocks represets 1 4 of the brow 8 cm rod, or 1 5 of the orage 10 cm rod, but it represets either 1 4 or 1 of the blue 9 cm rod. 5 E. If a fractio represets a divisio situatio, the it is a ame for a umber i our umber system. It represets a portio of oe uit. For example, 4 represets 4 2, 2 or the umber 2, which is two whole uits, ad 1 represets oe divided by 2 2, 1, or 05.. If the fractios represet umbers, the ay fractios ca be combied, 2 because they all represet parts of the same whole umber uit, ad ot parts of differet wholes, regios, or sets. Whe fractios are preseted without a stated cotext, they represet this ame for a umber meaig, ad ca be added or subtracted freely. Fractio strips are a coveiet represetatio of this cocept. (See the ext learig activity.) Number 105

132 Variatios: For more direct istructio, guide studets through specific models ad matchig additio ad subtractio seteces. Provide scaffoldig by supplyig studets with templates to complete. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. Suggestios for Istructio Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. Materials: BLM : Fractio Bars (copied o heavy paper or card stock) prepared sample of fractio strips ad matchig umber lies scissors resealable bags for storig pieces math jourals or otebooks magetic tape (optioal) Orgaizatio: Whole class, idividual Procedure: 1. Demostrate how fractio strips ca serve as coveiet models, as there are multiple fractio sizes for the same size of a whole. They ca be used to fid equivalet fractios. 2. Fractio strips are useful models for represetig fractios as umbers because they are parts of the uit umber 1. They represet fractios as poits o a umber lie, ad ca be joied together o the coordiatig umber lie to match fractio combiatios with correspodig sums or differeces. Demostrate how to use the model. 106 Grade 7 Mathematics: Support Documet for Teachers

133 3. Provide studets with card stock copies of BLM : Fractio Bars. Have studets make a set of fractio strips ad correspodig umber lies to use i various learig activities. Store the products i resealable bags. 4. Have studets geerate a list of additio ad subtractio questios, ad ask them to use their models to write several additio ad subtractio statemets, recordig them i their math jourals or otebooks. Variatios: Faste the fractio strips to magetic tape before cuttig them. This adds greater durability, ad the strips ca be stored ad used o a magetic surface to miimize difficulty maagig so may loose pieces. Prepare a list of additio ad subtractio statemets for studets to practise usig the fractio strips. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. r Use models to aid i the visualizatio of addig ad subtractig with fractios. Number 107

134 Suggestios for Istructio Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. Determie the sum of two positive fractios or mixed umbers with like deomiators. Determie a commo deomiator for a set of positive fractios or mixed umbers. Determie the sum of two positive fractios or mixed umbers with ulike deomiators. Simplify the solutio to a problem ivolvig the sum or differece of two positive fractios or mixed umbers. Solve a problem ivolvig the additio or subtractio of positive fractios or mixed umbers, ad determie if the solutio is reasoable. Materials: BLM 7.N.5.3A: Ace Aviatio: Addig Fractios BLM 7.N.5.4A: Represetig Recogizable Fractios ad Writig Additio Statemets (optioal) presetatio board maipulatives to represet fractios (e.g., BLM : Fractio Bars, patter blocks, fractio circles, Cuiseaire rods, grids, couters) fractio strips ad umber lies (made by studets) a referece list or a hadout of model illustratios, scearios, ad problems ivolvig addig ad subtractig fractios (optioal) idex cards (optioal) Orgaizatio: Small groups, whole class, idividual Procedure: 1. Provide studets with copies of BLM 7.N.5.3A: Ace Aviatio: Addig Fractios, ad have them respod to questios 1 to Discuss studets resposes to questios 1 to Discuss studets thikig regardig the additio i questio 4. Iclude the followig ideas: Discuss the models studets used to combie the tourists ad the vacatioers 1 1 with those visitig family or frieds +. These umbers ca be added 3 6 together, because each fractio represets parts of the same whole. That whole is all the airlie passegers. The priciples of addig apply. Discuss the beefits of makig a estimate before addig the fractios. The estimate helps establish a target zoe for the solutio, ad verifies whether or 108 Grade 7 Mathematics: Support Documet for Teachers

135 ot the aswer is reasoable. Bechmarks of close to, more or less tha, 0, 1 2, or wholes are helpful. Studets may use their model of a whole, or a umber lie, as a referece poit for bechmarks (e.g., 1 3 is less tha 1 1,ad is a little more 2 6 tha 0, ad together they must be close to 1. 2 I the models for 1 1 +, it is evidet the pieces cover 1 of the whole, but how to cout these pieces to equal 1 is ot as clear. As i combiig ay 2 measure, before we cout, the pieces must all have the same uit of measure. Whe coutig fractios, covertig the uits to the same measure is called fidig a commo deomiator. For example, 1 3 is equivalet to , + =, ad , =, 6 ad 6 is equivalet to 1 2. To represet this coversio with a cocrete model, replace the 1 3 piece with two 1 6 pieces, ad the replace the three 1 6 pieces with a 1 piece. The parts are differet ames for the same umbers. 2 Model writig the additio statemets with the coversios. Esure studets are comfortable with these cocepts. If the umbers represet differet types of parts, such as 1 3 of the passegers ad 1 of the crew, they 6 caot be added together, because they are parts of differet wholes. See Coceptual Uderstadigs i the Backgroud Iformatio for learig outcome 7.N Have studets represet the additio of other recogizable fractios ad write represetative additio statemets. Iclude improper fractios ad mixed umbers. Discuss how to reame the resultig improper fractios as mixed umbers. Solicit the addeds from the class, or have a list prepared as a hadout, such as BLM 7.N.5.4A: Represetig Recogizable Fractios ad Writig Additio Statemets. Variatios: Have studets ivet a passeger survey for a differet airlie, or prepare data for other scearios, ad write questios ad aswers based o their data. Supply studets with a umber of scearios requirig them to fid the total. Use friedly fractios ad iclude mixed umbers ad improper fractios. Have studets geerate a umber of scearios requirig addig friedly fractios, icludig mixed umbers ad improper fractios. Have them fid the sums. These ca be recorded o idex cards, with the sceario o oe side ad the aswer o the reverse. The cards ca be used later for review ad drill exercises. Number 109

136 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. r Determie the sum of two positive fractios or mixed umbers with like deomiators. r Determie a commo deomiator for a set of positive fractios or mixed umbers. r Determie the sum of two positive fractios or mixed umbers with ulike deomiators. r Simplify the solutio to a problem ivolvig the sum or differece of two positive fractios or mixed umbers. r Solve a problem ivolvig the additio or subtractio of positive fractios or mixed umbers, ad determie if the solutio is reasoable. Suggestios for Istructio Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. Determie the sum of two positive fractios or mixed umbers with like deomiators. Determie the differece of two positive fractios or mixed umbers with like deomiators. Determie a commo deomiator for a set of positive fractios or mixed umbers. Determie the sum of two positive fractios or mixed umbers with ulike deomiators. Determie the differece of two positive fractios or mixed umbers with ulike deomiators. Simplify the solutio to a problem ivolvig the sum or differece of two positive fractios or mixed umbers. Solve a problem ivolvig the additio or subtractio of positive fractios or mixed umbers, ad determie if the solutio is reasoable. 110 Grade 7 Mathematics: Support Documet for Teachers

137 Materials: BLM 7.N.5.3B: Ace Aviatio: Subtractig Fractios BLM 7.N.5.4B: Represetig Recogizable Fractios ad Writig Subtractio Statemets (optioal) BLM 7.N.5.5: Addig ad Subtractig Fractios (Scearios) (optioal) presetatio board maipulatives to represet fractios (e.g., BLM : Fractio Bars, patter blocks, fractio circles, Cuiseaire rods, grids, couters) fractio strips ad umber lies (made by studets) math jourals or otebooks a referece list or a hadout of model illustratios, scearios, ad problems ivolvig addig ad subtractig fractios (optioal) idex cards (optioal) Orgaizatio: Small groups, whole class, idividual Procedure: 1. Provide studets with copies of BLM 7.N.5.3B: Ace Aviatio: Subtractig Fractios, ad have them respod to questios 1 to Have studets share their resposes to the subtractio questios. Examie studets models for represetig removig the busiess travellers. 3. Model differet subtractio scearios usig umber lies ad fractio strips. 4. Have studets make ay ecessary revisios to their work, or add ay otes they cosider useful. Number 111

138 5. Oce agai, discuss the beefits of makig a estimate before subtractig the fractios. The estimate helps establish a target zoe for the solutio, ad verifies whether or ot the aswer is reasoable. Recall that i order to subtract a quatity, you must begi with a amout that is equal to or larger tha the portio you subtract. Bechmarks of close to, more or less tha, 0, 1, or wholes are helpful. 2 Studets may use their model of a whole, or a umber lie, as a referece poit for bechmarks. 6. Have studets model ad record subtractio of other recogizable fractios ad mixed umbers, such as those o BLM 7.N.5.4B: Represetig Recogizable Fractios ad Writig Subtractio Statemets. 7. Ask studets to create scearios to match the fractios. Have them write subtractio statemets for each model. Esure they uderstad that the fractioal part must be equal to or less tha the fractioal part it is beig take away from. Solicit ad reiforce ideas about borrowig from the wholes ad cuttig up the ew piece to form a improper fractio from which to subtract. Iclude related additio statemets if you wish. 8. Reassemble as a class ad have studets preset a few models ad subtractio statemets to esure everyoe is o the right track. Preset models ad subtractio scearios, ad ask studets to write subtractio statemets to represet the models ad scearios, recordig them i their math jourals or otebooks. Iclude related additio statemets if you wish. Discuss the resposes. 9. Have studets complete a selectio of additio ad subtractio problems ad arithmetic questios. Iclude improper fractios ad mixed umbers i the problems ad questios. Solicit the addeds from the class, or have a list prepared o a hadout, such as those preseted o BLM 7.N.5.5: Addig ad Subtractig Fractios (Scearios). Variatios: Have studets ivet a passeger survey for a differet airlie, or prepare data for other scearios, ad write questios ad aswers based o their data. Supply studets with a umber of scearios requirig them to fid the total or the differece. Use friedly fractios ad iclude mixed umbers ad improper fractios. Have studets geerate a umber of scearios requirig addig or subtractig friedly fractios, icludig mixed umbers ad improper fractios. Have them fid the sums ad differeces. These ca be recorded o idex cards, with the sceario o oe side ad the aswer o the reverse. The cards ca be used later for review ad practice exercises. 112 Grade 7 Mathematics: Support Documet for Teachers

139 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Model additio ad subtractio of positive fractios or mixed umbers usig cocrete represetatios, ad record symbolically. r Determie the sum of two positive fractios or mixed umbers with like deomiators. r Determie the differece of two positive fractios or mixed umbers with like deomiators. r Determie a commo deomiator for a set of positive fractios or mixed umbers. r Determie the sum of two positive fractios or mixed umbers with ulike deomiators. r Determie the differece of two positive fractios or mixed umbers with ulike deomiators. r Simplify the solutio to a problem ivolvig the sum or differece of two positive fractios or mixed umbers. r Solve a problem ivolvig the additio or subtractio of positive fractios or mixed umbers, ad determie if the solutio is reasoable. Suggestios for Istructio Determie a commo deomiator for a set of positive fractios or mixed umbers. Materials: maipulatives presetatio board a list of fractios for studets to add ad subtract, some with obvious solutios ad some with solutios that require fidig commo deomiators grids ad couters (optioal) paper, markers, ad other art supplies (for makig posters or brochures) Orgaizatio: Whole class, idividual Number 113

140 Procedure: 1. Preset studets with sets of fractios to add ad subtract. Ask them what makes some types of questios ivolvig fractios easier to aswer tha other types. The questios that have friedly fractios with commo deomiators, or fractio sizes that relate easily to each other, are easier to model ad to solve. If the more difficult questios could be reamed to have commo deomiators, they would be easier to aswer. 2. Ask studets to explore fidig a way to reame fractios with commo deomiators. Fidig a commo deomiator is the term used for covertig fractios to commo uits. Studets may use grids ad couters (as explaied i the Backgroud Iformatio for learig outcome 7.N.5), or geerate a list of equivalet fractios, ad look for a patter. All deomiators i the equivalet fractios are multiples of the origial deomiators. Therefore, the commo deomiator must be a multiple of both the origial deomiators. Studets ca use their prior kowledge of fidig multiples ad the lowest commo multiple, ad their ability to simplify fractios, to develop a strategy to fid commo deomiators. Extesio: Have studets geerate a list of the best hits ad strategies for addig ad subtractig fractios, ad preset them as small posters or brochures. Variatios: Guide studets through a series of steps to fid commo deomiators. Provide studets with hadouts cotaiig sets of fractios. Have studets show how they geerated a commo deomiator for the set, ad how they reamed the fractios. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie a commo deomiator for a set of positive fractios or mixed umbers. r Make coectios betwee determiig a commo deomiator ad their prior kowledge regardig factors ad multiples. 114 Grade 7 Mathematics: Support Documet for Teachers

141 Suggestios for Istructio Simplify the solutio to a problem ivolvig the sum or differece of two positive fractios or mixed umbers. Solve a problem ivolvig the additio or subtractio of positive fractios or mixed umbers, ad determie if the solutio is reasoable. Materials: BLM 7.N.5.6: Problems Ivolvig Fractios (or other sample fractio problems) markig pes Orgaizatio: Idividual, whole class Procedure: 1. Distribute copies of BLM 7.N.5.6: Problems Ivolvig Fractios or other fractio problems. 2. Ask studets to solve the problems, provide estimates, ad simplify aswers. 3. After studets have had sufficiet time to complete the tasks, discuss their solutios to the problems. 4. Have studets use a markig pe to make ay ecessary correctios or add ay otes to their work. Variatio: Have studets set criteria for problems ivolvig fractios, ad ask them to create a set umber of problems ad a aswer key. Photocopy their problem sheets. Studets trade sheets, solve the problems, ad the reassemble as a group ad review the solutios together. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Simplify the solutio to a problem ivolvig the sum or differece of two positive fractios or mixed umbers. r Solve a problem ivolvig the additio or subtractio of positive fractios or mixed umbers, ad determie if the solutio is reasoable. Number 115

142 N o t e s 116 Grade 7 Mathematics: Support Documet for Teachers

143 Number (7.N.6) Edurig Uderstadig(s): The priciples of operatios used with whole umbers also apply to operatios with decimals, fractios, ad itegers. Number sese ad metal mathematics strategies are used to estimate aswers ad lead to flexible thikig. Geeral Learig Outcome(s): Develop umber sese. Specific Learig Outcome(s): 7.N.6 Demostrate a uderstadig of additio ad subtractio of itegers, cocretely, pictorially, ad symbolically. [C, CN, PS, R. V] Achievemet Idicators: Explai, usig cocrete materials such as iteger tiles ad diagrams, that the sum of opposite itegers is equal to zero. Illustrate, usig a horizotal or vertical umber lie, the results of addig or subtractig egative ad positive itegers (e.g., a move i oe directio followed by a equivalet move i the opposite directio results i o et chage i positio). Add two give itegers usig cocrete materials or pictorial represetatios, ad record the process symbolically. Subtract two give itegers usig cocrete materials or pictorial represetatios, ad record the process symbolically. Solve a give problem ivolvig the additio ad subtractio of itegers. Number 117

144 Prior Kowledge Studets should be able to do the followig: Q Q (4.N.3) Demostrate a uderstadig of additio of umbers with aswers to ad their correspodig subtractios (limited to 3- ad 4-digit umerals) by Q Q Q Q Q Q usig persoal strategies for addig ad subtractig estimatig sums ad differeces solvig problems ivolvig additio ad subtractio (4.N.5) Describe ad apply metal mathematics strategies, such as skip-coutig from a kow fact usig doublig or halvig usig doublig ad addig oe more group usig patters i the 9s facts usig repeated doublig to develop recall of basic multiplicatio facts to 9 9 ad related divisio facts. (5.N.2) Apply estimatio strategies icludig frot-ed roudig compesatio compatible umbers i problem-solvig cotexts. (6.N.7) Demostrate a uderstadig of itegers, cocretely, pictorially, ad symbolically. Related Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q (7.PR.1) Demostrate a uderstadig of oral ad writte patters ad their correspodig relatios. (7.PR.2) Costruct a table of values from a relatio, graph the table of values, ad aalyze the graph to draw coclusios ad solve problems. (7.PR.3) Demostrate a uderstadig of preservatio of equality by modellig preservatio of equality, cocretely, pictorially, ad symbolically applyig preservatio of equality to solve equatios (7.PR.5) Evaluate a expressio, give the value of the variable(s). (7.PR.6) Model ad solve problems that ca be represeted by oe-step liear equatios of the form x + a = b, cocretely, pictorially, ad symbolically, where a ad b are itegers. 118 Grade 7 Mathematics: Support Documet for Teachers

145 Q Q Q Q (7.SP.1) Demostrate a uderstadig of cetral tedecy ad rage by determiig the measures of cetral tedecy (mea, media, mode) ad rage determiig the most appropriate measures of cetral tedecy to report fidigs (7.SP.2) Determie the effect o the mea, media, ad mode whe a outlier is icluded i a data set. Backgroud Iformatio Itegers: Defiitio ad Notatio Itegers are the set of umbers cosistig of the atural umbers (1, 2, 3,...), their opposites ( 1, 2, 3,...), ad zero. They are also referred to as the whole umbers ad their opposites. Itegers idicate both a quatity ad a directio from zero. Positive itegers are greater tha zero. They are represeted by a positive symbol (+) before the iteger, such as (+5). Negative itegers are less tha zero. They are represeted by a egative symbol ( ) before the iteger, such as ( 3). There are two commo otatios for itegers. The symbols are writte either as superscripts precedig the iteger, as i + 5, 3, or the symbol ad the iteger are both eclosed withi paretheses, as i (+5), ( 3). The paretheses are commoly used i studet materials to avoid ay cofusio betwee the iteger sig ad the otatios for additio ad subtractio. I the equatio (+5) ( 3) = (+8), the paretheses idicate the umbers iside are itegers ad distiguish the iteger symbols from the subtractio symbol. Iteger Use Uderstadig ad workig with itegers is importat i daily life. Itegers are regularly ecoutered i cotexts such as fiaces, ivestmets, temperatures, elevatios, time relevat to evets, ad sports. Proficiecy with addig ad subtractig itegers will be importat i studets future algebra work, ad is a useful metal mathematics strategy for multi-digit subtractio. Kowledge of itegers provides a laguage for studets to express their thikig whe they use umbers less tha zero. Example: To subtract , Thik: ( ) + (20 70) + (6 9) (+200) + ( 50) + ( 3) (+150) + ( 3) Equals: (+147) Number 119

146 Represetig Iteger Operatios with Models I Grade 7, studets exted their uderstadig of itegers acquired i Grade 6 as they lear to add ad subtract positive ad egative umbers. Provide studets with may opportuities to represet itegers (cocretely, pictorially, ad symbolically) to develop their uderstadig. Ecouragig studets to use a variety of maipulatives ad strategies will help them to develop cofidece i determiig ad applyig geeral rules for both addig ad subtractig itegers. Cocrete Models Cocrete models iclude the followig: 1. Algebra tiles: Oe face of a algebra tile is oe colour, ad the opposite side is a differet colour. Use oe side to represet positive itegers, ad the reverse side to represet egative itegers. 2. Sets of couters i two differet colours: Choose oe colour to represet positive itegers, ad the other to represet egative itegers. Matchig sets of both colours represet zero. For example, blue chips represet egative itegers, while red chips represet positive itegers. To solve the problem of (+3) + ( 7), set out three red chips ad seve blue oes. Physically match up pairs of red ad blue chips to equate them to zero, ad remove the remaiig chips. The remaiig four blue chips represet the solutio, ( 4). 3. Computer models: May computer simulatios allow studets to pull the represetative positive ad egative couters ito a collectio bi. Studets match opposite represetatios to represet zero, ad the couters disappear. The aswer remais i the bi to be couted. Sample Website: For a example of a computer model, refer to the followig website: Utah State Uiversity. Number ad Operatios (Grades 3 5). Natioal Library of Virtual Maipulatives < Select Color Chips Additio or Color Chips Subtractio from the list of virtual maipulatives provided. 4. Number lies: A thermometer is a atural umber lie, ad ca be viewed vertically or horizotally. The distace of a iteger from zero represets the quatity of the iteger, ad the directio from zero represets whether the iteger is positive or egative. O a vertical umber lie, the distace above zero represets positive itegers. Distaces below zero represet egative itegers. Values always icrease up the lie, ad decrease dow the lie. 120 Grade 7 Mathematics: Support Documet for Teachers

147 O a horizotal umber lie, positive itegers are represeted to the right of zero, ad egative itegers are represeted to the left of zero. Values always icrease from left to right, ad decrease from right to left. A iteger s quatity may also be represeted with vector models or the legth of a arrow from zero to the iteger. A arrow poitig to the right idicates a positive value, ad a arrow to the left idicates a egative value. The legth of the combied arrows idicates the combied value. These arrows are typically placed ed to ed. Example: Arrows poitig i opposite directios are laid o top of (o a horizotal umber lie) or beside (o a vertical umber lie) each other. The begiig of oe arrow is matched with the ed of the other arrow. Example: The combiatio of itegers may also be represeted by jumps o a umber lie. Jumps to the right represet additio, ad jumps to the left represet subtractio. A egative iteger moves i the opposite directio. Use both vertical ad horizotal umber lies to represet chages i temperatures, elevatios, ad distaces travelled. Number 121

148 Geeralizatios about Itegers As studets work with differet maipulatives ad use differet strategies, they will likely come to the followig geeralizatios about itegers. Rather tha explicitly teachig the geeralizatios as rules, provide studets with opportuities to discover these geeralizatios. The Zero Priciple The sum of opposite itegers (sometimes called the zero pairs) is always zero. Addig equal positive ad egative umbers to a quatity does ot chage the et value of the quatity. Addig Itegers The sum of two positive itegers is always positive (e.g., (+2) + (+3) = (+5)). The sum of two egative itegers is always egative (e.g., ( 2) + ( 3) = ( 5)). The sum of oe egative iteger ad oe positive iteger may be either egative or positive, depedig o the sig of the umber that is farthest from zero (i.e., subtract the absolute values of the itegers ad use the sig of the iteger with the greater absolute value) (e.g., (+2) + ( 3) = ( 1)). Subtractig Itegers Subtractig a iteger is equivalet to addig its opposite (e.g., (+4) ( 2) = (+4) + (+2) = (+6)). If both itegers have the same sig ad the miued is further away from zero tha the subtrahed, fid the differece ad keep the sig (e.g., (+7) (+3) = (+4) or ( 7) ( 3) = ( 4)). If both itegers have the same sig ad the subtrahed is further away from zero tha the miued, fid the differece ad use the opposite sig (e.g., (+2) (+6) = ( 4) or ( 2) ( 6) = (+4)). (Equivalet to addig the opposite.) If the sigs are differet, add the values ad use the sig of the miued (e.g., ( 5) (+3) = ( 8) or (+3) ( 5) = (+8)). (Equivalet to addig the opposite.) Mathematical Laguage absolute value* iteger miued egative iteger positive iteger sig subtrahed zero priciple * Note: Teachers may model correct use of absolute value, but it is ot a expectatio for Grade 7 studets. 122 Grade 7 Mathematics: Support Documet for Teachers

149 Learig Experieces Assessig Prior Kowledge Materials: research resources, such as magazies, ewspapers, pamphlets almaacs olie databases Statistics Caada. Learig Resources. < scissors, glue, markers, ad poster board (for makig posters) Orgaizatio: Whole class, small groups Procedure: 1. As a class, braistorm cotexts i which itegers are used i daily life. Situatios may iclude the followig: depths/levels of oceas, lakes, ad rivers levels of tides, moutais, ad cities levels of tall buildigs, udergroud parkig garages, ad mie shafts temperatures fiaces, savigs ad spedig, loas ad debts value of ivestmets (e.g., share prices, stocks, mutual fuds) sports ad player statistics 2. Divide the class ito small groups, ad have each group research ad compare statistics related to a selected theme. Themes may iclude elevatios of various cities or moutais, depths of lakes or oceas, river levels i times of flood ad drought, temperature extremes i various cities, ad player statistics i various sports leagues (e.g., +/ differetials i hockey, ad par i golf). 3. Each group the fids a creative way to preset their research fidigs i a collage or o a poster with appropriate titles. For example, studets may create a illustratio of cities i order of lowest to highest elevatio, or coldest to warmest cities, or highest ad lowest poit o each cotiet, or worst to best performig stocks for a give period, or the performace of sports teams or players. Number 123

150 Variatio: Each group has a geeral thematic focus. Studets cut out headlies, diagrams, charts, graphs, ad illustratios of cotexts i which itegers are represeted i daily life, ad use the clippigs to create a collage. They add appropriate headigs. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Describe cotexts i which itegers are used. r Order itegers correctly. Assessig Prior Kowledge Materials: paper ad pes or demostratio board (to keep track of poits) space for each pair of teams to use as a pitchig moud, home plate, ad bases Orgaizatio: Oe to four teams, depedig o class size Procedure: 1. Have studets form teams to play iteger baseball. 2. Choose oe team to bat, oe team to pitch, ad a scorekeeper. 3. Each team lies up at the home plate or o the pitchig moud. 4. The first pitcher pitches to the first batter a situatio or a actio phrase that may be represeted by a iteger. 5. The batter replies with the represetative iteger. The waitig pitchers determie whether the aswer is correct. If the batter is correct, he or she has a hit, ad begis rotatig through the bases. A error couts as a out. The bases are cleared, ad the players retur to the ed of the battig lie. 6. The previous pitcher goes to the ed of the pitchig lie, ad the ext pitcher pitches a ew situatio, ad play cotiues. 7. Whe a player returs to home plate, a ru is scored. At three outs, the teams switch places. 124 Grade 7 Mathematics: Support Documet for Teachers

151 8. Suggested rules: No situatios may be repeated. Pitchers ad batters must respod withi five secods. No players ca steal bases; they must be hit to the ext base. If the pitchers make a error i judgmet, the battig team scores a home ru, ad ay players o bases are hit home, each scorig a ru. Variatios: Have small teams or pairs sit i groups ad rotate bases o a paper field. Switch or vary the actios. Pitchers pitch itegers, ad batters describe a matchig cotextual situatio. Supply a list of situatios for pitchers to use. Use paper-ad-pecil or studet-geerated tasks requirig studets to write the iteger that matches a situatio ad/or describe a situatio that may match a iteger. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Describe cotexts i which itegers are used. r Use itegers to represet cotexts. Number 125

152 Suggestios for Istructio Explai, usig cocrete materials such as iteger tiles ad diagrams, that the sum of opposite itegers is equal to zero. Illustrate, usig a horizotal or vertical umber lie, the results of addig or subtractig egative ad positive itegers (e.g., a move i oe directio followed by a equivalet move i the opposite directio results i o et chage i positio). Materials: a log lie o the floor or playgroud (e.g., use a patter i the floor tiles, use lies paited o the gymasium floor or sports field, or create a lie with tape, chalk, or coloured dots) a measurig tool to create itervals low-tack stickers to label itervals (use a special marker for zero) a pile of cards labelled with itegers two arrows, oe labelled icreasig i value, ad the other labelled decreasig i value paper to create a umber lie (10 cm by 55 cm) (optioal) Orgaizatio: Small groups Procedure: 1. Create a umber lie o the classroom floor, the gymasium floor, or outdoors, usig tape, chalk, or coloured dots. Number the lie ( 20) to (+20). The lie ca be used later for other learig experieces. 2. Whe the umber lie is complete, have studets draw a iteger card from the pile, show where the umber would be o the lie by pacig the distace from zero, ad the stad at the spot o the umber lie that represets the iteger. The first studet compares his or her umber to zero: is (greater or less) tha zero. The ext perso compares the size of his or iteger to that of a eighbour already o the lie: is (greater or less) tha. 3. Note that the farther a umber is from zero, the larger or smaller its value is, depedig o its directio from zero. Place arrows o the lie to label icreasig values or decreasig values. 126 Grade 7 Mathematics: Support Documet for Teachers

153 Variatios: Whe studets state their compariso resposes, they could add a value to the greater or less tha statemet. For example, istead of sayig, ( 8) is less tha (-6), they could say, ( 8) is 2 less tha ( 6). Have studets create ad use both vertical ad horizotal umber lies. Make a persoal umber lie (10 cm wide ad 55 cm log) o paper, label the itegers ( 20) to (+20), ad use small cards to represet the above actios. This lie may be used for other learig experieces. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create a accurate umber lie. r Order umbers correctly o a umber lie. Suggestios for Istructio Explai, usig cocrete materials such as iteger tiles ad diagrams, that the sum of opposite itegers is equal to zero. Illustrate, usig a horizotal or vertical umber lie, the results of addig or subtractig egative ad positive itegers (e.g., a move i oe directio followed by a equivalet move i the opposite directio results i o et chage i positio). Materials: demostratio board BLM 7.N.6.1: Cetimetre Number Lie BLM 5 8.9: Cetimetre Grid Paper coloured paper or markers (Decide whether to use the same colour of maipulatives to represet positive ad egative itegers for every activity.) scissors a directioal toy (e.g., a perso, aimal, car) (for each studet or group) 20 couters of two differet colours (for each studet or group) math jourals or otebooks Orgaizatio: Whole class, idividual or small groups Number 127

154 Procedure: The purpose of this learig activity is to determie the sum of opposite itegers. 1. Remid studets that itegers measure quatity ad directio from zero. 2. Ask studets to use BLM 5 8.9: Cetimetre Grid Paper to measure ad cut five strips of paper the legth of five differet positive itegers. The strips should be of the same colour ad the same height. Have studets mark zero at the left ed of the strip, the iteger at the right ed, ad the vector arrow the legth of the strip ad poitig to the right, idicatig that the strip represets a positive iteger. Example: Ask studets to use aother colour to create strips represetig the egative or opposite of each chose iteger. This time, have studets label zero at the right ed of the strip, the iteger at the left ed, ad the vector arrow poitig to the left, idicatig that the strip represets a egative iteger. Example: 3. Demostrate how to combie the opposite strips o the umber lie to represet addig opposite itegers: (+5) + ( 5) =. Place the begiig or zero poit of the vector arrow of the strip represetig the first iteger at zero, ad place the begiig or zero poit of the vector arrow of the strip represetig the secod iteger at the ed or iteger value of the first strip. The resultig ed poit of the secod strip ca be read o the umber lie. I this case, it is zero. 4. Ask studets to use their strips to model five combiatios of differet itegers ad their opposites, ad to write a geeral statemet about the sum of a iteger ad its opposite, recordig their work i their math jourals or otebooks. 5. After givig studets sufficiet time to work o their combiatios, ask studets what they discovered. Record each of their combiatios o the demostratio board as equatios. Ask studets to make a geeral statemet about the sum of a iteger ad its opposite. 128 Grade 7 Mathematics: Support Documet for Teachers

155 6. Ask studets to imagie that the umber lie is a elevator shaft. Use of a vertical umber lie is realistic for this situatio. Have a studet demostrate eterig at level zero, go up three floors, ad the come back dow three floors. The elevator will be back where it started. There has bee zero chage. 7. Demostrate the combiatio (+5) + ( 5) = as jumps o a umber lie. If you are usig the large lie, have a studet begi at zero ad take 5 positive jumps, facig to the positive right. The ext actio is addig or combiig, so the studet cotiues to face right, ready to jump o. The ext iteger is egative though. It is the opposite of 5, so the studet must face the same directio ad jump backwards to show the opposite of 5. The umber to which the studet jumps idicates the sum. Have studets demostrate several of their opposite combiatios. Be sure to act out some combiatios begiig with egative itegers. Also act out the same combiatios begiig at ay floor i the elevator sceario, or begiig at ay umber o the umber lie. Addig oe value ad the its opposite results i zero or o et chage to the origial positio or umber. If you do ot have access to a large umber lie, have studets use a directioal toy with a frot (e.g., a car, a aimal) to act out the situatios o their idividual umber lies. Number lies measurig 10 cm 55 cm are hady ad ca be used for may learig experieces. Review the geeralizatios about addig itegers ad their opposites, as discussed i the Backgroud Iformatio for learig outcome 7.N.6. Iform studets this is called the zero priciple. 8. Distribute two colours of couters to studets. State the colour that will represet positive itegers, ad the colour that will represet egative itegers. Remid studets of the zero priciple they have just established, ad ask them how they could use the couters to illustrate that the sum of a iteger ad its opposite is zero. Circulate amog studets ad, after sufficiet time, have studets share their ideas. Liste for the idea that matchig a positive ad a egative couter equals zero, so the pair ca be withdraw. If all the itegers match up as opposite pairs, there is othig remaiig, ad the value of the leftovers is zero. Example: This example represets (+3) + ( 3), ad each (+1) + ( 1) pair ca cacel, leavig 0. Computer applets may also be used to illustrate that the sum of a iteger ad its opposite is zero. Number 129

156 Sample Website: Computer applets are available o the followig website: Utah State Uiversity. Natioal Library of Virtual Maipulatives < Select Number ad Operatios (Grades 3 5), ad the select Color Chips Additio. 9. Studets record the zero priciple i their math jourals or otebooks ad draw diagrams ad the correspodig umber seteces to illustrate the geeralizatio. Ecourage them to draw both horizotal ad vertical umber lies. Variatios: Supply umber lies ad cut strips for studets to work with. Supply templates o which studets ca record the zero priciple. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Explai, usig cocrete materials such as iteger tiles ad diagrams, that the sum of opposite itegers is equal to zero. r Illustrate, usig a horizotal or vertical umber lie, the results of addig or subtractig egative ad positive itegers (e.g., a move i oe directio followed by a equivalet move i the opposite directio results i o et chage i positio). 130 Grade 7 Mathematics: Support Documet for Teachers

157 Suggestios for Istructio Illustrate, usig a horizotal or vertical umber lie, the results of addig or subtractig egative ad positive itegers (e.g., a move i oe directio followed by a equivalet move i the opposite directio results i o et chage i positio). Add two itegers usig cocrete materials or pictorial represetatios, ad record the process symbolically. Subtract two itegers usig cocrete materials or pictorial represetatios, ad record the process symbolically. Solve a problem ivolvig the additio ad subtractio of itegers. Materials: demostratio board umber lies (a large physical umber lie such as the oe used i the previous learig activity, or a persoal umber lie) small directioal toy (e.g., a perso, a aimal, a car) 20 couters of two differet colours (for each studet or group) cotaiers or paper boudaries i which to place the couters (for each studet or group) idex cards rulers, pecils, ad colours display area (to post completed scearios) math jourals or otebooks Orgaizatio: Whole class (for demostratio), small groups (for ivestigatios) Procedure: 1. Remid studets that i a previous learig experiece they modelled addig opposite itegers usig a umber lie by comparig distaces, ad represetig moves or jumps o a umber lie. They also modelled makig zero pairs usig two colours of couters. I this learig activity, studets will exted their modellig to represet addig or subtractig ay itegers ( 20) to (+20). Ask studets to use their umber lies or couters to model scearios, ad use itegers to write correspodig additio ad subtractio statemets to represet the actio i the scearios. Ecourage studets to use both the umber lies ad the couters to model scearios. Iclude a subtractio sceario that requires addig more itegers usig zero pairs. Number 131

158 Example: I this example, the dotted chip represets egative ad the solid chip represets positive. To model (+3) (+5), start with +3. The remove +5 (but there are oly 3 to remove, so it is ecessary to add eough zero pairs so that there are 5 positives to take away). You are left with 2, so (+3) (+5) = ( 2). 2. Preset sample scearios, such as those suggested below. I the samples, have studets model the actio both o the umber lies ad with couters to gai experiece with both models. Record the represetative equatio usig itegers. Sample Scearios: Luciee put $8 i a evelope i the morig. Later i the day, he put $2 i the evelope. How much moey is i the evelope? (+8) + (+2) = (+10) Ricki was i a cyclig derby. She rode 5 km, ad realized she missed the tur by the oak grove, which was 2 km back. How much of the course has she completed? (+5) + ( 2) = (+3) Here, studets may begi to otice that subtractig the positive iteger ad addig the egative iteger are equivalet. It was a dry summer i Okitow. The river was 2 m below its ormal level. Durig August, there was o rai, ad the water level wet dow aother metre. How far is the river below the ormal level ow? ( 2) + ( 1) = ( 3) Aisley owed his dad $12. His dad cacelled $5 of the debt. How much debt remais? ( 12) ( 5) = ( 7) Ravi had a collectio of model cars. He sold three cars to frieds at school, ad used the moey to purchase a ew model. What is the resultig chage i the umber of cars i his collectio? ( 3) + (+1) = ( 2) Remid studets to use itegers to represet the quatities. 132 Grade 7 Mathematics: Support Documet for Teachers

159 3. Whe studets have developed sufficiet proficiecy i modellig scearios, have them work i small groups to write scearios, act them out, ad idetify the correspodig equatios usig itegers. Remid studets to vary the actio models they use, sometimes usig umber lies ad sometimes usig couters. Whe a sceario is completed, they record the situatio o oe side of a idex card. O the reverse side, they draw a pictorial represetatio of the solutio, ad write the correspodig iteger equatio(s). Aim for a variety of additio ad subtractio scearios combiig positive ad egative itegers. As studets work together i groups, ask them to look for geeralizatios or rules they ca apply whe addig or subtractig itegers. As groups complete cards, have them verify their correctess, write their group ame o each card, ad post the cards i the desigated area so classmates have access to them. 4. Whe groups have completed five cards, they ca select a few scearios from their classmates ad write the solutios i their math jourals or otebooks. Ask them to iclude diagrams of maipulatives used, ad to write a applicable equatio. 5. Meet together as a class ad have studets share ay geeralizatios or methods that were helpful to them. Studets may record useful geeralizatios i their math jourals or otebooks. Variatios: Provide studets with scearios istead of havig them create their ow. Provide a hadout with ecessary supports for solvig the scearios. Iclude olie computer applets of iteger couters as maipulatives for studets to use while solvig their scearios. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Illustrate, usig a horizotal or vertical umber lie, the results of addig or subtractig egative ad positive itegers (e.g., a move i oe directio followed by a equivalet move i the opposite directio results i o et chage i positio). r Add two give itegers usig cocrete materials or pictorial represetatios, ad record the process symbolically. r Subtract two give itegers usig cocrete materials or pictorial represetatios, ad record the process symbolically. r Solve a give problem ivolvig the additio ad subtractio of itegers. r Visualize the itegers to assist i symbolically addig ad subtractig itegers. Number 133

160 Suggestios for Istructio Solve a problem ivolvig the additio ad subtractio of itegers. Materials: a deck of regular playig cards umber lies couters paper ad pecils (for fidig solutios) Orgaizatio: Pairs or small groups (of three or four studets) Procedure: I this learig activity, studets play a game requirig them to calculate the value of iteger cards. 1. Decide which cards (red or black) will represet positive itegers, ad which cards will represet egative itegers. Aces will have a value of 1. Jokers may be icluded as zero cards. Decide whether to iclude the face cards as values 11, 12, ad 13, or whether to remove them ad work with itegers 0 to Have studets form pairs or small groups. The deal all the cards evely amog the players. Players put their cards i a pile face dow. O the dealer s sigal, all players flip over their top cards, makig them easily visible to all. All players calculate the value of the cards. The first perso to say the correct value takes the up-tured cards ad puts them i his or her wi pile. The dealer sigals for the ext roud, ad play cotiues. Whe someoe s pile is depleted, the player shuffles his or her wi pile ad cotiues playig with it. Oce someoe has o remaiig cards, that player becomes a referee. Play cotiues util a set time is called, or util oly oe player has all the cards. The player with the most cards wis. Variatio: Use the cards to play iteger baseball, usig the process outlied i the Assessig Prior Kowledge learig experiece. This time, the pitcher has two card piles. The top cards are tured over, ad the batter returs the combied value. A error results i a out. 134 Grade 7 Mathematics: Support Documet for Teachers

161 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a give problem ivolvig the additio ad subtractio of itegers. r Apply metal mathematics strategies whe addig ad subtractig itegers. Suggestios for Istructio Illustrate, usig a horizotal or vertical umber lie, the results of addig or subtractig egative ad positive itegers (e.g., a move i oe directio followed by a equivalet move i the opposite directio results i o et chage i positio). Solve a problem ivolvig the additio ad subtractio of itegers. Materials: BLM 7.N.6.2: Iteger Football blak game cards markers ad tokes paper or card stock (for a football field) scoreboard rulers, scissors, ad tape word processor ad priter (optioal) Orgaizatio: Pairs or small groups Procedure: 1. Ask studets to work i pairs or i small groups to develop a football game requirig players to aswer iteger problems. Have them prepare rules of play ad ay required materials, such as those listed below. I desigig their games, studets may wish to refer to BLM 7.N.6.2: Iteger Football. Game Suggestios: Draw a paper football field with the yards ad ed zoes marked off. Provide a toke (for each team) ad a ball. Prepare sets of game cards, icludig ru cards ad pass cards, each with appropriate iteger statemets or problems, ad the solutios o the reverse or uder a fold. (The solutios would be the yards gaied or lost o the play.) Number 135

162 Idetify rules of play, icludig pealties or iterceptios for icorrect challeges. Provide a scoreboard. 2. Have studets play each other s games. Variatios: Supply the materials ad cards ad have studets play the game. Have studets develop ay other game requirig players to aswer iteger problems to score poits or to advace a play. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a give problem ivolvig the additio ad subtractio of itegers. 136 Grade 7 Mathematics: Support Documet for Teachers

163 Number (7.N.7) Edurig Uderstadig(s): Percets, fractios, decimals, ad ratios are differet represetatios of the same quatity. Number sese ad metal mathematics strategies are used to estimate aswers ad lead to flexible thikig. Geeral Learig Outcome(s): Develop umber sese. Specific Learig Outcome(s): 7.N.7 Compare ad order fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals. [CN, R, V] Achievemet Idicators: Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. Idetify a umber that would be betwee two umbers i a ordered sequece or o a horizotal or vertical umber lie. Idetify icorrectly placed umbers i a ordered sequece or o a horizotal or vertical umber lie. Positio fractios with like ad ulike deomiators from a set o a horizotal or vertical umber lie, ad explai strategies used to determie order. Order the umbers of a set by placig them o a horizotal or vertical umber lie that cotais bechmarks, such as 0 ad 1 or 0 ad 5. Positio a set of fractios, icludig mixed umbers ad improper fractios, o a horizotal or vertical umber lie, ad explai strategies used to determie positio. Number 137

164 Prior Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q (4.N.8) Demostrate a uderstadig of fractios less tha or equal to oe by usig cocrete ad pictorial represetatios to ame ad record fractios for the parts of a whole or a set compare ad order fractios model ad explai that for differet wholes, two idetical fractios may ot represet the same quatity provide examples of where fractios are used (5.N.7) Demostrate a uderstadig of fractios by usig cocrete ad pictorial represetatios to create sets of equivalet fractios compare fractios with like ad ulike deomiators (5.N.8) Describe ad represet decimals (teths, hudredths, thousadths) cocretely, pictorially, ad symbolically. (5.N.9) Relate decimals to fractios (teths, hudredths, thousadths). (5.N.10) Compare ad order decimals (teths, hudredths, thousadths) by usig bechmarks place value equivalet decimals (6.N.1) Demostrate a uderstadig of place value for umbers greater tha oe millio less tha oe-thousadth (6.N.4) Relate improper fractios to mixed umbers. (6.N.6) Demostrate a uderstadig of percet (limited to whole umbers), cocretely, pictorially, ad symbolically. Related Kowledge Studets should be able to do the followig: Q Q (7.N.2) Demostrate a uderstadig of the additio, subtractio, multiplicatio, ad divisio of decimals to solve problems (for more tha 1-digit divisors or 2-digit multipliers, the use of techology is expected). Q Q (7.N.3) Solve problems ivolvig percets from 1% to 100%. Q Q (7.N.4) Demostrate a uderstadig of the relatioship betwee repeatig decimals ad fractios, ad termiatig decimals ad fractios. 138 Grade 7 Mathematics: Support Documet for Teachers

165 Q Q Q Q Q Q (7.N.5) Demostrate a uderstadig of addig ad subtractig positive fractios ad mixed umbers, with like ad ulike deomiators, cocretely, pictorially, ad symbolically (limited to positive sums ad differeces). (7.N.6) Demostrate a uderstadig of additio ad subtractio of itegers, cocretely, pictorially, ad symbolically. (7.SP.4) Express probabilities as ratios, fractios, ad percets. Backgroud Iformatio Comparig ad Orderig Fractios, Decimals, ad Itegers To be efficiet at comparig ad orderig fractios, decimals, ad itegers, studets must uderstad the values of these umbers i our umber system ad their various represetatios. They must realize that fractios ad decimals are iterchageable ames for the same quatity ad must be able to covert oe to the other. They must be proficiet at reamig ad simplifyig fractios ad use multiple strategies for comparig them. Fractios Fractios are used to ame quatities betwee whole umbers. The part betwee whole umbers ca be divided ito ay umber of equal parts. The umber of equal parts i oe whole becomes the deomiator of the fractio, ad the umber of parts referred to forms the umerator of the fractio. The larger the deomiator is, the smaller the fractio pieces are. The smaller the deomiator is, the larger the fractio 1 1 pieces are. For example, is less tha. As the umeric value of the umerator approaches the umeric value of the deomiator, the umber gets closer to oe whole. Numerators with umeric values larger tha their deomiators represet improper fractios with values greater tha oe whole. Equivalet fractios have differet umerators ad deomiators, but represet the same portio of a whole. (The learig experieces suggested for learig outcomes 7.N.3, 7.N.4, ad 7.N.5 cotai iformatio about, ad strategies for, reamig equivalet fractios ad mixed umbers. They are recommeded for studets who eed to review these skills.) Number 139

166 Decimals Decimal umbers represet fractio quatities usig the base-10 umber system. Each successive decimal place represets a teth of the previous place value. All fractios ca be reamed as decimals. Oe way to reame a fractio as a decimal is to fid a equivalet fractio with a deomiator that is a power of 10. Examples: is equal to, writte as is equal to, writte as Aother way to reame a fractio as a decimal is to divide the umerator by the deomiator. Example: 12 = = Likewise, most decimal umbers ca be reamed as fractios. Termiatig decimals have a defiite umber of digits ad ca easily be reamed as fractios with deomiators that are powers of 10. The digits i the decimal umber form the umerator of the fractio, ad the deomiator is 1, followed by a umber of zeros 623 equal to the umber of digits to the right of the decimal umber e.g., = ad =. Repeatig decimals ca be reamed as fractios accordig to characteristic 10 patters explored i relatio to learig outcome 7.N.4 (a sigle repeatig digit has a 7 deomiator of 9, so 07. =. Decimals that are both o-repeatig ad o-termiatig 9 are irratioal umbers, ad, therefore, caot be reamed as fractios e.g., π, 2. Itegers ( ) Itegers comprise positive umbers, egative umbers, ad zero. Positive itegers refer to the regular coutig umbers, ad egative itegers refer to umbers less tha zero. Negative itegers are the opposite of their positive couterparts. The greater the umeric value of the egative iteger is, the farther it is from zero, ad, therefore, the smaller is the value of the umber. For example, ( 9) is smaller tha ( 1). 140 Grade 7 Mathematics: Support Documet for Teachers

167 Strategies for Comparig Relative Size Strategies for comparig the relative size of fractios, decimals, ad itegers iclude the followig: Associate the umbers with bechmarks such as 0, 1, ad 1, ad place them o a 2 umber lie. For closer comparisos, use bechmarks of 0, , 2, 4, ad 1. All egative itegers are less tha 0. The greater the umeric value is, the smaller the umber is. If fractios have the same umerator, the it is ecessary to compare oly the deomiators. The larger the deomiator is, the smaller each piece is. If the umerator is 1, the the larger the deomiator is, the closer the fractio is to zero (e.g., 1 9 is closer to zero tha 1 5 is). If fractios have the same deomiator, the it is ecessary to compare oly the umerators. The larger the umerator is, the more pieces there are, ad so the larger the fractio is. If the umerator is close to half of the deomiator, the the fractio is close to oe-half (e.g., 3 8 is a little less tha 1 5,ad is a little more tha If the umerator is close to the deomiator, the the umber is close to oe whole. A larger deomiator idicates smaller pieces. The smaller the piece is, the closer it is to 1, ad the larger the fractio is. Compare decimal umbers to the decimal equivalets for the bechmark fractios, 0.25, 0.5, 0.75, ad 1.0. Use a uderstadig of place value to compare decimal umbers. First, compare the uits i the largest place value. Teths are larger tha hudredths, which are larger tha thousadths (e.g., is a little larger tha 0.54, which is a little larger tha 0.5). Rewritig decimal umbers as equivalets with the same umber of digits helps make this cocept clear (e.g., is larger tha 0.540, which is larger tha 0.500). Move flexibly betwee the various represetatios (e.g., 7 10 because is 0.7, ad is or 0.68) is larger tha Number 141

168 Mathematical Laguage ascedig bechmark deomiators descedig equivalet fractios horizotal improper fractios mixed umbers umerators proper fractios repeatig decimal sequece termiatig decimal ulike deomiators verify vertical Learig Experieces Assessig Prior Kowledge Materials: BLM 7.N.7.1: Equivalet Fractios ad Decimals BLM 7.N.7.2: Equivalet Fractios, Decimals, ad Percets maipulatives for represetig fractios ad decimals (e.g., couters, fractio bars, umber lies, base-10 blocks) math jourals or otebooks calculators (optioal) Orgaizatio: Idividual or pairs, whole class 142 Grade 7 Mathematics: Support Documet for Teachers

169 Procedure: 1. Select a BLM for studets to work with (e.g., BLM 7.N.7.1: Equivalet Fractios ad Decimals or BLM 7.N.7.2: Equivalet Fractios, Decimals, ad Percets). 2. Provide studets with copies of the selected BLM, ad have them complete the tasks idividually or i pairs, usig maipulatives or calculators as eeded. 3. After a desigated time has passed, reassemble as a class, ad have studets share their resposes ad the strategies they used to arrive at their aswers. 4. Have studets use their math jourals or otebooks to record helpful strategies for covertig fractio ad decimal umbers. They ca add to these strategies i future learig activities. Variatios: Have studets create pictorial, equivalet fractio, decimal, ad percet represetatios for fractios of their choice. Provide template squares, ad combie the differet represetatios as a class display. Have studets create their ow scearios ad word problems ivolvig covertig decimal ad fractio otatios. Observatio Checklist Note: May of the learig experieces for learig outcome 7.N.7 ca be used to assess the followig competecies o the Grade 7 Numeracy Assessmet: Studet orders fractios. Studet orders decimal umbers. Studet uderstads that a give umber may be represeted i a variety of ways. Referece: Maitoba Educatio, Citizeship ad Youth. Middle Years Assessmet: Grade 7 Mathematics: Support Documet for Teachers: Eglish Program. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at <www. edu.gov.mb.ca/k12/assess/ support/math7/>. Liste to ad observe studets resposes to determie whether studets ca do the followig: r Covert decimal umbers to fractios (to thousadths), ad vice versa. r Relate improper fractios to mixed umbers. Number 143

170 Assessig Prior Kowledge Materials: blak paper pes of differet colours Orgaizatio: Idividual, small groups or whole class Procedure: 1. Ask studets to create a braistormig web etitled Everythig I Kow about Fractios. 2. Have studets participate i a group or class discussio to share iformatio from their webs. 3. Have studets use pes of differet colours to add ew ideas to their ow webs as they liste to the shared ideas of classmates. Variatio: Studets keep the webs so that they ca revisit ad add to them as their coceptual uderstadig of fractio grows. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Represet a variety of fractios i various ways, such as pictorially, as parts of a whole or a set a ratio a divisio statemet improper fractios ad mixed umbers equivalet fractios expressed as a decimal expressed as a percet 144 Grade 7 Mathematics: Support Documet for Teachers

171 Suggestios for Istructio Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. Positio a set of fractios, icludig mixed umbers ad improper fractios, o a horizotal or vertical umber lie, ad explai strategies used to determie positio. Materials: BLM 7.N.7.3: Comparig Fractio ad Decimal Equivalets BLM : Base-Te Grid Paper BLM : Fractio Bars maipulatives for represetig fractios ad decimals (e.g., couters, umber lies, base-10 blocks) calculators (optioal) math jourals or otebooks Orgaizatio: Idividual or pairs Procedure: 1. Distribute copies of BLM 7.N.7.3: Comparig Fractio ad Decimal Equivalets, ad have studets fid solutios to the give problems, usig maipulatives or calculators as eeded. 2. After a desigated time has passed, have studets share their resposes ad the strategies they used to arrive at their aswers. 3. Have studets use their math jourals or otebooks to record helpful strategies for covertig ad comparig fractio ad decimal umbers. They ca add to these strategies i future learig activities. Variatios: Ask studets to create their ow scearios ad word problems that ivolve covertig decimal ad fractio otatios or that require comparig fractios ad decimal umbers. Have studets share their scearios with either the class or a small group. Have idividuals preset ad explai their solutios, ad have other group members verify the solutios. Number 145

172 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Covert decimal umbers to fractios (to thousadths), ad vice versa. r Relate improper fractios to mixed umbers. r Compare ad order fractios ad decimals. Suggestios for Istructio Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. Idetify a umber that would be betwee two umbers i a ordered sequece or o a horizotal or vertical umber lie. Idetify icorrectly placed umbers i a ordered sequece or o a horizotal or vertical umber lie. Materials: a collectio of library books labelled with decimal umbers o the spie (If possible, have three books per studet. Choose books with whole umbers relatively close together, ad a variety of decimal umbers i betwee.) math jourals or otebooks card stock (optioal) Orgaizatio: Small groups (two to four studets), whole class Procedure: 1. Choose a variety of ways to form groups quickly (e.g., combiatios of colours wor, begiig letters i first ames). 2. Have two or three studets combie their library books ad sort them i ascedig order accordig to the decimal umbers o the spies. (Groups of four ca work as two groups of two.) Have group members verify that their order is correct ad explai their reasoig to oe aother. 3. Have studets reclaim their books, form ew groups, sort the books, ad verify the results oce agai. Repeat as ofte as the learig activity seems useful. 4. Meet as a class ad have studets share strategies for orderig decimal umbers. 5. Have studets use their math jourals or otebooks to record strategies for orderig decimal umbers. 146 Grade 7 Mathematics: Support Documet for Teachers

173 Variatios: Have oe or two studets order the books icorrectly, ad have a third studet idetify the misplaced books ad explai why the books belog elsewhere. Use fewer books. Desigate the learig activity as oe statio of a set of rotatig learig activities. Istead of usig library books, use card stock to create a set of book spies labelled with titles ad decimal umbers, or provide studets with a hadout cotaiig images of book spies labelled with decimal umbers. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. r Idetify a umber that would be betwee two umbers i a ordered sequece or o a horizotal or vertical umber lie. r Idetify icorrectly placed umbers i a ordered sequece or o a horizotal or vertical umber lie. Suggestios for Istructio Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. Materials: BLM 7.N.7.4: Orderig Decimal Numbers Orgaizatio: Idividual Procedure: 1. Defie ascedig order ad descedig order. 2. Distribute copies of BLM 7.N.7.4: Orderig Decimal Numbers, ad have studets complete the tasks. Studets place the give decimal umbers i ascedig order. They the choose six of the umbers ad write them i descedig order. Number 147

174 Variatios: Have studets add decimal umbers that would fit betwee the give umbers. Create other umber sets. Cosider restrictig the type of umbers preseted i each set (e.g., use umbers cotaiig oly hudredths or oly thousadths). Have studets create their ow umber sets, exchage sets with a parter, order their parter s set, ad the verify their parter s orderig of the sets. Repeat the learig activity with fractios or combiatios of fractios ad decimals. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. Suggestios for Istructio Idetify a umber that would be betwee two umbers i a ordered sequece or o a horizotal or vertical umber lie. Idetify icorrectly placed umbers i a ordered sequece or o a horizotal or vertical umber lie. Positio fractios with like ad ulike deomiators from a set o a horizotal or vertical umber lie, ad explai strategies used to determie order. Order the umbers of a set by placig them o a horizotal or vertical umber lie that cotais bechmarks, such as 0 ad 1 or 0 ad 5. Materials: BLM 7.N.7.5: Sequetial Fractios ad Their Decimal Equivalets calculators 45 small blak cards markers large area to create a umber lie Orgaizatio: Idividual, whole class 148 Grade 7 Mathematics: Support Documet for Teachers

175 Procedure: 1. Distribute copies of BLM 7.N.7.5: Sequetial Fractios ad Their Decimal Equivalets. Have studets follow the directios to fid the decimal equivalets for the sequetial fractios, compare sizes usig decimal ad fractio otatio, idicate equivalet fractios, ad make geeralizatios about comparig fractios. 2. Distribute all the blak cards to studets. Assig fractios to studets, ad ask them to write the give fractios o the blak cards. Specify a size of fot so that the cards look similar. Esure that each card cotais a fractio. 3. Have studets create a umber lie by orderig the fractio cards. Ask them to explai why they have placed umbers i a particular order. 4. Discuss strategies studets used to determie the order of the fractios. 5. Choose two of the fractios o the umber lie, ad ask studets to suggest a umber betwee the two fractios. Discuss the strategies used to determie that umber. Variatios: Prepare umber cards ahead of time, distribute them to studets, ad ask studets to order the cards. Call upo groups of studets to order the fractio cards they have. Call for all studets with cards ear bechmarks or betwee particular umbers to order their cards. Have studets play games with the cards (e.g., call two to four studets, ad the studet with the largest or smallest fractio wis). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify a umber that would be betwee two umbers i a ordered sequece or o a horizotal or vertical umber lie. r Idetify icorrectly placed umbers i a ordered sequece or o a horizotal or vertical umber lie. r Positio fractios with like ad ulike deomiators from a set o a horizotal or vertical umber lie, ad explai strategies used to determie order. r Order the umbers of a set by placig them o a horizotal or vertical umber lie that cotais bechmarks, such as 0 ad 1 or 0 ad 5. Number 149

176 Suggestios for Istructio Idetify a umber that would be betwee two umbers i a ordered sequece or o a horizotal or vertical umber lie. Materials: grid paper rulers (to draw umber lies with bechmarks) math jourals or otebooks bolts of various sizes ad a socket set (optioal) Orgaizatio: Pairs, whole class, idividual Procedure: 1. Select the type of umbers to work with (e.g., fractios, mixed umbers, decimals, itegers, combiatios). 2. Select the size ad the orietatio of the umber lie (e.g., 0 1, ( 5) (+5), horizotal, vertical). Draw the umber lie ad mark some bechmarks. 3. Assig roles to pairs of studets. For example, player A will write above a horizotal umber lie, or to the left of a vertical umber lie, ad player B will write below a horizotal umber lie, or to the right of a vertical umber lie. Choose which player will play first. 4. Player A marks a poit o the lie, ad, writig above the lie, draws a arrow to the poit ad labels the poit with a approximate value. 5. Player B marks aother poit o the lie, ad, writig below the lie, draws a arrow to the poit ad labels the poit with a approximate value. 6. Player A the marks ad labels a poit that ca be foud betwee the last two marked poits. 7. Player B the marks ad labels a poit betwee the last two marked poits. 8. Play cotiues util oe of the players ca o loger place poits o the umber lie. 9. Whe studets have had sufficiet time for the learig activity, have them reassemble as a class ad discuss strategies they used to fid a umber betwee two fractios, decimals, or itegers. (See Backgroud Iformatio for strategies.) 10. Have studets use their math jourals or otebooks to record strategies for fidig a umber betwee two fractios, decimals, or itegers. 150 Grade 7 Mathematics: Support Documet for Teachers

177 Variatios: Provide studets with pre-marked umber lies. Play the game as a class, selectig studets to place umbers o a demostratio umber lie. 1 3 Provide studets with two umbers e.g., ad or 14ad 1 3,. ad ask them to idetify a umber betwee the give umbers. Discuss the strategies they used to idetify the umber. Desigate the learig activity as oe statio of a set of rotatig learig activities. Use a assortmet of bolt sizes ad a socket set as a alterative to a umber lie. Have studets fid a bolt betwee two specified bolt sizes. Use the sockets to verify the order. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify a umber that would be betwee two umbers i a ordered sequece or o a horizotal or vertical umber lie. Suggestios for Istructio Order the umbers of a set by placig them o a horizotal or vertical umber lie that cotais bechmarks, such as 0 ad 1 or 0 ad 5. Positio a set of fractios, icludig mixed umbers ad improper fractios, o a horizotal or vertical umber lie, ad explai strategies used to determie positio. Materials: BLM 7.N.7.6: Relatig Numbers to Bechmarks pes or pecils of differet colours Orgaizatio: Idividual, small groups, whole class Procedure: 1. Distribute copies of BLM 7.N.7.6: Relatig Numbers to Bechmarks, ad have studets complete the sheet idividually. They place umbers (e.g., words, pictures, symbols, proper fractios, improper fractios, mixed umbers, decimals, itegers, percets) i the appropriate boxes labelled as follows: less tha oe-half, equal to oe-half, ad greater tha oe-half. They the place eight selected umbers o a umber lie by drawig a poit ad a label for each umber, explaiig their placemet choices. Number 151

178 2. Have studets form small groups of two to four. Ask them to share examples of their umbers, umber lies, ad strategies. If they wish, they may make revisios to their sheets, usig a differet coloured pe or pecil. 3. Meet as a class ad have studets share examples ad strategies. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Order the umbers of a set by placig them o a horizotal or vertical umber lie that cotais bechmarks, such as 0 ad 1 or 0 ad 5. r Positio a set of fractios, icludig mixed umbers ad improper fractios, o a horizotal or vertical umber lie, ad explai strategies used to determie positio. Suggestios for Istructio Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. Positio a set of fractios, icludig mixed umbers ad improper fractios, o a horizotal or vertical umber lie, ad explai strategies used to determie positio. Materials: BLM 7.N.7.7: Orderig Numbers ad Verifyig the Order sets of six to te umbers, icludig combiatios of proper ad improper fractios, mixed umbers, itegers, decimal umbers, ad percets presetatio board blak paper blak cards (optioal) Orgaizatio: Whole class, idividual Procedure: 1. Preset studets with a set of umbers ad ask them to place the umbers i order. Specify whether you would like ascedig or descedig order. 2. Choose studets to demostrate the order ad to explai strategies they used to place the umbers. 152 Grade 7 Mathematics: Support Documet for Teachers

179 3. Repeat the process with other umber sets, or ask studets to suggest umbers. Each time, discuss strategies studets used to place the umbers. 4. Distribute copies of BLM 7.N.7.7: Orderig Numbers ad Verifyig the Order. Have studets draw a vertical or a horizotal umber lie, place a set of umbers o the lie, ad explai the strategies they used. Variatio: Write umber sets o cards, distribute the cards, ad have studets lie up with their cards i ascedig or descedig order. Classmates ca share how they kow a umber is i the correct order, or why it is out of place. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Order the umbers of a set that icludes fractios, decimals, or itegers i ascedig or descedig order, ad verify the result usig a variety of strategies. r Positio a set of fractios, icludig mixed umbers ad improper fractios, o a horizotal or vertical umber lie, ad explai strategies used to determie positio. Number 153

180 N o t e s 154 Grade 7 Mathematics: Support Documet for Teachers

181 G r a d e 7 M a t h e m a t i c s Patters ad Relatios

182

183 Patters ad Relatios (Patters, ad Variables ad Relatios) (7.PR.1, 7.PR.2, 7.PR.3, 7.PR.4, 7.PR.5, 7.PR.6, 7.PR.7) Edurig Uderstadig(s): Words, tables, graphs, ad expressios are differet represetatios of the same patter. Preservatio of equality is used to solve equatios. The priciples of operatios used with whole umbers also apply to operatios with decimals, fractios, ad itegers. Number sese ad metal mathematics strategies are used to estimate aswers ad lead to flexible thikig. Geeral Learig Outcome(s): Use patters to describe the world ad solve problems. Represet algebraic expressios i multiple ways. Note: Backgroud Iformatio for the Patters ad Relatios strad is preseted i a slightly differet order tha i other strads. This variatio is iteded to accommodate learig experieces that itegrate achievemet idicators from learig outcomes i both the Patters substrad ad the Variables ad Equatios substrad. Some achievemet idicators related to Variables ad Equatios are developed usig studet experieces devoted to explorig Patters. Specific Learig Outcome(s): 7.PR.1 Demostrate a uderstadig of oral ad writte patters ad their equivalet relatios. [C, CN, R] Achievemet Idicators: Formulate a relatio to represet the relatioship i a oral or a writte patter. Provide a cotext for a relatio that represets a patter. Represet a patter i the eviromet usig a relatio. (cotiued) Patters ad Relatios 3

184 Specific Learig Outcome(s): 7.PR.2 Costruct a table of values from a relatio, graph the table of values, ad aalyze the graph to draw coclusios ad solve problems. [C, CN, R, V] 7.PR.3 Demostrate a uderstadig of preservatio of equality by modellig preservatio of equality, cocretely, pictorially, ad symbolically applyig preservatio of equality to solve equatios [CN, R, V] 7.PR.4 Explai the differece betwee a expressio ad a equatio. [C, CN] 7.PR.5 Evaluate a expressio give the value of the variable(s). [CN, R] Achievemet Idicators: Create a table of values for a relatio by substitutig values for the variable. Create a table of values usig a relatio ad graph the table of values (limited to discrete elemets). Sketch the graph from a table of values created for a relatio, ad describe the patters foud i the graph to draw coclusios (e.g., graph the relatioship betwee ad 2 + 3). Describe the relatioship show o a graph usig everyday laguage i spoke or writte form to solve problems. Match a set of relatios to a set of graphs. Match a set of graphs to a set of relatios. Model the preservatio of equality for additio, subtractio, multiplicatio, or divisio usig cocrete materials or usig pictorial represetatios, explai the process orally, ad record it symbolically. Solve a problem by applyig preservatio of equality. Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. Explai what a variable is ad how it is used i a expressio. Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. Substitute a value for each ukow i a expressio ad evaluate the expressio. (cotiued) 4 Grade 7 Mathematics: Support Documet for Teachers

185 Specific Learig Outcome(s): 7.PR.6 Model ad solve problems that ca be represeted by oe-step liear equatios of the form x + a = b, cocretely, pictorially, ad symbolically, where a ad b are itegers. [CN, PS, R, V] 7.PR.7 Model ad solve problems that ca be represeted by liear equatios of the form ax + b = c ax = b x = b, a 0 a [CN, PS, R, V] Achievemet Idicators: Represet a problem with a liear equatio ad solve the equatio usig cocrete models. Draw a visual represetatio of the steps required to solve a liear equatio. Solve a problem usig a liear equatio. Verify the solutio to a liear equatio usig cocrete materials or diagrams. Substitute a possible solutio for the variable i a liear equatio to verify the equality. Model a problem with a liear equatio ad solve the equatio usig cocrete models. Draw a visual represetatio of the steps used to solve a liear equatio. Solve a problem usig a liear equatio ad record the process. Verify the solutio to a liear equatio usig cocrete materials or diagrams. Substitute a possible solutio for the variable i a liear equatio to verify the equality. Prior Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q (4.N.5) Describe ad apply metal mathematics strategies, such as skip-coutig from a kow fact usig doublig or halvig usig doublig ad addig oe more group usig patters i the 9s facts usig repeated doublig to develop recall of basic multiplicatio facts to 9 9 ad related divisio facts. (4.PR.1) Idetify ad describe patters foud i tables ad charts, icludig a multiplicatio chart. (4.PR.2) Reproduce a patter show i a table or chart usig cocrete materials. (4.PR.3) Represet ad describe patters ad relatioships usig charts ad tables to solve problems. Patters ad Relatios 5

186 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q (4.PR.4) Idetify ad explai mathematical relatioships usig charts ad diagrams to solve problems. (4.PR.5) Express a problem as a equatio i which a symbol is used to represet a ukow umber. (4.PR.6) Solve oe-step equatios ivolvig a symbol to represet a ukow umber. (5.PR.1) Determie the patter rule to make predictios about subsequet elemets. (5.PR.2) Solve problems ivolvig sigle-variable (expressed as symbols or letters), oe-step equatios with whole-umber coefficiets, ad whole-umber solutios. (6.N.3) Demostrate a uderstadig of factors ad multiples by determiig multiples ad factors of umbers less tha 100 idetifyig prime ad composite umbers solvig problems ivolvig factors or multiples (6.N.5) Demostrate a uderstadig of ratio, cocretely, pictorially, ad symbolically. (6.N.7) Demostrate a uderstadig of itegers, cocretely, pictorially, ad symbolically. (6.PR.1) Demostrate a uderstadig of the relatioships withi tables of values to solve problems. (6.PR.2) Represet ad describe patters ad relatioships usig graphs ad tables. (6.PR.3) Represet geeralizatios arisig from umber relatioships usig equatios with letter variables. (6.PR.4) Demostrate ad explai the meaig of preservatio of equality cocretely, pictorially, ad symbolically. (6.SS.8) Idetify ad plot poits i the first quadrat of a Cartesia plae usig whole-umber ordered pairs. (6.SP.1) Create, label, ad iterpret lie graphs to draw coclusios. (6.SP.3) Graph collected data ad aalyze the graph to solve problems. Related Kowledge Studets should be able to do the followig: Q Q Q Q (7.N.1) Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. (7.N.5) Demostrate a uderstadig of addig ad subtractig positive fractios ad mixed umbers, with like ad ulike deomiators, cocretely, pictorially, ad symbolically (limited to positive sums ad differeces). 6 Grade 7 Mathematics: Support Documet for Teachers

187 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q (7.N.6) Demostrate a uderstadig of additio ad subtractio of itegers, cocretely, pictorially, ad symbolically. (7.N.7) Compare ad order fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals (7.PR.1) Demostrate a uderstadig of oral ad writte patters ad their equivalet relatios. (7.PR.2) Costruct a table of values from a relatio, graph the table of values, ad aalyze the graph to draw coclusios ad solve problems. (7.PR.3) Demostrate a uderstadig of preservatio of equality by modellig preservatio of equality, cocretely, pictorially, ad symbolically applyig preservatio of equality to solve equatios (7.PR.4) Explai the differece betwee a expressio ad a equatio. (7.PR.5) Evaluate a expressio give the value of the variable(s). (7.PR.6) Model ad solve problems that ca be represeted by oe-step liear equatios of the form x + a = b, cocretely, pictorially, ad symbolically, where a ad b are itegers. (7.PR.7) Model ad solve problems that ca be represeted by liear equatios of the form ax + b = c ax = b x = b, a 0 a cocretely, pictorially, ad symbolically, where a, b, ad c are whole umbers. (7.SS.1) Demostrate a uderstadig of circles by describig the relatioships amog radius, diameter, ad circumferece of circles relatig circumferece to pi determiig the sum of the cetral agles costructig circles with a give radius or diameter solvig problems ivolvig the radii, diameters, ad circumfereces of circles (7.SS.2) Develop ad apply a formula for determiig the area of triagles parallelograms circles Patters ad Relatios 7

188 Q Q Q Q Q Q Q Q (7.SS.4) Idetify ad plot poits i the four quadrats of a Cartesia plae usig ordered pairs. (7.SS.5) Perform ad describe trasformatios of a 2-D shape i all four quadrats of a Cartesia plae (limited to itegral vertices). (7.SP.1) Demostrate a uderstadig of cetral tedecy ad rage by determiig the measures of cetral tedecy (mea, media, mode) ad rage determiig the most appropriate measures of cetral tedecy to report fidigs (7.SP.3) Costruct, label, ad iterpret circle graphs to solve problems. Backgroud Iformatio Patters The world is full of patters. They are foud i multiple cotexts both i ature ad i the creatios of people. Patters are prevalet i plat ad aimal life, as well as i the physical world. They are evidet i the arts, music, structures, movemet, time, ad space. Our umber system is rooted i patter, ad a uderstadig of patter is the basis of mathematical cocepts i every strad of mathematics. As studets iterpret patters ad geeralize the relatioships represeted by them, they develop algebraic thikig ad reasoig skills that eable them to apply mathematics i everyday situatios. These geeralizatios ad the equatios ad formulas derived from them are powerful tools for makig predictios ad solvig problems. They make mathematics meaigful, ad are also importat for the mathematics that studets will study i later grades. Someoe who possesses the ability to idetify a ew patter ad its symbolic relatio ca solve a problem that previously seemed isurmoutable. That perso may make a ew relatioal discovery leadig to a ew advacemet i sciece or techology. Such was the case with Dmitri Medeleev s work o the periodic table, Albert Eistei s formulatio of the E = mc 2 equatio, ad the more recet work o the relatioships betwee prime umbers ad quatum mechaics. Of more iterest to Middle Years studets will be the schoolroom mathematics story commoly attributed to the Germa mathematicia Carl Gauss. Aroud 1787, Carl Gauss s teacher directed his class to fid the sum of the cosecutive umbers 1 to 100. The 10-year-old Carl promptly submitted a aswer of Whe questioed how he could possibly perform the calculatio so quickly, Carl explaied he did ot add the umbers from 1 to 100. Rather, he saw a patter. He paired the largest ad the smallest umbers ( = 101, = 101, = 101), ad determied the set cotaied 50 such pairs, each totallig 101. Thus, the total sum was , or Presetig the same task to studets i today s classroom will likely ucover some iterestig discussio regardig patters, ad reveal that there are may ways to view a problem ad its solutio. 8 Grade 7 Mathematics: Support Documet for Teachers

189 The process of makig coectios i patters is developed i the learig outcomes of previous grades. It ivolves covertig patters to umeric values, extedig the patters, graphig the patters, ad explaiig the mathematical relatioship betwee the quatities. The relatioship is expressed i mathematical symbols usig the laguage of algebra ad exteded to a equatio or formula. The Grade 7 learig outcomes focus o matchig symbolic relatios, i the form of algebraic expressios ad equatios, with patter cotexts ad the represetatios of these relatios as tables of values ad graphs. The relatios are the used to solve problems. The ability to exted patters ad represet them as tables, graphs, ad equatios is assessed i the Grade 7 Numeracy Assessmet. Therefore, backgroud iformatio regardig patters, how to represet them, ad how to idetify the relatioships withi them is provided, begiig with a discussio of the categories of patters. Categories of Patters For the purposes of this documet, there are two mai categories of patters: repeatig patters ad growig patters. Repeatig patters: Repeatig patters cosist of repeated sequeces or arragemets of items about which predictios ca be made. I a repeatig patter, a set of elemets, referred to as a core, appears i a set order over ad over agai, or appears as a trasformatio over ad over agai. Colour patters, shape patters, rhyme patters, ad tessellatios are examples of repeatig patters. Numerals may also be arraged i repeatig patters. Examples: Growig patters: Growig patters (also called sequeces or umber patters) cosist of a series of steps called figures or terms. Each term is related to the previous term accordig to a patter. The terms are umbered accordig to their sequetial order, ad each term is assiged a correspodig umeric term value, which is determied by the umber of items i that term. Growig patters may grow or shrik, depedig o whether the umeric values of the terms icrease or decrease. The growth may occur at a costat or o-costat rate. Patters ad Relatios 9

190 Example: Pictorial represetatio of a patter Growig Patters Term umber (the positio of the term i the sequece) Term value (the umber of items i the term) Growig patters, which are emphasized i Grade 7, are further classified as arithmetic sequeces or geometric sequeces. Arithmetic sequeces: I arithmetic sequeces, the rate of chage is costat. Each term value chages by a fixed amout i relatio to the previous term value. A particular value is added to or subtracted from the previous term value. A example is the patter 2, 4, 6,..., where each successive term value icreases by a value of 2. Arithmetic sequeces are geerally easier to idetify tha geometric sequeces. Geometric sequeces: I geometric sequeces, the ext term value is a multiple of the previous term value. Patter growth is ot costat. I the example 2, 4, 8, 16,..., each term value is twice the previous value. I the example 900, 300, , each term value is 1 the amout of the 3 previous value. I the Fiboacci (Leoardo Pisao) umber series (1, 1, 2, 3, 5, 8, 13,...), each umber is the sum of the two previous umbers. It is either arithmetic or geometric. Iterestig patters occur i squares, cubic umbers, ad triagular umbers, as well as i Fiboacci umbers ad fractals. While the patters are iterestig to explore, keep i mid that Grade 7 learig outcomes are limited to liear relatios, ad do ot iclude powers ad expoets. Studets may be able to describe recursive relatioships i these patters, but ot the explicit rules with relatioships that describe the equatio of the patters. Patters are all aroud us, ragig from four legs o a chair to the patters foud i geometry, art, architecture, ad music. The legths ad diameters of pipes i a pipe orga provide a example of relatioships i pitches ad harmoics ad differet octaves. 10 Grade 7 Mathematics: Support Documet for Teachers

191 Pleasat patters ofte correspod to the ratio of the golde mea or rectagle. The patters i soud waves ad light waves are coverted to umeric values, ad the relatioships are used i producig CDs, DVDs, ad other digital techology. Two Types of Relatioships i Patters Two types of relatioships i patters are recursive ad explicit relatioships. Recursive relatioships: These relatioships explai how each term i the patter compares or relates to the precedig term i the patter. They are useful for extedig patters ad for completig tables of values. I the patter 2, 4, 6,..., each successive term value icreases by a value of 2. The words, begi at 2 ad add 2 to each term value, or the term value plus 2 equals the ext term value, or the expressio, t + 2, where t is the previous term value all describe the recursive relatioship i the patter. This relatioship is limitig whe the eed arises to determie term values for terms that are ot close to those kow. It also represets a misuse of the variable t. Explicit relatioships (or rule): These relatioships relate the term umber to the term value for each term. They are used to predict the th value i a patter, ad to express the patter as a equatio or formula. The words, the term umber multiplied by 2 describes the explicit relatioship i the patter 2, 4, 6,.... Recall that multiplicatio is repeated additio; thus, the relatioship is also described as, the term umber multiplied by 2, or 2, where is the term umber. Placig the umbers i a table of values makes it easier to compare the term ad term value umbers. Term umber Term value Whe assessig studet performace, keep i mid that Grade 7 learig outcomes are limited to discrete elemets ad liear relatios, ad do ot iclude powers ad expoets. Studets may be able to describe recursive relatioships i some geometric patters, but may ot be able to articulate symbolic relatios to describe the explicit rules i these patters. Whe askig studets to represet the patter algebraically, the expectatio is that studets will be able to represet the explicit patter, 2. Patters ad Relatios 11

192 Represetig Patters ad Idetifyig Relatioships Patters ca be represeted i differet ways. The recursive ad explicit relatioships betwee the elemets exist i the differet represetatios of a patter. Geeralizig explicit relatioships ca be challegig, ad may require persistece. Each type of represetatio provides a differet view ad a differet way to thik about the relatioships. Ecourage studets to work toward idetifyig the relatioship i each type of represetatio. To icrease studets ability to thik symbolically, begi with more obvious relatios ad teach studets to ask themselves icreasigly complex questios about the relatios (e.g., What remais the same? What chages? By how much does it chage? Is this true for every term i the sequece? How ca that idea be represeted? What happes if...?). The followig process for represetig patters flows from the cocrete or pictorial to the symbolic. It is importat to guide studets through the process. Cocrete or pictorial represetatio: The cotext of the patter exists i the physical patter itself, ad ca be represeted cocretely or pictorially. Studets ca examie the physical terms to determie what remaied the same i each term ad what chaged. Playig with colour or arragemet of patters ca ofte help studets to see the costat ad the chagig aspects of a patter. Example: Pictorial represetatio of a patter I this example, two dots are added to the top of each ew term. The term value = the previous term value + 2. Charts or tables of values: Charts or tables of values display umeric represetatio of the patter values, ad may also be used to record the recursive chages betwee the terms. These represetatios, as illustrated i the followig example, facilitate umeric comparisos. They ca be preseted i horizotal or vertical format. Example: Term umber () (the positio of the term i the sequece) Term value (v) (the umber of items i the term) Grade 7 Mathematics: Support Documet for Teachers

193 Read the table i oe directio (across i the table above) to idetify the recursive relatioship (each term value icreases by 2). Read i the opposite directio (dow i the table above) to get iformatio about the explicit relatioship. Ask, How ca the term umber be chaged to get the term value? Here, multiply the term umber by 2 ad add 1. Express the relatioship symbolically as the expressio, or equatio = v, where = the term umber ad v = the term value If the relatioships are ot evidet i the charts or tables, examie the other represetatios for clues. Studets will have a easier time seeig umeric relatioships if they have a good grasp of additio ad multiplicatio facts. Graphs: A graph provides a cocrete picture of the relatioships i the patter. It provides clear evidece of whether the values are icreasig or decreasig, how quickly the chage is happeig, ad the icremet of chage. Relatioships ca be described by articulatig these chages. Each poit o the graph represets a (x, y) pair. Try to express both umbers i terms of the x-value (e.g., (x, x + 2)). If the poits lie up alog a 45º iclie, the relatio may have oly a costat (e.g., it has a coefficiet of 1). If the iclie is steeper, the relatio will have a coefficiet greater tha 1, ad it may have a costat. If it is less steep, the relatio will have a coefficiet less tha 1 ad greater tha zero, ad it may have a costat. Note: I the previous equatio, v = 2 + 1, ad 2 is the coefficiet ad 1 is the costat. Example: Note: The Grade 7 learig outcomes deal with discrete data. Sice ad v refer specifically to the term umber ad term value, they are represeted by atural umbers (1, 2, 3, 4,...). As a result, o lie should be draw through the poits. I this example, the plotted poits lie i a straight lie. Therefore, this patter is a liear relatio. As x icreases, so does y. So the patter is a icreasig liear relatio. Patters ad Relatios 13

194 Evaluatig the recursive relatioship shows that the value of the first term is 3, ad the subsequet terms icrease by 2. Goig to the first term, ad the goig back oe step from the first term, ad the removig 2 from the first term value results i 1. Evaluatig the explicit term could begi with thikig that somethig must be added to 1 to equal each term value. I the symbolic relatio, + 1 is preset i every term. It is referred to as a costat. Through experimetatio, studets ca determie that the y-value, whe x is zero, represets the costat this is represeted by the place where the relatio crosses the y-axis o a graph. O the graph, for each icrease i 1 horizotally (each icrease i 1 by x, or each time the term umber icreases by 1) the vertical icrease is 2 (y icreases by 2, or the term value icreases by 2). This ca be expressed usig the relatio 2x, where 2 is referred to as the umerical coefficiet. Mathematical laguage (words): Mathematical laguage ca be used to describe the patter as it appears i the physical represetatio, chart, or graph. A clear descriptio of a patter ca help studets to recogize the explicit relatioship. If studets describe a patter as icreasig by two dots each time, they may see the recursive relatioship. If, however, they ca be more descriptive ad say, the patter starts with three dots, ad two dots are added o top each time, they may begi to otice that there is a explicit relatioship. Mathematical symbols: Mathematical symbols ca be used to create expressios, formulas, ad equatios to represet the relatioships. The patter i the previous table ca be represeted symbolically by the expressio or equatio = v, where = the term umber ad v = the term value A blacklie master for recordig these represetatios is available i BLM 7.PR.1: Patters: A Process. The process flows from the cocrete to the pictorial to the symbolic, ad it is importat to guide studets through it. 14 Grade 7 Mathematics: Support Documet for Teachers

195 The Meaig of Variables Variables are symbols used to take the place of umbers. Variables provide the ability to express geeralizatios without ay attachmet to a specific value. I writig relatios, studets will gai experiece workig with all three applicatios i which variables are used. Uderstadig the differet applicatios ca help reduce cofusio over what a variable actually stads for. Cautio: Whe usig variables i relatios, keep i mid the followig: Ay visual ca serve as a variable, but it is covetioal to use lowercase letters. For example, ad 2x + 1 represet the same situatio. Both ad x are variables that take the place of whatever umber. I earlier grades, studets ofte use pictures or shapes to represet a variable. It is importat that studets uderstad mathematical covetios, ad regularly use lower case letters for variables. Usig a variety of variables with studets is importat. Be careful, however, ot to select variables that may be cofused with uits to represet a sceario where uits may be preset. The variable x is ofte cofused with the multiplicatio symbol. It is appropriate for studets to begi represetig multiplicatio usig paretheses, 2(x), the multiplicatio dot 2 x, or, i its simplified form, 2x. Applicatios i Which Variables Are Used Variables ca be used for the followig purposes: To represet a value that chages. I the precedig example of the chart of values, the variable represets the term umber of the patter. The term umber is a value that chages for each term. The variable ca represet ay particular term umber, but oly oe term value for a give term umber. Ay variable ca be chose to represet the term umber (e.g., for the expressio ad for a term umber of 3, there ca oly be oe term value, 7). To solve for a specific ukow. The explicit rule to determie a term value i the above example is 2 + 1, where is the term umber. Substitutig ay term umber for ad simplifyig the expressio will reveal the ukow term value for that particular term umber. To simplify a expressio. I the above example, represets the term umber, ad represets the term value. Together, (, 2 + 1) represet the x- ad y-coordiates used for each term o a graph. The variable y represets the simplified expressio for each term value (y = 2 + 1). Patters ad Relatios 15

196 Equatios ad Expressios I the above example, y = 2x + 1, the expressio, 2x + 1 represets the term value. The variable y also represets the same term value. The expressios 2x + 1 ad y are differet ames for the same value. They are equivalet expressios. A equal sig (=) is used i a equatio to show that both expressios represet the same value, y = 2x + 1. All equatios cotai a equal sig betwee two differet represetatios of the same value. Expressios are relatios that do ot cotai a equal sig. They provide oly oe descriptio of the value referred to. The expressio 2x + 1 represets a value, ad y is a separate expressio that represets the same value. Watch for cofusio i studets about the use of equal sigs. Whe studets see a equal sig, they sometimes iterpret it as a directio to do somethig with the umbers precedig it. They forget the equal sig s role as a symbol of equality. For example, if studets see 3 4 = 6, some might thik they are beig asked to multiply 3 4, which would result i = 12. If were 12, the statemet would say 3 4 = 12 6 or 12 = 72, ad that is ot a true statemet. Variables ad Equatios Recall that i the equatio, 2x + 3 = y, 2 is the umerical coefficiet, x ad y are variables, + 3 is a costat, ad the equal sig idicates a equatio. Altogether, they represet a equatio made up of two equivalet expressios, 2x + 3 ad y. I previous grades, studets model the preservatio of equality cocretely, pictorially, ad orally, ad represet ad verify equivalet forms of a equatio. I Grade 7, studets exted this ability to usig preservatio of equality to solve equatios ad problems. Strategies for Solvig Liear Equatios Studets may apply various strategies, such as the followig, to solve liear equatios: Use ituitio. Studets will be able to solve some liear equatios ituitively, by recallig a related arithmetic fact or by recogizig a relatio (e.g., doublig, 1 more or less). Substitute a value for a variable. The substitutio is essetially a guess-ad-check strategy that is verified through substitutio. It is good practice to verify all solutios by substitutig the solutio for the variable ad workig through the equatio. Graph the equatio. Create a table of values ad use a graphical represetatio of the equatio to read the iformatio required. (This strategy is outlied i relatio to learig outcome 7.PR.2.) 16 Grade 7 Mathematics: Support Documet for Teachers

197 Use couters to model the equatio. This strategy works well for whole umbers. For example, to model 3 = 24, a studet could distribute 24 couters ito 3 equal groups ad cout 8 i each group. To model = 22, a studet could distribute 22 couters so that there are a equal umber i 3 groups ad a group of 4 by themselves. There would be 6 i each group of 3, so = 6 i this equatio. Studets may exted this strategy to workig backwards. Work backwards through the equatio. I the previous example, = 22, take 22 couters. The last directio i the equatio is to add 4, so do the reverse ad remove 4, which would leave 18 couters. Some umber multiplied by 3 is 18. The opposite of multiplyig is dividig, so divide 18 by 3. The result is 6. Take care that studets do ot misiterpret the equal sig as a directio to do some operatio rather tha as a symbol that separates equal quatities. Use algebra tiles. The rectagular tile is used to represet x. The small squares are used to represet uits. Oe colour represets positive itegers, ad aother colour represets egative itegers. A vertical rod is used to represet the equal sig. Whe workig with algebra tiles, it is importat to be cosistet about what each maipulative represets. Differet sources sometimes have differet represetatios. Example: Patters ad Relatios 17

198 Use a balace scale model. This model is based o the priciple that a equatio represets two equal expressios separated by a equal sig. The equal sig represets the fulcrum or balace poit of a scale, ad the expressios o either side represet masses placed i either pa of the balace. The expressios are equal both represet the same value ad ca symbolize equal masses. Example: I the balace scale metaphor, chagig the mass o oe side of the fulcrum will tip the scale. Makig a idetical chage o the opposite side of the fulcrum will rebalace the scale. I the cocrete model, a balace scale is used alog with idetical objects, such as blocks, cubes, or marbles, to represet umbers, ad paper bags or polystyree cups, to represet variables. Desigated objects are added to the bags evely, ad idetical chages are made to both sides of the scale util balace is achieved. The objects i the bag could be couted to obtai the value of the variable, or the items ca be maipulated util oe bag is isolated o oe side of the scale. The quatity it represets is isolated o the opposite side, ad the scale is at equilibrium. I the symbolic represetatio of the model, the equatio is solved by performig idetical operatios o either side of the equal sig, util a variable remais o oe side ad a value o the other. (This method is developed i the learig experieces suggested for learig outcome 7.PR.3.) 18 Grade 7 Mathematics: Support Documet for Teachers

199 Example: Represet = 11 as a balace. represets a chip represets a bag cotaiig a ukow umber of chips = 11 Show this cocretely (or pictorially) = Maitaiig balace, remove 3 chips from each side. 2 = 8 Simplify Determie the umber of chips that would be i each bag. = 4 Simplify = 11 (?) = 11 (?) 11 = 11 (4) Check. Patters ad Relatios 19

200 Be sure to arrage learig experieces i such a way that studets have ample opportuity to work with a variety of cocrete materials whe solvig liear equatios through preservatio of equality, to explai the process orally, to represet it pictorially, ad to record it symbolically. The skills studets develop i solvig liear equatios through preservatio of equality, ad the experiece they gai represetig patters ad cotextual situatios as relatios ad liear equatios, ca be combied to solve problems with ease. Mathematical Laguage algebraic expressio costat coordiates core elemet equatio equivalet evaluate explicit relatioship expressio graph liear relatio umerical coefficiet patter recursive relatioship relatio solutio solve substitutio table of values term (step umber, figure umber) value variable 20 Grade 7 Mathematics: Support Documet for Teachers

201 Learig Experieces Assessig Prior Kowledge Materials: BLM 7.PR.1: Patters: A Process BLM 7.PR.2: Sample Patters demostratio board maipulatives (e.g., cubes, patter blocks) grid paper math jourals or otebooks Orgaizatio: Whole class, idividual Procedure: 1. Itroduce the topic of patters with a class discussio. a) Ask studets to share patters they have see, or preset them with samples of patters. Ask them to describe the patters. Ask whether there are other ways to represet the same patter. Review patter-related vocabulary as opportuity arises durig the discussio. b) Select a example of a patter from BLM 7.PR.2: Sample Patters, or use a studet example, icludig three or four terms of a basic growig patter. Record the example o the demostratio board. Studets may build the patter ad/or record it i their math jourals or otebooks. (Cautio: Avoid a triagular umber patter of addig a additioal row oe item loger for each term.) c) Have studets exted the patter aother three terms. Ask them to share the rule they used for extedig the patter usig words, ad the usig a mathematical expressio. Record the recursive rules or relatios, ad have studets do the same. d) Have studets complete a chart of the term umbers ad term values. Record the chart. The complete a table of values, with oe colum beig the term umber (x), ad the other beig the term value (the relatio to x) or (y). Patters ad Relatios 21

202 e) Trasfer the values i the table to a grid plot. Review the coordiate plae, ad label the x-axis ad y-axis as you do so. Iclude term umber ad term value i the labels. Ask studets whether the poits should be joied. Note: Learig outcomes i Grade 7 (ad i previous grades) deal with discrete data. Sice Grade 7 learig outcomes limit graphig to discrete data, the poits should ot be coected. f) Ask studets to describe the explicit relatioship i words ad as a symbolic relatio. Discuss strategies used to determie the explicit relatioship. For example, evaluatig whether the patter is icreasig or decreasig, ad by how much, iforms studets about the operatio used i the expressio. (For a discussio o Represetig Patters ad Idetifyig Relatioships, see the Backgroud Iformatio for learig outcomes 7.PR.1 ad 7.PR.2.) Have studets record strategy tips. g) Coect the x- ad y-variables to the term umber ad the term values ad to the coordiate poits. Write a equatio to represet y i terms of x. I the example (x, 2x + 1), y = 2x + 1. These terms ca be added to the graph labels. (This topic is explored further i a later learig activity about coectig relatios to oral ad writte patters.) 2. Choose a patter for studets to work with idepedetly. Distribute copies of BLM 7.PR.1: Patters: A Process ad BLM 7.PR.2: Sample Patters. Ask studets to complete the five represetatios of oe or more patter examples. Variatios: Studets may choose or create their ow patter to represet i the five ways. Or they may choose or create a patter for a classmate to represet. If studets are havig difficulty focusig o extedig patter variatios, have them begi by usig cocrete materials to represet the patters ad to exted them. 22 Grade 7 Mathematics: Support Documet for Teachers

203 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Exted a patter. r Create a chart ad a table of values represetig the patter. r Represet the patter as a graph, ad label the graph. r Describe a recursive relatioship to represet the patter with words ad with a symbolic expressio. r Idetify the explicit relatioship i the patter as words, ad represet it as a symbolic expressio ad/or a equatio. Suggestios for Istructio Formulate a relatio to represet the relatioship i a oral or a writte patter. Provide a cotext for a relatio that represets a patter. Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. Explai what a variable is ad how it is used i a expressio. Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. Substitute a value for each ukow i a expressio ad evaluate the expressio. Materials: BLM 7.PR.3: Directios for Playig a Relatios Game math jourals or otebooks Orgaizatio: Whole class, small groups, idividual Procedure: 1. Itroduce studets to a relatios game, such as that foud o BLM 7.PR.3: Directios for Playig a Relatios Game. 2. Demostrate the game to the class. 3. Divide the class ito small groups ad have them play the game. Patters ad Relatios 23

204 4. Provide studets with the followig problems ad have them record their resposes i their math jourals or otebooks: a) Amada puts a 3 ito the fuctio machie ad gets out a 7. Use symbols or words to show three differet rules the fuctio machie could be followig. b) The fuctio machie cotiues to use the same rule, but this time, Amada puts i a 6 ad gets out a 13. Use symbols ad words to show oe rule that you thik the fuctio machie is followig. c) The fuctio machie cotiues to use the same rule. Predict what the output will be if the iput is 5. Explai how you kow this. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Predict a elemet i a patter based o a patter rule. r Describe a recursive relatioship to represet the patter with words ad with a symbolic expressio. r Idetify the explicit relatioship i the patter as words, ad represet it as a symbolic expressio ad/or a equatio. Suggestios for Istructio Formulate a relatio to represet the relatioship i a oral or a writte patter. Provide a cotext for a relatio that represets a patter. Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. Explai what a variable is ad how it is used i a expressio. Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. Substitute a value for each ukow i a expressio ad evaluate the expressio. Materials: BLM 7.PR.1: Patters: A Process (completed from Assessig Prior Kowledge learig activity) BLM 7.PR.4: Uderstadig Cocepts i Patters ad Relatios demostratio board Ve diagram (optioal) textbook glossaries, mathematics dictioaries, ad/or other refereces (optioal) 24 Grade 7 Mathematics: Support Documet for Teachers

205 Orgaizatio: Pairs, whole class, idividual Procedure: Part A 1. Activate studets backgroud kowledge about algebra ad algebraic terms usig a Thik-Pair-Share strategy i which studets thik about a questio idividually, ad the share their ideas, first with a parter ad the with the whole class. The followig questios ad commets are offered as a guide. a) What is algebra? As studets share resposes with the class, iclude the cocept that algebra is the laguage of symbols used to represet the relatioships i patters. b) What are some of the symbols used i algebra ad what do they represet? As opportuity arises durig the sharig ad durig the ext steps of the learig activity, develop studets uderstadig of the followig vocabulary terms: relatio, variable, umerical coefficiet, costat term, expressio, ad equatio. Ecourage studets to use proper termiology as they explore patters. Record terms o the demostratio board as they arise. 2. Use studets experieces with relatios i patters to develop the meaig ad use of the vocabulary terms relatio ad variable. Record terms o the demostratio board as they arise. Have studets examie their completed work from the Assessig Prior Kowledge learig activity (BLM 7.PR.1: Patters: A Process). Some guidig questios ad commets are suggested below. Also coect vocabulary terms to visible ad familiar cotexts. a) Relatio: Examie the sectio (o BLM 7.PR.1: Patters: A Process) where studets describe the patter i their ow words, ad compare it to the algebraic expressio ad the equatio. Ask studets how these are related. The algebraic statemets are symbolic represetatios of the words used to describe the patters. The opposite is true as well. The equatio gives directios to perform some operatio o the term umber to come up with the term value. It dictates what to do with the x-value to get the y-value. I a relatio, oe umber i a pair is used to idetify the other umber, or the related umber, i the pair. I the equatio 2x = y, the expressio 2x relates x to y. The expressio 2x is a relatio. I everyday life, the relatio could, for example, describe the umber of chairs at each table: 2x = total # of chairs, where x = the # of tables. Ask studets to share their patters (from BLM 7.PR.1: Patters: A Process) ad idetify the matchig symbolic relatios, ad vice versa. Challege them to geerate examples of relatios to represet familiar cotexts ad to provide cotexts to match some relatios. Patters ad Relatios 25

206 b) Variable: Studets used variables i the expressios that represet the relatioships betwee term umbers ad term values ad i the algebraic equatios. Ask studets to share which variables they used i their represetatios, ad what the variables represet i each case. Icreasig studets awareess of the three differet uses for variables may help reduce some of the cofusio they may ecouter usig algebra. This discussio also provides a opportuity to discuss covetios ad cautios i choosig symbols. (See the Backgroud Iformatio.) You may wish to have studets use a variety of variables to express cotextual relatioships or to idetify possible cotexts for a relatio (e.g., 4g = s. 4 studets i a group, 4 # of groups = the # of studets). Practise substitutig values for the variables. Poit out how the umbers represeted by the variable vary or chage, depedig o which figure of the patter is beig referred to. This is oe way variables are used. Differet variables i oe equatio represet quatities of differet items. The same variable i oe equatio always represets quatities of the same thig. Replacig a variable with a umber i the expressio or equatio geerates the value for the other umber. Ask studets to geerate term values for specific term umbers, by substitutig the term umber for the variable i the relatio. Aother way variables are used is to fid a particular umber. Here, the particular umber is represeted by a relatio. Have studets examie the graph (o BLM 7.PR.1: Patters: A Process). How are the x-values ad the y-values for each poit related? The y-values are expressed i terms of the x-values i the relatioship rules. Both axes o the graphs may be labelled i terms of x. Cosider the algebraic equatios: y = (the relatio i terms of x), ad y is a simpler ame for the chage you made to x. A third use for variables is as a simplified ame for a relatio. Variables ca be used to make geeralized statemets about mathematical relatioships, such as l w = Area. 3. Have studets work with their parters to practise geeratig some cotextual relatios usig variables or to match a cotext to a relatio. Practise substitutig values i the relatios. Reassemble as a class ad share a few resposes to verify studets uderstadig. 4. Provide studets with copies of BLM 7.PR.4: Uderstadig Cocepts i Patters ad Relatios. Ask them to defie ad provide a example of the terms variable ad relatio. Part B 5. Cotiue usig the work completed i Part A as a referece. Use studets experieces with relatios i patters to review the vocabulary terms relatio ad variable, ad to develop the meaig ad use of the vocabulary terms expressio, equatio, costat term, ad umerical coefficiet. Record each term o the demostratio board as it arises. Some guidig questios ad commets follow. 26 Grade 7 Mathematics: Support Documet for Teachers

207 a) Expressios ad equatios: The previous discussio about expressios ad equatios i relatios provided some backgroud experiece with these terms. Now, compare ad differetiate the terms expressios ad equatios. Expressios cotai variables ad operatios that represet oe ame for a value. There is o equal sig i a expressio. A equatio cotais two expressios that are equal to each other. Studets commoly misuderstad the equal sig as a directive. (Refer to the discussio regardig expressios ad equatios i relatios i Part A of this learig experiece ad to the Backgroud Iformatio.) b) Costat term: A represetatio of the costat term is readily see o a graph by examiig the poit at which the relatio meets the y-axis (x = 0). It shows what quatity is at the base of each step i the patter, ad, therefore, must be added (or subtracted) each time you calculate a term value. Costat terms are separated from variables with a additio or a subtractio symbol. Ask studets to idetify costat terms i ay of their relatios. c) Numerical coefficiet: I some patters, the term umber (variable) is multiplied by the same amout i each term. Ask studets to idetify a umerical coefficiet for ay of the variables i their relatios. The slope of the lie i the graphical represetatio of patters provides a clue to the presece of a coefficiet i a relatio. Discuss covetios of otatio. 6. Provide several examples of relatios i words, expressios, ad equatios, or have studets create their ow examples. Have studets work with their parters to idetify relatios, variables, umerical coefficiets, costat terms, expressios, ad equatios i the examples, ad substitute values for the variables. Verify the accuracy of studets resposes. 7. Provide studets with copies of BLM 7.PR.4: Uderstadig Cocepts i Patters ad Relatios or a Ve diagram. Ask them to defie ad provide a example of each of the followig terms: expressio, equatio, costat term, ad umerical coefficiet. If you choose to use the Ve diagram, ask studets to compare expressio with equatio ad costat term with umerical coefficiet. Variatios: Have studets create ad/or complete crossword puzzles of the vocabulary terms. Have studets create a begiig algebra booklet, icludig the terms, defiitios, ad examples. This booklet could be used as a appedix for aother booklet idea i a culmiatig learig activity. Patters ad Relatios 27

208 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Formulate a relatio to represet the relatioship i a oral or writte patter. r Provide a cotext for a relatio that represets a patter. r Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. r Explai what a variable is ad how it is used i a expressio. r Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. r Substitute a value for each ukow i a expressio ad evaluate the expressio. Suggestios for Istructio Formulate a relatio to represet the relatioship i a oral or a writte patter. Provide a cotext for a relatio that represets a patter. Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. Explai what a variable is ad how it is used i a expressio. Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. Substitute a value for each ukow i a expressio ad evaluate the expressio. Materials: BLM 7.PR.5: Possible Word Patter Cotexts to Match a Relatio BLM 7.PR.6: Formulatig Relatios to Match Word Descriptios of Patters demostratio board card stock (optioal) Orgaizatio: Whole class, idividual 28 Grade 7 Mathematics: Support Documet for Teachers

209 Procedure: Remid studets, as oted i the previous learig activity, that patters are represeted i may equivalet forms. The patter ca be recogized i each of the forms, ad oe form ca be traslated ito aother. Each differet represetatio provides a differet view of the same patter. The more views studets see, the greater their uderstadig of the patter will be. A word descriptio of the patter ca be used to make a physical represetatio of the patter, as well as a chart, a table of values, ad/or a graph. The x- ad y-values of the graph are also represeted i the word descriptio of the patter, ad ca be used to form a algebraic equatio. Part A 1. Help studets establish a process of writig a relatio to match a word descriptio of a patter or cotext by workig backwards from a relatio. a) Begi by aalyzig a relatio such as x + 1 = y. Dissect the terms of the relatio ad idetify their symbolism. There are two expressios: x + 1 ad y. The equatio tells us that the two expressios are equal to each other. There are two variables: x represets some umber of thigs ad y is a umber of other thigs. There is oe operatio, a costat term of + 1. Together, the two expressios idicate that there is oe more y thig tha there are x thigs. To fid the umber of y thigs there are, use the quatity that x represets ad add 1. b) Create a possible cotext or situatio that this relatio could represet. Imagie somethig of which x could be a quatity. The costat will be a uchagig quatity of that same thig, ad y will be the combied quatity. Example 1: Note: The variables x ad y are used throughout this learig activity to reiforce the coectio of variables to term umber charts ad/or term value charts ad the graphical represetatios of patters o a coordiate grid. If you wish, emphasize that ay symbol or ame ca be used to represet the variable, or develop that cocept i a later learig activity. The relatio could represet people: the umber of guests attedig your party (x), you, who will be there o matter what (+ 1), ad the total umber of people at your party (y). Substitute some values for x, ad solve the equatio to get values for y. Cosider whether or ot the values make sese i relatio to each other. If ot, somethig eeds fixig. For example, if 3 people come + 1 (you), there will be 4 people at the party. That is reasoable. Reiforce that x + 1 is a expressio that tells us how to fid the value of y. I this example, x + 1 is oe ame for the umber of people at the party. Ad the umber y represets is aother ame for the umber of people at the party. So, x + 1 ad y represet the same umber. They are equal. The two expressios are combied as the equatio x + 1 = y. Patters ad Relatios 29

210 Example 2: The relatio could represet moey: the umber of dollars you decide to take from your piggy bak (x), the oe-dollar coupo you have (+ 1), ad the dollars you ca sped at the restaurat (y). Verify the relatio through substitutio. If you withdraw $10 ad add the $1, you will have $11. That is reasoable. 2. Complete a few examples of differet relatios together with studets. Also iclude examples where x = y, ad where y decreases as x icreases. Have studets idepedetly create possible word situatios to match a particular relatio. BLM 7.PR.5: Possible Word Patter Cotexts to Match a Relatio provides a framework for recordig these. You may wish to assig the relatios, or studets may create their ow. Remid studets to substitute values to verify the reasoableess of their relatios. After allowig sufficiet time for idividual work, have studets share some cotext ad relatio matches with the class so you ca verify their uderstadig. Assig more idepedet or parter practice if it seems beeficial. Part B 3. Reassemble as a class, ad establish a procedure to reverse the above process, eablig studets to formulate a relatio to match the word descriptio of a patter. Do this by aalyzig the word descriptio of the patter to fid parts to represet x, y, ad costats or umerical coefficiets. Combie the parts to write a matchig symbolic relatio. A chart such as the followig ca serve as a orgaizatioal tool. Writig a Symbolic Relatio Oe quatity that ca be represeted by a variable (similar to the term umber) Represet with x A operatio that tells what to do to x (the term umber) to get y (the term value) Record as a costat (+ or ) or umerical coefficiet ( or ) A quatity that will be represeted by the y-variable (similar to the term value) Represet with y 4. Illustrate how to aalyze the descriptio ad fid the parts to represet x, y, ad costats or umerical coefficiets by usig examples such as the followig: Example 1: A girl ows three horses. She purchases more horses at a auctio; cosequetly, she ow has more horses. Look for some quatity that acts like a term umber, i that it ca chage i a step-by-step fashio. That umber will be represeted by the x-variable. The girl may buy 1, or 2, or 3, or 4, or... horses. I this case, the x-variable will be the umber of horses she buys. 30 Grade 7 Mathematics: Support Documet for Teachers

211 Next, idetify which quatity will be represeted by the y-variable. This is similar to a term value. The term value depeds o the term umber. The umber of horses she eds up with depeds o how may she buys, so the y-variable will be the umber of horses she eds up with. The, cosider the presece of a costat or a umerical coefficiet. If there is a quatity to start with, or oe to remove at the ed, there will be a costat to add or subtract. If a variable is beig multiplied or divided, there will be a umerical coefficiet to coect to the variable. I this example, the girl starts with three horses. She has three horses o matter how may she buys. These three horses are represeted by the costat + 3. Put the pieces together i the relatio x + 3 = y. Reiforce this is a equatio. It cotais two equal expressios. Note the covetio to place the variable first ad the costat after. Establish these covetios as examples arise. Example 2: A boy sells hats at the fair for $5 each. He pays $25 for a daily vedor licece. At the ed, he has some moey. The umber of hats sold could be 1, or 2, or 3, or 4, or.... The umber of hats sold will be represeted by the x-variable (term umber). I the ed, the boy will have made some moey. That amout of moey is represeted by the y-variable. He will receive $5 for each hat he sells. To fid out how much moey he receives, multiply the umber of hats sold by $5. So, $5 is the umerical coefficiet. Five times the umber of hats equals the moey received (5x). The boy must pay the $25 fee, o matter how may hats he sells. That will the costat ( $25). The moey he eds up with will be the y-value. The relatio is 5x 25 = y. 5. As a class, complete a few examples of differet relatios. Iclude examples where x = y, where there are umerical coefficiets, ad where there are positive ad egative costats. 6. Distribute copies of BLM 7.PR.6: Formulatig Relatios to Match Word Descriptios of Patters. Have studets idividually aalyze word descriptios of patters to formulate relatios. After givig studets sufficiet time to formulate relatios, have them use their relatios to represet the cotexts. Verify their uderstadig, ad assig more idepedet practice if it seems beeficial. Variatio: Use cards or a master sheet of patter descriptios ad relatios to coduct a quiz game such as Relatio Baseball. Read a patter descriptio or a relatio, ad have a studet respod with either a matchig relatio or a patter descriptio. (For directios o, ad variatios of, playig a similar baseball game, refer to the Assessig Prior Kowledge suggestio for learig outcome 7.N.6.) Patters ad Relatios 31

212 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Formulate a relatio to represet the relatioship i a oral or writte patter. r Provide a cotext for a relatio that represets a patter. r Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. r Explai what a variable is ad how it is used i a expressio. r Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. r Substitute a value for each ukow i a expressio ad evaluate the expressio. Suggestios for Istructio Formulate a relatio to represet the relatioship i a oral or a writte patter. Provide a cotext for a relatio that represets a patter. Represet a patter i the eviromet usig a relatio. Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. Explai what a variable is ad how it is used i a expressio. Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. Substitute a value for each ukow i a expressio ad evaluate the expressio. Materials: a list of word descriptios of patters for which studets ca formulate relatios or word cards from previous learig activity (optioal) math jourals or otebooks three-colum charts for recordig relatios ad descriptios Orgaizatio: Whole class, idividual, small group 32 Grade 7 Mathematics: Support Documet for Teachers

213 Procedure: For this learig activity, have studets act as detectives, with the goal of ucoverig the relatios that represet word descriptios of patters. 1. Together with studets, develop a strategy for fidig clues from which to form the relatios. Workig through examples may be helpful. Suggested Strategies: Ucover clues that would idicate a term umber or a term value. Use the variable x to represet the umeric value of the term umber. Use y to represet the umeric value of the term value. Ucover directios about how to fid the term value from the term umber. Is the operatio applied additio, subtractio, multiplicatio, or divisio, or a combiatio of operatios? Use a costat term to represet additio or subtractio. Use a umerical coefficiet to represet multiplicatio or divisio. 2. Work through some examples together with studets. Iclude cotexts with various costats ad umerical coefficiets, ad combiatios of them. Example 1: Determie the umber of traffic lights if there is oe red light for every gree light. Clues: There are two objects that ca be quatified. Each ca be represeted by a variable. Let r represet the umber of red lights. Let g represet the umber of gree lights. The umber of gree lights is the same as the umber of red lights. Relatio: r = g There are o costats or umerical coefficiets. Patters ad Relatios 33

214 Example 2: Determie the umber of girls ad boys i a class if there are two more girls tha boys. Clues: There are two terms to quatify, boys ad girls. There are two more girls tha boys. This clue is importat to esure the operatio is performed o the correct variable. There are more girls tha boys, so the umber of boys + 2 = the umber of girls. The umber of boys acts as the term umber. So b represets the umber of boys. The umber of girls depeds o the umber of boys. The umber of girls acts as the term value, so g represets the umber of girls. The expressio b + 2 results i the umber of girls whe there are b umber of boys. Relatio: b + 2 = g Example 3: Determie the umber of wheels preset i a collectio of cars if each car has four wheels. Clues: There are two terms to quatify, the umber of cars ad the umber of wheels. Let x represet the umber of cars. Let y represet the umber of wheels. Every car has four wheels. Multiply the umber of cars by 4. 4 is the coefficiet for x. Relatio: 4x = y Use the above examples as a opportuity to discuss covetios such as writig 4x, rather tha x 4 or x 4, to avoid cofusig variables ad multiplicatio sigs. 3. After workig through several examples as a group, have studets describe, i their math jourals or otebooks, a strategy for fidig a relatio to match the descriptio of a patter. 34 Grade 7 Mathematics: Support Documet for Teachers

215 4. Have pairs of studets take turs doig the followig: Studet A: Describe a cotext i which patters are foud ad record it i the middle colum of a three-colum chart. The record the relatio to the patter cotext descriptio i the first colum, cover it (or fold it so that the relatio is hidde), ad pass the chart to Studet B. Example: Relatio (Studet A) Fid the total umber of ties (t) i a sectio (s). t = s + 1 Patter Cotext Descriptio A railroad track leaves 1435 mm betwee railway ties. The first 1435 mm sectio has two ties, ad each 1435 mm sectio after that adds oe more tie. Relatio (Studet B) Studet B: Write the relatio for the patter cotext descriptio i the third colum of the chart. Example: Patter Cotext Descriptio A railroad track leaves 1435 mm betwee railway ties. The first 1435 mm sectio has two ties, ad each 1435 mm sectio after that adds oe more tie. Relatio (Studet B) Let y represet each 1435 mm sectio. y Studets compare the relatios they formulated to the aswer keys, ad discuss ad resolve ay discrepacies. They test the relatios by substitutig the variable ad evaluatig the expressio. They verify that the relatios make sese whe compared to the descriptios. Patters ad Relatios 35

216 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Formulate a relatio to represet the relatioship i a oral or a writte patter. r Provide a cotext for a relatio that represets a patter. r Represet a patter i the eviromet usig a relatio. r Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. r Explai what a variable is ad how it is used i a expressio. r Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. r Substitute a value for each ukow i a expressio ad evaluate the expressio. Suggestios for Istructio Formulate a relatio to represet the relatioship i a oral or a writte patter. Provide a cotext for a relatio that represets a patter. Represet a patter i the eviromet usig a relatio. Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. Substitute a value for each ukow i a expressio ad evaluate the expressio. Materials: BLM 7.PR.7: Creatig Word Descriptios of Patters ad Matchig Relatios Orgaizatio: Idividuals or pairs Procedure: 1. As i the previous learig activities, review the cocept that patters have multiple represetatios ad oe represetatio ca be used to formulate aother. I this learig activity, studets will fid patters i their surroudigs ad the represet the patters as relatios to play a I Spy game. They will eed to substitute values ad evaluate expressios to fid the patters represeted by the relatios. 36 Grade 7 Mathematics: Support Documet for Teachers

217 2. Distribute copies of BLM 7.PR.7: Creatig Word Descriptios of Patters ad Matchig Relatios. Have studets survey the classroom to idetify patters they could represet usig relatios, ad complete the chart provided o the BLM. Examples could iclude patters i furiture, brick, tile, decoratio, clothig, supplies, ad so o. 3. Decide who will be the first studet to offer a relatio. That studet says, I spy a patter that is represeted with the relatio. Ca you guess what I see? Other studets will eed to look for a patter they thik matches the relatio. The substitute a value for the variable, ad evaluate the expressio to esure studets suggested patter matches the relatio. If idividuals have a match, they raise a had, ad, whe called upo, offer their suggestio to the oe who spies. Studets substatiate their proposal by substitutig a value, evaluatig the expressio to verify x ad y. If someoe is correct, she or he becomes the oe who spies, ad offers a relatio to the group. Choosig variables that begi with the first letter of the objects i the patter makes the game easier. Variatios: To icrease participatio, play i small groups rather tha i a large group. Chage the eviromet by takig studets to a ew area idoors or outdoors to play the game. To limit the optios, prepare a list of scearios, ad have studets spy from the list. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Represet a patter i the eviromet usig a relatio. r Formulate a relatio to represet the relatioship i a oral or a writte patter. r Provide a cotext for a relatio that represets a patter. r Idetify ad provide a example of a costat term, a umerical coefficiet, ad a variable i a expressio ad a equatio. r Substitute a value for each ukow i a expressio ad evaluate the expressio. r Commuicate mathematically. Patters ad Relatios 37

218 Suggestios for Istructio Create a table of values for a relatio by substitutig values for the variable. Create a table of values usig a relatio ad graph the table of values (limited to discrete elemets). Match a set of relatios to a set of graphs. Match a set of graphs to a set of relatios. Substitute a value for each ukow i a expressio ad evaluate the expressio. Materials: math jourals or otebooks pes or markers of differet colours demostratio board examples of relatios (These ca be teacher-geerated or created i previous learig activities, such as completed BLMs, cards, or lists.) a selectio of the followig: grid paper templates for tables of values cards with grids prited o them cards with blak tables of values blak cards scissors ad glue or tape display board graphig software or graphig calculators (optioal) Orgaizatio: Idividuals or pairs, whole class Procedure: The procedure suggested for this learig activity (Part A) will be cotiued i the ext two learig activities (Part B ad Part C). 38 Grade 7 Mathematics: Support Documet for Teachers

219 Part A Preset a progressio of relatios for studets to work with, begiig with x 2, followed by 2x, the 2x 2, ad the 2(x 1). The followig procedure uses 2(x 1) as a example. You may prefer to use relatios with additio, such as (x + 2), 2x, 2x +2, ad 2(x + 1). 1. Preset studets with a sample relatio such as the coordiate pair (x, 2(x 1)) or the equatio 2(x 1) = y. Challege studets to represet the relatio as a graph i their math jourals or otebooks, ad to record the steps they followed to do so. If or whe it seems appropriate, ote that the relatio is equivalet to 2x Whe studets have had sufficiet time to work idividually, reassemble as a class ad ask studets to share their ideas about how to go about represetig a relatio as a graph. As studets offer suggestios, record the process o the demostratio board. Ask guidig questios, ad supply prompts to esure the process is complete ad uderstood. Suggest studets make adjustmets to their math joural etries as the discussio reveals steps they had ot cosidered previously. Makig additios i a differet colour highlights for you ad for studets what studets are learig durig the sharig. Steps to Iclude: a) Create a table of values. b) The x-value is similar to a term umber, as show i previous work with represetig patters. Supply data for the table by substitutig values for the variable. Ay umber may be used to represet x, but begiig with a small umber, ad icreasig i a cosecutive fashio by eve icremets, will geerate umbers that are most helpful for x-value 2(x 1) = y y-value 2(x 1) viewig relatioships. If your class is familiar with addig ad subtractig egative umbers (see learig outcome 7.N.6), iclude egative values, because itegers appear i everyday situatios ivolvig moey, depth below sea level or udergroud, lost time, ad so o. Avoid usig egative umbers with umerical coefficiets, as multiplyig itegers is a Grade 8 learig outcome. Solvig the expressio with a particular x-value will geerate a value equivalet to y. This is the value referred to as the term value i previous work. The relatio 2(x 1) explais the relatio betwee x ad y. Evaluatig the relatio geerates the y-value. Patters ad Relatios 39

220 c) Examie the rage of umbers i the table of values ad select a appropriate scale for the x- ad y-axes. Draw ad label the axes with x below the grid, ad the relatio that ames y alog the y-axis. Write umbers to idicate the scale of each axis. Remid studets to alig the cetre of the umbers with the grid lies. If studets are familiar with the four quadrats of the Cartesia plae (learig outcome 7.SS.4), they will be able to represet values with egative itegers. Iclude a title for the graph. I this case, the graph represets the relatio betwee x ad 2(x 1), so that is a appropriate title. d) Plot the coordiate pairs o the graph. Sice Grade 7 learig outcomes limit graphig to discrete data, the poits should ot be coected. 3. Distribute grid paper, templates for tables of values, ad blak cards for writig the titles of the graphs. Studets will use these supplies to create a iteractive display. Assig relatios to studets, or have them create their ow, or use previously geerated relatios. Ask studets to create tables of values, as well as graphs to represet the relatios. Remid studets to choose appropriate scales ad to label the axes. 4. Have studets share their completed graphs. Verify their correctess. The mout the graphs o the display board ad distribute the table of values ad the labels. Have studets attach the pieces to the matchig graphs o the display board. Studets may have difficulty matchig some labels. The ext learig activity (Part B) provides strategies to make matchig easier. Alterately, the tables ad titles ca be mouted, ad studets ca match the correspodig graph. Leave room i the display for descriptios that will be made i the followig learig activities (Part B ad Part C). 40 Grade 7 Mathematics: Support Documet for Teachers

221 Variatios: Cotrol the complexity ad variety of relatios by assigig specific relatios to studets. Work with oe-step liear equatios (e.g., 3x or x + 1), before movig to 1 two-step liear equatios e.g., 2x 2 or x If studets made cards with matchig relatios ad word descriptios of patters i a previous learig activity, they could add to their card sets by creatig matchig tables of values cards ad graphs. The teacher or studets ca use computer software to geerate the tables ad graphs to represet particular relatios. Have studets fid matches either i hard copies or electroically, if the techical skills ad software are available. If the techology ad kow-how are available, studets ca use graphig calculators to ivestigate relatios ad see what types of graphs the relatios produce. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create a table of values for a relatio by substitutig values for the variable. r Create a table of values usig a relatio ad graph the table of values (limited to discrete elemets). r Match a set of relatios to a set of graphs. r Match a set of graphs to a set of relatios. r Substitute a value for each ukow i a expressio ad evaluate the expressio. r Reaso mathematically i order to make mathematical coectios. Patters ad Relatios 41

222 Suggestios for Istructio Sketch the graph from a table of values created for a relatio, ad describe the patters foud i the graph to draw coclusios (e.g., graph the relatioship betwee ad 2 + 3). Describe the relatioship show o a graph usig everyday laguage i spoke or writte form to solve problems. Materials: math jourals or otebooks pes or markers of differet colours demostratio board tables of values, graphs, labels, ad relatios from the previous learig experiece cards for recordig descriptios of graphs (blak file cards or the type of cards used i previous learig experieces) display board Orgaizatio: Idividual or pairs, whole class Procedure: Part B This is a cotiuatio of the previous learig activity. Studets use the products created i Part A ad exted their prior kowledge of graphical represetatios of patters to describe the relatioships show i a graph ad i expressios ad equatios. Describe the relatioships i each of the graphs from Part A: (x - 2), (2x), (2x 2), ad 2(x 1). The followig procedure uses the graph for 2(x 1) as a example. 1. Have studets examie the sample graphs they created i the previous learig activity (Part A). Challege studets to decipher the coded iformatio the graphs cotai about the relatioship betwee x ad 2(x 1), ad to write their discoveries i their math jourals or otebooks. 2. After studets have had sufficiet time to work o their ow or with a parter, reassemble as a class to debrief. Have studets share their iterpretatios of their graphs, ad add pertiet iformatio to their math jourals. Makig additios i a differet colour highlights what studets are learig durig discussio with others. Below are some commets you may wish to iclude i the discussio. (For additioal iformatio about graphs, see Represetig Patters ad Idetifyig Relatioships i the Backgroud Iformatio for learig outcomes 7.PR.1 ad 7.PR.2.) Discussio Ideas: Liear relatio: All the poits o the graph lie i a straight lie. Verify this by placig a ruler alog the poits. Whe all the coordiate pairs of a relatio lie i a straight lie, the relatio is called a liear relatio. 42 Grade 7 Mathematics: Support Documet for Teachers

223 Icreasig liear relatio: The lie goes up to the right. As the x-value icreases, the y-value icreases as well. The graph tells us that as we get more of whatever x represets, we will also get more of whatever y represets. The relatio is described as a icreasig liear relatio. Recursive relatio i words ad symbols: A graph tells us how much the icrease will be. The icremetal steps from oe poit i the graph to the ext may be described as move oe to the right ad move up two. I this relatio, for every additioal x, there will be two additioal ys. Each time x icreases by 1, 2(x 1) or y icreases by 2. Explicit relatio: The explicit relatio is also show i the graph if we wish to decipher it. Costat i the explicit relatio: If the lie o which the poits lie is followed backwards to the y-axis, we will otice that the y-value, whe the relatio hits the y-axis, is 2. Whe we use the distributive property to expad 2(x 1) to 2x 2, we ca see where the 2 is preset i the liear relatio. This 2 is referred to as the costat. Numerical coefficiet i the explicit relatio: The slope, or iclie, of the lie upo which the poits lie idicates there is also a small umerical coefficiet i this term. It ca be discovered i a variety of ways. For example, by examiig the graph, we ca see that for every icrease of 1 i the x directio, there is a icrease of 2 i the y-directio. This is represeted by the coefficiet i the relatio, 2 (2(x 1)). Complete explicit relatio: The explicit relatio cotais 2x ad 2. The combied explicit relatio for the patter is 2x 2, or the equatio 2x 2 = y. Equivalet expressios: The graph shows 2(x 1) ad 2x 2 are equivalet expressios. This provides a opportuity to talk about equatios beig two expressios for the same value, ad to review the distributive property of multiplicatio. Obtaiig values that are ot plotted: Extedig the lie upo which the poits lie, or lookig at poits betwee the oes that are plotted, ad the readig the coordiate pairs, provides iformatio about values that are ot listed i a table of values, ad provides aswers to questios based o the relatio. 3. Ask studets to idetify some cotexts that may be represeted by liear relatios that icrease i value. Examples may iclude the purchase of multiple sigle-priced items (e.g., bottles of water at $2 per bottle), a quatity discout (e.g., the first item costs $3 ad each additioal item costs $2), moey eared for hours of babysittig, distace travelled i relatio to time, volume of drik required i relatio to umber of people beig served, ad so o. 4. Illustrate, or have studets graph, a decreasig liear relatio where the poits lie i a lie that goes dow as you move to the right. Each time x icreases i this relatio, the y-value decreases (e.g., 6 x, 20 2x). This time, the graph idicates there is a iitial quatity that decreases i relatio to the x-value. Patters ad Relatios 43

224 5. Geerate some cotexts to represet decreasig liear relatios. Examples could iclude a certai amout of savigs i relatio to the amout spet at a regular rate, a quatity of items i relatio to the quatity used at a regular rate (e.g., There are 20 cas of cat food i the cupboard. The cat eats 2 cas every day.), ad so o. Variatios: Have studets work idepedetly or i pairs to write descriptios of the graphs they created i the previous learig activity (Part A). Record descriptios o cards, usig both everyday laguage ad algebraic laguage. Circulate withi the class to verify studets are o the right track. Whe the cards are complete, post the relatios, the tables of values, the graphs, ad the descriptios o the display board. Havig studets post complete sets of their ow work provides a assessmet opportuity. Later, oe or more idividual parts of the display may be distributed amog studets, ad reassembled with the correct matches. Play a versio of the game Pictioary. Oe studet illustrates a graph ad the others guess the matchig relatio or descriptio. Have studets explore the relatioships betwee relatios ad their represetative graphs through a iquiry activity. Studets create a list of relatios that differ i a systematic way (e.g., icreasig coefficiets, costats, or egative costats). They make graphical represetatios of the relatios, compare the two, ad describe the chages. The geeralized descriptios may be used to draw coclusios about the relatioship betwee a relatio ad its represetative graph. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Sketch the graph from a table of values created for a relatio, ad describe the patters foud i the graph to draw coclusios (e.g., graph the relatioship betwee ad 2 + 3). r Describe the relatioship show o a graph usig everyday laguage i spoke or writte form to solve problems. r Match a set of relatios to a set of graphs. r Match a set of graphs to a set of relatios. r Reaso mathematically i order to make coectios. 44 Grade 7 Mathematics: Support Documet for Teachers

225 Suggestios for Istructio Sketch the graph from a table of values created for a relatio, ad describe the patters foud i the graph to draw coclusios (e.g., graph the relatioship betwee ad 2 + 3). Describe the relatioship show o a graph usig everyday laguage i spoke or writte form to solve problems. Materials: BLM 7.PR.8: Template for Creatig ad Solvig Problems Usig Iformatio from a Graph idex cards or card stock a set of cotextual problems ad sketches of graphs that represet relatios portrayed i the problems (optioal) studet-geerated graphs from the previous two learig activities (Part A ad Part B) (optioal) graphig techology (optioal) Orgaizatio: Idividual or pairs, whole class, idividual Procedure: Part C This learig activity may be used as a cotiuatio of the previous two learig activities (Part A ad Part B), or coducted as a sigle learig experiece. Studets may use the graphs they made previously or create ew graphs. 1. Review descriptios of graphs by havig studets work idepedetly or i pairs to geerate cotexts or situatios that result i icreasig or decreasig liear relatios. Havig studets make rough graphical sketches of the relatios will reiforce the coectio betwee the descriptios ad the graphs. After givig studets sufficiet time to work o their ow or with a parter, have studets share examples with the class. Verify studets uderstadig by evaluatig their examples or by coductig a quick matchig game. Display a umber of sketched graphs. Read a problem ad have studets select the graph that represets the problem. Patters ad Relatios 45

226 2. Model how a graph that illustrates a cotextual situatio ca be used to solve problems. Preset a graph such as the followig. Example: Say that this graph represets the story of a cat ad her food. The title idicates that the graph tells the part of the story related to how much cat food is i the cupboard each day. The labels o the axes of the graph tell us that x represets the umber of days, ad y represets the umbers of cas of cat food. The story begis with a umber of cas of cat food i the cupboard. The poits lie o a lie, so the relatio is liear. Therefore, the chage will be costat. The lie is goig dow to the right, so the relatio is decreasig. We ca, therefore, coclude there are fewer cas each day. Likely this is because the cat is eatig the food. We ca fid out exactly how may cas the cat is eatig each day by lookig for the recursive relatioship i the graph. Each day, there are 2 fewer cas, so the recursive relatio is 2. We ca coclude that the cat eats 2 cas of food each day ( 2x). We ca fid out how may cas are i the cupboard o ay give day by readig the coordiate pairs. For example, o day eight, the correspodig y-value is 4. There are 4 cas of food i the cupboard o the eighth day. Have studets geerate a list of questios that could be aswered usig this graph as a source of iformatio (e.g., How may cas are i the cupboard to begi with? How may cas are left o the th day? O which day will the cat ru out of food if o more is added to the cupboard?). We ca also fid the explicit relatioship i the graph. The graph begis at 20 ad decreases by 2 each day. The equatio 20 2x = y ca be used to aswer ay of the questios as well, by substitutig values for a variable ad solvig the equatio. (This will be the focus of subsequet learig activities.) 3. Next, have studets use their ow graphs to create word problems to share with classmates. BLM 7.PR.8: Template for Creatig ad Solvig Problems Usig Iformatio from a Graph may be used as a template for this purpose, ad for assessig studets work. Have studets choose a graph from the oes they created i the previous two learig activities (Part A ad Part B), or have them create a ew graph. The ew graph may be geerated from a table of values that was obtaied by substitutig values for x i a give relatio. 46 Grade 7 Mathematics: Support Documet for Teachers

227 The followig steps are recommeded: a) Idetify a story cotext. b) Become specific about the story ad write a title for the graph. c) Label the x- ad y-axes. d) Describe the graph. e) Idetify the recursive relatioship. f) Idetify the explicit relatio. g) List questios that could be aswered by usig the graph as a source of iformatio, ad state the aswers. 4. Ask studets to choose oe or more of the questios from their list ad write iterestig word problems that ca be solved usig their graph as a source of iformatio. Have them write a good copy of the problems o oe side of idex cards, ad the solutios to the problems o the back of the cards, or uder a flap, or o whatever medium has bee chose. 5. Studets may wish to verify their work before displayig the problems. 6. Have studets share their problems, ad the matchig graphs, title, ad labels, for their classmates to practise aswerig. The questios may be preseted to the etire class, distributed amog small groups, or posted with the matchig graphs. Variatios: Use techology to create graphs ad matchig word problems, for presetatio or as iteractive questios. Problem cards may be combied with the other sets to play games (e.g., matchig games or quizzes). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Sketch the graph from a table of values created for a relatio, ad describe the patters foud i the graph to draw coclusios (e.g., graph the relatioship betwee ad 2 + 3). r Describe the relatioship show o a graph usig everyday laguage i spoke or writte form to solve problems. r Commuicate mathematically. Patters ad Relatios 47

228 Assessig Prior Kowledge Materials: BLM 7.PR.9: Associatig Clue Words with Operatios ad Expressios demostratio board Orgaizatio: Pairs or small groups, whole class Procedure: The itet of this learig activity is to have studets make geeralizatios that will help them iterpret mathematical scearios. Studets should ot be required to memorize the associatios made, but rather should gai cofidece i recogizig associatios. 1. Divide studets ito pairs or groups. Iform studets they will be workig i pairs or i small groups to complete a chart (e.g., BLM 7.PR.9: Associatig Clue Words with Operatios ad Expressios). The chart will show whether studets kow some clue words that may idicate which operatio to use ad whether they kow how to represet a give problem as a expressio or as a equatio (e.g., older tha, Geri is 4 years older tha Kasha, k + 4 or k + 4 = g). 2. Distribute copies of BLM 7.PR.9: Associatig Clue Words with Operatios ad Expressios, ad have studets work together i pairs or i small groups to complete the charts. 3. Whe studets have had sufficiet time to complete the charts, reassemble as a class. Verify studets uderstadig by havig some studets share a phrase, while others idetify the clue word, the operatio, ad the expressio or the equatio. Take time to verify resposes by substitutig values ad checkig for reasoableess. Variatio: After the charts are complete, divide the class ito two teams, ad have a cotest to see which team matches more phrases. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Associate clue words with correct operatios. r Represet phrases with expressios or equatios correctly. r Substitute values i the phrases ad test for reasoableess. 48 Grade 7 Mathematics: Support Documet for Teachers

229 Suggestios for Istructio Solve a problem usig a liear equatio. Solve a problem usig a liear equatio ad record the process. Materials: BLM 7.PR.10: Solvig Sigle-Variable Oe-Step Equatios Orgaizatio: Idividual, whole class Procedure: This learig activity ivites studets to use ad develop their curret strategies for solvig equatios. Later learig experieces will be devoted to solvig equatios through the preservatio of equality. 1. Activate studets backgroud kowledge by presetig a sigle-variable oe-step liear equatio, such as d + 9 = 15, ad askig studets to solve it. Poit out that here a variable is beig used to represet a ukow quatity. Ask studets to describe how they arrived at the aswer. Reiforce that there are multiple ways to solve problems. Cosult the Backgroud Iformatio for more iformatio. Provide or solicit a questio for each operatio. 2. Distribute copies of BLM 7.PR.10: Solvig Sigle-Variable Oe-Step Equatios, ad have studets solve the equatios idividually. 3. Whe studets have had sufficiet time to solve the equatios, reassemble as a class, ad have studets share their solutios ad strategies. This is a good opportuity to assess studets repertoire of strategies, ad to have studets hear alterative strategies from their classmates. Ask studets how they would solve the equatios if the values were larger, ad less metal mathematics friedly. (This learig activity provides backgroud for BLM 7.PR.11: Writig Expressios ad Solvig Equatios That Match Word Descriptios, which will be used i the ext learig activity.) Variatios: Studets could write cotextual problems to match each expressio. Or, they could use the equatios as models to write additioal equatios. The problems/equatios could be added to a classroom problem/questio bak for games, Etry Slips or Exit Slips, ad so o. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve problems usig sigle-variable oe-step equatios. r Explai a strategy used to solve the problems. r Apply metal mathematics strategies to solve problems. Patters ad Relatios 49

230 Suggestios for Istructio Solve a problem usig a liear equatio ad record the process. Verify the solutio to a liear equatio usig cocrete materials or diagrams. Substitute a possible solutio for the variable i a liear equatio to verify the equality. Materials: BLM 7.PR.11: Writig Expressios ad Solvig Equatios That Match Word Descriptios Orgaizatio: Idividual, whole class Procedure: This learig activity ivites studets to use ad develop their curret strategies for solvig equatios. Later learig experieces will be devoted to solvig equatios through the preservatio of equality. 1. Distribute copies of BLM 7.PR.11: Writig Expressios ad Solvig Equatios That Match Word Descriptios, ad have studets complete the tasks idividually. 2. Whe studets have had sufficiet time to complete their work, have them reassemble as a class. Discuss studets resposes. Test the reasoableess of the expressios, ad substitute the solutios i the equatios to verify their correctess. Note: Questio 2(c) i the BLM requires applyig the order of operatios. Variatios: Studets write word problems that could be represeted by the descriptios i the problems. Studets add cards with descriptios represetig expressios or equatios, or word problems that ca be represeted with liear equatios, to the classroom problem/ questio bak for games, Etry Slips or Exit Slips, ad so o. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Represet word descriptios with correct expressios. r Write a equatio to match a word descriptio ad solve the equatio correctly. 50 Grade 7 Mathematics: Support Documet for Teachers

231 Suggestios for Istructio Represet a problem with a liear equatio ad solve the equatio usig cocrete models. Model a problem with a liear equatio ad solve the equatio usig cocrete models. Materials: demostratio board balace scales for each group of studets (or studet-made balaces) blocks or cubes (iterlockig optioal) small paper bags, weigh boats, or polystyree cups math jourals or otebooks BLM 7.PR.12A: Represetig Equivalet Expressios o a Balace Scale (Sample) (multiple copies optioal) BLM 7.PR.12B: Represetig Equivalet Expressios o a Balace Scale (Template) BLM 7.PR.12C: Represetig Equivalet Expressios o a Balace Scale Usig Variables for Ukows (Sample) BLM 7.PR.12D: Represetig Equivalet Expressios o a Balace Scale Usig Variables for Ukows (Template) BLM 7.PR.12E: Represetig Equivalet Expressios (Template) Orgaizatio: Whole class, small groups (of four) Procedure: 1. Activate studets backgroud kowledge by demostratig a example to the class before studets work i their small groups. This process is illustrated o BLM 7.PR.12A: Represetig Equivalet Expressios o a Balace Scale (Sample). Record the demostrated process o the demostratio board. a) Preset the balace scale to studets, ad ask them to describe the techology ad its purpose. Sketch a schematic balace scale o the demostratio board. b) Itroduce the idividual blocks as represetig a value of 1. Blocks ca be liked together to represet other quatities. Add a give umber of blocks to oe side of the scale. Cout the blocks out loud, ad arrage them eatly o the platform (e.g., 8). The pa will tip. Record the scale ad the quatity pictorially ad symbolically (8). c) Ivite a studet to rebalace the scale. Stipulate that the studet must use blocks, ad caot duplicate what is already o the opposite pa (e.g., use 7 joied blocks ad 1 sigle block). Record this additio ad the chage pictorially ad symbolically. Patters ad Relatios 51

232 d) Note that the two expressios are equivalet. The scale is balaced. Record the equatio (8 = 7 + 1). Poit out how the differet parts of the equatio symbolize the differet parts of the scale. e) Ivite someoe else to balace the scale i a differet way, ad record that equality. Iclude a third alterative. f) All the expressios are equivalet to 8. Therefore, they must be equivalet to each other. Test the equivalecy of the expressios by rearragig the 8 blocks o the balace scale ito the differet expressios. Record the equivalet expressios as sets or as a strig (e.g., 8 = = = 3 + 5). Note that all expressios betwee the equal sigs are differet ames for the same quatity. This is also a opportuity to revisit the commutative property of additio. 2. Have studets, workig i groups, repeat the demostrated process usig their ow values. Each studet will record actios pictorially ad symbolically. A suggested procedure ad template are foud o BLM 7.PR.12B: Represetig Equivalet Expressios o a Balace Scale (Template). Templates are supplied for use with ad without variables. Studets may also use their math jourals or otebooks. a) Oe studet i each group has the resposibility of settig a quatity o either side of the scale, ad the aouces the quatity, draws a scale, ad records the quatity. b) Other studets i the group take turs represetig equal quatities o the opposite side of the scale, or rearragig the quatities i a pa. They verbalize their actios as they perform them, ad record the process pictorially ad symbolically. 3. As soo as studets are ready, reassemble as a class, ad demostrate balacig the scale usig blocks or cubes i paper bags or cups. The cocealed quatities represet the ukow meaig of a variable. It may be ecessary to tare the scales to compesate for the mass of the empty bag or cup. a) Secretly add a umber of blocks to the bag (e.g., 3). Add the bag ad a umber of cubes to oe pa of the scale. Record the actio pictorially ad symbolically ((b + 4 ), where b represets the quatity i the bag). b) Ivite a studet to balace the scale with a umber of blocks, coutig them i the process. Record the pictorial represetatio ad the liear relatio (b + 4 = 7). c) Ivite a studet to reame 7 i terms of b. (Use two bags with 3 blocks i each bag ad 1 sigle block.) d) Aother studet ca rearrage the blocks agai, or idetify the umber of blocks i the bag. e) Demostrate a ubalaced solutio with a empty bag ad some blocks o oe pa ad some blocks o the other pa. Ask studets how to fid the value of b ad rebalace the scale. (Cout blocks ito the bag util the scale balaces.) 52 Grade 7 Mathematics: Support Documet for Teachers

233 4. Have studets work i groups to repeat the process with variables. Oce agai, have studets share roles to model, represet, ad record equality i their math jourals or o the BLM template(s). This learig activity prepares studets for demostratig preservatio of equality ad usig preservatio of equality to solve equatios. Variatio: Arrage blocks without a scale. Cout to verify equivalecy. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Use a balace scale to model equivalet expressios. r Write equivalet expressios without variables as equatios. r Write equivalet expressios with variables as equatios. r Model ad record the commutative property of additio. r Reaso mathematically. Suggestios for Istructio Substitute a value for each ukow i a expressio ad evaluate the expressio. Materials: BLM 7.PR.13: Evaluatig Expressios, Give a Value for the Variable markig pes Orgaizatio: Whole class, idividual Procedure: 1. As a class, activate studets backgroud kowledge by askig idividual studets to a) defie a variable ad provide a example of oe b) defie a expressio ad provide a example of a expressio usig the variable provided c) suggest a value for the variable d) substitute that value for the variable ad evaluate the expressio e) offer a differet expressio usig the same variable f) evaluate the expressio usig the same value for the variable g) provide a differet value ad evaluate the expressio usig the ew value Patters ad Relatios 53

234 2. Whe you are satisfied that studets are able to complete this task idividually, distribute copies of BLM 7.PR.13: Evaluatig Expressios, Give a Value for the Variable. 3. Whe studets have had sufficiet time to evaluate the expressios, ask them to reassemble as a class, share their aswers, discuss ay discrepacies, ad use a markig pe to make ay otes, correctios, or additios to their sheets. The sheets ca be used for study otes at a later date. Variatio: Have studets create some additioal expressios of their ow, usig a ew variable ad/or values to substitute for the ew variable i their expressios. Studets could exchage their expressios ad have a classmate assess them. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Substitute a value for each ukow i a expressio ad evaluate the expressio. r Apply metal mathematics strategies to solve problems. Suggestios for Istructio Model the preservatio of equality for additio, subtractio, multiplicatio, or divisio usig cocrete materials or usig pictorial represetatios, explai the process orally, ad record it symbolically. Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. Materials: demostratio board balace scales for each group of studets (or studet-made balaces) blocks or cubes (iterlockig optioal) small paper bags or polystyree cups poster paper math jourals or otebooks algebra tiles (optioal) pes or markers of differet colours (optioal) 54 Grade 7 Mathematics: Support Documet for Teachers

235 Orgaizatio: Whole class (with three recorders ad oe voice ), small groups, idividual Procedure: 1. Activate studets backgroud kowledge by havig a group of studets use a balace scale to model ad record chages to a equatio. Oe studet may provide the cocrete model, while two others record the pictorial ad symbolic represetatios of the equatio o the demostratio board. Oe studet ca act as the voice, modellig self-talk durig the ivestigatio. a) Model a equatio with o variables. b) Have studets predict the outcome of addig 1 to the pa o oe side of the balace. c) Perform the actio. (The equatio is ubalaced.) Record the actio symbolically with less tha (<) or greater tha (>) symbols. d) Ask what will happe if 1 is added to the expressio o the other pa. (Balace is restored.) e) Play a game, askig studets to predict balace or tilt, ad the directio of tilt, if differet quatities are added to either or both of the pas. Have studets model a equatio, record the actio, ad commet o what is happeig for each sceario as it is performed. Iclude equatios with variables i the form of bags or cups. f) Ask studets to formulate a coclusio about preservig equality i a equatio whe usig additio. (Addig the same amout to each side of the equatio preserves equality.) 2. Have studets work i small groups to coduct the same ivestigatio ad draw a coclusio for each of the other operatios: subtractio, multiplicatio, ad divisio. Be sure to have them model cocretely, represet pictorially, record symbolically, talk through ad explai the process, ad formulate coclusios. 3. Studets ca record coclusios i their math jourals, or prepare persoal posters about preservig equality i a equatio. Ask them to iclude pictorial ad symbolic represetatios, as well as a explaatio of why the same operatio must be applied to each expressio to maitai equality. I this ivestigatio, studets have repeatedly used the terms expressio ad equatio. Have them iclude a statemet i their math jourals that explais how expressios ad equatios are similar ad how they are differet. This learig activity prepares studets for solvig a problem usig preservatio of equality i the ext learig experiece. Variatios: Use algebra tile models i place of, or i additio to, balace scale models. Ivestigate graphs as cocrete represetatios of equivalet expressios. Graph the equivalet expressios i differet colours o the same axes. Patters ad Relatios 55

236 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Model the preservatio of equality for additio, subtractio, multiplicatio, or divisio usig cocrete materials or usig pictorial represetatios, explai the process orally, ad record it symbolically. r Provide a example of a expressio ad a equatio, ad explai how they are similar ad differet. Suggestios for Istructio Solve a problem by applyig preservatio of equality. Draw a visual represetatio of the steps required to solve a liear equatio. Verify the solutio to a liear equatio usig cocrete materials or diagrams. Substitute a possible solutio for the variable i a liear equatio to verify the equality. Materials: BLM 7.PR.14A: Solvig Liear Equatios: Pictorial ad Symbolic Represetatios BLM 7.PR.14B: Solvig Liear Equatios with Costats: Applyig the Preservatio of Equality BLM 7.PR.14C: Solvig Equatios with Numerical Coefficiets: Applyig the Preservatio of Equality BLM 7.PR.14D: Solvig Liear Equatios with Costats ad Numerical Coefficiets: Applyig the Preservatio of Equality demostratio board balace scales for each group of studets (or studet-made balaces) blocks or cubes (iterlockig optioal) small paper bags or polystyree cups math jourals or otebooks couters ad algebra tiles (optioal) Orgaizatio: Whole class, pairs or small groups 56 Grade 7 Mathematics: Support Documet for Teachers

237 Procedure: The previous learig experiece, i which studets modelled ad explaied the preservatio of equality for each of the four operatios, provides a foudatio for this learig experiece. The emphasis here is o havig studets develop a strategy to solve a liear equatio. Part A 1. Represet a liear equatio with a costat. Begi with a simple cocrete represetatio of a oe-step equatio with a sigle variable. Revisit the example of 3 blocks or cubes i a bag, combied with 4 sigle blocks, o oe pa of the scale, balaced by 7 cubes o the other pa. Record the pictorial represetatio of the balace ad the liear relatio b + 4 = 7, where b represets the quatity i the bag. 2. Represet the solutio usig the preservatio of equality. Studets kow there are 3 blocks i the bag. Kowig basic umber facts makes this a easy equatio to solve. Propose that sometimes equatios are ot easy to solve, ad it would be helpful to have a strategy to fid the solutio. Easy questios assist with developig strategies, because ideas ad errors are more obvious i easy questios. Ask studets to suggest some strategies. Empty the bag. Studets will observe that the balace tilts. Ask studets how to fid the value of b ad rebalace the scale. Add blocks to the bag util the scale balaces, coutig the blocks i the process. Coutig the blocks that balace the scale provides a cocrete model to verify a solutio. Challege studets to prove there are 3 blocks i the bag by applyig the priciples they leared about the preservatio of equality. Applyig idetical operatios to both expressios will preserve the equality of the relatio. Remove 4 from each side of the balace. Edig up with the bag o oe pa ad 3 blocks o the other pa idicates there are 3 blocks i the bag. Whe the equatio is solved, verify the solutio symbolically by substitutig 3 i the origial equatio. Alteratively, verify the solutio cocretely by opeig the bag ad coutig the blocks. As studets make suggestios, record their suggestios pictorially ad symbolically. The strategy is outlied with both a balace scale ad algebra tiles i the Backgroud Iformatio. 3. Develop ad test a strategy. Whe studets are sufficietly prepared, suggest that they work with parters or i small groups to test more equatios cotaiig a variable ad a costat. For each equatio they test, have studets talk through steps as they proceed, ad verify the solutio. Patters ad Relatios 57

238 Apply the strategy to equatios i which the arithmetic is ot so easy (e.g., b + 17 = 42). Iclude egative costats (e.g., b 9 = 7). It may be ecessary to review the priciples of addig ad subtractig itegers. 4. Record the process. Whe studets have a process that works, ask them to record it o BLM 7.PR.14A: Solvig Liear Equatios: Pictorial ad Symbolic Represetatios or i their math jourals. Studets iclude the liear equatio ad the pictorial ad symbolic represetatios of the steps used to solve the equatio. They also verify the solutio, ad articulate a streamlied process for solvig a liear relatio with a costat. The process of solvig a liear equatio may iclude the followig: Aim to isolate the variable o oe side of the equatio ad a quatity o the other side. Remove the costat from the variable by addig its opposite to each side of the equatio (similar to zero pairs refer to Backgroud Iformatio for learig outcome 7.N.6). Equate the variable with a quatity i the fial equatio. Verify the solutio by substitutig the quatity for the variable i the equatio. 5. Apply the strategy. Part B Ask studets to test their process by applyig it to liear equatios with costats, such as those icluded o BLM 7.PR.14B: Solvig Liear Equatios with Costats: Applyig the Preservatio of Equality. 1. Review the process for solvig a liear equatio that studets created i Part A. Have four voluteers work together to model ad solve a liear equatio with oe variable ad a costat. Oe voluteer talks through the steps, oe models the cocrete represetatio, oe models the pictorial represetatio, ad oe models the symbolic represetatio. 2. Have studets work with their parters or small groups to cotiue the ivestigatio of Part A. I Part B, challege studets to outlie a process to solve liear equatios with a umerical coefficiet (e.g., 4c = 36), ad the progress to a process to solve liear equatios with a combiatio of coefficiets ad costats (e.g., 2b + 6 = 14). 3. For each equatio studets work through, have them articulate the actio they take, ad verify their solutio cocretely or with substitutio. Ask them to record both pictorial ad symbolic steps to solve the equatios, usig either their math jourals or BLM 7.PR.14A: Solvig Liear Equatios: Pictorial ad Symbolic Represetatios. 4. Have studets apply the process for solvig liear equatios by completig BLM 7.PR.14C: Solvig Equatios with Numerical Coefficiets: Applyig the Preservatio of Equality ad BLM 7.PR.14D: Solvig Liear Equatios with Costats ad Numerical Coefficiets: Applyig the Preservatio of Equality. 58 Grade 7 Mathematics: Support Documet for Teachers

239 5. Whe studets have completed their exploratio, have idividual studets create a persoal or class poster outliig the steps to solve liear equatios usig the preservatio of equality. Remember to iclude a verificatio step. Variatios: Use a variety of cocrete materials. Model solutios for the same or differet problems usig couters ad/or commercial or studet-made algebra tiles. Studets beefit from beig familiar with multiple represetatios. Exted the learig activity by havig studets create cotextual problems to match the liear equatios they work with. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a problem by applyig the preservatio of equality. r Draw a visual represetatio of the steps required to solve a liear equatio. r Verify the solutio to a liear equatio usig cocrete materials or diagrams. r Substitute a possible solutio for the variable i a liear equatio to verify the equality. Patters ad Relatios 59

240 Suggestios for Istructio Represet a problem with a liear equatio ad solve the equatio usig cocrete models. Model a problem with a liear equatio ad solve the equatio usig cocrete models. Draw a visual represetatio of the steps required to solve a liear equatio. Solve a problem usig a liear equatio. Solve a problem usig a liear equatio ad record the process. Verify the solutio to a liear equatio usig cocrete materials or diagrams. Substitute a possible solutio for the variable i a liear equatio to verify the equality. Materials: demostratio board a collectio of problems that ca be represeted by liear equatios (use those completed i previous learig activities, those completed for the classroom questio box, a collectio of teacher-prepared problems, or BLM 7.PR.15: Problems to Represet with Liear Equatios ad with Cocrete Materials) cocrete materials to represet the preservatio of equality (balace scales, tiles, couters) oe set of materials at each learig statio recordig booklets (two sheets of paper folded i half ad stapled) oe booklet for each group Orgaizatio: Learig statios with oe or two problems at each statio (oe more statio tha the umber of groups i the class), groups of four studets Procedure: 1. Set up learig statios i the classroom with oe set of cocrete materials ad oe or two problems at each statio. Decide how may statios studets must visit ad how may problems studets must complete. 2. Record the followig four studet roles o the demostratio board: a) Read the problem aloud. Record the liear equatio that matches the problem. b) Explai the steps to follow i solvig the problem. Model the solutio to the problem usig the cocrete materials at the statio. c) Record a diagram of the steps followed to solve the problem. Write the symbolic represetatio of the solutio. d) Verify the solutio, first by settig up the cocrete materials i a balaced fashio, ad the by substitutio. 60 Grade 7 Mathematics: Support Documet for Teachers

241 3. Studets work together to fid solutios to the problems, ad take turs performig each of the four roles. They use the group s booklet to record equatios, steps, solutios, ad verificatio. Have studets iitial their etries i the booklet. Variatios: Offer studets choice regardig the problems to be solved, or cotrol the questios or type of questios studets must aswer. The problems above cotai small quatities to facilitate modellig with cocrete materials. For a give umber of problems, have studets write similar problems usig larger umbers, ad solve them usig diagrams ad/or symbolic represetatios. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Represet a problem with a liear equatio ad solve the equatio usig cocrete models. r Model a problem with a liear equatio ad solve the equatio usig cocrete models. r Draw a visual represetatio of the steps required to solve a liear equatio. r Solve a problem usig a liear equatio. r Solve a problem usig a liear equatio ad record the process. r Verify the solutio to a liear equatio usig cocrete materials or diagrams. r Substitute a possible solutio for the variable i a liear equatio to verify the equality. Patters ad Relatios 61

242 Suggestios for Istructio Represet a problem with a liear equatio ad solve the equatio usig cocrete models. Model a problem with a liear equatio ad solve the equatio usig cocrete models. Draw a visual represetatio of the steps required to solve a liear equatio. Solve a problem usig a liear equatio. Solve a problem usig a liear equatio ad record the process. Verify the solutio to a liear equatio usig cocrete materials or diagrams. Substitute a possible solutio for the variable i a liear equatio to verify the equality. Materials: access to research materials assorted materials to create questios ad aswers, game boards, ad support materials booklets (for recordig solutios) cocrete materials for represetig the preservatio of equality (balace scales, blocks ad bags, couters, algebra tiles) Orgaizatio: Small groups, idividual Procedure: Iform the class that groups of studets will develop evets or games, such as the followig. Aroud the World i 10 Equatios Studets collect passport stamps as they move from oe city to aother o a regioal or world map. To obtai trasportatio from oe destiatio to the ext, travellers must use a liear equatio to solve a problem about iformatio related to the regio. Each traveller records equatios ad solutios i his or her passport book. Correct solutios ear travellers a passport stamp ad a ticket to the ext destiatio. Studets receive a souveir upo completig the jourey. Relatios Regatta Studets eter a boat race. They collect strips to represet the distace completed as they progress through the course. To cover distace i the course, competitors must use liear equatios to solve problems related to marie life, autical vessels, ad so o. Each competitor records the equatio ad solutio i his or her logbook. Correct solutios ear participats a distace strip ad a evet pass to the ext sectio of the course. Studets receive a trophy for completig the race. 62 Grade 7 Mathematics: Support Documet for Teachers

243 Part A Studets develop a evet or a game. 1. Form groups ad choose which evet to host. 2. Select the topics o which to base questios. 3. Assig topics to idividuals. 4. Idividuals research their respective topics to collect iformatio from which to create problems. 5. Each studet creates two or three problems that ca be represeted by a liear equatio, ad tests the solutio to the problems based o the preservatio of equality. 6. Iclude oe questio with a costat, oe with a umerical coefficiet, ad oe with both a umerical coefficiet ad a costat. 7. Groups decide o the format for presetig the problems ad cocealig the solutios. 8. Idividuals prepare good copies of the problems ad cocealed solutios. Solutios must iclude pictorial ad symbolic represetatios of the solutios, ad verificatio of the solutios. 9. Decide o the details required for the physical presetatio to play the game ad receive tokes. 10. Assig resposibilities, ad create the product. 11. Test the game, ad address ay problem areas. Part B Studets play the games created by others ad collect the rewards. 1. Studets decide which evet they will participate i, ad form groups to play each game. 2. Studets progress through the game idepedetly. Each idividual reads the problems, ad uses his or her booklet to represet each problem as a liear equatio, ad to record the steps used to solve the problem usig the preservatio of equality. 3. Studets collect the rewards at the ed of the game. Patters ad Relatios 63

244 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Draw a visual represetatio of the steps required to solve a liear equatio. r Solve a problem usig a liear equatio. r Solve a problem usig a liear equatio ad record the process. r Verify the solutio to a liear equatio usig cocrete materials or diagrams. r Substitute a possible solutio for the variable i a liear equatio to verify the equality. 64 Grade 7 Mathematics: Support Documet for Teachers

245 Puttig the Pieces Together Relatios Stories Itroductio: Studets create a book, graphic ovel, or cartoo strip emphasizig patters, their relatios, ad related vocabulary terms. Purpose: I this ivestigatio, studets have the opportuity to demostrate ay or all of the followig abilities (coectios to learig outcomes are idetified i paretheses): Correlate oral ad writte patters ad liear relatios. (7.PR.1) Costruct ad aalyze a table of values ad graphs to solve problems based o liear relatios. (7.PR.2) Apply the preservatio of equality to solve equatios. (7.PR.3) Differetiate betwee expressios ad equatios. (7.PR.4) Evaluate expressios, give the value of the variable(s). (7.PR.5) Solve oe-step liear equatios. (7.PR.6) Solve problems represeted by liear equatios. (7.PR.7) Relate radii, diameters, ad circumfereces of circles ad solve problems ivolvig the measuremet of circles. (7.SS.1) Apply formulas to determie the area of triagles, parallelograms, ad circles. (7.SS.2) Studets will also demostrate some or all of the followig mathematical processes: Commuicatio Coectios Metal Mathematics ad Estimatio Problem Solvig Reasoig Techology Visualizatio Patters ad Relatios 65

246 Materials/Resources: BLM : My Success with Mathematical Processes books based o problems related to patters, relatios, ad liear equatios, such as the followig: Gravett, Emily. The Rabbit Problem. Lodo, UK: Macmilla Childre s Books, McCallum, A. Rabbits Rabbits Everywhere: A Fiboaci Tale. Illus. Gideo Kedall. Watertow, MA: Charlesbridge Publishig Ic., Neuschwader, Cidy. Sir Cumferece ad the Isle of Immeter: A Math Adveture. Illus. Waye Geeha. Watertow, MA: Charlesbridge Publishig Ic., book-makig supplies computer access (optioal) Orgaizatio: Idividual, pairs Procedure: 1. The world aroud us abouds with patters ad relatios. Patters ca be described, ca provide a iterestig source of iformatio ad ivestigatio, ad ca be used to create ad solve mysteries. Example: Mathematicia Leoardo Fiboacci lived i Italy aroud the year He itroduced Hidu-Arabic umbers to Europe, ad revealed a iterestig umber patter i a ivestigatio of the rate at which a sigle pair of rabbits multiplies. Iterestig umber patters are prevalet i ature. Sample Website: Examples of patters ad relatios (e.g., Fiboacci umbers, Pascal s triagle, fractals) ca be viewed o websites such as the followig: World-Mysteries.com. Fiboacci Numbers i Nature ad the Golde Ratio. Sciece Mysteries < 2. Liste to your teacher read The Rabbit Problem by Emily Gravett. While you are listeig, ote refereces to Fiboacci ad to patters ad relatios. 3. Note the book s orgaizatio aroud Fiboacci s questio, ad how its presetatio as a caledar matches the ivestigatio period of oe year. Note the author s subtle refereces. 4. Liste to your teacher read Sir Cumferece ad the Isle of Immeter by Cidy Neuschwader. This book solves area problems usig relatioships betwee area ad the sides of rectagles ad relatioships betwee radius ad circumferece. 66 Grade 7 Mathematics: Support Documet for Teachers

247 5. Create your ow Advetures i Algebra series of stories ivolvig patters, relatios, variables, ad equatios. Stories may be pattered after The Rabbit Problem ad preseted as a caledar usig a scee for each moth of the year, or stories may be preseted as books, graphic ovels, or cartoo strips. Stories may iclude mai characters or a hero such as the Master of Relatios ad his sidekick the Vari Able Geerator. Stories should preset patters, equatios, or mysteries to solve, which may be desig-orieted, or focus o music, or ivolve quatity or measuremets such as time, distace, or area. The story compoets ad assessmet criteria are outlied i My Plaig Sheet for Relatios Stories (see ext page). 6. Oce you have decided o a pla, share your idea with your teacher, usig My Plaig Sheet for Relatios Stories. 7. Begi work o your project. 8. Share your stories with peers, or with youger studets, or as part of a authors ight, or i a library display, ad so o. 9. Have a coversatio with your teacher about your success i demostratig your learig ad usig the mathematical processes. Record your success usig BLM : My Success with Mathematical Processes. Assessmet: 1. Studets will demostrate their learig i the differet categories idetified i Assessmet of Relatios Stories (see last two pages of Patters ad Relatios), based o how they choose to complete the project. Have a coversatio with each studet about which learig he or she will demostrate to you through the process of desigig the product. 2. Work with studets to develop assessmet criteria i each of the idetified categories. 3. The fial assessmet of each category should be based o a studet s recet cosistet demostratio of learig. 4. Distribute copies of BLM : My Success with Mathematical Processes, ad have each studet record his or her success with the mathematical processes. Patters ad Relatios 67

248 My Plaig Sheet for Relatios Stories Name Date What will I show? (Check at least four boxes i Sectio 1 ad two boxes i Sectio 2) How do I kow that I have bee successful? How will I show it? Sectio 1: Kowledge ad Uderstadig of Mathematical Cocepts I ca give examples of patters ad I ca explai those patters usig math equatios (like d = 2c + 1). (7.PR.1) I gave oe or more examples of patters ad I used math equatios to show aother way to explai the patters. I ca look at a patter ad show the steps of a patter usig a T-chart ad a graph. (7.PR.2) I used a T-chart ad a graph to show how a patter chages. I kow that both sides of a equatio are equal, ad this helps me to solve equatios. (7.PR.3) I solved oe or more equatios ad showed that I remembered to keep both sides of the equatio equal whe solvig it. I kow the differece betwee a equatio ad a expressio. (7.PR.4) I showed a expressio ad a equatio ad I showed how they are differet. I ca replace a variable with a umber to solve a expressio. (7.PR.5) I replaced a variable with a umber to solve a expressio. I kow how to solve equatios that ca be solved i oly oe step (like 2y = 8 or p + 2 = 1). (7.PR.6) I solved equatios that eeded oly oe step to fid the aswer. I ca solve circle problems (like radius, circumferece, ad diameter). (7.SS.1) I showed that I uderstad the differet measuremets i circles. (cotiued) 68 Grade 7 Mathematics: Support Documet for Teachers

249 My Plaig Sheet for Relatios Stories (cotiued) Name Date What will I show? (Check at least four boxes i Sectio 1 ad two boxes i Sectio 2) How do I kow that I have bee successful? How will I show it? Sectio 1: Kowledge ad Uderstadig of Mathematical Cocepts (cotiued) I ca figure out the area of triagles, parallelograms, ad circles usig formulas. (7.SS.2) I figured out the area of triagles, parallelograms, ad circles usig a formula I kow or a formula that I figured out. I ca use math laguage whe describig patters. I used math laguage that I already kew ad math laguage that I was learig while showig what I kew about patters. Sectio 2: Metal Mathematics ad Estimatio I ca use metal math ad estimatio to help me solve expressios (like b + 5). (7.PR.5) I used metal math to solve expressios ad to check that my aswers were correct. I estimated to make sure my aswers made sese. I ca use metal math ad estimatio to help me solve ad check equatios that ca be solved i oly oe step (like 2y = 8 or p + 2 = 1). (7.PR.6) I used metal math to solve equatios ad to check that my aswers were correct. I estimated to make sure my aswers made sese. I ca use estimatio whe solvig circle problems (radius, circumferece, or diameter). (7.SS.1) I estimated to make sure my aswers made sese. I ca figure out the area of triagles, parallelograms, ad circles usig formulas. (7.SS.2) I used metal math to figure out the area of triagles, parallelograms, ad circles. I estimated to make sure my aswers made sese. Patters ad Relatios 69

250 Assessmet of Relatios Stories Criteria Not Demostrated (ND) Kowledge ad Uderstadig of Mathematical Cocepts Provide examples or scearios of patters ad liear relatios. (7.PR.1) makes coectios amog patter examples/scearios ad their liear relatios demostrates a good uderstadig of how to coect patter examples/scearios ad their liear relatios coects basic patter examples/scearios ad their liear relatios requires support to coect basic patter examples/scearios ad their liear relatios does ot coect patter examples/ scearios ad their liear relatios Create tables of values ad graphs based o liear relatios. (7.PR.2) accurately represets liear relatios as graphs ad tables of values Use preservatio of equality to solve equatios. (7.PR.3) demostrates a uderstadig of preservatio of equality whe solvig equatios Differetiate expressios from equatios. (7.PR.4) differetiates betwee expressios ad equatios Evaluate expressios, give the value of the variable(s). (7.PR.5) substitutes a value for the variable i order to solve a expressio Solve oe-step liear equatios. (7.PR.6) solves oe-step liear equatios cocretely, pictorially, or symbolically Solve problems ivolvig measuremets of circles. (7.SS.1) demostrates a uderstadig of measuremet related to circles (radius, circumferece, ad diameter) (cotiued) 70 Grade 7 Mathematics: Support Documet for Teachers

251 Assessmet of Relatios Stories (cotiued) Criteria Not Demostrated (ND) Kowledge ad Uderstadig of Mathematical Cocepts (cotiued) Determie the area of triagles, parallelograms, ad circles by applyig formulas. (7.SS.2) develops/applies a formula for determiig the area of triagles, parallelograms, ad/ or circles Use related vocabulary. demostrates a uderstadig ad applicatio of mathematics vocabulary Metal Mathematics ad Estimatio Evaluate expressios, give the value of the variable(s). (7.PR.5) applies metal mathematics strategies to solve expressios Solve oe-step liear equatios. (7.PR.6) applies metal mathematics strategies to solve oe-step liear equatios ad to check the accuracy of the solutios Solve problems ivolvig measuremets of circles. (7.SS.1) makes reasoable estimates whe solvig problems ivolvig the measuremets of circles Determie the area of triagles, parallelograms, ad circles by applyig formulas. (7.SS.2) applies metal mathematics strategies to determie the area of triagles, parallelograms, ad circles, ad makes reasoable estimates to determie the accuracy of the solutios Patters ad Relatios 71

252 N o t e s 72 Grade 7 Mathematics: Support Documet for Teachers

253 G r a d e 7 M a t h e m a t i c s Shape ad Space

254

255 Shape ad Space (Measuremet) (7.SS.1) Edurig Uderstadig(s): Circle graphs show a compariso of each part to a whole usig ratios. May geometric properties ad attributes of shapes are related to measuremet. Geeral Learig Outcome(s): Use direct or idirect measuremet to solve problems. Specific Learig Outcome(s): 7.SS.1 Demostrate a uderstadig of circles by describig the relatioships amog radius, diameter, ad circumferece of circles relatig circumferece to pi determiig the sum of the cetral agles costructig circles with a give radius or diameter solvig problems ivolvig the radii, diameters, ad circumfereces of circles [C, CN, R, V] Achievemet Idicators: Illustrate ad explai that the diameter is twice the radius i a circle. Illustrate ad explai that the circumferece is approximately three times the diameter i a circle. Explai that, for all circles, pi is the ratio of C the circumferece to the diameter, ad d, its value is approximately Explai, usig a illustratio, that the sum of the cetral agles of a circle is 360. Draw a circle with a give radius or diameter with or without a compass. Solve a cotextual problem ivolvig circles. Prior Kowledge Studets should be able to do the followig: Q Q (3.SS.5) Demostrate a uderstadig of perimeter of regular ad irregular shapes by estimatig perimeter usig referets for cetimetre or metre measurig ad recordig perimeter (cm, m) Q Q costructig differet shapes for a give perimeter (cm, m) to demostrate that may shapes are possible for a perimeter (6.N.5) Demostrate a uderstadig of ratio, cocretely, pictorially, ad symbolically. Shape ad Space 3

256 Q Q Q Q (6.SS.1) Demostrate a uderstadig of agles by idetifyig examples of agles i the eviromet classifyig agles accordig to their measure estimatig the measure of agles usig 45, 90, ad 180 as referece agles determiig agle measures i degrees drawig ad labellig agles whe the measure is specified (6.SS.3) Develop ad apply a formula for determiig the perimeter of polygos area of rectagles volume of right rectagular prisms Related Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q Q Q (7.PR.2) Costruct a table of values from a relatio, graph the table of values, ad aalyze the graph to draw coclusios ad solve problems. (7.PR.5) Evaluate a expressio give the value of the variable(s). (7.PR.6) Model ad solve problems that ca be represeted by oe-step liear equatios of the form x + a = b, cocretely, pictorially, ad symbolically, where a ad b are itegers. (7.PR.7) Model ad solve problems that ca be represeted by liear equatios of the form ax + b = c ax = b x = b, a 0 a cocretely, pictorially, ad symbolically, where a, b, ad c, are whole umbers. (7.SS.2) Develop ad apply a formula for determiig the area of triagles parallelograms circles (7.SP.3) Costruct, label, ad iterpret circle graphs to solve problems. 4 Grade 7 Mathematics: Support Documet for Teachers

257 Backgroud Iformatio Agles ad Circles Studets come to Grade 7 with a backgroud i classifyig ad measurig agles. Agles are formed from two rays emaatig from a commo poit, ad circles are all the poits equidistat from a commo poit. Agles are measured as fractios of circles, 1 each degree beig of a circle. The umber 360 is said to be related to Sumeria ad 360 Babyloia observatios of trackig the movemet of astroomical objects through the sky for the 360 days i their year. The umber 360 is coveiet because it has multiple factors. I the learig experieces that follow, the close relatio betwee agles ad circles is used as a startig poit to develop a uderstadig of circle cocepts. A circle is the full rotatio of a agle. The ability to look for ad describe patters with variables ad equatios is used to discover the relatioships ad ratios withi circles, ad these ratios are used to solve cotextual problems. Learig experieces that ivolve describig the relatioships i circles ad solvig problems ivolvig circles correspod well with the Variables ad Equatios substrad of the Patters ad Relatios strad. The first cocept developed i the followig learig experieces is the sum of cetral agles. Agles are used to develop cocepts related to radius, circle, ad circumferece. The cocept of radius is used to costruct circles of a give size, ad studets measure circumferece i terms of radius i their first ivestigatio about relatioships i circles. The term diameter is built from coectig two radii, ad further relatioships are determied betwee diameter ad circumferece. Studets develop accurate measurig skills, ad discover the ratio betwee the circumferece ad the diameter of circles. The C ratio is approximated as 3.14, ad is commoly referred to as pi (p). The relatio d is a very importat mathematical costat. There is evidece of its use i Aciet Egypt, Aciet Babylo, Aciet Israel, ad Aciet Idia. The Aciet Greeks studied the relatioship very carefully ad represeted it as 22. For every circle, the circumferece 7 divided by the diameter is a o-termiatig o-repeatig decimal. I the 1700s, it was give a special ame, pi. The ame was chose because pi is the first letter i the Greek phrase for perimeter/diameter. It is commo to use the approximate value 3.14 for pi. May people are fasciated with the umber that represets pi. There is eve a special Pi Day celebrated March 14 (3/14). You may wish to have studets research pi ad share the iformatio they fid. It is importat that studets recogize that pi is ot so much a special umber as it is a special relatioship (the relatioship of the circumferece of a circle divided by its diameter). Shape ad Space 5

258 Teachers are ecouraged to provide hads-o learig activities ad group work as a meas for studets to develop skills ad to explore ad discover the cocepts ad relatioships withi circles. Guide studets i their learig ad provide vocabulary to describe the cocepts, while allowig studets to make discoveries. Based o the relatioships they discover, studets ca develop cotextual problems for oe aother to solve. Mathematical Laguage agle arc cetral agle circle circumferece classificatios of agles (acute, obtuse, reflex, right, ad straight) diameter pi radius Learig Experieces Assessig Prior Kowledge Materials: BLM 7.SS.1.1: Assorted Agle Cards (or diagrams of various agles of differet sizes, icludig acute, obtuse, reflex, right, ad straight agles) BLM 7.SS.1.2: Agle Classificatios, Agle Estimatios ad Measures, ad Perimeter display board tacks, tape, or magets to hold agle cards o a display protractors Orgaizatio: Whole class, idividual Procedure: 1. Advise studets they will review what they already kow about agles by sortig agle cards ito the differet classificatios of agles. Give each studet oe of the cards from BLM 7.SS.1: Assorted Agle Cards. Ivite studets to post their agle cards (or sketch the represeted agles) with the similar agles o the display board. 6 Grade 7 Mathematics: Support Documet for Teachers

259 2. Studets ca critique the agle display ad share ay adjustmets they would like to make to correct it. Review defiitios of the differet classificatios of agles (acute, obtuse, reflex, right, ad straight agles). 3. Retur the cards to studets. Ivite studets to use their estimatio skills to idetify a agle that matches a approximate measuremet. They may show their respose by displayig a card or by sketchig a agle. Ask studets how to verify the measuremet. Review the use of protractors, ad have studets measure their agles usig protractors. 4. Review the cocept of perimeter by havig studets estimate the measure of the perimeter of desigated surfaces (e.g., a tissue box, desk surface, classroom door, widow, ceilig), ad havig them justify their resposes. Tracig the edge of objects with a figer reiforces that the perimeter is the distace aroud the outside of the object. 5. Distribute copies of BLM 7.SS.1.2: Agle Classificatios, Agle Estimatios ad Measures, ad Perimeter, ad have studets respod to the questios provided. Variatios: Use the agle cards to play games i which studets collect a set of oe classificatio of agles or a set of each classificatio of agles (acute, obtuse, reflex, right, straight). Or use the cards to play other types of games (e.g., Cocetratio, Pit). Play a I Spy game to develop studets uderstadig of perimeter (e.g., I spy a perimeter close to... ). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Classify agles of differet sizes as acute, obtuse, reflex, right, or straight agles. r Estimate the size of agles ad cofirm the estimate by measurig the agles with a protractor. r Demostrate a uderstadig of perimeter. Shape ad Space 7

260 Suggestios for Istructio Explai, usig a illustratio, that the sum of the cetral agles of a circle is 360. Materials: BLM 7.SS.1.3: Cut-outs for Agles of Differet Measures (oe of each of the six types per group, copied o card stock or o bod paper, plus extra copies if studets will make posters) demostratio board large cut-outs of agles (two copies of 45 ad 90 agles, oe copy of 180 agle) (optioal) card stock or bod paper scissors protractors math jourals file cards resealable bags for storage (optioal) poster paper (optioal) glue (optioal) computer software (optioal) Orgaizatio: Whole class, small groups Procedure: 1. Guide studets through a class discussio while buildig a circle usig agles. A sample procedure follows: Draw or use a large cut-out of a 45 agle, ad have studets estimate the measure of the agle. Ask what the diagram resembles if the eds of the rays are coected with a arc Grade 7 Mathematics: Support Documet for Teachers

261 Stack two agles, oe ext to the other, with the vertices meetig. Ask for a approximatio of the agle represeted (90 ). Coect the rays with aother arc, ad ask what the image resembles ow Lie up a reflectio of aother 90 agle alogside the image. Cotiue to ask for a estimatio of the measure of the agle (180 ), ad what the agle resembles Complete the task by addig a 180 agle below the image. Solicit a estimatio of the measure ad a descriptio of the image (360 circle). 360 Poit out to studets (either ow or followig some more ivestigatio) that the vertices of all the agles meet at oe poit i the cetre of the circle. The measure of each of the agles is take from the cetre of the circle. These agles are called cetral agles. Each cetral agle has its vertex at the cetre of the circle, ad each ray radiates to a differet poit o the edge of the circle or circumferece. Shape ad Space 9

262 2. Distribute copies of BLM 7.SS.1.3: Cut-outs for Agles of Differet Measures, ad have groups carry out the followig ivestigatio: Variatios: Each group will eed oe of each of the differetly partitioed circles (halves, thirds, quarters, sixths, eighths, ad twelfths). Studets ca share the circles withi their group to eve out the umber of pieces each studet will work with (e.g., thirds ad quarters, halves ad sixths). Each studet accurately measures the agles i the sectios ad eatly records the agle measures iside each sectio. Studets the carefully cut out each sectio. Have studets combie pieces with agles of differet sizes to form circles, ad calculate the sum of the agles of the sectios that complete the circles. The repeat the process, costructig a umber of circles with cetral agles of differet sizes. After a appropriate time, reassemble as a class, ad ask studets to share what they have discovered durig this ivestigatio. If you did ot itroduce the term cetral agle earlier, do so ow. There are iterestig relatioships i circles. Ask studets what they ca coclude about the measures of cetral agles i a circle (e.g., the sum of the cetral agles is 360 ). Note the coectio betwee agles ad circles. Agles are measured as fractios of circles. Each degree is 1 of a circle. Have studets add otes to their math jourals. 360 If the techology ad skills are available, use computer software to draw, measure, ad calculate the sum of agles. Computer applets or games may also be used. Sample Website: Computer applets are available o the followig website: Math Ope Referece. Agles < Circles < Have studets fid the measure of agles betwee differet poits o marked circles used i everyday objects (e.g., degrees betwee poits o a compass, miutes or hours o a clock, positios o dials to measure temperature or speed). Results could be displayed i posters, with the measures of the agles betwee the poits, ad the sum of all the agles. Play a game i which studets fid the missig agle. Preset a circle with oe or more cetral agles. Studets determie the measure of the remaiig agle. The questios ca also be used as Etry Slips or Exit Slips. 10 Grade 7 Mathematics: Support Documet for Teachers

263 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Explai, usig a illustratio, that the sum of the cetral agles of a circle is 360. r Visualize cetral agles. r Use metal mathematics ad estimatio strategies to solve problems. Suggestios for Istructio Explai, usig a illustratio, that the sum of the cetral agles of a circle is 360. Solve a cotextual problem ivolvig circles. Materials: BLM 7.SS.1.4: Hige Templates for Makig Agles scissors push-pis (to faste the higes) corrugated cardboard (o which to pi the cetre of the circle) protractors umber cubes, multi-sided umber cubes, or spiers blak paper pecils strig (optioal) maskig tape or chalk (optioal) computer software (optioal) Orgaizatio: Pairs, whole class Shape ad Space 11

264 Procedure: 1. Have pairs of studets take turs buildig a circle from joied agles ad calculatig the sum of the agles. A suggested procedure follows: a) Studets throw a umber cube to determie who makes the first agle, ad the throw it agai to idicate the umbers of agles (turs) to make. Studets place blak paper o the cardboard, mark a cetre poit for the circle, ad push a push-pi through the hige to keep it i the cetre of the circle. b) The first perso marks the edge of the hige, opes the agle to a desired size, marks the positio of the agle, ad uses a protractor to measure the agle formed. c) The secod perso records the agle ad keeps a ruig sum of the agle measures. d) The parters the switch roles. The secod perso begis from the last mark, forms a agle, marks the edge poit, ad measures the agle. The first perso records the agle measure ad adds the measure to the sum of the agles. e) Parters cotiue switchig roles util they have retured to the startig poit or have formed the desigated umber of agles. f) Studets record the sum of the agles ad perform aother roud. 2. Meet as a class, ad have studets report o the sums of the cetral agles that were obtaied durig the ivestigatio. Compare the sums of the agles for each circle. Ask why the sums are close to, but ot exactly, 360, ad what chages could be made to the procedure to icrease accuracy. Sources of error could iclude errors i liig up ad markig the higes, i usig the protractor, or i makig calculatios, variatios i the thickess of pecil lies or the positio o the hige used for markig, ad so o. If there is sufficiet iterest, ad time is available, challege studets to repeat the ivestigatio ad try to elimiate the sources of error. 3. Preset the followig problem for studets to solve: Variatio: A pizza was sittig o top of the stove. Jack cut out a piece of pizza ad ate it. The cetral agle of the missig piece was 45. Lisa came by, sliced some pizza, ate it, ad left. The cetral agle of the remaiig pizza was 90. How much of the pizza did Lisa eat? Ivestigate the sum of cetral agles usig computer applets. Sample Website: Computer applets are available o the followig website: Math Ope Referece. Cetral Agle. Circles < Drag a poit o the circumferece to make cetral agles of differet sizes. 12 Grade 7 Mathematics: Support Documet for Teachers

265 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Explai, usig a illustratio, that the sum of the cetral agles of a circle is 360. r Solve a cotextual problem ivolvig circles. r Reaso mathematically to solve problems. Suggestios for Istructio Illustrate ad explai that the diameter is twice the radius i a circle. Illustrate ad explai that the circumferece is approximately three times the diameter i a circle. Draw a circle with a give radius or diameter with or without a compass. Part A: Materials: math jourals materials to mark a trail as a large circle is formed (e.g., sidewalk chalk for hard surfaces, a stick for gravel or soil, a bag of flour, puffed rice, or popcor for laws, books, self-stick otes, or scraps from a hole pucher for idoor floors) Orgaizatio: Whole class, small groups (of five studets) Procedure: Review that the sum of cetral agles is 360, ad develop defiitios of the terms radius, circle, ad circumferece by costructig circles, as described below. 1. Have studets do the followig: a) sketch a agle, makig the rays fairly log. Label the agle ÐABC. Write the approximate measure of the agle ear the vertex. A B 45 C Shape ad Space 13

266 b) Add two adjacet agles that each share a arm of the origial agle ad the vertex B. To label the ew agles, add the poits D ad E. There are ow four adjacet agles. D A B C E c) Estimate the measure of each agle. Add the measures of the cetral agles. Determie whether they are close to 360, ad if ot, explai why ot. 2. Have studets draw a circle that passes through the arms of each agle. The have them ask a parter to rate the roudess of the circle o a scale of 1 to Together with studets, describe the criteria for a perfect circle. For example, the distace from the cetre to the outside of the circle must always be same. Iform studets that distace is called a radius. The measure for every radius of the same circle is idetical. I fact, a circle is the set of poits o a flat surface equal distaces from a fixed poit. The distace aroud those poits (or the perimeter of the circle) is called the circumferece. 4. Have each studet make a math joural etry to defie the terms radius, circle, ad circumferece. 5. Place studets ito groups of five. Have each group make a pla for creatig a large early perfect circle (outdoors, i the gymasium, or wherever space is available). Five studets may make a circle with a diameter of 12 m or more, so esure there is sufficiet space for the class. Iform groups of the materials that will be available to them, ad discuss the cleaup requiremets. If studets are havig difficulty gettig started, or thik oly of holdig hads ad spreadig out, suggest they thik of formig a radius. Studets holdig hads at arm s legth ca create quite a log radius. The cetre perso must be achored securely to avoid beig pulled out of positio as the studets formig the radius pivot aroud the cetre. The last perso ca leave a trail for the circumferece. Alteratively, make oe large circle, have five or six studets form the radius, ad ask the other studets to stad as markers aroud the circumferece as the circle is formed. 14 Grade 7 Mathematics: Support Documet for Teachers

267 6. Whe studets have completed the circle, have them carefully step outside their circle ad evaluate it. Recallig the power ad beauty that ca exist i discoverig patters, ask studets to relate the radius of the circle to its circumferece. Measurig the circumferece i terms of the radius is a good way to do this. Have studets do the followig: Clearly mark a startig poit o the circle. Lie up the radius alog the circumferece, begiig at the startig poit. Fold the radius over o itself util it returs to the startig poit, coutig the umber of radii at each fold. Have studets share how may legths of the radius the circumferece of their circles is (approximately six). Have studets make larger ad smaller circles ad test the relatioship betwee the circumferece ad the radius. 7. Retur idoors ad have studets make a math joural etry about what they leared i this learig activity. Part B: Materials: math jourals corrugated cardboard, large paper, or other media o which to draw large circles strig or light-gauge wire push-pis or ails with a large head compasses ad pecils rulers, metre sticks, tape measures, ad trudle wheels Orgaizatio: Pairs or small groups Procedure: I Part B of this learig experiece, studets build o what they leared i Part A to eable them to draw circles of a give radius. They also lear about diameter ad its relatio to the radius ad the circumferece of a circle. 1. Review what studets have writte i their math jourals about radius, circle, circumferece, ad how to build a perfect circle. Have them use their math jourals to sketch a rough circle ad draw two radii for that circle. Poit out that they have draw a cetral agle. By drawig more radii, they add more cetral agles. Ask studets to draw a circle with two radii that are perfectly lied up with each other. They form a agle of 180. This arragemet of radii is the diameter of the circle. Have studets do the followig: Describe the diameter of a circle. It is a straight lie passig through the cetre of the circle. Describe the relatioship betwee the radius ad the diameter. The diameter is twice as log as the radius, d = 2r. (I Part A, the relatioship betwee the radius ad the circumferece was foud to be ~6r = C.) Shape ad Space 15

268 Predict the relatioship betwee the diameter ad the circumferece. Fid a way to test the predictio. Make a math joural etry to defie diameter. 2. Direct studets to the materials available to make large circles. Explai that their task is to develop a method to draw circles of a give radius or diameter. The circle size ca be determied by the teacher, by a parter, or by the studets themselves. Have studets make small ad large circles. Whe studets have had some time to explore, ask them to share their success stories ad frustratios, so they ca help each other ad refie their methods. Some studets may use strig, ad may eed to lear to tie a slip kot. Some studets may use a strip of cardboard with a hole for the cetre ad hole for a pecil. They may have creative ideas. 3. Cotiue the exploratio. Aim for producig circles of a specific radius or diameter. Have studets compare the size of the circle draw to the iteded size by measurig the radius ad/or diameter with a ruler or a metre stick. If possible, also ivestigate the relatioship betwee the legth of the diameter ad the circumferece to test the predictios that were made. 4. After studets have had sufficiet time to explore, cogratulate them o their success. They will have iveted some effective methods to draw circles accurately, especially large oes. 5. Itroduce studets to the compass i their geometry kits, ad show them how to use it. The metal poit is held at the cetre of the circle, ad the pecil poit will form marks alog the circumferece as it revolves aroud the cetre poit. The distace betwee the metal poit ad the pecil poit will be the radius. Have them draw multiple circles of differet sizes to perfect the techique. 6. To ed the class, have studets draw two circles i their math jourals. Specify the radius for oe of the circles, ad the diameter for the other. Ask them to commet o aythig they foud out about the relatioship betwee the diameter ad the circumferece. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Illustrate ad explai that the diameter is twice the radius i a circle. r Illustrate ad explai that the circumferece is approximately three times the diameter i a circle. r Draw a circle with a give radius or diameter with or without a compass. 16 Grade 7 Mathematics: Support Documet for Teachers

269 Suggestios for Istructio Illustrate ad explai that the circumferece is approximately three times the diameter i a circle. Explai that, for all circles, pi is the ratio of the circumferece to the C diameter, ad its value is approximately d Materials: compasses ad pecils rulers, metre sticks, tape measures, ad trudle wheels measurig tape strig, ribbo, or light-gauge wire push-pis corrugated cardboard circular objects or cyliders tool for fidig the cetre of circles (optioal) BLM 7.SS.1.5: A Table to Compare Measures of Circles (optioal) computers ad spreadsheets (optioal) Orgaizatio: Small groups or whole class Procedure: I this learig experiece, studets focus o comparig the radius, diameter, ad circumferece of circles i a effort to discover pi, the ratio of the circumferece to the C diameter d. 1. Have studets quickly review what they have leared about cetral agles, radius, diameter, ad circumferece, ad their relatioships, ad about how to draw circles. 2. Explai that studets will ow use a variety of circles ad circular objects ad accurately measure their radius, diameter, ad circumferece to fid a famous ad useful relatioship betwee the measures. Iform studets there are several ways to measure circumferece, such as the followig: Wrap a measurig tape aroud the circumferece of a circle. Wrap a strig or a ribbo aroud the circumferece, ad the measure the strig or ribbo. Mark a startig poit o the circle ad roll the circle alog a ruler, returig to the startig poit. Shape ad Space 17

270 Roll the circle alog a paper ad mark the startig ad edig poits. Coect the poits with a lie, ad measure the lie. To fid the cetre of a circle, studets ca put the vertex of a right agle at a poit alog the circumferece, ad mark where the arms of the right agle cross the circumferece. Joiig these two marks creates a diameter of the circle. Repositioig the right agle approximately a quarter way aroud the circle ad repeatig the process will create aother diameter. The poit where the two diameters cross is the cetre of the circle. Kowig the cetre of the circle allows studets to measure the radius ad the diameter. Note: Rollig out the lie of the circumferece is a useful strategy, as studets ca the measure the diameter ad physically place the diameter over the lie ad see how may diameters log the lie is. This requires o calculatio. 3. Distribute copies of BLM 7.SS.1.5: A Table to Compare Measures of Circles, or have studets create their ow tables. 4. Ask studets to fid circular objects of various sizes i the classroom or school, accurately measure the radius, diameter, ad circumferece of the objects, ad record the required data. Have them iclude large circles (e.g., the cetre circle o the basketball court) available i the school. Have studets record ad calculate the ratios ad look for ay cosistet relatioships. 18 Grade 7 Mathematics: Support Documet for Teachers

271 5. Reassemble as a class ad discuss studets fidigs. Studets measuremets will ot be completely accurate, so the calculatios to decimal places will ot be cosistet. Nevertheless, studets should have foud that, regardless of the size of the circle, the circumferece is always a little loger tha three diameters of that circle (C = 3d ad a little more). If studets have measured very carefully, they should C have calculated values betwee 3.1 ad 3.2. Explai that this relatio is a very d importat mathematical costat. It is commoly approximated as 3.14, ad is termed pi (p). It is importat that studets recogize that pi is ot so much a special umber as it is a special relatioship, the relatioship of the circumferece of a circle divided by its diameter. (For more iformatio o pi, refer to the Backgroud Iformatio.) Note: Remember to celebrate Pi Day (March 14). Variatio: Use spreadsheets to eter required data ad calculate the value of the relatios. Sample Website: A spreadsheet to calculate the ratio of circumferece ad diameter ad average the results to approximate pi is available at the followig website: North Cato City Schools. Explorig Pi. Excel Activities for the Classroom. < explorig_pi.html>. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Illustrate ad explai that the circumferece is approximately three times the diameter i a circle. r Explai that, for all circles, pi is the ratio of the circumferece to C the diameter, ad its value is approximately d Shape ad Space 19

272 Suggestios for Istructio Solve a cotextual problem ivolvig circles. Materials: coloured paper o which to prit problems scissors tape math jourals computers for publishig (optioal) Orgaizatio: Small groups, whole class Procedure: I this learig activity, studets work i groups to review what they have discovered about the relatioships i circles, what the relatioships mea, ad the differet otatios for recordig those relatioships. 1. Challege studets to record the relatioships i circles i as may equivalet ways as they ca. They ca iclude words, diagrams, ad mathematical symbols. Cosiderig equivalet values usig opposite operatios will be helpful. 2. Have groups create small posters with ideas such as the followig: The sum of the cetral agles always equals 360. If I kow the radius, I ca kow the diameter too, because d = 2r. 1 If you tell me the diameter, I ca tell you the radius r= d. C 2 = a costat pi (p), ad pi has a approximate value of d C d» 3.14, so 3.14 d» C ad C» dd.. If I kow the circumferece, I ca fid the 314. diameter, ad vice versa. 2r ca replace d i every relatio, so r» C or 6.28r» C. I ca figure out the circumferece of a circle if I kow the radius of the circle. 3. After givig groups sufficiet time to work o their posters, post their work ad have a Gallery Walk, givig studets a opportuity to compare the represetatios i the differet posters. Reassemble as a class ad share observatios. 20 Grade 7 Mathematics: Support Documet for Teachers

273 4. Supply studets with the followig sample problem, ad ask them to work i their groups to solve it: Your sister is preparig for her weddig. You re at the store with your mom, who is pickig up last-miute items that are eeded to fiish the decoratig. The list icludes a special lace border to go aroud the roud table o which the desserts will be displayed. The measuremet give for the table is 1 metre i diameter. Oh o! your mother sighs. I ve bee give the wrog iformatio. The border goes aroud the table, ot across it. How am I supposed to kow how much lace border to buy? Never fear, mother, you say. The math costat pi holds the aswer to your problem. Give me a miute, ad I will tell you the legth of the border you eed to buy to go aroud a table with a diameter of 1 metre. What is the aswer your mother eeds? 5. Have studets use the relatioships o their posters to help them prepare cotextual problems about fidig the various measuremets of circles whe oly oe other measuremet is give. Ask groups to be imagiative ad creative i the problems they write ad the ways i which they preset the problems. Have them supply the solutios to their problems, cocealed uder a flap or usig some other method. Groups may wish to create problems cetred o a particular theme or evet. Remid studets that circles ca be uravelled. They ca roll a circle to measure its circumferece. The circumferece of the circle equals the distace the outside of a circle ca travel i oe revolutio. So, studets ca compare the distace travelled i oe revolutio for circles of differet sizes, or calculate the revolutios required to travel a certai distace usig circles of differet sizes. If a circle is peeled i layers, each successive strip will be a little smaller. If the thickess of the strips remais costat, studets could solve problems such as figurig out the diameter of a rolled hose compared to its legth. 6. Have studets solve the problems created by their classmates, ad verify their solutios. Have studets write a math joural etry commetig o their success i solvig the problems. Also have them commet o which types of problems they preferred, ad which problems they foud more difficult. Variatios: Provide scaffoldig i the form of templates for studets who may have trouble composig appropriate problems. Have studets publish their problems usig computer software. Serve a variety of pie at the pi celebratio (e.g., pizza pie, spiach pie). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a cotextual problem ivolvig circles. Shape ad Space 21

274 N o t e s 22 Grade 7 Mathematics: Support Documet for Teachers

275 Shape ad Space (Measuremet) (7.SS.2) Edurig Uderstadig(s): May geometric properties ad attributes of shapes are related to measuremet. The area of a rectagle ca be used to develop the formula for the area of other shapes. Geeral Learig Outcome(s): Use direct or idirect measuremet to solve problems. Specific Learig Outcome(s): 7.SS.2 Develop ad apply a formula for determiig the area of triagles parallelograms circles. [CN, PS, R, V] Achievemet Idicators: Illustrate ad explai how the area of a rectagle ca be used to determie the area of a triagle. Geeralize a rule to create a formula for determiig the area of triagles. Illustrate ad explai how the area of a rectagle ca be used to determie the area of a parallelogram. Geeralize a rule to create a formula for determiig the area of parallelograms. Illustrate ad explai how to estimate the area of a circle without the use of a formula. Apply a formula for determiig the area of a circle. Solve a problem ivolvig the area of triagles, parallelograms, or circles. Prior Kowledge Studets should be able to do the followig: Q Q (4.SS.3) Demostrate a uderstadig of the area of regular ad irregular 2-D shapes by recogizig that area is measured i square uits selectig ad justifyig referets for the uits cm 2 or m 2 estimatig area by usig referets for cm 2 or m 2 determiig ad recordig area (cm 2 or m 2 ) Shape ad Space 23

276 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q costructig differet rectagles for a give area (cm 2 or m 2 ) i order to demostrate that may differet rectagles may have the same area (5.PR.2) Solve problems ivolvig sigle-variable (expressed as symbols or letters), oe-step equatios with whole-umber coefficiets, ad whole-umber solutios. (5.SS.1) Desig ad costruct differet rectagles give either perimeter or area, or both (whole umbers), ad draw coclusios. (5.SS.2) Demostrate a uderstadig of measurig legth (mm) by selectig ad justifyig referets for the uit mm modellig ad describig the relatioship betwee mm ad cm uits, ad betwee mm ad m uits (5.SS.5) Describe ad provide examples of edges ad faces of 3-D objects, ad sides of 2-D shapes, that are parallel itersectig perpedicular vertical horizotal (5.SS.6) Idetify ad sort quadrilaterals, icludig rectagles squares trapezoids parallelograms rhombuses accordig to their attributes. (6.PR.1) Demostrate a uderstadig of the relatioships withi tables of values to solve problems. (6.PR.2) Represet ad describe patters ad relatioships usig graphs ad tables. (6.PR.3) Represet geeralizatios arisig from umber relatioships usig equatios with letter variables. (6.SS.1) Demostrate a uderstadig of agles by idetifyig examples of agles i the eviromet classifyig agles accordig to their measure estimatig the measure of agles usig 45, 90, ad 180 as referece agles determiig agle measures i degrees drawig ad labellig agles whe the measure is specified 24 Grade 7 Mathematics: Support Documet for Teachers

277 Q Q Q Q Q Q (6.SS.3) Develop ad apply a formula for determiig the perimeter of polygos area of rectagles volume of right rectagular prisms (6.SS.4) Costruct ad compare triagles, icludig scalee isosceles equilateral right obtuse acute i differet orietatios. (6.SS.5) Describe ad compare the sides ad agles of regular ad irregular polygos. Related Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q Q Q Q Q (7.PR.1) Demostrate a uderstadig of oral ad writte patters ad their correspodig relatios. (7.PR.2) Costruct a table of values from a relatio, graph the table of values, ad aalyze the graph to draw coclusios ad solve problems. (7.PR.3) Demostrate a uderstadig of preservatio of equality by modellig preservatio of equality, cocretely, pictorially, ad symbolically applyig preservatio of equality to solve equatios (7.PR.4) Explai the differece betwee a expressio ad a equatio. (7.PR.5) Evaluate a expressio give the value of the variable(s). (7.PR.6) Model ad solve problems that ca be represeted by oe-step liear equatios of the form x + a = b, cocretely, pictorially, ad symbolically, where a ad b are itegers. (7.PR.7) Model ad solve problems that ca be represeted by liear equatios of the form ax + b = c ax = b x b, a 0 a cocretely, pictorially, ad symbolically, where a, b, ad c, are whole umbers. Shape ad Space 25

278 Q Q Q Q (7.SS.1) Demostrate a uderstadig of circles by describig the relatioships amog radius, diameter, ad circumferece of circles relatig circumferece to pi determiig the sum of the cetral agles costructig circles with a give radius or diameter solvig problems ivolvig the radii, diameters, ad circumfereces of circles (7.SS.3) Perform geometric costructios, icludig perpedicular lie segmets parallel lie segmets perpedicular bisectors agle bisectors Backgroud Iformatio Calculatig Area Before eterig Grade 7, studets costructed rectagles of a give area, ad geeralized a rule for determiig the area of a rectagle. I Grade 7, studets build o their kowledge of rectagles to geeralize rules for determiig the area of a triagle ad of a parallelogram. They also use their familiarity with rectagles to fid the area of a circle. Determiig area is a useful skill for idetifyig ad comparig the sizes of objects of differet shapes, whether they are circular, rectagular, or triagular. There are may practical applicatios for determiig area (e.g., idetifyig which size or shape of pizza is a better buy, determiig the cost of floorig). Determiig area is also a prerequisite skill for determiig the volume of prisms ad cyliders i later grades. To make geeralizatios regardig area, studets must have a good coceptual uderstadig of what area is, ad of how to fid the area of a rectagle. The formula l w is a way of coutig the squares cotaied i the area of a rectagle. It represets area as a array of squares, ad provides a way of coutig them. That is, the formula l w represets the umber of squares i each row multiplied by the umber of rows. If ecessary, rebuild these uderstadigs before havig studets develop rules regardig the area of triagles ad parallelograms. To begi, review types of quadrilaterals ad classificatios of triagles, to esure studets have vocabulary to commuicate clearly about their learig. The ability to idetify the base ad height of shapes is also importat for clear commuicatio about shapes. Idetifyig base ad height is also required to develop ad apply formulas. Ay flat side of a object ca serve as its base. The base i ay situatio depeds o the orietatio of the object. Height is measured i relatio to the base; therefore, a object may have differet heights, depedig o its orietatio. Height is the distace betwee the highest poit of the object ad its base. It is measured alog a lie that is perpedicular to the base. 26 Grade 7 Mathematics: Support Documet for Teachers

279 Provide studets with opportuities to explore methods of fidig the areas of figures ad to discover the relatioships betwee the figures ad rectagles. As studets gai a uderstadig of the cocepts ad see the relatioships, the geeralizatio of b w will become apparet as the geeralized formula for fidig the area of a parallelogram. As they observe that two idetical triagles form a parallelogram, they will uderstad that 1 2 b h idetifies the area of a triagle. Whe a rectagle is halved alog a diagoal, all the resultig triagles are right triagles. It is clear that their area is oe-half of the area of the rectagle, or 1 2 l w. The legth ad width are obtaied by measurig the sides of the rectagle. For a right triagle, the area ca be calculated by measurig the sides of the triagle ext to the right agle. Whe a parallelogram is halved alog a diagoal, two triagles of differet sizes ad shapes may be created, depedig o the diagoal used to halve the parallelogram. It is still evidet that the area of either triagle is oe-half the area of the parallelogram, but either the area of the parallelogram or the triagle ca be calculated by measurig the sides. The base ad the height of the figures must be determied i order to calculate the area. Shape ad Space 27

280 Relatig the properties of parallelograms to those of rectagles is a way of establishig the cocept of base height as the way to determie the area of a parallelogram. Ay two idetical triagles ca be arraged to form a parallelogram. The triagle is related to the parallelogram, ad the parallelogram is related to the rectagle. There is a advatage to studyig the area of parallelograms before studyig the area of triagles, because the use of base ad height i relatio to area has already bee established ad there is o eed to separate right triagles from other triagles. Whe a circle is sectioed through the cetre, ad the pieces are rearraged so that the orietatio of the cetre alterates betwee poitig up ad poitig dow, the pieces will form a approximate rectagle. The approximate rectagle ca be used to estimate the area of a circle, ad to explai the formula for fidig the area of a circle. Whe studets build their ow geeralizatios from their ow experieces, they will uderstad the coectios i formulas. Buildig these coectios provides the best opportuities for studets to remember formulas ad to apply them correctly. If studets forget a formula, they are i a positio to rebuild it, test it, ad carry o usig it. The suggested learig experieces that follow are closely related. Studets are set o three missios to discover methods of calculatig areas of parallelograms, triagles, ad circles. Each edeavour is based o kowledge of the rectagle. The fial learig activity egages studets i a problem-solvig party. Mathematical Laguage area rectagle base square uits formula triagle (obtuse, right, acute, scalee, equilateral, isosceles) height vertex horizotal vertical itersectig width legth parallel parallelogram perpedicular polygo quadrilateral (square, rectagle, rhombus, trapezoid, parallelogram) 28 Grade 7 Mathematics: Support Documet for Teachers

281 Learig Experieces Assessig Prior Kowledge Materials: BLM 7.SS.2.1: The Area of Rectagles (Assessig Prior Kowledge) demostratio board paper for writig clues (oe-quarter sheets) two log pecils, pes, or straws (for each studet) BLM : Isometric Dot Paper (optioal) BLM : Dot Paper (optioal) geoboards (optioal) Orgaizatio: Groups of varyig sizes, whole class, idividual Procedure: 1. Review the vocabulary terms perpedicular, parallel, itersectig, vertical, ad horizotal by playig a Simo Says type of game. The game ca be played with a group of ay size. Choose a group size that is best for your class situatio. As Simo gives directios, studets use their arms or a pair of pecils, pes, or straws to demostrate the formatio. The phrase Simo says must preface the directio. For example, Simo says, show parallel lies. Ayoe ot showig the lies correctly is out for the roud. If the phrase Simo says does ot preface the directio, ayoe performig the demostratio is out. The last perso remaiig i the game wis, or becomes the ew Simo. 2. Review types of quadrilaterals ad classificatios of triagles by playig with riddles. The game ca be played with a group of ay size. Choose a group size that is best for your class situatio. Iclude the followig quadrilaterals ad triagles: quadrilaterals: rectagles, squares, trapezoids, parallelograms, ad rhombuses triagles: scalee, isosceles, equilateral, right, obtuse, ad acute Shape ad Space 29

282 Assig each studet oe or more quadrilaterals ad/or triagles (differet oes for each studet). Ask studets to cosider the attributes of a give shape, ad the write four clues from which their classmates ca idetify the shape. Studets fold a sheet of paper i half. They write the clues outside the fold, ad the ame of the shape, accompaied by a accurate diagram, iside the fold. Have studets take turs offerig their clues to the group. Whe idividuals idetify a shape correctly, they get possessio of the card, or ear x umber of poits, ad so o. 3. Through a whole-class discussio focused o quadrilaterals ad triagles, review the cocepts of base ad height usig objects ad diagrams. 4. To review what studets kow about calculatig the area of rectagles, distribute copies of BLM 7.SS.2.1: The Area of Rectagles (Assessig Prior Kowledge), ad have studets complete the tasks idividually. Variatios: Have studets review shapes with a Pictioary type of game. The desiger draws the desigated quadrilateral or triagle, ad group members guess its ame. The studet who guesses correctly becomes the ext desiger. Add competitio to the game by supplyig a set of cards with the shape ames, ad havig groups compete to be the first to complete desigs of the set of shapes. Use geoboards, geoboard templates, or dot paper to demostrate the differet shapes ad to explore various sizes of rectagles with the same area. (See BLM : Isometric Dot Paper ad BLM : Dot Paper.) 30 Grade 7 Mathematics: Support Documet for Teachers

283 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify differet classificatios of triagles ad quadrilaterals accordig to their attributes. r Idetify parallel, perpedicular, itersectig, horizotal, ad vertical lies. r Idetify the base ad height of diagrams. r Demostrate uderstadig that area is measured i square uits. r Calculate the area of rectagles. Suggestios for Istructio Illustrate ad explai how the area of a rectagle ca be used to determie the area of a parallelogram. Geeralize a rule to create a formula for determiig the area of parallelograms. Solve a problem ivolvig the area of triagles, parallelograms, or circles. Materials: grid paper rulers scissors tape calculators math jourals geoboards ad elastics (optioal) BLM : Dot Paper (optioal) BLM 5 8.9: Cetimetre Grid Paper (optioal) Orgaizatio: Ivestigative teams, whole class, idividual or pairs Shape ad Space 31

284 Procedure: 1. Assig studets the missio of usig their ivestigative ad mathematical reasoig skills to discover ad provide the world with a method that will quickly determie the area of ay parallelogram, of ay dimesio, i ay situatio. Allow studets to coduct their ow ivestigatio, but provide whatever hits are required to keep them o track. 2. The first step is to idetify a strategic pla for the missio. Prelimiary stages could iclude the followig: a) Cosider how to idetify a parallelogram, varieties of parallelograms, ad similarities ad differeces betwee parallelograms. Idetify the attributes of the parallelograms for which to collect data. b) Examie the iitial fidigs of areas for differet parallelograms, icludig parallelograms with the same area but differet shapes. Idetify the attributes of the parallelograms to measure ad ivestigate. c) Look for coectios betwee the kow ad the ukow. How to determie the area of a rectagle is a kow. Is there a similarity betwee rectagles ad parallelograms that may help guide the ivestigatio? (A parallelogram ca be divided ad reassembled as a rectagle. The reassembly does ot chage the legth or height of the figure. The base is removed from oe ed of the figure ad reattached at the opposite ed. The height of the figure does ot chage, just the iterior agles chage.) d) Idetify relatioships betwee a parallelogram s attributes ad its area, ad search for a patter. Look for a coectio betwee the legth of the base ad the height that equals the cout of the squares. 3. After idetifyig a method to determie the area of ay parallelogram, of ay dimesio, i ay situatio, studets test it for a variety of parallelograms. Ecourage studets to base their formulas o legths of the base ad the height. If their formula works cosistetly, the missio is accomplished. 4. Hold a debriefig sessio with the class, askig studets to share their strategies ad fidigs. a) I the discussio, iclude situatios i which the area of a parallelogram is kow ad the legth of the base or height of the parallelogram eeds to be determied. 32 Grade 7 Mathematics: Support Documet for Teachers

285 b) Discuss the value of kowig the legth of the side, ad whether the measuremet is useful i fidig the area. c) Ask whether a parallelogram ca be accurately reproduced give the area ad the legth of the base or height. 5. Have studets work idividually, or i pairs, to create oe or two parallelogram problems for their parters to solve. 6. Give studets a set of problems, with solutios cocealed. Workig out a solutio verifies that the creator of the problem uderstads what he or she is doig, ad that the problem works. Attemptig to work out a solutio may idicate the problem requires revisio, ad provides feedback for the problem solver. Examples of Problems: Variatios: Draw two differet parallelograms that have a area of x square uits, or a base of x uits ad a area of x uits. Provide drawigs of parallelograms from which to calculate area. Provide dimesios for two parallelograms ad ask which parallelogram has the larger area. If studets become stuck i their ivestigatio, provide them with a diagram of a parallelogram o cetimetre grid paper ad provide eough prompts for the discovery to be made. (See BLM 5 8.9: Cetimetre Grid Paper.) Geoboards provide a easy way to ivestigate parallelograms of the same base ad height, but with differet iterior agles. Ivestigate the chage from a rectagle to may differet parallelograms with the same area by systematically shiftig the base oe uit to the right of the top. This looks iterestig o paper as well, ad it is self-evidet that the dimesios for the base ad height have ot chaged. It also provides a illustratio of circumstaces i which the height of the parallelogram is ot obvious, ad must be measured outside the parallelogram. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Illustrate ad explai how the area of a rectagle ca be used to determie the area of a parallelogram. r Geeralize a rule to create a formula for determiig the area of parallelograms. r Solve a problem ivolvig the area of triagles, parallelograms, or circles. Shape ad Space 33

286 Suggestios for Istructio Illustrate ad explai how the area of a rectagle ca be used to determie the area of a triagle. Geeralize a rule to create a formula for determiig the area of triagles. Solve a problem ivolvig the area of triagles, parallelograms, or circles. Materials: file from the ivestigatio of parallelograms grid paper rulers scissors tape calculators math jourals geoboards ad elastics (optioal) BLM : Dot Paper (optioal) Orgaizatio: Ivestigative teams (from previous learig activity), whole class, idividual or pairs Procedure: This learig experiece o determiig the area of a triagle is desiged to follow the previous learig activity o determiig the area of parallelograms. 1. Iform studets they have bee assiged a ew missio that requires ivestigators with mathematical reasoig skills. This time, the world is i eed of a method or formula that will quickly determie the area of ay triagle of ay classificatio, of ay dimesio, i ay situatio. Their missio is to discover the formula. Oce agai, allow studets to coduct their ow ivestigatio, but provide whatever hits are required to keep them o track. A geeral suggested protocol is outlied below. For more detail, refer to the procedure for fidig the area of parallelograms (i the previous learig experiece). 2. The first step is to develop a strategic pla. This may iclude the followig: a) Idetify classificatios of triagles, their similarities ad differeces, ad the attributes for which to collect data. b) Collect some iitial data about area for differet triagles, icludig triagles with the same area, but differet shapes. Idetify how to measure the height of a triagle. Choose which attributes to measure for the ivestigatio. 34 Grade 7 Mathematics: Support Documet for Teachers

287 c) Look for coectios betwee the kow ad the ukow. Look for similarities betwee triagles ad parallelograms ad rectagles. Similarities provide clues to help guide the ivestigatio. Example: Ay two cogruet triagles ca be arraged to form a parallelogram. Therefore, their area must be 1 2 the area of the parallelogram. Triagles also have bases ad heights. A triagle ca be eclosed withi a rectagle of the same height ad sharig the same base. A perpedicular lie ca be draw from the highest vertex of the triagle to its base. The lie divides both the triagle ad the rectagle i two. The sides of the triagles become the diagoals of the two rectagles. Studets ca explore fidig the area of differet triagles, usig computer applets. Sample Website: Computer applets are available o the followig website: Cut the Kot. Area of Triagle. Geometry Articles, Theorems, Problems < Choose the colour optio to make the relatio more obvious. The area of each triagle is 1 the area of each rectagle; therefore, the area 2 of the origial triagle is 1 the area of the origial rectagle. The squares i 2 the area of the triagle equal 1 the umber of squares i a rectagle with the 2 same base ad height. Usig a array model to cout the squares leads to the formula 1 2 b h. Shape ad Space 35

288 3. Look for a relatioship betwee the base ad height of triagles, ad their areas. Look for a mathematical relatioship that will equal the area for a give triagle. 4. Idetify a formula for determiig the area of a triagle, ad test it o a variety of triagles. If the formula works cosistetly, aother useful formula has bee discovered, ad is ready for use. 5. Debrief as a class to discuss fidigs ad strategies. For example, dividig a parallelogram through the diagoal creates two cogruet triagles, ad dividig through the opposite diagoal creates two differet cogruet triagles. Studets may wish to discuss why that is so. What value is there i measurig the legth of the sides of the triagle? Ca a triagle be reproduced by kowig oly the base ad the height? Ca a triagle be reproduced by kowig oly the area, or the area ad the base? 6. Have studets work idividually or i pairs to create problems ad solutios related to determiig the area of a triagle, ad the exchage problems ad fid the solutios. These problems ad solutios may be used as Exit Slips. Variatios: Explore the effect of varyig the height ad base measuremets of a triagle or the area of the triagles. Also explore creatig differet triagles with the same area. Geoboards, paper templates of geoboards, or dot paper may be used. (See BLM : Dot Paper.) Explore the relatioships betwee the height, base, ad area of triagles usig computer applets. Sample Website: Computer applets are available o the followig website: Math Ope Referece. Area of a Triagle. Triagles < This applet allows studets to chage the legth of the base, the height of the triagle, or the measures of the agles, while the applet costatly measures the matchig area. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Illustrate ad explai how the area of a rectagle ca be used to determie the area of a triagle. r Geeralize a rule to create a formula for determiig the area of triagles. r Solve a problem ivolvig the area of triagles, parallelograms, or circles. 36 Grade 7 Mathematics: Support Documet for Teachers

289 Suggestios for Istructio Illustrate ad explai how to estimate the area of a circle without the use of a formula. Apply a formula for determiig the area of a circle. Solve a problem ivolvig the area of triagles, parallelograms, or circles. Materials: BLM 7.SS.2.2: Circles for Estimatig Area math jourals grid paper rulers scissors glue or tape markers of differet colours calculators Orgaizatio: Idividual or pairs, whole class, idividual Procedure: I this learig experiece, studets ivestigate the area of a circle. Be prepared to use more tha oe class for the ivestigatio ad applicatio, depedig o the amout of time the studets use for explorig ad the depth of their exploratio. 1. As a itroductory problem, or as a closig exercise, preset roud pizzas (cut ito eve wedges) ad rectagular pizzas (close i area, but ot too close) or cut-outs of these, ad ask studets to compare the two pizzas ad fid out which shape of pizza is larger. Have studets record evidece for their decisio i their math jourals. 2. Iform studets their curret missio is to fid effective methods to approximate the area of circles. 3. Preset studets with oe or more circles ad ask them to approximate the area. 4. Whe studets have a good estimate, ask them to use their math jourals to ote the procedure they followed. The ask them to attempt to fid more methods that work for them. Ecourage studets to steer their ow ivestigatio, coectig the kow with the ukow. 5. As studets work o their ivestigatio, circulate amog the class ad observe. If studets experiece difficulty i their work, ask guidig questios or suggest a actio. Their ivestigative methods may iclude the followig: a) Trace the circle oto grid paper ad cout the umber of squares. Shape ad Space 37

290 b) Draw the smallest square the circle could fit ito, calculate the area of the square, ad determie some amout to deduct to compesate for the area of the square that does ot iclude part of the circle. c) Draw the smallest square to cotai the circle, ad the largest square iside the circle (use the diagoals of the large square to idicate the positio of the corers). The thik, the circle is larger tha the small square ad smaller tha the large square, so the area of the circle must cover a area betwee the areas of both squares. The umber midway betwee is a good approximatio. d) Cosider whether the circle could be trasformed ito a rectagle. Sectio a circle ito four pieces ad try arragig the pieces to approximate a rectagle, as illustrated below. (Colourig the circumferece of the circle before begiig helps idetify the orietatio of the pieces.) Alteratig the cetre poits betwee poitig up ad poitig dow elogates the circle. Halve each of the four pieces, ad arrage the pieces to approximate a rectagle agai. It becomes evidet that the more times the sectios are halved, the more the elogatio resembles a parallelogram. Measurig the base ad the height of this parallelogram will approximate the area of the circle. 6. After studets have had sufficiet time to develop their ivestigative methods ad have made some discoveries, reassemble as a class for a debriefig sessio. Discuss studets fidigs regardig the areas of the circles, ad the methods they foud most useful, or most accurate. 38 Grade 7 Mathematics: Support Documet for Teachers

291 Optioal: 7. Ask the class whether ayoe has come up with a formula for determiig the area of a circle,* as they did for the parallelograms ad triagles. If studets have explored sectioig the circle, use their ivestigatio as the basis for guidig them to the formula pi r r. If they have ot tried arragig the sectios, recommed the strategy to them, ad let them explore for a while. The hads-o applicatio of this idea provides a powerful cocrete model of the formula. 8. Use the cut-up circle arraged like a parallelogram as a model to guide studets through a explaatio of the formula. * Note: The procedure described for poits 1 to 6 idicates how studets ca use their method for determiig the area of a circle to uderstad how the formula ca be developed. The achievemet idicators for learig outcome 7.SS.2 do ot suggest that studets develop the formula for the area of a circle. Cosider the procedure for poits 7 to 10 as optioal, but it may ehace studets uderstadig of the formula ad its applicatio. a) Ask what measure of the circle forms the height of the parallelogram. If studets have difficulty with this cocept, ask them to colour the parts of the circle before begiig. This helps to idetify the orietatio of the pieces ad to idetify what part of the circle is represeted where i the parallelogram. Colour the circumferece oe colour, several radii a differet colour, ad a diameter aother colour. The height is the radius. b) Determie which measure of the circle forms the base of the parallelogram. The base is half the circumferece, ad the top is the other half of the circumferece. The circumferece ca be expressed i a umber of ways. Begi with the approximate measures. C» 3d, i terms of the radius C» 6r 6r was the estimatio used for circumferece i the first learig experiece suggested for learig outcome 7.SS.1. Sice half of the circumferece forms the base of the parallelogram, half of 6r is 3r, so the base is approximately 3r. c) To fid the approximate area of the parallelogram that represets the circle, we could apply the formula b h, or (3 r) (r). Because 3 was a approximatio for pi, we ca replace 3 with 3.1, which is a closer approximatio. Or, to be more precise, we could replace 3 with the symbol pi, ad express the formula as (p r) r or pr 2. Everyoe will eed a chace to play with this formula. 9. Have studets apply the formula to the circles they worked with earlier, ad compare the ew results with their approximatios. Iquire about how close their approximatios were. Idetify some of the sources of error. 10. Preset a problem that ca be solved usig the formula for fidig the area of a circle. Compare the amout of pizza i a small 10² pizza, a medium 12² pizza, ad a large 15² pizza. Discuss studets aswers. Shape ad Space 39

292 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Illustrate ad explai how to estimate the area of a circle without the use of a formula. r Apply a formula for determiig the area of a circle. r Solve a problem ivolvig the area of triagles, parallelograms, or circles. Suggestios for Istructio Solve a problem ivolvig the area of triagles, parallelograms, or circles. Materials: rulers grid paper protractors paper for publishig problems calculators math jourals or otebooks, or logs of completed problems art or craft supplies (optioal) sacks for the party (optioal) computers for publishig problems (optioal) Orgaizatio: Idividual, pairs, or small groups Procedure: 1. Plaig Have studets thik of cotexts where parallelograms, triagles, ad circles may appear i everyday situatios (decide whether or ot to exclude rectagles). Examples for parallelograms may iclude logo desigs, persoal flags, geometric art desigs, optical illusios, tagrams, oe half of gable roofs, ad architectural desigs. 40 Grade 7 Mathematics: Support Documet for Teachers

293 Example: The followig website has a photograph of a buildig i the shape of a parallelogram alog the Elbe River i Hamburg, Germay. Stich, Mike. Parallelogram. 22 May flikr. < 2. Preparig the Problems a) Give studets a opportuity to create diagrams, desigs of structures, logos, artwork, puzzles, ad so o, that ivolve oly parallelograms, triagles, or circles, or combiatios of these shapes. b) Whe the desig projects are complete, have studets use their desigs as the subject to write problems whose solutios require calculatig area ad to provide a solutio for each problem created. Examples: Determie the square metres of glass required for widows i a buildig. Determie the square metres of material required for the differet shapes i a flag, or i a piece of artwork. Create three of these thigs belog questios, where three parallelograms or triagles with differet base ad height combiatios, or differet iterior agles, have the same area, ad oe has a differet area. Create problems for combiatios of all three shapes. The challege is to fid the shape with the differet area. Iclude problems that require calculatig area, ad problems that supply area ad require calculatig bases, heights, radii, or circumferece. 3. Solvig the Problems a) Whe the projects are complete, studets ca share them with oe aother at a party. The focus of the party becomes solvig various problems. b) Perhaps completed problems could be exchaged for driks or sacks at the party (sig off each problem that has bee exchaged). c) Assig greater value to problems that are more challegig. d) Agree o a defiite umber of problems to complete, or solve problems with a aim to accumulate a target umber of square uits, or have a cotest to accumulate the greatest umber of square uits i problem aswers. e) Studets will be busy calculatig may areas. Have fu. Shape ad Space 41

294 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a problem ivolvig the area of triagles, parallelograms, or circles. 42 Grade 7 Mathematics: Support Documet for Teachers

295 Shape ad Space (3-D Objects ad 2-D Shapes) (7.SS.3) Edurig Uderstadig(s): May geometric properties ad attributes of shapes are related to measuremet. While geometric figures are costructed ad trasformed, their proportioal attributes are maitaied. Geeral Learig Outcome(s): Describe the characteristics of 3-D objects ad 2-D shapes, ad aalyze the relatioships amog them. Specific Learig Outcome(s): 7.SS.3 Perform geometric costructios, icludig perpedicular lie segmets parallel lie segmets perpedicular bisectors agle bisectors [CN, R, V] Achievemet Idicators: Describe examples of parallel lie segmets, perpedicular lie segmets, perpedicular bisectors, ad agle bisectors i the eviromet. Idetify lie segmets o a diagram that are parallel or perpedicular. Draw a lie segmet perpedicular to aother lie segmet, ad explai why they are perpedicular. Draw a lie segmet parallel to aother lie segmet, ad explai why they are parallel. Draw the bisector of a agle usig more tha oe method, ad verify that the resultig agles are equal. Draw the perpedicular bisector of a lie segmet usig more tha oe method, ad verify the costructio. Prior Kowledge Studets should be able to do the followig: Q Q (4.SS.5) Describe ad costruct rectagular ad triagular prisms. Q Q (5.SS.2) Demostrate a uderstadig of measurig legth (mm) by selectig ad justifyig referets for the uit mm Shape ad Space 43

296 Q Q Q Q Q Q Q Q Q Q Q Q modellig ad describig the relatioship betwee mm ad cm uits, ad betwee mm ad m uits (5.SS.5) Describe ad provide examples of edges ad faces of 3-D objects, ad sides of 2-D shapes, that are parallel itersectig perpedicular vertical horizotal (5.SS.6) Idetify ad sort quadrilaterals, icludig rectagles squares trapezoids parallelograms rhombuses accordig to their attributes. (6.SS.1) Demostrate a uderstadig of agles by idetifyig examples of agles i the eviromet classifyig agles accordig to their measure estimatig the measure of agles usig 45, 90, ad 180 as referece agles determiig agle measures i degrees drawig ad labellig agles whe the measure is specified (6.SS.2) Demostrate that the sum of iterior agles is 180 i a triagle 360 i a quadrilateral (6.SS.4) Costruct ad compare triagles, icludig scalee isosceles equilateral right obtuse acute i differet orietatios. (6.SS.5) Describe ad compare the sides ad agles of regular ad irregular polygos. 44 Grade 7 Mathematics: Support Documet for Teachers

297 Related Kowledge Studets should be able to do the followig: Q Q (7.SS.2) Develop ad apply a formula for determiig the area of triagles parallelograms circles Backgroud Iformatio The cocepts of perpedicular ad parallel surroud us i everyday life. I Grade 5, i their study of 2-D shapes ad 3-D objects, studets idetified examples of perpedicular ad parallel sides, edges, ad faces. They also idetified examples of perpedicular ad parallel lie segmets i the eviromet. Because Grade 5 studets lack experiece with measurig agles, the agle formed by perpedicular lies was idetified as havig square corers. I Grade 7, studets will create parallel ad perpedicular lie segmets ad bisectors, as well as agle bisectors, usig geometric costructios. Geometric Costructios Geometric costructios are coected to the Aciet Greeks ad Euclidea geometry. They are differet tha drawigs i that the oly tools used i creatig geometric costructios are a straightedge, a compass, ad a pecil. Iterestig coectios exist betwee geometric costructios, art, ad architecture i may differet cultures. Parallel ad perpedicular lies are also importat to surveyors, desigers, egieers, cotractors, ad people buildig just about aythig. Studets may ejoy recreatig geometric costructios. Kowig about various applicatios may provide studets with a icreased purpose ad motivatio for usig a straightedge ad a compass to create lies ad bisectors. Sample Website: For directios o recreatig geometric costructios, such as rose widows commoly see i cathedrals, refer to the followig website: Scheider, Michael S. Geometry of the North Rose Widow of Chartres Cathedral. Costructig the Uiverse. < html>. Shape ad Space 45

298 Lies, Rays, ad Lie Segmets Lies, rays, ad lie segmets are made up of sets of poits that are straight ad oedimesioal. Their oly dimesio is legth. A lie is a set of poits that exteds idefiitely i opposite directios. Example: The lie AB : A ray is a set of poits that exteds idefiitely i oe directio. Example: The ray CD : A lie segmet is a set of poits alog a lie with two fiite edpoits. Example: The lie segmet EF : Lies, rays, ad lie segmets ca be parallel or itersectig. Lies that itersect at right agles are perpedicular lies. Parallel lies ever meet; they are always the same distace apart. I diagrams, idicate parallel lies by markig a arrow o the lie. Example: The symbol idicates that the lies are parallel, as i AB CD. 46 Grade 7 Mathematics: Support Documet for Teachers

299 Perpedicular lies itersect at 90 agles. I diagrams, idicate perpedicular lies by drawig a small square where the lies joi. Example: The symbol ^ idicates that the lies are perpedicular, as i AB ^ CD. Lies ad agles ca be bisected. I the word bisectors, bi meas two ad sect meas to cut. Whe a lie or a agle is bisected, it is cut ito two pieces of equal size. We could say it is divided i half or divided dow the middle. Studets will perform geometric costructios, icludig costructios of perpedicular bisectors ad agle bisectors: A perpedicular bisector is a lie, ray, or lie segmet that divides a lie segmet ito two equal segmets ad is perpedicular to the origial lie, ray, or segmet. A agle bisector is a lie or ray that divides a agle ito two agles of equal size. Methods for creatig each of these costructios are outlied i the learig experieces suggested for learig outcome 7.SS.3. Mathematical Laguage agle agle bisector arc bisect bisector itersectig lies lie lie segmet perpedicular perpedicular bisector perpedicular lies parallel parallel lies ray Shape ad Space 47

300 Learig Experieces Assessig Prior Kowledge Materials: BLM 7.SS.3.1: Parallel ad Perpedicular Lies (Assessig Prior Kowledge) skippig ropes or other types of ropes (two ropes per group of six studets) math jourals or otebooks Orgaizatio: Small groups (of six studets), idividual Procedure: 1. Form studets ito groups of six studets two to hold oe rope, two to hold the other rope, oe to verify that the lies are parallel or perpedicular, ad oe to record the group s actio. 2. Ask studets to use two or more skippig ropes to demostrate a variety of parallel lies ad provide evidece to verify that the lies are parallel. Evidece may iclude slidig a object betwee the ropes to demostrate the ropes are the same distace apart at each poit alog the ropes, or measurig the distace betwee the ropes at several itervals. 3. Oce studets have perfected makig parallel lies i several positios, repeat the process. This time, have them demostrate a variety of perpedicular lies ad provide evidece that the lies are perpedicular. As evidece, they may place a right-agle object, such as a book, at the itersectio of the lies, or they may form the lies agaist a object with right agles, such as the side ad frot edges of a desk or a table. 4. Challege studets to fid the middle of a lie, prove it is the middle, ad record their demostratio i their math jourals or otebooks. 5. Have studets demostrate a agle, usig their ropes. Ask them to add aother lie that would divide the lie i half ad create two equal agles. Challege them to verify that the agles are equal, ad have them record the process. 6. Whe the groups have completed the demostratios, provided evidece, ad recorded their work, distribute copies of BLM 7.SS.3.1: Parallel ad Perpedicular Lies (Assessig Prior Kowledge), ad have studets complete the tasks idividually. 48 Grade 7 Mathematics: Support Documet for Teachers

301 Variatios: Play a versio of Simo Says, as outlied i the Assessig Prior Kowledge learig experiece suggested for learig outcome 7.SS.2. Ask studets to create a simple drawig composed of a certai umber of parallel ad perpedicular lies ad write directios for each lie formed. Whe directios are complete, have studets joi a parter, ad either exchage directio sheets or give oral directios to make the compositio. Whe the drawigs are fiished, studets compare them to the origial drawigs ad aalyze ay discrepacies. Alteratively, give directios to the whole class, ad compare the products with a projectio of the origial drawig. Compare studets drawigs to the origial ad discuss ay differeces. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify parallel lies. r Idetify perpedicular lies by usig a square corer. r Draw ad ame agles, lies, rays, ad lie segmets. r Fid the middle of a lie. r Divide a agle i half. Suggestios for Istructio Describe examples of parallel lie segmets, perpedicular lie segmets, perpedicular bisectors, ad agle bisectors i the eviromet. Materials: magazies poster paper electroic or other display medium Iteret access (optioal) cameras (optioal) projector (optioal) Orgaizatio: Whole class, pairs or idividuals Shape ad Space 49

302 Procedure: 1. Itroduce the class to the term bisector. Verify studets uderstadig of the term by askig them to idetify or illustrate examples of perpedicular lies that bisect aother lie, or lies that bisect a agle. 2. Tell studets they will participate i a treasure hut to fid examples of parallel lie segmets, perpedicular lie segmets, perpedicular bisectors, ad agle bisectors. They ca work idividually or i pairs. 3. Together with studets, set criteria for the umber of each lie to fid. Places to search could iclude the classroom, lockers or desks, the hallways, the gymasium, the playgroud, ad other areas of the school. Alteratively, have studets search magazies or the Iteret for examples. (Examples may cosist of musical istrumets, such as keyboards ad strig istrumets, the fulcrum o a balace scale, widow frames, door frames, the separatio i a double door, hallways, sidewalk cracks, brick patters, goalposts, court markigs i the gymasium or outdoors, laes i a swimmig pool, streets or paths, the path of tires or skis, mitred corers o a box, logos ad emblems, the alphabet.) 4. After the search criteria are established, have studets udertake their treasure hut. They ca record their fidigs as sketches ad labels, or take photographs. 5. They may wish to cotiue their search for homework. 6. Whe studets have collected a sufficiet umber of treasures, have them orgaize their examples accordig to the type of lies represeted, ad choose a format for presetig their fidigs. They may choose a collage, a poster, a large classroom display, or a electroic display or slide show. 7. Arrage for studets to share their fidigs with each other. Variatios: Usig a projector, show examples of the differet lies i various cotexts. Ask studets to idetify ad describe the various lies they see i the examples. Play a versio of I Spy, askig studets to idetify parallel lie segmets, perpedicular lie segmets, perpedicular bisectors, ad agle bisectors i the classroom or i aother eviromet. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Describe examples of parallel lie segmets, perpedicular lie segmets, perpedicular bisectors, ad agle bisectors i the eviromet. 50 Grade 7 Mathematics: Support Documet for Teachers

303 Suggestios for Istructio Idetify lie segmets o a diagram that are parallel or perpedicular. Materials: schematic diagrams i books, blueprits, or drawigs rulers grid paper tracig paper (optioal) highlighters or pecil crayos of differet colours (optioal) computer drawig program (optioal) umber cubes (optioal) photographs with examples of parallel or perpedicular lies (optioal) Orgaizatio: Whole class, idividual Procedure: 1. Itroduce studets to schematic diagrams through books, blueprits, or drawigs. 2. Itroduce them to the symbols that idicate parallel ad perpedicular lies. 3. Have studets create a schematic diagram showig the iterior structure of a woodframed buildig. Ask them to iclude top plates, bottom plates, widow ad door frames, ad headers. Have them mark arrows ad square corers o their diagrams to idicate the parallel ad perpedicular lie segmets. 4. Have studets make some geeral statemets, at the bottom of their diagrams, regardig perpedicular ad parallel lie segmets that appear i their diagrams. Variatios: Istead of creatig a schematic diagram of a wood-framed buildig, studets could diagram the shell of a bus, plae, ship, sport court, sewig patter, road map, or airport ruway. Have studets create artwork composed of coloured lies ad agles (perhaps similar to the work of Piet Modria) ad labelled poits. Ask them to create a key to go with their artwork that lists the parallel ad perpedicular lies. The pictures ad keys ca be posted for display. Before idicatig the parallel ad perpedicular lie segmets i the diagrams described above, studets could exchage diagrams with their parters, who would mark idicator lies o their drawigs, ad the retur them to the creators to verify, discussig ay discrepacies. Have studets build a origami figure or shape, ufold it, trace the lies, ad idicate parallel ad perpedicular lies i the fold lies. Shape ad Space 51

304 Have studets use a computer drawig program to create diagrams of itercoected lies. They ca idetify the parallel ad perpedicular lies o their ow diagrams, ad exchage work with parters. Studets could use their diagrams (or a supplied diagram) to play a game with a parter. Studets each choose a differet colour of highlighter or light-coloured pecil crayo. They pick either odd or eve umbers o a umber cube to represet parallel or perpedicular lies. Studets take turs shakig the umber cube ad markig either a set of parallel lies or a set of perpedicular lies with their selected colour. If studets caot fid a set of lies, they forfeit their tur. The player with the greatest umber of sets of lies wis. Provide studets with photographs o which they ca mark the parallel or perpedicular lies represeted, or have them create a schematic diagram usig symbols to idicate the parallel ad perpedicular lie segmets. Have studets idetify parallel ad perpedicular lies i capital letters of the alphabet. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify parallel or perpedicular lie segmets o a diagram. Suggestios for Istructio Draw a lie segmet perpedicular to aother lie segmet, ad explai why they are perpedicular. Materials: records from Assessig Prior Kowledge learig activity (optioal) math jourals or otebooks geometry sets with straightedges or rulers, right triagles, protractors, ad compasses Miras tracig paper pes or markers of differet colours spiers or umber cubes (optioal) BLM 7.SS.3.2: Creatig Perpedicular Lies (optioal) Orgaizatio: Whole class, small groups, idividual 52 Grade 7 Mathematics: Support Documet for Teachers

305 Procedure: 1. Ask studets to share the methods they used i the Assessig Prior Kowledge learig activity to create perpedicular lies. Studets will likely have used square corers, some may have used protractors to measure 90 agles, ad some may have carefully folded paper ad used the folds as guidelies for perpedicular lies. 2. Challege studets, workig i groups, to thik of multiple methods that could be used to draw perpedicular lies. Ecourage studets to use the tools i their geometry kits, Miras, ad tracig paper. Have each studet use his or her math joural or otebook to record the differet methods their group thiks of. For each method, studets draw a example ad explai their thikig. Ecourage studets to make geometric drawigs carefully, labellig poits ad idicatig perpedicular lies with a square isert i the corer. Have studets iclude commets (e.g., I kow AB ^ CD because.... The reaso this method works is because.... Suggestios for avoidig errors whe usig this method are.... Situatios for which this method is recommeded iclude....). 3. Whe studets have had sufficiet time for their group work, reassemble as a class ad have studets share their ideas ad explaatios regardig the methods that could be used to draw perpedicular lies. Have studets add ay ew ideas to their math joural etries, usig aother colour to highlight ew learig. 4. Ecourage studets to thik critically about ad commet o the ideas that are shared i class. They may ackowledge ideas they agree with, express appreciatio for the way ideas are explaied, ask how or why questios, or offer further suggestios or support for a idea. If the class has ot addressed a specific method whe the sharig is fiished, provide some guidig questios or hits, ad sed studets back to work to develop aother idea. Methods of drawig perpedicular lies may iclude the followig: a) Use a square corer. Draw a lie segmet usig a straightedge. Place the rightagle triagle with oe side of the right-agle corer lied up with the lie segmet. Trace a vertical lie segmet alog the adjoiig side of the triagle. The lie segmet is perpedicular because both lie segmets itersect at 90º agles, or form a square corer. Followig the same priciple, artists ad desigers use T-squares to make perpedicular lie segmets. Carpeters use carpeter squares to do the same thig. b) Use a protractor. Draw a lie segmet usig a straightedge. Mark a poit o the lie segmet. Alig the poit ad the base lie segmet with the cross lies o the protractor. Mark the 90º measure. Use a straightedge to coect the origial poit ad the 90º mark. The resultig lie segmet is perpedicular because the agle betwee the two lie segmets is 90º. Shape ad Space 53

306 c) Use a straightedge ad a compass. Draw a lie segmet usig a straightedge. Mark ay two poits alog the lie segmet. Use the compass to draw a circle aroud oe of the poits that has a radius greater tha half the distace betwee the poits. That radius is ecessary for the circles to itersect. Maitai the same radius ad draw a circle aroud the secod poit. The circles will itersect at two poits. Use a straightedge to coect those two poits. The agle betwee the origial lie segmet ad the resultig lie segmet measures 90º; therefore, the lie segmets are perpedicular to each other. 54 Grade 7 Mathematics: Support Documet for Teachers

307 With this method, it is ot ecessary to draw the etire circle; oly a arc eeds to be draw. However, the circle emphasizes that the poits joied to form the perpedicular lie segmets are radii of cogruet circles, ad, therefore, are the same legth. Itroduce studets to the use of this method i art ad medieval architecture, ad how it ca be used to determie large-scale perpedicular lies outdoors or i the sky. d) Use a Mira. Use a straightedge to draw a lie segmet. Lay the Mira across the lie segmet ad adjust its positio util the reflectio of the lie segmet i the Mira lies up o top of the lie segmet itself. Trace a lie segmet alog the edge of the Mira. That lie segmet is perpedicular to the origial lie segmet because the agles at the itersectio of the lie segmets are right agles. e) Use tracig paper. Use a straightedge to draw a lie segmet. Carefully fold the paper across the lie segmet so that the portio of the lie o the top paper lies o top of the lie segmet uder the folded paper. Whe the lie segmets are aliged, crease the fold i the paper. Ope it up ad use a straightedge to trace the lie segmet alog the crease. The lie segmets are perpedicular because the agle formed at the itersectio measures 90º. 5. Have studets practise drawig lie segmets usig each of the methods. The ask them to make a math joural etry commetig o which method they prefer, which method they believe to be most accurate, ad which applicatios each method is best suited for. Variatios: Provide studets with directios to produce specific lie segmets usig specific methods. For example, draw lie segmet YZ 6 cm log. Use a straightedge ad a protractor to draw a perpedicular lie segmet that crosses YZ at poit X 4 cm away from Y. Create a spier idetifyig the five differet methods for drawig perpedicular lies, or assig the umbers o a umber cube to the five differet methods, with the sixth umber represetig a missed tur. Studets take turs spiig or rollig, ad the drawig perpedicular lie segmets usig the desigated method. The first perso to draw perpedicular lie segmets successfully usig all five methods wis. Studets ca use BLM 7.SS.3.2: Creatig Perpedicular Lies to record progress. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Draw a lie segmet perpedicular to aother lie segmet, ad explai why they are perpedicular. Shape ad Space 55

308 Suggestios for Istructio Draw the perpedicular bisector of a lie segmet usig more tha oe method, ad verify the costructio. Materials: math jourals or otebooks geometry sets with straightedges or rulers, right triagles, protractors, ad compasses tracig paper Miras Number cubes or spiers (optioal) BLM 7.SS.3.3: Creatig Perpedicular Bisectors (optioal) Orgaizatio: Small groups, whole class, idividual or pairs Procedure: Be prepared to be flexible regardig the time required for this learig experiece. Esure studets have ample opportuity to explore ad develop their uderstadig of differet methods of drawig perpedicular bisectors before begiig the presetatios. 1. Ask studets to defie perpedicular bisector ad to explai how a perpedicular bisector differs from a perpedicular lie segmet. 2. Discuss why ad whe people may use a perpedicular bisector. For example, a perpedicular bisector may be used to fid the best place to put a sigle support uder a beam, to fid the divisio i a drawig, a desig, or a buildig from which to build formal symmetry, to divide a piece of property ito equal portios, to fid a lie that is the same distace from two poits. The last applicatio may be importat i a variety of cotexts, ragig from plaig a city or a meetig spot to settig up a lemoade stad. 3. Remid studets that i the previous learig experiece, they worked with five differet methods to draw perpedicular lies. Ask studets to work i groups to review those methods ad to determie whether or ot each method could be used as is, or with some modificatio, to draw a perpedicular bisector. As i the previous learig experiece, have studets provide proof that the perpedicular bisectors are perpedicular ad bisect the lie segmet. They should also cosider hits to esure success i usig a give method, ideas o how chagig the techique affects the outcome, ad explaatios for why a method works. 4. Have each studet make a math joural etry discussig the methods for creatig perpedicular bisectors, icludig explaatios ad poiters regardig techiques ad applicatios. 56 Grade 7 Mathematics: Support Documet for Teachers

309 5. Whe studets have had sufficiet time for their group exploratio, reassemble as a class ad call upo differet groups to preset their modificatios for oe of the methods to draw a perpedicular bisector. Discuss their presetatios. 6. Ecourage studets to thik critically about ad commet o the ideas that are shared i class. They may ackowledge ideas they agree with, express appreciatio for ideas that are explaied, ask how or why questios, or offer further suggestios or support for a idea. If the class has ot addressed a specific method whe the sharig is fiished, provide some guidig questios or hits, ad sed them back to work to develop aother idea. The methods used for drawig perpedicular lies could also be used for drawig perpedicular bisectors, but with the followig modificatios: a) Use a square corer. Modificatio: Before begiig, fid the midpoit of the lie. Iclude ideas o how to fid the midpoit. Place the right agle at the midpoit. b) Use a protractor. Modificatio: Oce agai, fid the midpoit of the lie ad measure the 90º agle from the midpoit. c) Use a straightedge ad a compass. Modificatio: Whe drawig the circle, use the edpoits of the lie segmet as the cetres for the circles. Coect the itersectios of the circles to create the perpedicular bisector. d) Use a Mira. Modificatio: Whe adjustig the Mira across the lie segmet, adjust it so the reflectio of oe edpoit of the lie segmet i the Mira lies up o top of the other edpoit of the lie segmet. Iclude hits for usig the Mira successfully. e) Use tracig paper. Modificatio: Whe foldig the paper across the lie, fold it carefully to esure the half of the lie o the top paper lies o top of the lie uder it, ad that the edpoits of the lie segmet lie exactly o top of each other. Shape ad Space 57

310 f) Use a ruler to create a rhombus. Creatig a rhombus aroud the lie segmet will create the perpedicular bisector of the segmet because the diagoals of a rhombus are perpedicular bisectors of each other. Lay a straightedge at a agle across the lie segmet. Adjust the agle of the straightedge util each edpoit just touches the straightedge. The edpoits will be o opposite sides of the straightedge. Trace a lie alog both the top ad the bottom of the straightedge. Rotate the straightedge oe-quarter tur util the ed that was above the lie is ow below the lie, ad the ed of the straightedge that was below is ow above. Oce agai, adjust the straightedge so oe poit is above it ad oe poit is below it, ad trace both edges of the straightedge. Remove the straightedge. The itersectig lies create the vertices of the perpedicular bisector. 58 Grade 7 Mathematics: Support Documet for Teachers

311 7. Have studets practise usig each of the six methods to create perpedicular bisectors. Idividuals ca create their ow lie segmets or create lie segmets for their parters, or the teacher ca assig lie segmets. After studets have had sufficiet practice i usig the methods, ask them to make a math joural etry commetig o their preferred method, the method they thik is most useful or most accurate, applicatios for the differet methods, ad so o. Variatios: Play a game i which studets practise drawig perpedicular lie segmets. Studets roll two umber cubes to determie the legth of a lie segmet. They spi a spier, or roll a umber cube, to determie the method to use to draw the bisector. They draw the perpedicular bisector or the lie segmet, ad the verify that their drawig is correct by checkig that the agles created measure 90º ad that the lie segmets o either side of the bisector are equal i legth. Parters work idepedetly to be the first to create a bisector usig all six methods. They ca use BLM 7.SS.3.3: Creatig Perpedicular Bisectors to record their progress. Ivestigate drawig perpedicular bisectors for each side of a triagle. Draw a circle usig the poit of itersectio of the perpedicular bisectors as the cetre of the circle ad the radius as the distace from the cetre to oe of the vertices of the triagle. Cotiue this ivestigatio for a variety of triagles, parallelograms, or other polygos. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Draw the perpedicular bisector of a lie segmet usig more tha oe method, ad verify the costructio. Shape ad Space 59

312 Suggestios for Istructio Draw a lie segmet parallel to aother lie segmet, ad explai why they are parallel. Materials: math jourals or otebooks geometry sets with straightedges or rulers, right triagles, protractors, ad compasses tracig paper Miras straws, cardboard strips, stir sticks push-pis Orgaizatio: Whole class, small groups Procedure: 1. Ask studets to recall the Assessig Prior Kowledge learig activity i which they created parallel lies usig ropes. Review what parallel lies are ad how to test whether a set of lies are parallel. Effective testig methods iclude the followig: Reflect lies i a Mira. If the Mira is placed perpedicular to parallel lies, both the lies will reflect o themselves. Fold paper. If the paper is folded alog a lie perpedicular to the lie segmets, the folded lies will lie o top of each other. Idetify perpedicular lies betwee the two parallel lies, ad measure to verify that the lies are the same distace apart. 2. Ask groups of studets to idetify as may methods as they ca for creatig parallel lie segmets. Have them test each of their methods to verify that the lie segmets created are parallel. Have studets use their math jourals to record the methods that work. If studets eed a hit, ask them whether perpedicular lies are parallel to each other. 3. Reassemble as a class, ad have studets share the methods they foud. Discuss the advatages of each method ad uder what circumstaces each method may be best to use. Have studets add ay ew methods to their math jourals. Methods for creatig parallel lie segmets may iclude the followig: Trace both edges of a straightedge, such as a ruler. 60 Grade 7 Mathematics: Support Documet for Teachers

313 Diagoals of a rectagle are the same legth ad they bisect each other. If the diagoals are coected i the cetre to form a X, all four arms of the X are equal to each other. Use two straws (or cardboard strips, stir sticks, ad so o) to represet the diagoals, mark the midpoit of each, ad coect them with a push-pi. Lay the X o a paper ad coect two of the arms with a straightedge. Trace a lie segmet. Mark the edpoits of the two remaiig arms of the X ad coect the poits with a straightedge. The result is two parallel lie segmets. Stretch or collapse the X to adjust the distace betwee the parallel lie segmets. Use a right-agle triagle, or some other square corer, to draw a lie segmet. Place a straightedge alog the lie. Set the base of the right agle o the straightedge, ad trace the side. This creates a lie segmet perpedicular to the origial lie segmet. Slide the right agle alog the straightedge to ay desired positio, ad trace the side of the right agle. All the perpedicular lie segmets are parallel to oe aother. Coect poits that are 90º ad equidistat from the lie segmet. Use a right triagle or a protractor to draw two lies that are perpedicular to the origial lie segmet. Measure ad mark the same distace up each perpedicular lie. Coect the marks to create a parallel lie segmet. The perpedicular lies are also parallel. Use a Mira to draw a lie segmet. The use the Mira to draw perpedicular lies, whose reflectios are i lie with the origial lie segmet. The perpedicular lies are all parallel to each other. Fold a piece of paper carefully with the corers matchig. Fold the paper agai. Crease the folds well. Ope the paper ad, usig a straightedge, trace lie segmets alog the creases. The resultig lie segmets are parallel. Use a compass ad a straightedge. Draw a lie segmet AB. Use a compass to draw a circle aroud poit A ad a circle aroud poit B with the same radius. Shape ad Space 61

314 Mark the highest (or lowest) poits of each circle ad coect them with a lie. This lie will be parallel to the origial lie segmet AB. 4. Have studets practise drawig parallel lie segmets of differet legths ad differet distaces apart. Parters, teachers, or the toss of a umber cube ca determie the legth or distace, or studets ca choose their ow measuremets. After tryig several methods to draw desigated parallel lie segmets, studets ca choose the method they prefer ad write about it i their math jourals. Variatios: Create optical illusios by geeratig a set of parallel lie segmets, ad the decoratig each lie with differet colours or markigs at differet agles or legths or thickesses. Alteratively, divide the space betwee the parallel lie segmets with perpedicular lies of various widths ad colours. Create a display of the illusios with a title (e.g., Fid the Parallel Lies). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Draw a lie segmet parallel to aother lie segmet, ad explai why they are parallel. 62 Grade 7 Mathematics: Support Documet for Teachers

315 Suggestios for Istructio Draw the bisector of a agle usig more tha oe method, ad verify that the resultig agles are equal. Materials: math jourals or otebooks geometry sets with straightedges, protractors, ad compasses Miras tracig paper geometry software (optioal) Orgaizatio: Small groups, pairs or idividual, whole class Procedure: 1. Ask studets to explai what a agle bisector is, ad how they could test to see whether or ot a lie actually bisects a agle. 2. Have small groups work together to devise multiple ways of geeratig agle bisectors for a particular agle, ad prove that the method geerates a true bisector. Ask studets to eter successful methods i their math jourals. 3. Reassemble as a class to share studets methods ad discuss the beefits ad applicatios or use of the suggested methods. Have studets make additioal otes i their math jourals as eeded. Methods for geeratig agle bisectors may iclude the followig: a) Use a protractor to measure the origial agle. Divide the measuremet i half. Use the protractor to mark the ew measure, ad draw the bisector. b) Use tracig paper. Copy the agle oto tracig paper. Fold the paper so that the two origial rays lie o top of each other. Crease the paper alog the fold. Ope the paper ad use a straightedge to trace the crease. c) Use a Mira. Place the Mira so that part of the legth lies over the vertex of the agle. Adjust the agle of the Mira util each of the agle rays are reflected o top of each other. Trace the edge of the Mira. Shape ad Space 63

316 d) Use a ruler. Lie up the edge of the ruler alog oe of the rays i the agle so that the ruler lies iside the agle. Trace the side of the ruler ot o the ray. Trace the side of the ruler for the other ray. Coect the vertex of the agle with the itersectio of the lies just draw. 64 Grade 7 Mathematics: Support Documet for Teachers

317 e) Use a compass ad a straightedge. Place the compass poit o the vertex of the agle ad draw a arc across the two rays. Place the compass poit o oe of the itersectig poits, ad draw a circle or a arc aroud the cetre area of the agle. Keep the same radius settig, ad draw a circle aroud the other poit where the arc itersects the other ray. It is ot ecessary to draw the etire circles, just the itersectio of the arcs of the circle approximately where the bisector will be. Use the straightedge to coect the itersectio of the two circles with the vertex of the ray. The arc across the rays creates two poits equidistat from the vertex. Coect the two poits with a straight lie. Itersectig two circles with cetre poits o each of the ray poits creates the perpedicular bisector of the lie coectig them. Shape ad Space 65

318 4. Have studets practise drawig agle bisectors. Supply studets with agles or agle measures for which studets ca create bisectors, or have studets geerate their ow measures. Ask studets to idetify ad justify a preferred techique for bisectig agles ad write about it i their math jourals. Variatios: Provide studets with a hadout outliig ad illustratig the methods for creatig a agle bisector, alog with a compass ad templates for those studets who may eed them. Have studets create the agle bisectors of various triagles, ad compare the results. Have them draw circles with the cetre at the itersectio of the agle bisectors, ad the radii just touchig oe side of the triagle. Ask studets to ivestigate what happes whe they bisect the agles of parallelograms ad other polygos. Have studets use geometry software to practise ad ivestigate agle bisectors of differet shapes. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Draw the bisector of a agle usig more tha oe method, ad verify that the resultig agles are equal. r Draw a lie segmet perpedicular to aother lie segmet, ad explai why they are perpedicular. r Draw a lie segmet parallel to aother lie segmet, ad explai why they are parallel. r Draw the bisector of a agle usig more tha oe method, ad verify that the resultig agles are equal. r Draw the perpedicular bisector of a lie segmet usig more tha oe method, ad verify the costructio. 66 Grade 7 Mathematics: Support Documet for Teachers

319 Puttig the Pieces Together Maps, Floor Plas, or Desig Projects Itroductio: Studets use geometric costructios to replicate a floor pla, create a map, or desig a project. Purpose: I this ivestigatio, studets will demostrate the ability to do the followig (learig outcome coectios are idetified i paretheses): Costruct circles ad solve problems ivolvig radius, diameter, ad circumferece of circles. (7.SS.1) Perform geometric costructios, icludig perpedicular ad parallel lie segmets ad bisectors. (7.SS.3) Studets will also demostrate the followig mathematical processes: Commuicatio Coectios Problem Solvig Reasoig Materials/Resources: geometry kits cotaiig protractors, rulers, compasses, right triagles Miras tracig paper idividual project supplies Orgaizatio: Idividual, whole class Procedure: 1. Select a project, such as the followig: a) Replicate the floor pla for a sport facility (e.g., court, field, rik). b) Desig a map for a commuity, a fairgroud, a school campus, or a campgroud. Desig major services to be equidistat from strategic poits i the area. c) Create a desig for a fece, lattice, fabric patter, or piece of artwork. 2. As a class, set criteria for the followig: a) the type ad umber of lies that must be icluded i the project (e.g., circles or half circles, parallel lies, perpedicular lies, perpedicular ad agle bisectors) b) the equipmet that may be used to create the lies c) a scorig rubric based o the criteria idetified i (a) ad (b) Shape ad Space 67

320 3. Use what you have leared about agles, parallel ad perpedicular lies, ad bisectors to create your project. 4. Prepare a report o the types of lies icluded i your project, how you created the lies, methods you used to overcome challeges you faced creatig the lies, ad poits you are proud of. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Costruct circles ad solve problems ivolvig radius, diameter, ad circumferece of circles. r Draw a lie segmet perpedicular to aother lie segmet, ad explai why they are perpedicular. r Draw a lie segmet parallel to aother lie segmet, ad explai why they are parallel. r Draw the bisector of a agle usig more tha oe method, ad verify that the resultig agles are equal. r Draw the perpedicular bisector of a lie segmet usig more tha oe method, ad verify the costructio. 68 Grade 7 Mathematics: Support Documet for Teachers

321 Shape ad Space (Trasformatios) (7.SS.4) Edurig Uderstadig(s): The coordiate grid is used for plottig ad locatig poits o a plae. Geeral Learig Outcome(s): Describe ad aalyze the positio ad motio of objects ad shapes. Specific Learig Outcome(s): 7.SS.4 Idetify ad plot poits i the four quadrats of a Cartesia plae usig ordered pairs. [C, CN, V] Achievemet Idicators: Label the axes of a Cartesia plae ad idetify the origi. Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. Draw shapes ad desigs, usig ordered pairs, o a Cartesia plae. Create shapes ad desigs o a Cartesia plae ad idetify the poits used. Prior Kowledge Studets should be able to do the followig: Q Q (6.PR.2) Represet ad describe patters ad relatioships usig graphs ad tables. Q Q Q Q Q Q Q Q (6.SS.8) Idetify ad plot poits i the first quadrat of a Cartesia plae usig whole-umber ordered pairs. (6.SS.9) Perform ad describe sigle trasformatios of a 2-D shape i the first quadrat of a Cartesia plae (limited to whole-umber vertices). (6.SP.1) Create, label, ad iterpret lie graphs to draw coclusios. (6.SP.3) Graph collected data ad aalyze the graph to solve problems. Shape ad Space 69

322 Related Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q (7.PR.1) Demostrate a uderstadig of oral ad writte patters ad their correspodig relatios. (7.PR.2) Costruct a table of values from a relatio, graph the table of values, ad aalyze the graph to draw coclusios ad solve problems. (7.PR.6) Model ad solve problems that ca be represeted by oe-step liear equatios of the form x + a = b, cocretely, pictorially, ad symbolically, where a ad b are itegers. (7.PR.7) Model ad solve problems that ca be represeted by liear equatios of the form ax + b = c ax = b x = b, a 0 a cocretely, pictorially, ad symbolically, where a, b, ad c are whole umbers. (7.SS.5) Perform ad describe trasformatios of a 2-D shape i all four quadrats of a Cartesia plae (limited to itegral vertices). Backgroud Iformatio I Grade 6, studets graphed data, plotted poits from ordered pairs, ad drew shapes ad desigs i the first quadrat of a Cartesia plae. Experiece with horizotal ad vertical itegral umber lies i both Grades 6 ad 7 prepared studets to exted plottig skills to work i all four quadrats of a Cartesia plae. Plottig ordered pairs accurately is a importat skill for performig ad describig trasformatios i learig outcome 7.SS.5, ad for graphig equatios i Patters ad Relatios. The Cartesia Plae Reé Descartes, a Frech mathematicia, philosopher, physicist, ad writer who lived i the first part of the seveteeth cetury, developed the Cartesia plae. The Cartesia coordiate system allows geometric shapes to be expressed i algebraic equatios. 70 Grade 7 Mathematics: Support Documet for Teachers

323 To teach about the Cartesia plae, start with a umber lie ad exted it to the left to iclude egative itegers. This represetatio is oedimesioal (1-D). To make it two-dimesioal (2-D), take a secod umber lie ad make it perpedicular to the first, ruig it through 0, with positive umbers extedig above the 0 ad egative umbers below. You ow have a Cartesia plae. Patters ca be draw o a Cartesia plae. Therefore, they ca be also described by a algebraic equatio. To help studets coceptualize this cocept, show them a patter o some material (e.g., o a piece of cloth, wrappig paper, wallpaper), ad tell them that this patter ca be described by a algebraic equatio, ad the plotted o a Cartesia plae. The equatio of the selected patter might be too complex for Grade 7, but it is a equatio, evertheless. Cartesia Plae The Cartesia plae is formed by a horizotal axis ad a vertical axis, labelled the x-axis ad the y-axis respectively. It cotais quadrats 1 to 4 (the quadrats are ofte labelled usig Roma umerals I to IV). A practical applicatio of this cocept ca be foud i computer-aided desig (CAD), where equatios describig 2-D (as well as 3-D) Cartesia plaes are etered ito a computer. The computer the istructs a machie to draw, cut, or stitch lies or desigs oto wood, metal, textiles, ad so o. Amog its may other uses, CAD is used for embroidery ad iterior desig. For example, the Departmet of Textile Scieces at the Uiversity of Maitoba is equipped with a CAD laboratory. I the future, people might go to special boutiques ad eter body-scaig booths i order to take their measuremets. They could later sed their measuremets to a olie store, which would create custom-made clothig for them. The booths would use Cartesia plaes to calculate the perso s measuremets. You ca itegrate the Cartesia plae with the study of geography by usig the coordiates o a map of the world. The equator could be represeted as the x-axis, ad 0 logitude could be represeted as the y-axis. Usig the map, ask studets to determie the umber of degrees (i relatio to both the x-axis ad the y-axis) betwee two cities. Coordiates ca be determied with ay type of map. You could, for example, use a highway map or a topological map of the area aroud your school or commuity. Sice Grade 7 studets are ofte iterested i expressig themselves, you could have them create their ow flag or symbol o a Cartesia plae, icludig the coordiates, ad have them explai how that shape represets them. Begi by showig them a flag with a simple desig (e.g., Switzerlad s flag), ad ask them to determie its coordiates o a Cartesia plae. Shape ad Space 71

324 Mathematical Laguage axes Cartesia plae coordiates ordered pair origi x-axis y-axis Learig Experieces Assessig Prior Kowledge Materials: grid paper magetic surface ad magetic tape (optioal) Orgaizatio: Pairs Procedure: 1. Review the cocept of ordered pairs ad how to plot coordiates i the first quadrat of a Cartesia plae by playig a versio of Battleship, Private Detective, hide-ad-seek, or whatever title seems appropriate. Guidelies for the game follow: a) Divide the class ito pairs for this game. b) Each player prepares two grids (about 10 10) by labellig the axes ad umberig the grids accordig to the agreed-upo scale. Remid studets to iclude the origi ad to umber the lies, but ot the spaces. All the grids used i the same game must be idetical. 72 Grade 7 Mathematics: Support Documet for Teachers

325 c) Each player secretly hides the predetermied umber of items i oe of his or her grids (e.g., vessels for Battleship, a crook ad clues for Private Detective, a umber of hidig people for hidead-seek). Plot each item as either a vertical lie or a horizotal lie of three adjacet poits. d) Studets take turs amig ordered pairs i a attempt to ucover their parters hidde items. As they ame a ordered pair, they mark it o their ow guessig grid, ad the studet who has hidde the objects marks the guessed poit o the grid with the hidde items. If the guessig grid is visible to both studets, they ca limit errors i amig poits. If the poit that is amed is part of a item, the the hider says, hit, ad the guesser receives aother tur. Whe all poits have bee guessed, the idetity of the object is revealed. The first player to ucover all the hidde objects is the wier. Variatio: Create a reusable display cosistig of a coordiate grid o a magetic surface, with magetic tape o the back for use as hidig spots for various items (e.g., bushes, steps, shed, barrel, crate, tree, rock, car). Scatter the hidig spots at radom itersectios o the grid. I secret, hide the items behid hidig spots. Studets take turs amig ordered pairs to fid the hidig spots. The studet who discovers the hidig spot wis a roud. Play as a class, or allow small groups of studets to take turs usig the display. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify ad plot poits i the first quadrat of a Cartesia plae usig whole-umber ordered pairs. r Reaso mathematically i order to guess strategically. Shape ad Space 73

326 Suggestios for Istructio Label the axes of a Cartesia plae ad idetify the origi. Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. Draw shapes ad desigs, usig ordered pairs, o a Cartesia plae. Create shapes ad desigs o a Cartesia plae ad idetify the poits used. Materials: grid paper lists of ordered pairs that, whe plotted ad coected together, form a simple shape or a picture i quadrat I (optioal) Orgaizatio: Idividual, pairs or small groups Procedure: 1. Have each studet create a simple shape, such as a polygo, o quadrat I of a coordiate grid. The studet the uses the plot to geerate a list of ordered pairs. 2. Have studets exchage lists with a parter, plot the poits o the parter s list, ad coect the poits i the order give to create the polygo. Studets verify each other s work. Variatios: Have oe studet share his or her list orally with a larger group or with the whole class. As the studet reads the list, the other studets plot ad coect the poits. Whe the list is complete, the reader shows what the fiished product looks like, ad the group or class discusses ay discrepacies. Provide studets with a hadout of ordered pairs that create desigs or pictures. Iclude a plot of a desig or picture, ad have studets list the ordered pairs. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify ad plot poits i the first quadrat of a Cartesia plae usig whole-umber ordered pairs. 74 Grade 7 Mathematics: Support Documet for Teachers

327 Suggestios for Istructio Label the axes of a Cartesia plae ad idetify the origi. Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Materials: demostratio board grid paper pecils ad highlighters lists of ordered pairs that, whe plotted ad coected together, form a simple shape or a picture i quadrat I (optioal) Orgaizatio: Whole class Procedure: 1. Iform studets they will create a math story. The story will require a mai character who is able to move about freely (e.g., a dog i a field, a sightseeig tourist, a basketball player o a court, a balleria o a stage, a taxi driver i New York city, a fly o the wall). a) Ask studets to begi plaig the story by makig a mark o a poit ear the cetre of a grid paper. Ask them to imagie that the mark represets the mai character, ad the grid paper represets the area where the character moves about. Have studets label the origial mark with the letter O (for origial mark). b) Next, tell studets to poit their pecils to the O poit ad prepare to mark a trail o the grid as they follow their imagiary character o a adveture. Each time you call out a letter ame, studets form a poit at the earest itersectio o the grid ad label the poit with the letter that was called. c) As you call out the letters of the alphabet, studets track the movemets of the character. Wait about five secods betwee letters. Stop aroud letter J, ad have studets go back ad make their poits ad letters obvious by highlightig them. d) Provide a sample copy of the trail o the demostratio board. Use either a studet s copy or a copy that has bee prepared ahead of time. The trail represets the path the character followed. The letters were writte at regular itervals, so each poit represets where the character was at a give time. 2. Tell studets that a method is eeded to describe the locatio of the character at ay give time durig the story. The locatio is to be give i relatio to the startig poit. Provide sufficiet time for studets to thik, ad perhaps talk with a parter, about a method. Shape ad Space 75

328 a) If studets experiece difficulty i their work, ask them to cocetrate just o the first poits, ad describe where poit A is i relatio to poit O. Resposes may iclude descriptios such as these: to the left, to the right, above, or below. b) If studets do ot quatify the directio, ask them to make the descriptios more specific by idicatig how far above or below the startig poit a give poit is, or how far to the left or to the right. c) If studets do ot thik of umber lies, ask whether there are referece marks they could add to the grid paper to make the descriptios easier, such as a lie to delieate left ad right, ad a lie to separate poits above ad below the iitial positio. Addig umbers to the lie would idicate how far right, ad how far above. Negative umbers would idicate how far left ad how far below. This amouts to addig both a horizotal ad a vertical umber lie through the startig poit of O. 3. Tell studets about the history of the Cartesia plae. Reé Descartes, who lived i Frace i the early 1600s, is credited as beig the first perso to thik of coordiatig two itersectig umber lies. This type of grid is very importat to mathematicias, ad is amed for its ivetor. The ame Descartes meas from Cartes, so the adjective that describes his last ame would be Cartesia, just as somethig from Caada is Caadia. The Cartesia plae mixes algebra ad geometry ad allows mathematicias to graph equatios (which Grade 7 studets do i the Patters ad Relatios strad). The positios of the character i the story created i this learig activity could be described with a equatio, although it may require a rather complicated equatio to describe most of the patters. There is a story that Reé Descartes thought of the Cartesia plae to describe the movemets of a fly o the ceilig; however, this story is ot verified. 4. Explai the features of the Cartesia plae. The horizotal umber lie i the Cartesia umber lie is termed the x-axis, ad the vertical umber lie the y-axis. The poit foud at (0, 0) is called the origi. The scale is chose based o the umbers i the situatio. The four mai areas are amed quadrats. They are umbered 1 to 4 (ofte expressed as Roma umerals I to IV) i a couter-clockwise directio begiig with the positive coordiates with which we are most familiar. Example: 76 Grade 7 Mathematics: Support Documet for Teachers

329 5. If studets have ot already doe so, have them iclude the followig o their ow grid papers: add the horizotal ad vertical umber lies through the origi, label the x-axis ad the y-axis, ad label the quadrats. 6. Returig to the advetures of the character i the story, have studets sort the poits where the character was i each quadrat ad make a list. Ask them to iclude the quadrat, the ame of the poit, ad the ordered pair that describes each poit. 7. To complete the story about the character s advetures (where the character wet), have studets tur the grid ito a map. If studets are iterested i writig the story, perhaps the project ca be itegrated with Eglish laguage arts. Variatios: Provide scaffoldig (e.g., pre-plotted grid paper, a pre-umbered grid, a list of questios to guide coclusios) for those requirig it. Elimiate the math story ad the studet-created poits. Provide a piece of grid paper with oe poit labelled as origi (O), ad 10 poits labelled A to J. Explai the quadrat system, ad have studets add the axes, scale, ad quadrats. Have them list the ordered pairs of the poits i each quadrat. Photocopy ay type of map ad add a grid system that aligs with some sigificat poits, icludig a origi ad a x-axis ad a y-axis. Distribute copies of the map to studets, ad have them trace the axes ad add labels. The have studets use ordered pairs to describe the positio of sigificat poits i relatio to the origi, or idetify the locatio for a poit of ordered pairs. The lik betwee coordiate grids ad mappig the Earth with lies of logitude ad latitude provides a itegratio poit for mathematics ad social studies. Sample Website: The followig olie resource, writte for childre, provides some history o mappig the Earth with lies of logitude ad latitude: The Math Drexel. Aciet Greek Maps. Chameleo Graphig < Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Label the axes of a Cartesia plae ad idetify the origi. r Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Shape ad Space 77

330 Suggestios for Istructio Label the axes of a Cartesia plae ad idetify the origi. Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. Materials: Cartesia plae grid paper rulers stickers (optioal) magetic board ad magetic tape (optioal) Orgaizatio: Whole class (divided ito two teams), small groups (of three studets) Procedure: 1. Review the features of the Cartesia plae ad the ways i which the coordiate grid ca be labelled with differet scales, depedig o the umbers beig worked with. 2. Usig a Cartesia plae, plot several poits ad label them with letter ames. Ask studets to ame the ordered pair that idetifies the locatio of a poit for a particular letter, or ask studets to ame a letter i a quadrat, or the quadrat that matches a letter. Divide the class ito two teams, ad have studets complete this exercise as a competitive game. 3. Whe studets have demostrated a level of competecy, ask them to prepare a similar plot of eight letter poits ad a key that idetifies the ordered pairs that represet the poits of each letter. Iform studets that the plot will be used for a group game, so they eed to make it large eough to be see ad keep the key separate for quick referece. 4. Have studets work i groups of three. Oe studet takes the role of leader. The leader displays the plot studets made, ad asks the other group members questios related to readig the plot (e.g., What ordered pair ames the locatio of X? What letter is at the poit of a particular ordered pair?). The group must decide whether the leader will alterate questios betwee the cotestats or whether the cotestats will compete to be the first to aswer. The roud is over whe the poits have all bee idetified. At the ed of the roud, the wier becomes the ext leader. 78 Grade 7 Mathematics: Support Documet for Teachers

331 Variatios: Studets ca play olie games idetifyig poits o a Cartesia plae. Two examples are available o the followig websites: FuBased Learig. Medium Versio of Graph Mole. Graphig < I this game, a mole pops up at radom poits ad the player must select the ordered pair that idetifies the spot. The game has three levels, each successive level requirig a faster respose. Math-Play.com. The Coordiate Plae. Coordiate Plae Game. < Coordiate%20Plae%20Game.html>. I this game, players match a poit o a Cartesia plae with x- or y-coordiates, ordered pairs, or the quadrat i which the poit is located. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Label the axes of a Cartesia plae ad idetify the origi. r Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. r Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. Suggestios for Istructio Label the axes of a Cartesia plae ad idetify the origi. Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. Materials: Cartesia plae grid paper rulers Orgaizatio: Small groups (of five studets) Shape ad Space 79

332 Procedure: 1. Studets play a modified group versio of the game Coect Four or Four i a Row. A group ca cosist of five studets oe host ad four participats. 2. Usig grid paper (1 cm or larger), each studet creates a game sheet cosistig of a Cartesia plae with a x-axis ad a y-axis, each havig five egative ad five positive divisios. Do ot label the scale. 3. Each participat chooses a easy-to-draw symbol (e.g., #,,, ), ad the host chooses the scale to be used for the roud ad records it at the top of the game sheet. 4. The participats take turs amig a ordered pair that matches the scale, ad the host writes that participat s symbol o the matchig poit the participat ames. The participats practise idetifyig the ordered pairs, ad the host practises plottig the poits. 5. The participats try to get four of their symbols i a row. Whe they achieve this, they become the host. 6. Each participat will eed to moitor that the host is plottig the ordered pairs correctly. If participats ame a poit already marked by a symbol, they lose their tur. Variatios: Icrease the size of the grid ad the umber of symbols to alig. Use stickers istead of drawig symbols. Or play o a magetic board ad have studets affix magetic tape to the back of their symbol pieces. Play the game i larger groups ad have studets cooperate to pla strategies. Hold a touramet. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Label the axes of a Cartesia plae ad idetify the origi. r Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. r Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. 80 Grade 7 Mathematics: Support Documet for Teachers

333 Suggestios for Istructio Label the axes of a Cartesia plae ad idetify the origi. Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. Draw shapes ad desigs, usig ordered pairs, o a Cartesia plae. Materials: BLM 7.SS.4.1: Plottig Poits o a Cartesia Plae rulers Orgaizatio: Whole class, idividual, pairs Procedure: 1. As a class, review the axes, origi, ad quadrats of a Cartesia plae. 2. Distribute copies of BLM 7.SS.4.1: Plottig Poits o a Cartesia Plae, ad have studets idividually label the axes with the appropriate scales, plot the poits, ad idetify the quadrats i which the figures are located. 3. Have studets, workig i pairs, exchage their completed plots with their parters. They compare plots ad quadrats ad discuss ay discrepacies. Variatios: Have studets choose a scale, ad label the axes of a Cartesia plae accordigly. Next, they draw a simple shape, such as a polygo or some other figure, i ay quadrat or combiatio of quadrats, ad use the plot to geerate a list of ordered pairs. Have studets exchage lists with a parter, plot the poits o the parter s list, ad coect the poits i the order give to create the figure. Studets ca verify each other s work. Provide studets with desigs plotted o a Cartesia plae, ad ask them to create a list of the ordered pairs to create the desig. Shape ad Space 81

334 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Label the axes of a Cartesia plae ad idetify the origi. r Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. r Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. r Draw shapes ad desigs, usig ordered pairs, o a Cartesia plae. Suggestios for Istructio Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Materials: BLM 7.SS.4.2: Cartesia Plae Quadrat Cards scissors grid paper rulers blak card templates (optioal) Orgaizatio: Small groups (of two to four studets) Procedure: 1. Have studets form small groups. 2. Distribute oe set of card sheets to each group of studets ad have them cut the sheets to separate the cards. (See BLM 7.SS.4.2: Cartesia Plae Quadrat Cards.) 3. After givig studets sufficiet time to separate the cards, call the groups together, ad review Cartesia plaes, axes, origi, labellig scales, ad quadrats. 4. Give the followig directios to studets: a) Mix up the quadrat cards ad put them i a stack. b) Lay four ordered pair cards face up o the table. c) The first player draws a quadrat card ad matches it to the set of ordered pairs that would create a triagle i that quadrat or set of quadrats. 82 Grade 7 Mathematics: Support Documet for Teachers

335 d) If studets agree the match is correct, the player keeps the set, ad aother ordered pair card is tured face up. If it is ot a match, the quadrat card goes back i the deck. Some plottig o grid paper may be ecessary to verify resposes. e) Play cotiues to pass to the ext player. f) The studet who makes the most sets wis. Variatios: Studets ca create additioal ordered pair cards that create other figures. Studets or the teacher ca create additioal ordered pair cards. The cards ca be used to play make a set card games fashioed after Go Fish!, Rummy, Pit, ad so o. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. r Use reasoig ad visualizatio to help determie the placemet of poits. Suggestios for Istructio Label the axes of a Cartesia plae ad idetify the origi. Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. Draw shapes ad desigs, usig ordered pairs, o a Cartesia plae. Create shapes ad desigs o a Cartesia plae ad idetify the poits used. Materials: BLM 7.SS.4.3: Plot This Picture grid paper ruler patter blocks (optioal) Orgaizatio: Idividual Shape ad Space 83

336 Procedure: 1. Provide studets with grid paper ad copies of BLM 7.SS.4.3: Plot This Picture, ad ask them to plot ad coect the specified poits to create desigs. 2. Ask studets to create their ow desigs o grid paper ad list the ordered pairs. If they eed help, suggest usig patter blocks to build a desig, ad the tracig it oto grid paper. 3. Display studets creatios. Variatio: Provide studets with the fiished plot ad ask them to create a list of ordered pairs that match the picture. Sample Website: For a display of coordiate picture desigs ad a studet gallery of completed pictures, refer to the followig website: Plottig Coordiates.com. CoordiArt News < Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Label the axes of a Cartesia plae ad idetify the origi. r Idetify the locatio of a poit i ay quadrat of a Cartesia plae usig a ordered pair. r Plot the poit correspodig to a ordered pair o a Cartesia plae with uits of 1, 2, 5, or 10 o its axes. r Draw shapes ad desigs, usig ordered pairs, o a Cartesia plae. r Create shapes ad desigs o a Cartesia plae ad idetify the poits used. 84 Grade 7 Mathematics: Support Documet for Teachers

337 Shape ad Space (Trasformatios) (7.SS.5) Edurig Uderstadig(s): While geometric figures are costructed ad trasformed, their proportioal attributes are maitaied. Geeral Learig Outcome(s): Describe ad aalyze the positio ad motio of objects ad shapes. Specific Learig Outcome(s): 7.SS.5 Perform ad describe trasformatios of a 2-D shape i all four quadrats of a Cartesia plae (limited to itegral vertices). [C, CN, PS, T, V] Achievemet Idicators: (It is iteded that the origial shape ad its image have vertices with itegral coordiates.) Idetify the coordiates of the vertices of a 2-D shape o a Cartesia plae. Describe the horizotal ad vertical movemet required to move from a give poit to aother poit o a Cartesia plae. Describe the positioal chage of the vertices of a 2-D shape to the correspodig vertices of its image as a result of a trasformatio or successive trasformatios o a Cartesia plae. Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. Describe the image resultig from the trasformatio of a 2-D shape o a Cartesia plae by comparig the coordiates of the vertices of the image. Prior Kowledge Studets should be able to do the followig: Q Q Q Q (5.SS.7) Perform a sigle trasformatio (traslatio, rotatio, or reflectio) of a 2-D shape, ad draw ad describe the image. (5.SS.8) Idetify a sigle trasformatio (traslatio, rotatio, or reflectio) of 2-D shapes. Shape ad Space 85

338 Q Q Q Q Q Q Q Q Q Q (6.SS.6) Perform a combiatio of trasformatios (traslatios, rotatios, or reflectios) o a sigle 2-D shape, ad draw ad describe the image. (6.SS.7) Perform a combiatio of successive trasformatios of 2-D shapes to create a desig, ad idetify ad describe the trasformatios. (6.SS.8) Idetify ad plot poits i the first quadrat of a Cartesia plae usig whole-umber ordered pairs. (6.SS.9) Perform ad describe sigle trasformatios of a 2-D shape i the first quadrat of a Cartesia plae (limited to whole-umber vertices). (6.SP.3) Graph collected data ad aalyze the graph to solve problems. Related Kowledge Studets should be able to do the followig: Q Q Q Q (7.SS.3) Perform geometric costructios, icludig perpedicular lie segmets parallel lie segmets perpedicular bisectors agle bisectors (7.SS.4) Idetify ad plot poits i the four quadrats of a Cartesia plae usig ordered pairs. Backgroud Iformatio Trasformatios Movemet is ubiquitous i our world; eve the Earth itself is i costat motio. Movemets occur as chages i size, shape, or positio. The area of mathematics that brigs geometry ad algebra together to describe these chages is the study of trasformatios. The three trasformatios that Middle Years studets study are traslatios, reflectios, ad rotatios. Iformally, these trasformatios are referred to as slides, flips, ad turs, ad each relates to chages i positio ad/or orietatios i a 2-D plae. I previous grades, studets performed ad described sigle trasformatios of 2-D shapes. I Grade 7, they exted their skills to work with successive trasformatios i all four quadrats of the plae. The learig experieces suggested for learig outcome 7.SS.5 will help studets to develop their uderstadig ad appreciatio of the trasformatios existig aroud them, ehace their problem-solvig skills ad spatial sese, ad prepare them for further studies i algebra ad geometry. 86 Grade 7 Mathematics: Support Documet for Teachers

339 Studets regularly ecouter 2-D trasformatios represeted i desig patters ad computer graphics. They are evidet o logos, fabric patters, frieze patters, wallpaper, architectural desig, ladscape desig, ad so o. Trasformatios ca be used to create iterestig symmetrical patters. I additio, 2-D trasformatios o Cartesia plaes ca be used to represet physical movemets i a sigle plae (e.g., sports plays, rides at a fair, traffic routes). Movie aimatio is created usig motio geometry. May desigs are symmetrical trasformatios of a core. Examples ca be viewed ad created olie. Sample Websites: For a rage of samples of rotated shapes, refer to the followig website: Wolfram Demostratios Projects. Symbol Rotatio Patters. Cotributed by Daielle Nogle. Based o a program by Chris Carlso < The followig website allows studets to create desigs with up to eight lies of symmetry: MathIsFu. Symmetry Artist. Geometry < Trasformatios are studied usig 2-D shapes, or pre-images, ad their images. The shapes are amed by their vertices (ABC) ad the images are labelled (A B C ), read as A prime, B prime, C prime, ad so o. Successive images are labelled with additioal prime marks (A²B²C²), ad so o. Traslatios, Reflectios, ad Rotatios I traslatios, reflectios, ad rotatios, the shapes ad their images are cogruet, but their orietatio o the plae ad/or their locatio o the plae may chage, depedig o the shape ad the type of trasformatio. Traslatios (slides): Each poit i the shape moves the same distace ad the same directio to create the image. The orietatio of the shape ad its image remai the same; oly the locatio o the plae chages. Demostrate traslatios o a coordiate grid by copyig the shape Note: As studets advace i grades, it is importat for them to be familiar with ad use mathematical laguage. Ecourage studets to use the terms traslatio, reflectio, ad rotatio, rather tha slide, flip, ad tur. oto grid paper of the same size, cuttig it out, ad physically slidig the copy o the grid. Aother method is to cout the horizotal ad vertical moves of each vertex. A slide arrow idicates the directio of a traslatio. Shape ad Space 87

340 Example: A B A B C traslatio arrow idicatig 3 left ad 2 up C Traslatios are also commoly described usig coordiate otatio with square brackets (e.g., 3 left, 2 up is [ 3, 2]). Reflectios (flips): The poits i the shape ad the matchig poits i the image are equal distaces from a lie of reflectio. The lie of reflectio may be iside or outside the shape. The orietatio of the figure flips over the lie of reflectio to create a mirror image i a ew locatio. Demostrate reflectios by physically flippig a copy of the shape over the lie of reflectio, by placig a Mira alog the reflectio lie, or by coutig the perpedicular distace of each poit from the mirror lie. The lie of reflectio is idicated by markig a mirror lie o the grid. Example: E F D G D G lie of reflectio E F 88 Grade 7 Mathematics: Support Documet for Teachers

341 Rotatios (turs): The poits of the shape are rotated the same umber of degrees or fractio of a tur clockwise (cw) or couter-clockwise (ccw) aroud a poit termed the cetre (or poit) of rotatio. The cetre of rotatio may be ay poit iside or outside the figure. The orietatio of the image will deped o the directio ad the agle of rotatio. The chage i locatio of the image varies greatly, depedig o the locatio of the cetre of rotatio ad the directio ad agle of rotatio. Demostrate rotatios by physically rotatig the shape the specified umber of degrees aroud the cetre of rotatio, or by tracig the shape ad the cetre of rotatio oto tracig paper, ad the matchig the cetre of rotatio ad physically rotatig the tracig paper. Rotatig the side of the shape rather tha just oe poit may help reduce accidetal slidig durig the rotatio. Aother techique to locate a rotatio requires the use of a protractor to measure the agle ad a compass to copy the lie legth. Specify rotatios with a curved arrow that idicates the directio of the rotatio, ad write the umber of degrees or the fractio of a tur for the rotatio. Rotatios are commoly described usig a degree measure ad directio (e.g., 90 ccw). Example: S Q Q S R 90 cw T R poit of rotatio Trasformatioal chages ca be described by idetifyig the type of trasformatio, the chages i orietatio or positio of the vertices, the horizotal ad vertical movemet, or the ew x- ad y-coordiates of the vertices of the image, or by statig the chage i x- ad y-coordiates betwee the shape ad its image. Shape ad Space 89

342 Mathematical Laguage Cartesia plae cetre of rotatio clockwise cogruet coordiates couter-clockwise image lie of reflectio quadrat reflectio rotatio shape trasformatio traslatio vertex vertices Learig Experieces Assessig Prior Kowledge Materials: large space (gymasium or outdoors) oe skippig rope (or aother lie) for each group math jourals (optioal) Orgaizatio: Small groups (of six studets), whole class Procedure: 1. Form groups, with six studets i each group. 2. Explai to studets that they will be workig together to demostrate three types of trasformatios. 90 Grade 7 Mathematics: Support Documet for Teachers

343 3. Review the cocept that trasformatios are about movig ad chagig positio. a) The three types of trasformatios studets will demostrate are traslatios (slides), reflectios (flips), ad rotatios (turs). Remember to iclude a mirror lie for the reflectio ad reflect the etire shape. Rotatios require a cetre of rotatio, a directio, ad a agle of rotatio (e.g., 90º, 180º, 270º clockwise or couter-clockwise). Cetres of rotatio ad mirror lies ca be iside or outside the shape. b) To show the pre-image ad the resultig image, studets ca pair up as oe poit of the shape. To trasform, they ca divide, leavig oe poit (studet) i the origial locatio ad oe poit (studet) trasformed to the ew locatio. If this will cofuse studets, leave out this aspect ad move the etire pre-image to the image. 4. Have group members work together to come up with a way to demostrate the trasformatios usig their bodies ad a skippig rope. Later they will regroup to show their demostratio to their classmates ad explai the trasformatio ad its importat features, icludig the pre-image, image orietatio, ad locatio. 5. As studets are practisig, circulate amog them ad provide hits ad ecouragemet where eeded. 6. Reassemble as a class ad have studets share their learig. Below are some ideas that may be icluded i the presetatios: a) Traslatios may iclude triagle or rectagle arragemets. Directios are give to slide so may steps forward, backward, right, or left. The image has chaged locatio oly. The poits are still facig the same directio. The orietatio remais the same. The image is still the same size or cogruet to the pre-image. Shape ad Space 91

344 b) Reflectios may iclude studets lyig alog a mirror lie, ad mirrorig oe aother at ay distace. Each poit ad its parter are equal perpedicular distaces to the mirror lie. The orietatio of the pre-image has bee flipped from the origial. c) Rotatios require studets to select a cetre of rotatio, a directio of rotatio, ad the amout or agle of rotatio. Studets may be i a lie with the group rotatig aroud oe ed, the middle, or ay poit withi the lie, or they may rotate aroud aother poit away from the lie. They may also rotate i the formatio of a triagle or a rectagle. Variatios: Assig oe trasformatio to each group. Each group prepares a thorough presetatio, complete with multiple examples. Have studets work iside the classroom i pairs, performig the demostratios with multiple objects rather tha with their bodies. I place of presetatios, studets idividually draw their example(s) ad write a explaatio i their math jourals. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Perform ad describe traslatios. r Perform ad describe reflectios. r Perform ad describe rotatios. 92 Grade 7 Mathematics: Support Documet for Teachers

345 Assessig Prior Kowledge Materials: grid paper (three for each studet) polygo cut-outs from grid paper or patter blocks, or some other small agular flat objects math jourals Miras (optioal) tracig paper (optioal) push-pis (optioal) Orgaizatio: Whole class, idividual or pairs Procedure: 1. Have studets practise demostratig trasformatios. Direct studets to place a object o the grid paper ad demostrate differet types of trasformatios. a) Traslatios (slides up, dow, right, or left, ad diagoally) b) Reflectios (flips over differet lies of reflectios i differet directios) Use Miras to perform or cofirm the reflectios. c) Rotatios about a poit of rotatio Vary the poits of rotatio to iclude vertices ad poits iside ad outside the object. Tracig paper ad push-pis may be used to perform or cofirm the rotatios. 2. Review otatios. Model recordig the various trasformatios, ad review correct labellig. a) Label the vertices of the pre-image with capital letters (ABC) ad label the correspodig vertices of the image with the letter, followed by a prime mark (A B C ). Successive images are labelled with additioal prime marks (A²B²C²), ad so o. b) Iclude symbols for slide arrows, lies of reflectio or mirror lies, the cetre of rotatio, ad the directio ad amout of rotatio. c) Traslatios are commoly writte usig slide arrows or usig coordiate otatio with square brackets (e.g., 3 left, 2 up is writte as [ 3, 2]). d) Reflectios are writte as reflected i the lie (e.g., reflected i the lie x = 3). e) Rotatios are commoly writte usig a degree measure ad directio (e.g., 90 ccw). Shape ad Space 93

346 3. Model descriptios of movemet. Have studets describe the chage betwee the pre-image ad the image. Examples: The triagle was traslated 3 to the right, ad so the pre-image ad image hold the same orietatio. The square was reflected through the lie x = 1, ad so the image looks as though a mirror was held o the right-had side of it. The image looks iverted. The rectagle was rotated 90 couter-clockwise aroud poit B, ad so the image has B i the same place, but ow A is below B. 4. Have studets record some trasformatios. Ask them to use oe grid paper for each type of trasformatio ad iclude a few examples of each. a) Remid studets to iclude mirror lies ad cetres of rotatio. b) They may record combiatios or successive trasformatios usig the same pre-image, or use various pre-images ad separate trasformatios. c) Have studets iclude descriptios of the trasformatios they record. d) Some studets may repeat successive trasformatios of oe preimage to create desig patters, as demostrated o the followig website. Sample Website: Wolfram Demostratios Projects. Symbol Rotatio Patters. < SymbolRotatioPatters/>. 5. Post studets displays whe they are complete. 6. Meet as a class for a quick debriefig. a) Discuss what studets leared or were remided of durig this learig experiece, as well as ay difficulties they faced ad how they overcame them. b) Have studets make a record of their learig i their math jourals. 94 Grade 7 Mathematics: Support Documet for Teachers

347 Variatios: Provide grid paper with some shapes already sketched o the paper, ad provide specificatios for some trasformatios. Have studets perform the trasformatios ad label the images. I place of the sketches, provide the ordered pairs for the vertices of the pre-images ad have studets plot the pre-images ad images. Have studets sketch pre-images, specify trasformatios, prepare a key, ad exchage papers with a parter, who will create the images. The studets retur the papers to each other, verify resposes, ad discuss ay discrepacies. Various games ad exercises ivolvig trasformatios are available olie. Sample Websites: To play a game of golf ivolvig trasformatio, refer to the followig website: Maths Olie. Trasformatio Golf. < To explore reflectios i differet mirror lies or rotatios with lies of symmetry, refer to the followig website: MathIsFu.com. Symmetry Artist. Geometry < Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Plot poits i the first quadrat of a Cartesia plae. r Perform trasformatios of traslatios. r Perform trasformatios of reflectios. r Perform trasformatios of rotatios. r Record trasformatios. r Describe trasformatios. Shape ad Space 95

348 Suggestios for Istructio Describe the horizotal ad vertical movemet required to move from a give poit to aother poit o a Cartesia plae. Describe the positioal chage of the vertices of a 2-D shape to the correspodig vertices of its image as a result of a trasformatio or successive trasformatios o a Cartesia plae. Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. Materials: grid paper labelled as a Cartesia plae oe umber cube ad oe coi (or spiers with itegers) for each pair of studets BLM 7.SS.5.1: Comparig Poits Orgaizatio: Whole class, pairs Procedure: 1. Place studets i pairs. 2. Distribute copies of BLM 7.SS.5.1: Comparig Poits, alog with the umber cubes ad cois. 3. Demostrate the process of geeratig poits. a) Assig meaig to the maipulatives (e.g., heads represet positive, tails represet egative, six represets 0). b) Roll the umber cube ad toss the coi to determie a x-coordiate, ad the repeat the process for a y-coordiate. (Try to work without plottig the poits). Call the poit A, ad record the poit ad coordiates i the chart. c) The parter rolls ad tosses to idetify the ext coordiate, labels it B, ad records the coordiates. Compare the positio of the ew poit to the old poit. Discuss resposes ad justificatios. 4. Ask studets to follow the above method for geeratig poits as they complete BLM 7.SS.5.1: Comparig Poits. 5. Circulate amog studets as they work with their parters, ad moitor whether they are o track. 6. After studets have had sufficiet time to work i pairs, reassemble as a class ad discuss what studets leared. 96 Grade 7 Mathematics: Support Documet for Teachers

349 Variatios: Have studets cotiue the learig activity ad geerate a ew set of three to five poits, coect them ito a polygo, ad follow the patter of questio 3 (o BLM 7.SS.5.1) with the ew shape. Have studets choose ay trasformatio, or successive or combied trasformatios. Ask them to try predictig the locatio of the images ad the cofirm the locatio by plottig them. Studets ca create their ow Cartesia plaes o grid paper ad charts. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Describe the horizotal ad vertical movemet required to move from a give poit to aother poit o a Cartesia plae. r Describe the positioal chage of the vertices of a 2-D shape to the correspodig vertices of its image as a result of a trasformatio or successive trasformatios o a Cartesia plae. r Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. Suggestios for Istructio Describe the horizotal ad vertical movemet required to move from a give poit to aother poit o a Cartesia plae. Materials: BLM 7.SS.5.2: A Coordiate Map ad/or blak grid paper Orgaizatio: Idividual or pairs Procedure: 1. Decide whether or ot you will eed to discuss ad demostrate otatios, describig movemets as uits left, right, up, ad dow, ad describig chages i terms of chage i each coordiate, or if studets have sufficiet backgroud kowledge to proceed from the examples. 2. Itroduce BLM 7.SS.5.2: A Coordiate Map. Maps provide small visuals of much larger spaces. They show how locatios are related to oe aother ad they provide a way to commuicate about the locatios of differet places. Shape ad Space 97

350 Ecourage studets to be creative with their maps. They may represet ay place, ragig from their desks to aywhere i the world, actual or imagied. 3. Have each studet plot ad label several locatios o the grid ad create a key, followig the directios i step 1 of BLM 7.SS.5.2: A Coordiate Map. 4. I step 2, whe makig trips aroud the commuity, studets may work idividually or with a parter. Studets look at oe map at a time. The parter tells the mapmaker which trip to make. The mapmaker records the trip o the chart, determies the movemet, ad records it. The parter verifies whether the iformatio is correct. 5. Studets draw a Cartesia plae o their grid. Studets title the fial colum of the key ad the fourth colum of the grid Coordiates of Trip. They record the coordiates of the places o the map ad the trips. Parters verify each other s work. 6. Have studets complete the fial colum, describig the movemet betwee coordiates i terms of chage i the x- ad y-coordiates. Variatios: A positive chage i y is a upward movemet, ad a egative chage represets a dowward movemet. For the x-coordiate, a positive chage is right ad a egative chage is left. Studets ca prepare directios for a parter to plot. First they plot the poits, specify the trip, or describe movemets. Parters exchage papers ad complete the missig iformatio, ad the retur papers to each other ad verify each other s work. The teacher plots ad labels the poits o the grid, ad fills i some coordiates ad descriptios of movemets o the trip chart before distributig it to studets. Studets determie the trip destiatios. This allows the teacher a little more cotrol over the learig activity, ad provides experieces for studets to work the moves backwards. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Describe the horizotal ad vertical movemet required to move from a give poit to aother poit o a Cartesia plae. 98 Grade 7 Mathematics: Support Documet for Teachers

351 Suggestios for Istructio Idetify the coordiates of the vertices of a 2-D shape o a Cartesia plae. Describe the positioal chage of the vertices of a 2-D shape to the correspodig vertices of its image as a result of a trasformatio or successive trasformatios o a Cartesia plae. Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. Describe the image resultig from the trasformatio of a give 2-D shape o a Cartesia plae by comparig the coordiates of the vertices of the image. Materials: BLM 7.SS.5.3: Cartesia Plae Map ad UFO Templates BLM 7.SS.5.4: Explorig Trasformatios: UFO Pilot Traiig BLM 7.SS.5.5: Recordig Trasformatios: Travel Logbook Cartesia plae ad triagle shape to fit the grid (for demostratio) Miras tracig paper push-pis (optioal) scissors (optioal) Orgaizatio: Whole class, pairs or small groups, idividual Procedure: Preparatio 1. Tell studets they will be explorig trasformatios to gai experiece to pilot a uidetified flyig object (UFO) accurately o a Cartesia plae. 2. Distribute copies of BLM 7.SS.5.3: Cartesia Plae Map ad UFO Templates ad two triagles per studet, or have studets prepare their ow plaes ad triagles. Part A: Demostratio (whole class) 3. Demostrate, or have a studet demostrate, various trasformatios of the UFO o the grid provided. The purpose of the demostratio is to esure that studets have the skill to explore productively o their ow. Studets ca mimic the demostratio usig their ow grid. Ask a studet to idetify the coordiates of the shape. Tell studets what trasformatio to make. Remember to iclude lies of reflectio, ad poits, degree, ad directio of rotatio. Shape ad Space 99

352 Trasform, or have a studet trasform, the UFO ad idetify the coordiates of the image. Discuss the movemets ad descriptios that would compare the positio of the image to the positio of the shape or pre-image. Record the coordiates ad various descriptios as a model for later referece. Try varyig the iformatio give ad required. Examples: State coordiates of the shape, supply a descriptio of the movemet, ad ask studets to idetify the coordiates of the image ad/or a possible trasformatio. Have studets idetify the coordiates of a shape, give the coordiates of the image ad a descriptio of the trasformatio. Supply studets with coordiates of the shape ad the image, ad ask them to idetify a possible trasformatio. Iclude demostratios of successive trasformatios ad combiatios of trasformatios. More tha oe trasformatio ca match the same descriptio ad result i the same image. 4. Whe studets are ready to explore pilotig the UFO, move to Part B of this learig activity. Part B: Exploratio (pairs or small groups) 5. Distribute copies of BLM 7.SS.5.4: Explorig Trasformatios: UFO Pilot Traiig. 6. Tell studets they will explore trasformatios, with the aim of beig able to cotrol the UFO ad pilot it to ad from specific locatios. 7. Have studets work i pairs or i small groups to explore describig ad predictig the images created by trasformig the UFO. The BLM provides some remiders ad suggestios, as well as a chart for recordig observatios. 8. Whe studets feel competet, have them try some itetioal trips, usig differet trasformatios to move betwee two poits o the Cartesia plae map. Use the vertex O as the o/off poit that must touch the poits foud o the Cartesia plae map (BLM 7.SS.5.3: Cartesia Plae Map ad UFO Templates). 9. Studets ca challege each other to fid the most efficiet trasformatios, or alterate trasformatios, to travel betwee poits. 10. Whe they are successful, they are ready for Part C of this learig activity. 11. Issue UFO liceces if you wish. 100 Grade 7 Mathematics: Support Documet for Teachers

353 Part C: Trasformatio Travel (idividual) 12. Distribute copies of BLM 7.SS.5.5: Recordig Trasformatios: Travel Logbook. Studets will eed their copies of BLM 7.SS.5.3: Cartesia Plae Map ad UFO Templates ad their UFO triagles. 13. Decide o the umber of trips to be completed ad ay specific trasformatios, or alterate routes, the pilots must make to complete their missio (e.g., make it to poit F i less tha five trips usig at least two differet trasformatios). 14. Have studets label the plotted poits to represet locatio destiatios o their maps. 15. Have studets pilot their UFO from oe locatio to aother usig trasformatios, ad ask them to complete BLM 7.SS.5.5: Recordig Trasformatios: Travel Logbook. A trip takes vertex O (o/off) from oe locatio poit to the other. Variatios: Hold a UFO derby. Studets compete to see who ca travel from oe destiatio to aother usig the fewest trasformatios. (Traslatios cout as oe trasformatio per uit.) Studets ca explore ad practise trasformatios usig computer trasformatio software, applets, or games. Sample Websites: The followig websites allow studets to trasform squares, parallelograms, ad triagles. BBC. Shape, Space ad Measures. KS2 Bitesize < play.shtml>. I this computer game, studets choose a mirror lie or rotatio poits to reflect a petago house oto its shadow. Reed, Jim The Coordiate Plae. Grade 7: The Learig Equatio Math < Directios are displayed with the applet. Shodor. Trasmographer. Iteractive < Shape ad Space 101

354 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify the coordiates of the vertices of a 2-D shape. r Describe the horizotal ad vertical movemet required to move from a give poit to aother poit o a Cartesia plae. r Describe the positioal chage of the vertices that result from a trasformatio or from successive trasformatios. r Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. r Describe the image resultig from the trasformatio of a shape by comparig the coordiates of the vertices of the image to the vertices of the pre-image. Suggestios for Istructio Idetify the coordiates of the vertices of a 2-D shape o a Cartesia plae. Describe the positioal chage of the vertices of a 2-D shape to the correspodig vertices of its image as a result of a trasformatio or successive trasformatios o a Cartesia plae. Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. Describe the image resultig from the trasformatio of a give 2-D shape o a Cartesia plae by comparig the coordiates of the vertices of the image. Materials: BLM 7.SS.5.6: Creatig a Desig Usig Reflectios grid paper rulers art supplies (for desigs) display board file cards (optioal) trasformatio software, draw programs, computer applets (optioal) Orgaizatio: Idividual, whole class 102 Grade 7 Mathematics: Support Documet for Teachers

355 Procedure: 1. Distribute copies of BLM 7.SS.5.6: Creatig a Desig Usig Reflectios. Have studets follow istructios to plot a five-sided shape reflect the shape o the y-axis reflect the expaded shape o the x-axis to create the desig describe the chages betwee the iitial shape ad the images 2. Have studets create their ow symmetrical desig by creatig a basic shape ad oe or more trasformatios. Ask them to iclude a plot of the shape ad its images o a Cartesia plae a chart of the coordiates of the shape ad its images directios for creatig the image (writte o a file card) a descriptio of the chages betwee the shape ad its images 3. Have studets colour ad frame their desigs. Post the desigs ad the directios o a display board. 4. Hold a class debriefig sessio i which studets share their desigs, ad discuss difficulties ad solutios they ecoutered while creatig the desigs. Variatios: Explore creatig desigs usig trasformatio software, draw programs, or computer applets. Add the file cards with desig directios to a box from which studets ca draw cards ad follow the directios to create desigs. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify the coordiates of the vertices of a 2-D shape. r Describe the positioal chage of the vertices that result from a trasformatio or from successive trasformatios. r Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. r Describe the image resultig from the trasformatio of a shape by comparig the coordiates of the vertices of the image to the vertices of the pre-image. Shape ad Space 103

356 Suggestios for Istructio Idetify the coordiates of the vertices of a 2-D shape o a Cartesia plae. Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. Materials: BLM 7.SS.5.7: Which Plot Is Correct? umber cubes rulers grid paper Miras (optioal) tracig paper (optioal) computer software or applets (optioal) Orgaizatio: Idividual, whole class Procedure: 1. Iform studets that i this learig activity they will perform trasformatios o desigated shapes. 2. Radomly assig shapes ad trasformatios to each studet. A suggested method follows: Preset six possible shapes ad assig each a umber from 1 to 6. Examples: 104 Grade 7 Mathematics: Support Documet for Teachers

357 List six types of trasformatios you would like studets to practise, ad assig each a umber from 1 to 6. Suggestio: 1 traslatio 2 rotatio iside shape 3 reflectio iside the shape 4 rotatio outside the shape 5 reflectio outside the shape 6 combiatio Studets roll a umber cube to determie the shape ad trasformatio(s) with which they will work. 3. Studets idividually plot their assiged shape o grid paper, choosig their ow scale ad coordiates to create the vertices of their shape. 4. Studets choose details for their assiged trasformatio, ad perform it o the shape. 5. Studets write their ame, draw a small diagram of their shape, provide a list of poits ad coordiates for the shape ad image, ad write directios for the trasformatio o the grid paper. This product will be used later i this learig activity. 6. Provide each studet with a copy of BLM 7.SS.5.7: Which Plot Is Correct? Have studets use the shape ad trasformatio they made to create a multiple-choice puzzle o BLM 7.SS.5.7: Which Plot Is Correct? Plot the correct shape i oe of the locatios A to D. Plot the correct image i aother box A to D. Plot the shape ad image icorrectly i the two remaiig boxes. Suggest that studets try ot to be obvious as they create their errors. 7. Chroologically assig each studet a puzzle umber to write o his or her sheet, ad o the origial grid paper plot, which is the solutio key. 8. Have studets exchage papers with 10 or so people, ad aswer the puzzles i a chart like the oe o page 2 of BLM 7.SS.5.7: Which Plot Is Correct? 9. Studets retur all puzzles to their creators. 10. Ask studets to share the correct solutios to the puzzles by takig turs readig their puzzle umber, ad the correct letters for the shape ad the image. 11. As a class, discuss ay discrepacies idetified, ad refer to the origial grid paper to correct ay errors. 12. Store the puzzles ad origial plots i a folder for studets to solve at other times. Shape ad Space 105

358 Variatios: Prepare several puzzles, copy them, ad give the same sheet to each studet to solve. Provide studets with a list of coordiate poits ad trasformatio directios, ad have them produce the coordiates for the image. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify the coordiates of the vertices of a 2-D shape. r Perform a trasformatio or cosecutive trasformatios o a 2-D shape, ad idetify the coordiates of the vertices of the image. 106 Grade 7 Mathematics: Support Documet for Teachers

359 Puttig the Pieces Together Golf Touramet or Aimatios Itroductio: Studets create a golf course ad use trasformatios to move a ball through the course. Purpose: I this ivestigatio, studets will demostrate the ability to do the followig (learig outcome coectios are idetified i paretheses): Costruct circles with a give radius. (7.SS.1) Costruct perpedicular ad parallel lie segmets ad bisectors. (7.SS.3) Perform ad describe trasformatios of 2-D shapes i the four quadrats of a Cartesia plae. (7.SS.5) Studets will also demostrate the followig mathematical processes: Commuicatio Coectios Metal Mathematics ad Estimatio Problem Solvig Reasoig Materials/Resources: BLM : My Success with Mathematical Processes grid paper, large grid paper, ad poster paper ruler, compass, ad protractor calculator art supplies aimatio software (optioal) Orgaizatio: Idividual or pairs Procedure: 1. Create a pla for a golf course that icludes ie holes. The goal is to complete the course with the fewest possible trasformatio strokes. Specific features of the course iclude the followig: a) ie labelled tees ad matchig hole umbers (Each tee must be located alog a perpedicular bisector of a straight lie betwee the previous hole ad its tee.) b) two water traps with a 10 cm diameter c) two sad traps, with a radius of 7 cm ad 5 cm Shape ad Space 107

360 d) two treed areas, with a radius of 3 cm ad 4 cm e) two parallel rows of trees 20 cm by 2 cm 2. Create ad label a Cartesia plae o grid paper. Add a detailed pla of your golf course. Place the tees ad holes at coordiate itersectios. 3. Provide a list of coordiates, trasformatio directios, ad grid plots to show the fewest golf strokes required to complete the golf course. 4. Build the golf course ad test your directios. 5. Pla a golf touramet. Decide o the umber of pealty strokes for ladig i traps or i trees. 6. Play each other s golf courses ad see who completes a course with the fewest trasformatios. 7. Complete BLM : My Success with Mathematical Processes. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Costruct circles of a give radius. r Costruct perpedicular bisectors. r Costruct parallel lies. r Describe trasformatios i the four quadrats of a Cartesia plae. Extesio: Use aimatio software to create a aimatio based o trasformatios. Aimatios may iclude the followig: a ature scee i which the su comes up, travels through the sky, ad sets, while a creature visits differet places, fidig food, drikig water, or hidig. Iclude a combiatio of jumpig, slidig, turig, ad spiig movemets. a park scee with playgroud equipmet, a play area, a garde area, a picic area, etraces, ad so o a amusemet park with rides ad cocessios a city scee with a bak, a theatre, stores, restaurats, bus stops, medical offices, ad so o a basketball or hockey game with a court or a rik draw to scale 108 Grade 7 Mathematics: Support Documet for Teachers

361 G r a d e 7 M a t h e m a t i c s Statistics ad Probability

362

363 Statistics ad Probability (Data Aalysis) (7.SP.1, 7.SP.2) Edurig Uderstadig(s): Data ca be described by a sigle value used to describe the set. Geeral Learig Outcome(s): Collect, display, ad aalyze data to solve problems. Specific Learig Outcome(s): 7.SP.1 Demostrate a uderstadig of cetral tedecy ad rage by determiig the measures of cetral tedecy (mea, media, mode) ad rage determiig the most appropriate measures of cetral tedecy to report fidigs [C, PS, R, T] 7.SP.2 Determie the effect o the mea, media, ad mode whe a outlier is icluded i a data set. [C, CN, PS, R] Achievemet Idicators: Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Solve a problem ivolvig the measures of cetral tedecy. Aalyze a set of data to idetify ay outliers. Explai the effect of outliers o the measures of cetral tedecy for a data set. Idetify outliers i a set of data, ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Statistics ad Probability 3

364 Prior Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q (5.SP.1) Differetiate betwee first-had ad secod-had data. (5.SP.4) Compare the likelihood of two possible outcomes occurrig, usig words such as less likely equally likely more likely (6.SP.1) Create, label, ad iterpret lie graphs to draw coclusios. (6.SP.2) Select, justify, ad use appropriate methods of collectig data, icludig questioaires experimets databases electroic media (6.SP.3) Graph collected data ad aalyze the graph to solve problems. Related Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q (7.N.7) Compare ad order fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals (7.SP.3) Costruct, label, ad iterpret circle graphs to solve problems. (7.SP.4) Express probabilities as ratios, fractios, ad percets. (7.SP.5) Idetify the sample space (where the combied sample space has 36 or fewer elemets) for a probability experimet ivolvig two idepedet evets. (7.SP.6) Coduct a probability experimet to compare the theoretical probability (determied usig a tree diagram, table, or aother graphic orgaizer) ad experimetal probability of two idepedet evets. 4 Grade 7 Mathematics: Support Documet for Teachers

365 Backgroud Iformatio Studets live i a iformatio age that abouds with data. Various media cotiually offer iformatio o fashio, etertaimet, sports, fiaces, safety, health, ad world evets. Studets ecouter data regularly at school, i their marks, i sciece experimets, i social studies iformatio, ad so o. To be helpful, data eeds to be categorized ad uderstood. Statistics help reduce large quatities of data to sigle values. The sigle value makes it much simpler to coceptualize ad commuicate about the iformatio cotaied i the data. Statistics, however, are sometimes maipulated or preseted i a maer that uses facts to mislead people ad sway their opiios. By studyig statistics, studets develop their ability to uderstad ad evaluate iformatio preseted i advertisig, politics, ad ews reports, ad to commuicate their experiece with data. Measures of Cetral Tedecy I previous grades, studets collected data first had ad from electroic sources, ad leared whe to use each source. I Grade 7, studets are itroduced to three statistical measures of cetral tedecy: mea, media, ad mode. Each is a umeric value attemptig to represet a etire set of data. Each measure is a average with its ow focus, stregths, ad weakesses. The more symmetrical the set of data is, the closer the measures of cetral tedecy will be to oe aother. The more skewed the set of data is, the greater the differece betwee the values will be. The differet measures are best used i differet situatios, although sometimes all three measures provide meaigful represetatios of the data. The measures of cetral tedecy ad rage are discussed below: Mea: The arithmetic mea is commoly referred to as average, ad is commoly used to assig studet grades. The mea is the measure of cetral tedecy most affected by outliers; therefore, it is best used whe the rage of values i the set is arrow. To fid the mea, combie all the values i the set ad the evely redistribute them. The algorithm for calculatig the mea is to sum all values i the set, ad divide the combied value by the umber of values i the set. The mea ca also be foud by fidig the cetral balace poit o a umber lie. Example: Give the umbers 3, 4, 6, 3, 3, 9, 7, a) plot the umbers o a umber lie Statistics ad Probability 5

366 b) move the umbers toward the cetre, allowig oe move to the left for every oe move to the right c) cotiue this process util the umbers lie up o oe poit Media: The media is the middle value i a ordered set of data. The media is easy to uderstad ad easy to determie. To fid the media, place all the values i the set, icludig repeated umbers, i umerical order ad select the value i the middle. If there is o sigle middle value, add the two middle umbers together ad divide by two. Because the media is the middle value, half the values i a data set will be greater tha the media ad half the values will be less tha the media. The media represets the 50th percetile. (Note: Formal study of percetiles occurs i Grade 12 Essetial Mathematics.) The media is less affected by outliers; therefore, it is more stable. It is the most appropriate measure of cetral tedecy to represet a set of data cotaiig extreme values. Mode: The mode is the most commoly occurrig item i a set. A set of data may ot have a mode, or it may have oe mode, be bimodal, or have multiple modes. The mode may or may ot idicate the cetre of the data it represets. Geerally, outliers (extreme values at either the high or the low ed of the rage) do ot affect modes. Modes are very ustable, however, ad a small chage i the data ca drastically chage the mode. Because the mode idetifies the most typical item i a set, it is useful for predictig the case i a particular situatio. For example, if the mode for shirts sold is size 10, the buyers for a store ca use the mode to help them decide which sizes to stock i the store s ivetory. Rage: The rage describes a set of data by idetifyig the differece betwee the greatest value ad the least value i a data set. Uderstadig how ad whe to use the differet statistical values gives studets the ability to uderstad ad commuicate about data more clearly, ad to use data wisely to make iformed decisios. Whe plaig for studet learig experieces, choose learig activities that emphasize cocepts ad uderstadig. Have studets gather data for the purpose of aswerig questios. Allowig studets to ask their ow questios ad collect their ow data provides cotexts ad purposes for aalyzig data ad for explorig the differet statistics. Studets may, for example, wish to compare their classmates habits or physical skills, itegrate sciece or social studies cotet, or aswer questios about world coditios or treds. 6 Grade 7 Mathematics: Support Documet for Teachers

367 Sample Website: The followig website is a source of data ad iformatio for teachers ad studets: Statistics Caada. Learig Resources. < I the sidebar, go to Teachers ad select resources by subject area. Mathematical Laguage average data mea measure of cetral tedecy media mode outlier rage statistics Ve diagram Learig Experieces Assessig Prior Kowledge Materials: grid paper rulers Orgaizatio: Idividual or pairs Procedure: 1. Review the cocepts of formulatig questios, collectig first- or secod-had data, ad preparig bar graphs. 2. Have studets work idividually, or i pairs, to do the followig: a) Formulate a survey questio about peers that ca be aswered with umeric values. Sample Questios: How may sibligs are i your family? How may pets (or cell phoes, televisios) does your family have? Statistics ad Probability 7

368 How may times a week do you eat a particular food, watch a movie, or participate i physical activity? How may hours do you sleep per ight? How tall are you? How may pairs of mittes (or shoes, pats) do you ow? How may coutries have you visited? Compare the heights (or heart rates, legths of ames) of boys ad girls i the class. b) Gather the iformatio. c) Display the data i a bar graph. d) Formulate a questio about the populatio of the survey that could be aswered usig the iformatio from the graph. Iclude a aswer key to the questio. 3. Have studets preset ad display their work. These data sets ca be used for subsequet learig experieces. Variatios: Have studets choose a questio ad create bar graphs from data you provide or from the school cesus data available olie. Sample Website: Statistics Caada. Data ad Results. Cesus at School. 29 July < See Your Class Results for a example of results from a fictitious class. Rather tha havig studets coduct surveys, have them research sources to collect data to aswer specific questios (e.g., temperatures or raifall amouts over a certai period of time, sizes of farms i a regio, the price of a commodity). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Pose a survey questio that ca be aswered with umeric values. r Coduct a survey or search resources to obtai data. r Display data i a bar graph. 8 Grade 7 Mathematics: Support Documet for Teachers

369 Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Materials: bar graphs from the surveys coducted i the previous learig experiece (Assessig Prior Kowledge) presetatio board math jourals Orgaizatio: Idividual or pairs, whole class Procedure: 1. Remid studets that their surveys ad graphs reveal iterestig iformatio about their peers. 2. Ask studets to use the iformatio i their graphs to create a geeral statemet about a typical or average studet i the class, or grade, or whatever group they surveyed. Examples: How may sibligs does a typical Grade 7 studet have? How may times does the typical Grade 7 studet eat Frech fries i a week? 3. Have studets work idividually or with their parters to determie the best aswer to their questio, explai how they arrived at the aswer, ad explai why the aswer represets a typical studet. 4. Reassemble as a class ad have studets share their questios, aswers, explaatios, ad justificatios. 5. Durig the class discussio, ecourage studets to commet o ad ask questios about their classmates resposes, ad to preset alterative aswers. 6. Itroduce vocabulary related to statistics as topics preset themselves durig the sharig. 7. Record the vocabulary o a presetatio board. Iclude the three measures of cetral tedecy (mea, media, ad mode), the rage, ad outliers (if they are preset). a) If studets choose the most frequet respose to represet the average studet, itroduce mode. b) If they fid the arithmetic mea, or redistribute the items, itroduce mea ad discuss the methods they used to determie the value. c) If they use the middle value, itroduce media ad discuss the methods of fidig the media. Statistics ad Probability 9

370 d) If they discuss the rage of values, itroduce the rage as the differece betwee these values. e) If they metio aomalies such as a very high or very low value, itroduce outliers. 8. Iform studets that i usig oe value to represet a rage of data, they have bee explorig statistical measures of cetral tedecy. Measures of cetral tedecy will be studied i greater detail i the followig learig experieces. 9. Have studets record the ew vocabulary terms ad what they have leared about them i their math jourals. Variatios: Have studets combie the iformatio from several surveys to create a profile of the typical Grade 7 studet. Rather tha usig studet survey data, provide studets with several questios ad sets of data. For example, provide data for several styles of T-shirts, each sellig for a differet price. Ask what would be a fair price for the T-shirts if they all sold for the same price. Or supply data for the amout of moey idividual studets raised for a class trip. Ask how much moey a typical studet raised. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Demostrate a uderstadig of mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Demostrate a uderstadig of the rage of a set of data. 10 Grade 7 Mathematics: Support Documet for Teachers

371 Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Materials: BLM 7.SP.1.1: Fidig the Cetre of a Graph ad Comparig the Values bar graphs from the previous learig experiece or supplied data modellig clay, coloured cubes or blocks, ad/or couters grid paper (1 cm) trasparet rulers or grid strips presetatio board math jourals BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy (optioal) Orgaizatio: Idividual or pairs, whole class Procedure: 1. Distribute copies of BLM 7.SP.1.1: Fidig the Cetre of a Graph ad Comparig the Values, ad have studets work idividually or with a parter to complete it. The reassemble as a class ad discuss what studets discovered. 2. Alterately, guide the class through the steps ad have them record their learig i their math jourals. a) Have studets, workig idividually or i pairs, build a cocrete model of their ow graph, or a classmate s graph. Alteratively, supply studets with data ad have them prepare a graph ad a model for the data. Studets may use cubes or blocks, couters, ad/or 1 cm grid paper ad modellig clay. b) Whe studets have completed their graphs, ask them to do the followig: Idetify the rage for the data represeted i their graph, ad record it (subtract the least value from the greatest value). Rearrage the graph to emphasize the rage. Idetify ay outliers or extreme values i the graph. Fid the mode, or most frequet value, represeted i their graph, ad record it. Explai how the graph could be rearraged to emphasize the mode. Fid the media or middle value i their graph, ad record it. Explai how the graph could be rearraged to emphasize the media. c) Ask studets to explore, o their ow or with their parter, how to level the data ad fid its cetre, or balace poit. Emphasize that this is ot the middle value or media. Studets will be rearragig their graphs to emphasize the cetre of the data, or the mea. They record the mea. Statistics ad Probability 11

372 3. Have studets reassemble as a class ad share what they did to level the data, ad discuss ay questios or commets that arise. Strategies for levellig the data could iclude the followig: a) Compress the modellig clay graph, while holdig the sides ad surface firm. b) Rearrage the blocks by takig blocks from the loger bars ad placig them o the smaller bars util they are similar i height. If whole blocks caot be shared evely, it may be ecessary to share fractios of a block. c) Place a ruler perpedicular to the bars of the graph, ad adjust the positio of the ruler util there are a equal umber of blocks above the lie ad below the lie. It may be ecessary to positio the ruler withi a block if the mea is ot a whole umber. 4. Have studets compare their three values, mode, media, ad mea, ad determie whether they each represet the data well, or whether oe value represets the data better tha the others, ad why that may be. 5. Share studets reflectios ad discuss how each value is a measure of cetral tedecy or a way to represet the average value. Whe the data set has a small rage, the average values are similar, ad each represets the graph. Whe there are outliers i the data, or the rage is wide, the averages may be quite differet from each other, ad o average by itself represets the data well. Differet measures are better for differet situatios, ad sometimes more tha oe measure is eeded to represet the data. Variatios: Supply studets with graphs of data that cotrols the value of the averages (i.e., a whole-umber mea if studets are ot prepared to work with decimals or fractios), or cotrol the presece of modes or outliers. Have studets rearrage the values of the bars to create multiple bar graphs that have the same mea. Ask why the differet graphs have the same mea. Have studets explore what effect rearragig the values of the bars of the graphs have o each of the average values. See BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. r Use reasoig ad visualizatio to determie measures of cetral tedecy. 12 Grade 7 Mathematics: Support Documet for Teachers

373 Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Materials: demostratio board magets or self-stick otes umber lies Uifix or likig cubes BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy (optioal) Orgaizatio: Pairs, whole class Procedure: 1. Review the three types of averages (mode, media, ad mea), ad how studets foud these values usig graphs. 2. Preset a set of data such as the followig: 3, 4, 6, 3, 3, 9, 7. Ask studets to use a Thik-Pair-Share strategy (thik about the questio idividually, discuss ideas with a parter, ad the share resposes with the class) to do the followig: a) Idetify the rage i the data. b) Idetify ad explai how to determie the mea without makig a graph. c) Idetify ad explai how to determie the media without makig a graph. d) Idetify ad explai how to determie the mode without makig a graph. 3. Itroduce studets to usig a umber lie to fid the cetre of the data. a) Draw a umber lie o the demostratio board from 0 to 10. b) Place a square to represet each value o the correspodig poit of the umber lie. Magets or self-stick otes work well o a chalkboard or whiteboard. (I the data set specified above, there are three umber 3s, so place three squares o 3.) c) The goal is to fid the cetre of all these values. Systematically move the selfstick otes from each ed toward the cetre util all the otes are stacked up o oe poit (e.g., a move of two jumps from the right toward the cetre must be coutered by a move of two jumps from the left toward the cetre). d) For the above data set, the blocks will all lie up o the mea 5. Statistics ad Probability 13

374 Variatios: Have studets practise determiig the mea ad explore the effect of differet values o averages. Alter the values i the above data set, but maitai a set of seve digits with a sum of 35. Record the measures of cetral tedecy for each set, usig BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy. Compare the values for differet sets of data. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Solve a problem ivolvig the measures of cetral tedecy. Aalyze a set of data to idetify ay outliers. Explai the effect of outliers o the measures of cetral tedecy for a data set. Idetify outliers i a set of data, ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Materials: BLM 7.SP.1.3A: Simoe s Spellig Scores (Questios) BLM 7.SP.1.3B: Simoe s Spellig Performace Record Orgaizatio: Pairs or small groups, whole class (for Thik-Pair-Share) 14 Grade 7 Mathematics: Support Documet for Teachers

375 Procedure: 1. Distribute copies of BLM 7.SP.1.3A: Simoe s Spellig Performace (Questios) ad BLM 7.SP.1.3B: Simoe s Spellig Performace Record. 2. Preset a set of data such as Simoe s spellig quiz results, scored out of 10. Her scores for the first seve quizzes were: 8, 8, 7, 9, 6, 10, ad Ask studets what score best represets Simoe s spellig performace, ad why they believe it to be so. I this set of data, the mea, media, ad mode are all 8. The rage is Simoe writes three more quizzes, with scores of 3, 7, ad 8. Have studets idetify ad support her performace level ow (mea 7.4, media 8, mode 8, rage 7). 5. O the last three quizzes, Simoe receives scores of 9, 10, ad 0. Have studets idetify which oe umber will represet Simoe s spellig performace. Ask studets to support their choice usig measures of cetral tedecy ad rage (mea 7.2, media 8, mode 8, rage 10). 6. Discuss studets choices ad reasos. Iclude a discussio of a) the effect of outliers o the mea, media, ad mode b) the ifluece of the rage o the differet measures c) possible reasos for the outliers (e.g., did t study, called to the office, cheated, lost quiz) d) whether or ot the outliers should be icluded i the data Variatios: Use studets ow assigmet or test scores. Use a differet data set that does ot represet school scores (e.g., prices for jeas, party sizes at a pizza restaurat, sizes of shoes or clothes). Have studets research to obtai data to aswer a questio they pose. Have studets geerate radom data to explore the effects of large outliers, or rage size, o measures of cetral tedecy. Ask studets to aalyze their fidigs ad make a geeral statemet regardig circumstaces for which they recommed each measure of cetral tedecy. Statistics ad Probability 15

376 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. r Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. r Solve a problem ivolvig the measures of cetral tedecy. r Aalyze a set of data to idetify ay outliers. r Explai the effect of outliers o the measures of cetral tedecy for a data set. r Idetify outliers i a set of data, ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. r Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Solve a problem ivolvig the measures of cetral tedecy. Materials: BLM 7.SP.1.4: Usig Cetral Tedecy to Choose a Quarterback spiers (optioal) Orgaizatio: Idividual, small groups, whole class 16 Grade 7 Mathematics: Support Documet for Teachers

377 Procedure: 1. Remid studets that the mea, media, ad mode are all measures of cetral tedecy that represet a etire set of data. Studets will ow use these measures to choose a quarterback for a football game. 2. Distribute copies of BLM 7.SP.1.4: Usig Cetral Tedecy to Choose a Quarterback, ad ask studets to complete the page idividually. 3. The have studets meet i small groups to discuss their thikig. Each group will choose oe quarterback, ad a spokesperso will preset ad justify the group s choice to the class. 4. As groups preset ad defed their choices, ecourage studets to respod to presetatios with commets ad questios. 5. Discuss what to do with errors, ad whe each measure (mea, media, ad mode) is best used. Variatios: Have studets geerate additioal data by usig a spier with sectios for 0, 5, 10, 15, 20, ad 25 yards, ad the ask them to recalculate the measures ad re-evaluate their decisios. Have studets geerate data for additioal quarterbacks. Questio whether their decisios are based o the same measures each time. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. r Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. r Solve a problem ivolvig the measures of cetral tedecy. Statistics ad Probability 17

378 Suggestios for Istructio Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Materials: BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy previously completed record sheets of data sets ad measures of cetral tedecy, icludig the iformatio from the graphs produced i the Assessig Prior Kowledge learig experiece spiers or pairs of umber cubes (regular or multi-sided) Orgaizatio: Pairs or small groups Procedure: 1. Explai to studets that they will be ivestigatig sets of data to determie geeralizatios about which circumstaces best match each measure of cetral tedecy. 2. Have studets work with a parter or i a small group. 3. Ask studets to evaluate their previous records. If they require more data, or larger data sets, they ca do the followig: a) Radomly geerate ew data sets by spiig spiers or by rollig the umber cubes ad multiplyig the displayed umbers. b) Research the legitimate aswers to actual questios (e.g., salaries eared i specific compaies, umbers of differet sadwiches sold at a fast-food restaurat, flavours of ice cream sold, quatities of differet driks sold i the school cafeteria, teams wiig champioships). 4. Have studets geerate a list of guidelies for the types of data or circumstaces they recommed for each measure of cetral tedecy. 5. Discuss the guidelies as a class (refer to Backgroud Iformatio). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. 18 Grade 7 Mathematics: Support Documet for Teachers

379 Suggestios for Istructio Solve a problem ivolvig the measures of cetral tedecy. Idetify outliers i a set of data ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Materials: data sets from previous learig experieces paper Orgaizatio: Idividual, pairs or small groups Procedure: 1. Explai that i this learig activity studets will demostrate their uderstadig ad use of measures of cetral tedecy. 2. Have studets, idividually, create a realistic questio ad a accompayig data set o oe side of a sheet of paper, ad a detailed solutio to the questio o the reverse side of the paper. Questios may iclude outliers that would or would ot be used i determiig the cetral tedecy. Solutios require the rage, outliers, ad the mea, media, ad mode to be idetified. Ask studets to idetify the best measure to reflect the cetre of that data ad justify the choice. 3. Studets the share their questios with a parter or a small group ad demostrate their ability to use ad choose measures of cetral tedecy to solve problems. 4. Whe studets have solved a problem, they check their solutio ad discuss ay discrepacies with the creator of the problem. Variatios: Prepare additioal problems ad data sets for studets who require them. Provide studets with several problems ad data sets, ad ask them to provide solutios for each. Have a statistics challege i which idividual studets compete to solve the problems i frot of a classroom audiece, or i which teams of studets compete agaist oe aother to fid the best measure of cetral tedecy i each case. Statistics ad Probability 19

380 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a problem ivolvig the measures of cetral tedecy. r Idetify outliers i a set of data, ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. r Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. 20 Grade 7 Mathematics: Support Documet for Teachers

381 Puttig the Pieces Together Defiig the Average Potato Itroductio: Studets collect data, determie measures of cetral tedecy, cosider the rage ad the effect of outliers, ad aalyze circle graphs i order to defie the average potato. Purpose: I this ivestigatio, studets will demostrate the ability to do the followig (coectios to learig outcomes are idetified i paretheses): Determie measures of cetral tedecy (mea, media, ad mode) ad rage. (7.SP.1) Determie the most appropriate measures of cetral tedecy to report fidigs. (7.SP.1) Determie the effect o the mea, media, ad mode whe a outlier is icluded i the data. (7.SP.2) Costruct circle graphs to solve problems. (7.SP.3) Express probabilities as ratios, fractios, ad percets. (7.SP.4) Studets will also demostrate the followig mathematical processes: Commuicatio Coectios Metal Mathematics ad Estimatio Problem Solvig Reasoig Techology Materials/Resources: Each studet will require a potato a data sheet a way to label each potato (e.g., adhesive tape) Each group will require a tape measure or strig a ruler (cm) a mass scale Statistics ad Probability 21

382 a setup for measurig volume (e.g., water, a basi, a cotaier with a wide mouth filled to the brim that sits iside aother cotaier with a pour spout, a measurig cup to measure the overflow, towels for cleaig up spills) a calculator a compass a protractor Orgaizatio: Small groups (of four or five) Procedure: 1. Esure each member of your group has a potato, a assiged letter of the alphabet, ad a copy of the Data Sheet for Determiig the Average Potato. 2. Label your potato with the letter of the alphabet you were assiged. Give your potato a ame that begis with that letter, ad record the ame o your data sheet. Cout the umber of eyes o your potato, ad record the umber o your data sheet. 3. Measure ad record the legth aroud (cm), the breadth (cm), the mass (gr), ad the volume (ml) of your potato. 4. Whe the measurig is complete, have each perso i your group clearly read ad repeat his or her data aloud. Liste carefully to your classmates data ad record it accurately. 5. After all the data has bee collected, work as a group to defie the average potato. a) Have each group member take resposibility to calculate the measures of cetral tedecy for oe of the potato measures. b) Idetify the rage ad ay outliers, ad discuss the effect of the outliers o the measures of cetral tedecy. c) Prepare circle graphs for each potato measure to help defie the average potato. d) As a group, agree o a defiitio for a average potato. e) Determie the umber of potatoes that would fit your defiitio of a average potato. What is the experimetal probability that a potato i the group would be a average potato, accordig to your defiitio? Explai. f) Prepare to preset your group s defiitio ad a defece of that defiitio to the class. 6. Preset defiitios to the class, ad take part i a discussio. Does the class agree o a defiitio for a average potato? Explai. 22 Grade 7 Mathematics: Support Documet for Teachers

383 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie measures of cetral tedecy. r Determie the affect of outliers icluded i the data. r Determie the most appropriate measures of cetral tedecy to report fidigs. r Costruct circle graphs to solve problems. Extesio (Optioal): Studets desig a average-potato lottery ad compare the experimetal results to the theoretical probability of wiig. Purpose: I this extesio, studets will demostrate the ability to do the followig (coectios to learig outcomes are idetified i paretheses): Express probabilities as ratios, fractios, ad percets. (7.SP.4) Defie the sample space for the probability experimet. (7.SP.5) Compare the theoretical ad experimetal probability of two idepedet evets. (7.SP.6) Procedure: Use the collected data ad the agreed-upo defiitio to hold a potato lottery. Defie the criteria for the lottery. The wier could receive the potatoes. Radomly select five or six potatoes from your group. Calculate the theoretical probability of selectig a average potato from the group twice i a row, give the first potato is retured to the buch after it has bee selected. Determie the experimetal probability of the same evet by holdig a class lottery. The potatoes could serve as the lottery prize. Statistics ad Probability 23

384 Data Sheet for Determiig the Average Potato Name Legth aroud (cm) Breadth (cm) Mass (gr) Volume (ml) Number of Eyes A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Total Mea Media Mode 24 Grade 7 Mathematics: Support Documet for Teachers

385 Statistics ad Probability (Data Aalysis) (7.SP.3) Edurig Uderstadig(s): Circle graphs show a compariso of each part to a whole usig ratios. Percets, fractios, decimals, ad ratios are differet represetatios of the same quatity. Geeral Learig Outcome(s): Collect, display, ad aalyze data to solve problems. Specific Learig Outcome(s): 7.SP.3 Costruct, label, ad iterpret circle graphs to solve problems. [C, CN, PS, R, T, V] Achievemet Idicators: Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Create ad label a circle graph, with or without techology, to display a set of data. Fid ad compare circle graphs i a variety of prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Traslate percetages displayed i a circle graph ito quatities to solve a problem. Iterpret a circle graph to aswer questios. Prior Kowledge Studets should be able to do the followig: Q Q (5.SP.1) Differetiate betwee first-had ad secod-had data. Q Q Q Q (6.N.6) Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. (6.SS.1) Demostrate a uderstadig of agles by idetifyig examples of agles i the eviromet classifyig agles accordig to their measure estimatig the measure of agles usig 45, 90, ad 180 as referece agles determiig agle measures i degrees drawig ad labellig agles whe the measure is specified Statistics ad Probability 25

386 Q Q Q Q Q Q (6.SP.2) Select, justify, ad use appropriate methods of collectig data, icludig questioaires experimets databases electroic media (6.SP.3) Graph collected data ad aalyze the graph to solve problems. (6.SP.4) Demostrate a uderstadig of probability by idetifyig all possible outcomes of a probability experimet differetiatig betwee experimetal ad theoretical probability determiig the theoretical probability of outcomes i a probability experimet determiig the experimetal probability of outcomes i a probability experimet comparig experimetal results with the theoretical probability for a experimet Related Kowledge Studets should be able to do the followig: Q Q (7.N.3) Solve problems ivolvig percets from 1% to 100%. Q Q Q Q Q Q (7.SS.1) Demostrate a uderstadig of circles by describig the relatioships amog radius, diameter, ad circumferece of circles relatig circumferece to pi determiig the sum of the cetral agles costructig circles with a give radius or diameter solvig problems ivolvig the radii, diameters, ad circumfereces of circles (7.SP.4) Express probabilities as ratios, fractios, ad percets. (7.SP.6) Coduct a probability experimet to compare the theoretical probability (determied usig a tree diagram, table, or aother graphic orgaizer) ad the experimetal probability of two idepedet evets. 26 Grade 7 Mathematics: Support Documet for Teachers

387 Backgroud Iformatio The purpose of graphs is to display data. Studets come to Grade 7 with experiece i usig lie graphs to display cotiuous data, ad bar graphs, double bar graphs, ad pictographs to display discrete data. I Grade 7, studets are itroduced to circle graphs. Circle graphs are also referred to as pie charts. Circle Graphs (Pie Charts) Various media use circle graphs to display comparative data. The circle graph displays the distributio of data, ot the actual data values. The set of data is grouped ito categories, ad each category is expressed as a percet of the whole set of data. Each sector of the graph represets a part-to-whole ratio. Circle graphs emphasize the relatio betwee a category ad the whole set of data, as well as the relatio betwee differet categories withi the data set. Comparisos withi circle graphs are most clear whe the umber of categories is small ad whe there is a defiite variatio i the size of the categories. Example: Beverage Choice Data Number of Studets Percet Agle Size Juice % 166 Soda 75 23% 83 Milk 68 21% 75 Water 32 10% 36 Totals % 360 This circle graph shows that early half of the studets eatig i the school cafeteria choose juice as a luch beverage, ad that early equal umbers of studets choose milk or soda. Circle graphs may also be used to compare data sets of differet size, as circle graphs compare ratios rather tha defiite quatities. The ratios regardig studets choices of beverage i the example above ca be compared to choices made by studets i other schools or i other regios. The comparisos may be used to aswer questios or to solve problems (e.g., which school to target for a utritio educatio program). Circle graphs are also used effectively to display probability. Statistics ad Probability 27

388 Experiece with circles ad cetral agles (learig outcome 7.SS.1), a uderstadig of decimals, percets, ad fractios, ad the ability to perform calculatios with these values (learig outcomes 7.N.2, 3, 4, 5, ad 7) make it easier for studets to create ad iterpret circle graphs. A uderstadig of roudig is useful whe costructig circle graphs (e.g., if the majority of percets or agle sizes have bee rouded up or dow, adjustmets may be required to esure the sum of percets totals 100%, ad cetral agles represeted i the graph total 360 ). Ways to Create Circle Graphs There are may ways to create circle graphs. Several of these are described below. Each circle graph must have a descriptive title ad must be labelled with the category ames ad correspodig percets, or be accompaied by a leged. The percets represeted by the sectors must total 100%, ad the sum of the cetral agles must equal 360. Make cocrete represetatios. Divide studets ito categories, such as those who have pets, ad those who do ot have pets. Ask studets i each group to stad side by side, equidistat from each other, ad the have the two groups form a circle. Estimate the middle of the circle, ad draw a lie (perhaps usig a skippig rope) from the cetre of the circle to each poit where the two groups meet. Studets could also use tokes to represet the umbers i the two groups. The tokes could be evely spaced aroud a circle whose circumferece has bee divided ito percets, ad a lie could be draw from the cetre to the poits o the circumferece midway betwee adjacet groups. Joi bars from a bar graph. Create a bar to represet the quatity i each group. Colour the bars. The cut out each bar, ad joi the bars ed to ed with tape to create oe log strip. Brig the eds of the strip together to create a circle. Draw a lie from the cetre of the circle to each poit where a ew category begis. Use fractio circles. Choose a fractio circle that matches the umber of pieces of data. For example, if there are 10 marbles i a set, choose a circle divided ito teths. Each teth represets oe marble. If six of the marbles are blue, colour 6 of the circle 10 blue; if three of the marbles are yellow, colour 3 of the circle yellow; ad if the 10 remaiig marble is red, colour 1 of the circle red. 10 Draw lies from the cetre of the circle to the poit o the circumferece where the categories meet. 28 Grade 7 Mathematics: Support Documet for Teachers

389 Calculate percets ad use percet circles. Express the umber i each category as a fractio of the whole set. Covert each fractio to a decimal umber ad the to a percet. Use a circle divided ito 100ths, or ito 20ths, to represet itervals of 5% (see BLM : Percet Circle). Create sectors to represet the percet of each category. Calculate percets ad create cetral agles. Create a chart such as the oe below. Category Quatity Fractio of the Whole Percet of the Whole Percet Times 360 i the Circle Size of the Cetral Agle Draw a circle ad oe radius. Use the radius to measure oe of the cetral agles. Use the subsequet radii to create successive cetral agles. Mathematical Laguage agle circle graph key leged percet pie chart sectors sum sum of the cetral agles Statistics ad Probability 29

390 Learig Experieces Assessig Prior Kowledge Materials: grid paper markers rulers access to data sources (optioal) Orgaizatio: Whole class, idividual or pairs Procedure: 1. Use a class discussio to review the characteristics of graphs, icludig the visual display of data, descriptive titles, labellig of axes, scale, ad plots. 2. Ask studets to work idividually or i pairs to collect data o some topic, ad the display the data as a graph. Studets may obtai data through surveys or observatios, or they may research a topic (e.g., colour of clothes wor o a give day, movie, music, readig, or food prefereces, laguage(s) spoke, umber of sibligs, populatios, life spas). 3. Ask studets to write questios that ca be aswered usig the iformatio i their graphs, ad the have them write aswers to these questios. 4. Post studets graphs, alog with the accompayig questios ad aswers, aroud the room. 5. Have studets participate i a Gallery Walk to view the displayed graphs. As a class, discuss the purpose ad effectiveess of usig graphs to display iformatio about a topic. Variatios: Supply studets with data or prepared graphs, rather tha havig them collect their ow data. Supply prepared graphs ad related questios for studets to aswer. The discuss the characteristics ad purposes of graphs. 30 Grade 7 Mathematics: Support Documet for Teachers

391 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Select, justify, ad use appropriate methods of collectig data, icludig questioaires, experimets, databases, ad electroic media. r Graph collected data ad aalyze the graph to solve problems. Assessig Prior Kowledge Materials: BLM 7.SP.3.1: Calculatig the Percet of the Total idex cards (optioal) calculators (optioal) Orgaizatio: Idividual or pairs, whole class Procedure: 1. Review strategies for covertig fractios to percets. 2. Distribute copies of BLM 7.SP.3.1: Calculatig the Percet of the Total, ad have studets work idividually or i pairs to fid the percets preseted i the scearios. 3. Whe studets have had sufficiet time to respod to the questios, have them compare percets with a parter ad resolve ay discrepacies i their aswers. 4. Reassemble as a class ad discuss strategies studets used to express portios of a whole as percets. Variatios: Have studets create their ow scearios ad questios regardig percets. Ask them to record the scearios ad questios o oe side of a idex card, ad the solutios o the opposite side of the card. The cards may be used for drill games, learig activities, or Exit Slips. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. Statistics ad Probability 31

392 Assessig Prior Kowledge Materials: BLM 7.SP.3.2: Percet of a Circle protractors calculators (optioal) paper, compasses, ad multi-sided umber cubes or spiers (optioal) Orgaizatio: Idividual, whole class Procedure: 1. Review how to use a protractor to measure ad draw agles. 2. Distribute copies of BLM 7.SP.3.2: Percet of a Circle, ad have studets, workig idividually, idetify various percets of shaded circles, shade desigated percets of circles, ad draw agles to represet a percet of a circle. 3. Review ad correct studets resposes as a class, ad discuss ay questios that arise. Variatios: Provide studets with additioal practice i idetifyig various percets of shaded circles, shadig various percets of circles, ad drawig agles that correspod to a percet of a circle. Have studets use a olie computer game to idetify the percet of a circle that has bee shaded. Sample Website: Games are available o the followig website: Scweb4free.com. Circle Graphs Game < I this game, studets view segmeted circles ad select a multiple-choice respose to idetify the percet of studets who prefer hamburgers. Have studets play a game i pairs, usig multi-sided umber cubes, paper, a compass, ad protractors. Each studet uses the compass or template to draw a circle, mark its cetre, ad draw a radius from the cetre to the outside of the circle. The parters take turs rollig the umber cubes. The product of the two umbers rolled equals the percet of the circle to shade. The percet 360º idicates the size of agle to draw. Studets shade each sector they draw. The first studet to shade the etire circle wis. 32 Grade 7 Mathematics: Support Documet for Teachers

393 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. r Demostrate a uderstadig of agles by drawig ad labellig agles whe the measure is specified. Suggestios for Istructio Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Create ad label a circle graph, with or without techology, to display a set of data. Iterpret a circle graph to aswer questios. Materials: a large ope area with a marked cetre (e.g., the cetre of a basketball court, the pitcher s moud of a ball diamod) log cords or skippig ropes (about four) tape or pegs to hold dow oe ed of the cords grid paper markers scissors rulers demostratio board Orgaizatio: Whole class, idividual Statistics ad Probability 33

394 Procedure: Part A 1. As a class, review the cocept that graphs are visual ways to display data. Iform studets that for this learig activity they will use differet methods to create a graph called a circle graph. The circle graph eables them to divide a group ito differet categories, ad allows them to compare the size of each category to each other ad to the whole group. 2. Secure oe ed of the cords to the cetre of a circle i a ope area. 3. Ask studets to lie up i two categories, such as those who ate breakfast, ad those who skipped breakfast. Record the categories ad umbers i each category. 4. Have the lies follow their leader to form a circle aroud the cetre poit. 5. Have oe studet from where the two lies meet go to the cetre of the circle ad brig the ed of oe of the cords back to the circumferece of the circle. Note how the circle has bee divided ito two sectios, those who ate breakfast, ad those who did ot. 6. Talk about which sector is smaller, ad which is larger. Discuss whether most of the studets ate breakfast or whether most did ot. Estimate the percet of the circle represeted by each category. Discuss a descriptive title for the circle graph. 7. Repeat the procedure with other categories (e.g., favourite colours, umber of sibligs, ethic backgrouds). The four cords accommodate four categories. 8. Stop the exercise after sufficiet examples have bee explored, ad review what studets leared about circle graphs. 9. Post the categories ad umbers i each category for each of the graphs formed i Part A. Part B 10. Demostrate creatig a circle graph for oe set of data by colourig grids to represet each category, cuttig the coloured grids ito strips, tapig the strips ed to ed, ad the joiig the eds to form a circle. Trace the circle, estimate the cetre of the circle, ad mark a poit of the circumferece where differet colours meet. Use a ruler to coect the cetre of the circle ad the poits o the circumferece. Estimate the percet of the circle represeted by each sector, record the percet, ad label the sector. Title the graph. Use the data to write comparative statemets about the categories ad the whole set of data represeted by the graph. 11. Have studets, workig idividually, select oe data set ad the create their ow circle graph ad comparative statemets for that data set. Post studets graphs. 34 Grade 7 Mathematics: Support Documet for Teachers

395 Variatio: As a alterative to usig the ope area, solicit questios from the class, record the data o the demostratio board, ad have pairs of studets act out scearios usig couters ad circles (as described i the Backgroud Iformatio for learig outcome 7.SP.3). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% r Create ad label a circle graph, with or without techology, to display a set of data. r Iterpret a circle graph to aswer questios. Suggestios for Istructio Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Create ad label a circle graph, with or without techology, to display a set of data. Iterpret a circle graph to aswer questios. Materials: BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs BLM : Percet Circle fractio circles, available o the followig website: Maitoba Educatio. Middle Years Activities ad Games. Mathematics. < coloured couters (e.g., marbles, cubes, toy cars, or toy aimals, i bags) markers or pecil crayos compasses Statistics ad Probability 35

396 Orgaizatio: Whole class, idividual or pairs Procedure: This learig experiece will likely take more tha oe class ad is divided ito three parts. Use the same materials to create three graphs i Parts A to C. Part A 1. As a class, review the characteristics ad purposes of circle graphs. 2. Distribute fractio circles that have bee divided ito 10ths. 3. Have studets, workig idividually or i pairs, radomly choose 10 items (or the umber of divisios o the fractio circles) from a bag. The ask studets to do the followig: a) Sort the items accordig to colour. b) Colour adjacet segmets o a fractio circle to match the umber of items of each colour. c) Draw bold lies to divide the colours ad create sectors of each colour. 4. Ask studets to label each sector with the applicable colour ad the correspodig percet, or create a leged for the categories (e.g., if 6 of the 10 cubes selected were blue, the 6 or 60% of the cubes were blue) Have studets write a title for their graph, as well as comparative statemets relatig to the graph. 6. Have studets total the percets represeted i each sectio of their graph, ad record the totals. Ask studets to make a geeral statemet regardig the sum of the percets i each graph. Discuss why the sum is 100%. I the discussio, iclude the cocept that each category is a part of the whole set ad 100% represets the whole set. Part B 7. Distribute copies of BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs fractio circles that have bee divided ito 20ths or 100ths (see BLM : Percet Circle) 8. Have studets radomly select 5 to 30 coloured couters ad sort them ito colour groups. Studets the complete the followig process, usig BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs: a) Record the colours i the Category colum of the chart. b) Record the umber of couters of each colour i the Quatity colum. c) Calculate the total quatity. d) Write the quatity of each colour as a fractio of the total couters selected. e) The covert that fractio to a percet. 36 Grade 7 Mathematics: Support Documet for Teachers

397 f) Add the percets. If it was ecessary to roud some of the percets, it is possible that they will ot total 100%. If they do ot total 100%, determie which percets were rouded up ad which were rouded dow ad make adjustmets to the most appropriate values. (The fial two colums of the chart will be completed i Part C.) 9. Ask studets to use the percets to create a circle graph usig the percet circles. Each 100th mark represets 1% of the circle, or each 20th represets 5% of the circle. 10. Have studets label the graph, icludig a) a title for the graph b) a label for each category (if there is isufficiet room, a leged may be used istead of labels) c) the percet of the whole for each category 11. Have studets write comparative statemets related to the categories of the graph. Part C 12. As a class, review how to use a protractor ad how to draw agles of specific measures. 13. Demostrate to studets that each sector of the circle represets a agle measure there are 360º i a circle 14. Show studets how to fid the agle measure by fidig a percet of 360º. 15. Have studets calculate the agle measures ad record them o BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs. Some of the agle measures may eed to be rouded. Whe the agle measures are totalled, they may ot equal 360º. If this is the case, review the roudig, ad adjust the values up or dow as ecessary. 16. Ask studets to draw a circle graph usig the measures of the cetral agles for each category. a) Use a compass to draw a circle. b) Mark the middle of the circle. c) Draw oe radius for the circle. d) Use the radius as a startig poit to measure oe agle. e) Use subsequet radii to create successive cetral agles. 17. Have studets label the graph, icludig a) a title for the graph b) a label for each category (if there is isufficiet room, a leged may be used istead of labels) c) the percet of the whole for each category 18. Have studets write comparative statemets related to the categories of the graph. 19. As a class, discuss the applicatios for usig each method of creatig a circle graph. Statistics ad Probability 37

398 Variatios: Supply data for each graph, rather tha havig studets geerate their ow data. Have studets use previously collected data, or collect their ow data, to create circle graphs, usig BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs (optioal). Use techology to create circle graphs. For example, use graphig programs such as Graphical Aalysis, spreadsheet programs such as Excel or Numbers, or olie graphig programs. Sample Websites: Graphig programs are available o websites such as the followig: Math Playgroud. Circle Graphs. Math Maipulatives < U.S. Departmet of Educatio. Istitute of Educatio Scieces, Natioal Ceter for Educatio Statistics. Create a Graph. Kids Zoe, Learig with NCES. < Utah State Uiversity. Data Aalysis ad Probability. Natioal Library of Virtual Maipulatives < Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% r Create ad label a circle graph, with or without techology, to display a set of data. r Iterpret a circle graph to aswer questios. 38 Grade 7 Mathematics: Support Documet for Teachers

399 Suggestios for Istructio Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Fid ad compare circle graphs i a variety of prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Traslate percetages displayed i a circle graph ito quatities to solve a problem. Iterpret a circle graph to aswer questios. Materials: BLM 7.SP.3.4: Comparig Examples of Circle Graphs BLM 7.SP.3.5: Traslatig Percetages i a Circle Graph ito Quatities media sources (e.g., magazies, ewspapers, advertisemets, the Iteret) for examples of circle graphs scissors glue or tape Orgaizatio: Small groups Procedure: Part A 1. Divide the class ito small groups. Have the studets i each group search through various media sources to fid five examples of circle graphs. Have them prit or cut out the graphs, icludig ay titles, legeds, or captios that accompay the graphs. 2. Ask each group to aalyze their selected graphs to determie whether or ot each graph icludes the commo attributes of circle graphs. 3. Group members ca take turs recordig iformatio about each graph o the chart provided o BLM 7.SP.3.4: Comparig Examples of Circle Graphs. 4. Have the groups aalyze their graphs ad pose two or three questios that ca be aswered usig iformatio from their graphs. They record their questios o the first page of BLM 7.SP.3.5: Traslatig Percetages i a Circle Graph ito Quatities. Studets the prepare aswers to their questios ad record them o the secod page of the BLM. Statistics ad Probability 39

400 Part B 5. Preset to the class a graph prepared by oe of the groups. Demostrate how the percetages i the graph could be used to solve a problem. Example: A graph about teeage music prefereces shows the percetage of studets who prefer differet musical artists. A music store caterig to teeagers could use this iformatio to choose which products to stock i its store. If 62% of the studets prefer artist B, ad the store is spedig $9500 o ivetory this moth, how much moey should the store sped purchasig artist B s music? 6. Have studets work together to prepare problems that could be solved usig their graphs. Ask them to record the problems o the first page of BLM 7.SP.3.5: Traslatig Percetages i a Circle Graph ito Quatities. Have studets prepare solutios to their problems ad record them o the secod page of the BLM. 7. Post the completed pages for a Gallery Walk. Have studets solve their classmates problems ad verify their solutios. Variatio: Have studets respod to questios usig iformatio i circle graphs available olie. Sample Website: The followig website offers questios related circle graphs, alog with their solutios: icoachmath.com. Circle Graphs Solved Examples. Iowa Math Framework < Circle-GraphsStatistics_cstlqvXEMXDXMAFMXEDGFFXBMDKGXD.html> Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% r Fid ad compare circle graphs i a variety of prit ad electroic media, such as ewspapers, magazies, ad the Iteret. r Traslate percetages displayed i a circle graph ito quatities to solve a problem. r Iterpret a circle graph to aswer questios. 40 Grade 7 Mathematics: Support Documet for Teachers

401 Suggestios for Istructio Create ad label a circle graph, with or without techology, to display a set of data. Traslate percetages displayed i a circle graph ito quatities to solve a problem. Materials: BLM : Percet Circle fractio circles, available o the followig website: Maitoba Educatio. Middle Years Activities ad Games. Mathematics. < a variety of packages (of various sizes) cotaiig coloured items (e.g., coloured beads, cady) compasses protractors BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs (optioal) Orgaizatio: Small groups Procedure: 1. Have studets, workig i small groups, sort the cotets of the supplied packages ito various colours ad use the iformatio to create a circle graph. Studets may use BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs to help orgaize their aswers. They may create their graph with a compass ad a protractor, or usig fractio circles or percet circles provided. 2. Have studets propose a questio to be solved usig the iformatio from their graph. For example, if we buy 2000 cadies, how may ca we expect to be red? 3. Ask each group to exchage their graph with that of aother group, ad solve the ew problem. 4. After solvig the problem, studets retur it to its origiators ad discuss ay discrepacies. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create ad label a circle graph, with or without techology, to display a set of data. r Traslate percetages displayed i a circle graph ito quatities to solve a problem. Statistics ad Probability 41

402 Puttig the Pieces Together Geometric Flags Itroductio: Studets use iformatio from a group survey to create a flag represetative of the group. The they calculate the area of the spaces i the flag. Purpose: I this ivestigatio, studets will demostrate the ability to do the followig (coectios to learig outcomes are idetified i paretheses): Solve problems ivolvig percets. (7.N.3) Costruct circle graphs to solve problems. (7.SP.3) Perform geometric costructios, icludig parallel ad perpedicular lie segmets ad perpedicular ad agle bisectors. (7.SS.3) Perform ad describe trasformatios. (7.SS.5) Apply a formula for determiig the area of triagles, parallelograms, ad circles. (7.SS.2) Studets will also demostrate the followig mathematical processes: Commuicatio Metal Mathematics ad Estimatio Problem Solvig Reasoig Techology Materials/Resources: ruler compass protractor right triagle coordiate grid paper (for the flags) art supplies calculator (optioal) Mira (optioal) tracig paper (optioal) Orgaizatio: Small groups, idividual 42 Grade 7 Mathematics: Support Documet for Teachers

403 Procedure: 1. Survey a group of people regardig their preferred colour. You may survey groups withi your class, i differet classes, i your family, ad so o. If each perso i the group chooses a differet colour, you may wish to have the survey participats select from a list of three to five colours. You may have them rak the colours as first, secod, ad third choice. 2. Calculate the percet of the group that prefers each colour. 3. Costruct a circle graph to represet the prefereces. 4. Desig a flag for the group usig the group s preferred colours. a) Calculate the area of the flag to be covered with each preferred colour. b) Create your desig usig circles, triagles, parallelograms, ad trasformatios. Iclude parallel ad perpedicular lie segmets ad perpedicular ad agle bisectors. c) Adjust the size of each shape to match the area to be covered with each preferred colour. 5. Create a summary chart that shows the areas of the differet shapes i each preferred colour, icludig a) the total area represetig each colour b) the percet of the total area covered by each colour c) a circle graph that represets the percet of the area covered by each colour 6. Create a fial copy of your flag. 7. Prepare a presetatio about your flag. I the presetatio, prove that your flag represets the group fidigs because the area covered by each colour i the flag is the same as the percet of the group that preferred each colour. Highlight some features of your flag ad how you solved a problem i creatig the flag. Statistics ad Probability 43

404 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Calculate percets. r Costruct circle graphs. r Estimate ad calculate areas of triagles, parallelograms, ad circles. r Solve a problem ivolvig the percet of a area. r Perform ad describe a trasformatio i the flag. r Costruct ad idetify parallel ad perpedicular lie segmets i the flag. r Costruct ad idetify perpedicular ad agle bisectors i the flag. r Commuicate mathematical ideas effectively. Extesio: Sew large fabric replicas of each flag ad display them i the classrooms or i the school hallway. 44 Grade 7 Mathematics: Support Documet for Teachers

405 Statistics ad Probability (Chace ad Ucertaity) (7.SP.4, 7.SP.5, 7.SP.6) Edurig Uderstadig(s): Percets, fractios, decimals, ad ratios are differet represetatios of the same quatity. The priciples of the probability of a sigle evet apply to the probability of idepedet evets. Geeral Learig Outcome(s): Use experimetal or theoretical probabilities to represet ad solve problems ivolvig ucertaity. Specific Learig Outcome(s): 7.SP.4 Express probabilities as ratios, fractios, ad percets. [C, CN, R, T, V] 7.SP.5 Idetify the sample space (where the combied sample has 36 or fewer elemets) for a probability experimet ivolvig two idepedet evets. [C, ME, PS] Achievemet Idicators: Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Provide a example of two idepedet evets, such as spiig a four-sectio spier ad a eight-sided die tossig a coi ad rollig a twelve-sided die tossig two cois rollig two dice ad explai why they are idepedet. Idetify the sample space (all possible outcomes) for a experimet ivolvig two idepedet evets usig a tree diagram, a table, or aother graphic orgaizer. (cotiued) Statistics ad Probability 45

406 Specific Learig Outcome(s): 7.SP.6 Coduct a probability experimet to compare the theoretical probability (determied usig a tree diagram, a table, or aother graphic orgaizer) ad the experimetal probability of two idepedet evets. [C, PS, R, T] Achievemet Idicators: Determie the theoretical probability of a outcome for a experimet ivolvig two idepedet evets. Coduct a probability experimet for a outcome ivolvig two idepedet evets, with or without techology, to compare the experimetal probability to the theoretical probability. Solve a probability problem ivolvig two idepedet evets. Prior Kowledge Studets should be able to do the followig: Q Q Q Q Q Q Q Q Q Q Q Q (4.N.8) Demostrate a uderstadig of fractios less tha or equal to oe by usig cocrete ad pictorial represetatios to ame ad record fractios for the parts of a whole or a set compare ad order fractios model ad explai that for differet wholes, two idetical fractios may ot represet the same quatity provide examples of where fractios are used (4.N.9) Describe ad represet decimals (teths ad hudredths) cocretely, pictorially, ad symbolically. (5.N.7) Demostrate a uderstadig of fractios by usig cocrete ad pictorial represetatios to create sets of equivalet fractios compare fractios with like ad ulike deomiators (5.N.8) Describe ad represet decimals (teths, hudredths, thousadths) cocretely, pictorially, ad symbolically. (5.N.9) Relate decimals to fractios (teths, hudredths, thousadths). (5.N.10) Compare ad order decimals (teths, hudredths, thousadths) by usig bechmarks place value equivalet decimals 46 Grade 7 Mathematics: Support Documet for Teachers

407 Q Q Q Q Q Q Q Q Q Q Q Q (5.SP.3) Describe the likelihood of a sigle outcome occurrig, usig words such as impossible possible certai (5.SP.4) Compare the likelihood of two possible outcomes occurrig, usig words such as less likely equally likely more likely (6.N.5) Demostrate a uderstadig of ratio, cocretely, pictorially, ad symbolically. (6.N.6) Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. (6.N.8) Demostrate a uderstadig of multiplicatio ad divisio of decimals ivolvig 1-digit whole-umber multipliers 1-digit atural umber divisors multipliers ad divisors that are multiples of 10 (6.SP.4) Demostrate a uderstadig of probability by idetifyig all possible outcomes of a probability experimet differetiatig betwee experimetal ad theoretical probability determiig the theoretical probability of outcomes i a probability experimet determiig the experimetal probability of outcomes i a probability experimet comparig experimetal results with the theoretical probability for a experimet Related Kowledge Studets should be able to do the followig: Q Q (7.N.3) Solve problems ivolvig percets from 1% to 100%. Q Q (7.N.4) Demostrate a uderstadig of the relatioship betwee repeatig decimals ad fractios, ad termiatig decimals ad fractios. Statistics ad Probability 47

408 Q Q Q Q Q Q Q Q (7.N.7) Compare ad order fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals (7.SP.1) Demostrate a uderstadig of cetral tedecy ad rage by determiig the measures of cetral tedecy (mea, media, mode) ad rage determiig the most appropriate measures of cetral tedecy to report fidigs (7.SP.2) Determie the effect o the mea, media, ad mode whe a outlier is icluded i a data set. (7.SP.3) Costruct, label, ad iterpret circle graphs to solve problems. Backgroud Iformatio I our society, probability is used i makig weather forecasts to express the likelihood of precipitatio, i makig medical ifereces such as the likelihood of cotractig a ifectio or a disease, i makig correlatios betwee lifestyle habits ad health, i predictig electio results, i determiig the chaces of wiig a lottery or a draw, i uderstadig whether or ot a game is fair, ad so o. The study of probability begis i Grade 5, with studets describig the likelihood of a sigle evet occurrig ad comparig the likelihood of two possible outcomes usig the laguage of probability. Uless a evet is the oly possible outcome, or uless it is impossible for a outcome to occur, oe ca ever be certai of a outcome, or of the umber of times a outcome will occur i a give umber of trials. Thus, i Grade 6, studets determie ad compare theoretical probability ad experimetal results. Experimetal Probability Experimetal probability describes what actually did happe i a real situatio. Sometimes experimetal probability is called relative frequecy. Experimetal Probability of a Evet = Number of Observed Favourable Outcomes* Total Number of Trials *A favourable outcome is the outcome that the experimeter is lookig for. If, for example, a experimeter was examiig the probability of rollig a 4 whe rollig a regular umber cube, ad rolled the umber cube 50 times, ad 10 of those times he or she rolled 4s, the experimetal probability would be or 1 or 0.2 or 20% Grade 7 Mathematics: Support Documet for Teachers

409 Probability is a geeralized statemet used to predict future evets. A geeralizatio caot be trusted for makig predictios if it is based o a small umber of trials; however, studets ca icrease the umber of trials i their experimets by combiig trials coducted by differet studets, or by usig computer applets for geeratig large umbers of umber cube rolls, coi tosses, or spis. Sample Websites: Some applets are available o the followig websites: Moore, David S., ad George P. McCabe. Statistical Applets. Itroductio to the Practice of Statistics. 4th ed. New York, NY: W. H. Freema, Available o the W. H. Freema website at < html>. The statistical applets o the W. H. Freema website are iteded to help studets master cocepts addressed i Itroductio to the Practice of Statistics. The Probability applet o this website features a coi-toss simulator. Shodor. Experimetal Probability. Iteractive < Select the spier, or the type of die, ad the umber of trials desired. The simulator tallies ad calculates results. Utah State Uiversity. Data Aalysis ad Probability. Natioal Library of Virtual Maipulatives < I the Grades 6 to 8 sectio, select Coi Tossig or Spiers. A larger umber of trials will permit studets to make a geeralizatio i which they ca have cofidece. The larger the sample size is, the more similar the experimetal results ad the theoretical probability will be. Theoretical Probability Theoretical probability helps studets to predict what is likely to happe i a give circumstace, but does ot foretell what will happe for sure. To calculate the theoretical probability of a evet, the evets must occur radomly, ot iflueced by ay outside force, ad the evets must be equally likely, or have the same chace of occurrig. Theoretical Probability of a Evet = Number of Outcomes Favourable* to the Evet Total Number of Possible Equally Likely Outcomes *A favourable outcome is the desired outcome. If, for example, a experimeter was examiig the probability of rollig a 4 whe rollig a regular umber cube, he or she would kow the umber of favourable outcomes is 1, ad the umber of equally likely outcomes is 6, so the probability is 1 or 016. or 166. %. 6 Statistics ad Probability 49

410 Theoretical probability may be used to make predictios about future evets, whe evets are equally likely. If the evets are ot equally likely, as whe tossig a object ad predictig whether it will lad right-side up, upside dow, or o its side, experimetal probability may be used to make future predictios. I Grade 7, studets express probability as ratios, fractios, or percets. Probability rages betwee impossible 0% ad certai 100% (betwee 0 1 ad or betwee 0 ad 1). 1 1 A probability lie with bechmarks may be used to illustrate the equivalet expressios. Example: Probability Ivestigatios ad Problems Grade 7 studets exted ivestigatios of theoretical ad experimetal probability to iclude two idepedet evets. They also solve probability problems ivolvig two idepedet evets. To ivestigate probability with experimets, studets will eed access to devices that geerate radom results. These may iclude spiers, items to toss (e.g., cois, bicoloured tiles, umber cubes, tacks), ad items to draw from (e.g., cards from a deck, coloured blocks, tiles, marbles from a bag). Computer-simulated applets may also be used. Sometimes probability problems ivolve situatios that may be too dagerous, too expesive, or too difficult to experimet with. I these circumstaces, a simulatio ca be chose to mimic the experimet. For example, i a situatio that requires ivestigatig the possibility of whether a birth is male or female, the two outcomes ca be represeted with the two sides of a coi. 50 Grade 7 Mathematics: Support Documet for Teachers

411 Orgaizig Outcomes of Probability Ivestigatios Studets may begi studyig the probability of two idepedet evets with a cocrete ivestigatio, such as predictig the outcome of tossig two cois. The obvious outcomes are two heads, two tails, ad oe of each, with a likelihood of 1 for either 3 combiatio. A ivestigatio will geerate differet results, ad the differece may iterest studets i orgaizig possible outcomes i a systematic way. Outcomes of probability ivestigatios ca be orgaized with the use of a tree diagram ad a frequecy table or chart: Tree Diagram: A tree diagram lists the possible outcomes for each evet i two colums ad coects them with lies to form braches. The first colum (or row) lists all possible outcomes for the first evet. The secod colum (or row) lists all outcomes for the secod evet beside each of the outcomes i the first evet. A tree diagram is used to describe theoretical probability. Example: Below is a example of a horizotal tree diagram of the possible outcomes for the idepedet evets whe tossig two cois. Number of Favourable Outcomes The Probability ofay Evet = Number ofpossible Outcomes e.g., P T, T ( ) = 1 4 Statistics ad Probability 51

412 Frequecy table or chart: Whe used to predict possible outcomes of a evet, a frequecy table or chart orgaizes all possible outcomes for two idepedet evets. Each cell i the table idicates oe possible outcome. Example: Number of Favourable Outcomes The Probability ofay Evet = Number ofpossible Outcomes e.g., P( assumig orderdoes 1H, 1T) = 2 = 50% * ot matter * 4 Orgaizig outcomes helps to reveal hidde outcomes, as i the examples above, where studets may erroeously believe there are oly three possible outcomes: 2 heads, 2 tails, ad oe of each. To demostrate a thorough uderstadig of probability, studets should recogize the followig: Outcomes must be equally likely. For example, a spier that is 1 2 red, 1 yellow, ad blue has 2 possibilities for red, ot 1. 4 The probability of a outcome is equivalet whe tossig the same umber cube six times or whe tossig six umber cubes oe time each. Or, if there are six red marbles ad six white marbles i a bag, the chace of pullig either colour is the same as if there is oe red marble ad oe white marble i the bag. For idepedet evets, the probability of a outcome does ot chage based o the previous outcome. Whe a evet such as rollig a sum of 7 o two umber cubes has repeated itself several times, there is o icreased likelihood that the ext roll will or will ot be a sum of 7. The likelihood of rollig or ot rollig a sum of 7 remais the same, regardless of the outcome i the previous roll. Kowig the vocabulary terms is importat. For example, likely meas more tha 50% of the time, ot almost all the time. 52 Grade 7 Mathematics: Support Documet for Teachers

413 Mathematical Laguage certai evet depedet evet evet experimetal probability favourable outcome frequecy table or chart impossible evet idepedet evet likely outcome possible outcome probability radom relative frequecy sample size sample space theoretical probability tree diagram Statistics ad Probability 53

414 Learig Experieces Assessig Prior Kowledge Materials: a set of two multi-sided umber cubes or spiers per group calculators (optioal) BLM 7.SP.4.1: Recordig Sheet for Fractio Decimal Percet Equivalets (optioal) paper (optioal) Orgaizatio: Whole class, small groups (of three or five) Procedure: 1. As a class, review writig equivalet fractios, decimals, ad percets. 2. Orgaize the class ito groups, ad have each group desigate a Perso A, B, ad C. 3. Demostrate a few rouds of the game usig three voluteers. The aim is to move as quickly ad accurately as possible to create 10 equivalet fractios, decimals, ad percets, usig the followig routie: a) Perso A: Roll two umber cubes. b) Perso B: Use the umbers o the umber cube to create a proper fractio. c) Perso C: Express the fractio as a equivalet decimal umber. d) Perso A: Express the umber as a equivalet percet. e) Perso B: Roll two umber cubes. The routie cotiues. f) Use BLM 7.SP.4.1: Recordig Sheet for Fractio Decimal Percet Equivalets to record results. Perso A begis the sheet after rollig the umber cube. Perso A records what Perso B says, ad the passes the sheet to Perso B. Perso B records what perso C says, ad passes the sheet to perso C. The routie cotiues, with the sheet followig behid the perso who is aswerig. 4. Groups carry o with the game, racig to see who ca be the first group to create 10 equivalet fractios, decimals, ad percets correctly. 5. Play as may rouds as seems iterestig ad useful. 54 Grade 7 Mathematics: Support Documet for Teachers

415 Variatio: Play the game as a baseball game. The pitcher pitches two umbers. The batter gets to first base by amig the proper fractio, to secod base by amig the decimal, ad to third base by amig the percet. By fiishig withi a time limit, the batter gets to home base ad ears a ru. Failure to respod i a give time results i a out. The game ca be played as a whole class or i pairs, usig a paper baseball diamod ad markers. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Name part-to-whole ratios as fractios ad their decimal ad percet equivalets. Suggestios for Istructio Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. Q Q Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Materials: BLM 7.SP.4.2: What Is the Probability? a bag of letter tiles (B, R, I, D, G, E) (optioal) Orgaizatio: Idividual, whole class Procedure: 1. Distribute copies of BLM 7.SP.4.2: What Is the Probability? 2. Have studets aswer the questios o the sheet o their ow. The questios require studets to a) idetify outcomes ad probabilities b) compare experimetal ad theoretical probability c) idetify outcomes as impossible, certai, or more likely d) create ad aswer their ow questio related to probability 3. Reassemble as a class ad have studets share their resposes to the questios, correctig ay errors. Statistics ad Probability 55

416 Variatios: As a class, complete BLM 7.SP.4.2: What Is the Probability? ad discuss ay questios studets may have. Have studets work i small groups. Provide each group with the bag of letter tiles (B, R, I, D, G, E), ad have them ivestigate the probability of drawig each letter i 36 trials. Combie the results of the small groups, recalculate the experimetal probability, ad compare the probability for the combied results to the results of the idividual groups. Ask studets to discuss why the experimetal results ad the theoretical probability are differet from each other. Have studets, workig i pairs, choose their ow letter tiles (or other devices) ad prepare a similar questio sheet ad aswer key. Groups exchage sheets, complete ad correct the questios, ad resolve ay discrepacies betwee the resposes. Have studets desig spiers, or situatios, that would result i give probabilities. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Describe sigle evets as impossible, possible, certai, less likely, more likely, or equally likely. r Idetify possible outcomes i a probability sceario. r Differetiate betwee experimetal ad theoretical probability. r Determie theoretical ad experimetal probability. r Compare experimetal results with theoretical probability. 56 Grade 7 Mathematics: Support Documet for Teachers

417 Suggestios for Istructio Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. Q Q Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Materials: Oe set per pair of studets: six-sided pecils otebooks markers BLM 7.SP.4.3: Experimetal Probability Tally Sheet ad Probability of Outcomes (optioal) Orgaizatio: Pairs, whole class Procedure: 1. Have studets, workig i pairs, mark each side of a six-sided pecil with oe possible outcome (e.g., umbers, studet ames, letters to form a word). 2. Studets coduct their probability experimet, proceedig as follows: a) Oe studet rolls the pecil by rubbig it betwee his or her palms, ad the lays a had o a otebook (to muffle the soud) ad allows the pecil to roll dow the had oto the otebook. b) The studet aouces the outcome that laded facig up. c) The other studet records the result o a tally sheet. d) Studets cotiue rollig ad recordig util they have completed a set umber of rolls, or util a set time has passed. 3. Studets total their tally marks, ad record the ratios of tallies for each outcome to the total umber of tallies (outcome : total). 4. Studets the record the probability for each outcome, expressig it as a fractio, a decimal, ad a percet. 5. Whe groups have completed their experimet, reassemble as a class ad ask studets what they have discovered or leared from the experiece. a) Compare the probabilities obtaied by differet groups. If studets have used the same outcomes, cosolidate the group data ad calculate the probability for the combied results. b) Studets may ote the sum of the values i the decimal colum totals about 1 (depedig o roudig), ad the sum of the percets is close to 100% (depedig o roudig). Statistics ad Probability 57

418 c) Ask studets for a example of a outcome that is impossible for this experimet, ad for a evet that is certai. Variatios: Have all studets mark their six-sided pecils with the same outcomes, ad combie the class results to represet a larger trial. Evets that are ot equally likely, such as a paper cup ladig upright, upside dow, or o its side, may be used for this experimet. Compare studets results ad discuss differeces observed. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. r Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Suggestios for Istructio Q Q Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Materials: rulers (30 cm) two colours of pes or highlighters demostratio board Orgaizatio: Whole class, pairs 58 Grade 7 Mathematics: Support Documet for Teachers

419 Procedure: 1. Remid studets that probability is used to predict the likelihood that a evet will happe. Ask whether it is possible to predict ay evet with certaity. Aswer: The oly evets that ca be predicted with certaity are those with probabilities of 0% or 100%. 2. Ask studets for examples of evets with a probability of 0% (impossible) or 100% (certai). Example: If a draw box cotais oly odd umbers, the probability of drawig a eve umber is 0% ad the probability of drawig a odd umber is 100%. 3. Have studets create a probability lie similar to the Sample Probability Lie illustrated i the Backgroud Iformatio. Have studets iclude some or all of the followig vocabulary i their probability lie: impossible, less possible, more possible, certai less likely, equally likely, more likely less probable, more probable ever, sometimes, ofte, always 4. Have pairs of studets challege oe aother by describig situatios that could match the probability of a poit or regio o the probability lie. Example: Variatios: Studet A describes a situatio, ad asks for a evet that would match a probability descriptor: Give a bag cotaiig the letters of the alphabet, idetify a evet that would be less likely tha aother evet. Studet B replies: Give a bag cotaiig the letters of the alphabet, it is less likely you ll draw a vowel tha a cosoat. Studet B the challeges Studet A, ad may ask for a impossible evet i the same situatio. Studet A could reply: Give the letters of the alphabet, it is impossible to draw a umber. Provide a template of a probability lie with bechmarks, directioal arrows, ad blaks for studets to fill i. Q Q As a extesio, have studets idetify situatios with evets that have a 50% probability (or other value) of occurrig. Statistics ad Probability 59

420 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Suggestios for Istructio Provide a example of two idepedet evets, such as spiig a four-sectio spier ad a eight-sided die tossig a coi ad rollig a twelve-sided die tossig two cois rollig two dice ad explai why they are idepedet. Materials: BLM 7.SP.5.1: Which Coditios Affect Probability? cois colour couters bags ticket stubs with umbers (optioal) BLM 7.SP.5.2: Examples of Two Idepedet Evets (optioal) Orgaizatio: Small groups (of three or four studets), whole class Procedure: 1. Divide studets ito small groups. 2. Distribute copies of BLM 7.SP.5.1: Which Coditios Affect Probability? 3. Have each group determie the theoretical probability of each evet occurrig, decide whether specified coditios affect the theoretical probability of the evet, ad explai why or why ot. 4. Reassemble as a class ad discuss studets resposes. Probability is based o the umber of favourable outcomes ad the umber of possible outcomes. Oly coditios that alter the umber of outcomes (e.g., addig a extra item, ot replacig a item that has bee removed) will affect the theoretical probability of a evet. 60 Grade 7 Mathematics: Support Documet for Teachers

421 5. Iform studets of the followig: a) Whe oe evet does ot affect the probability of a outcome of aother evet, the evets are said to be idepedet evets. Ask studets to idetify the idepedet evets i the examples o BLM 7.SP.5.1. b) Whe oe evet does affect the probability of a outcome of aother evet, the evets are said to be depedet evets. Ask studets to idetify the depedet evets i the examples o BLM 7.SP.5.1. Have studets idetify other evets that would ot represet idepedet evets. 6. Ask studets whether rollig a umber cube before pullig a coloured block from a bag will have ay effect o the colour of the block draw. 7. Have studets retur to their small groups ad list pairs of idepedet evets, explaiig why the two evets are idepedet. Ask them to list some pairs of evets that are ot represetative of idepedet evets. Examples: Idepedet evets could iclude Variatios: choosig a card from a deck (or a marble, other couter, umber, or letter tile from a bag), returig it to the deck (or bag), ad drawig a secod item tossig two cois, umber cubes, or letter cubes radomly choosig two items (e.g., ames, umbers, meu items, clothig selectios, colours, movies, trasportatio methods, vacatio spots, games) from a selectio (The first choice must ot be removed from the set, for the evets to remai idepedet.) ay combiatio of the above items Provide cocrete opportuities for studets to experimet with determiig whether or ot probability is affected by certai coditios. For example, collect tickets or ames o slips ad coduct a draw to determie the probability of a particular wier. Coduct the experimet, first replacig a ame each time it is draw, ad the ot replacig a ame after it has bee draw. Provide a chart (such as BLM 7.SP.5.2: Examples of Two Idepedet Evets) for studets to list sets of two idepedet evets ad to explai why the evets are idepedet. Provide studets with a list of pairs of evets ad ask them to idetify whether or ot the pairs represet idepedet evets, ad to explai why the evets are or are ot idepedet. Statistics ad Probability 61

422 Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Provide a example of two idepedet evets, such as spiig a four-sectio spier ad a eight-sided die tossig a coi ad rollig a twelve-sided die tossig two cois rollig two dice ad explai why they are idepedet. Suggestios for Istructio Idetify the sample space (all possible outcomes) for a experimet ivolvig two idepedet evets usig a tree diagram, a table, or aother graphic orgaizer. Materials: cois (the same or differet deomiatios) demostratio board math jourals BLM 7.SP.4.3: Experimetal Probability Tally Sheet ad Probability of Outcomes (optioal) Orgaizatio: Pairs or small groups, whole class, idividual Procedure: 1. Make preparatios for havig studets ivestigate the probability of outcomes whe tossig two cois. a) Divide studets ito pairs or small groups. b) Iform studets they will be experimetig to determie the possible outcomes, ad the probability of each outcome, whe tossig two cois. Have studets create a recordig sheet for their experimet (or distribute copies of BLM 7.SP.4.3: Experimetal Probability Tally Sheet ad Probability of Outcomes). c) Tell studets they will idetify all possible outcomes ad predict the probability of each outcome. 62 Grade 7 Mathematics: Support Documet for Teachers

423 d) Discuss whether the procedure for tossig the cois (tossig both cois at oce, or tossig them oe after the other) will affect the results of the experimet. Neither will affect the probability; however, good experimetal techique esures that the same procedure is followed as closely as possible throughout a experimet. 2. Have studets coduct the ivestigatio i pairs or i small groups. a) Distribute two cois to each group. b) Have studets coduct their experimet for a give umber of coi tosses, or for a set period of time. c) Have studets total their results. d) Cosolidate studets data o the demostratio board. 3. Compare studets results to their predictios, ad discuss differeces or similarities. If studets predicted a probability of 1 for each outcome, the results will likely ot 3 support that predictio. This raises a opportuity for discussig how to determie the sample space (umber of possible outcomes) for two idepedet evets. 4. Calculate probability as the umber of favourable outcomes out of the umber of possible outcomes (the sample space). Number of Favourable Outcomes P Number of Possible Outcomes a) Agree that the favourable outcomes are two heads, two tails, or oe of each. b) Ask studets to idetify all possible outcomes. 5. Ask studets to demostrate, or demostrate for them, systematic ways to orgaize the sample space for two idepedet evets i a way that esures all possible outcomes are idetified. If studets do ot see the beefits of orgaizatioal charts, have them idetify the sample space for evets with multiple outcomes, such as rollig two umber cubes. 6. Orgaizers iclude the followig (see Backgroud Iformatio for examples): a) tree diagrams b) frequecy charts or tables c) orgaized lists 7. Have studets make a math joural etry illustratig methods to idetify the sample space (all possible outcomes) for two idepedet evets. Variatios: Use computer applets to simulate coi tosses. Replace the itroductory ivestigatio with a guided example of tossig two cois as two idepedet evets. Demostrate calculatig the sample space with a tree diagram, ad the with a frequecy chart or table, as illustrated i the Backgroud Iformatio. Statistics ad Probability 63

424 Demostrate usig orgaizers for idetifyig the sample space for a experimet ivolvig two idepedet evets, ad the have studets idetify such a experimet ad use two methods to idetify the correspodig sample space. Studets may use evets from the list they created i the previous learig experiece. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify the sample space (all possible outcomes) for a experimet ivolvig two idepedet evets usig a tree diagram, a table, or aother graphic orgaizer. Suggestios for Istructio Determie the theoretical probability of a outcome for a experimet ivolvig two idepedet evets. Coduct a probability experimet for a outcome ivolvig two idepedet evets, with or without techology, to compare the experimetal probability to the theoretical probability. Materials: maipulatives (e.g., cois, umber cubes, colour couters, marbles, letter tiles, bags) software programs or olie applets (such as those listed i the Backgroud Iformatio) BLM 7.SP.6.1: Frequecy Chart for Orgaizig Outcomes for Two Idepedet Evets (optioal) Orgaizatio: Idividual or pairs Procedure: 1. Iform studets they will desig ad coduct a probability experimet ivolvig two idepedet evets, as outlied i the followig steps. a) Choose two idepedet evets. Previous lists provide examples of evets to choose from. b) Determie the sample space (all possible outcomes) by drawig a tree diagram, a frequecy chart, or aother orgaizer. BLM 7.SP.6.1: Frequecy Chart for Orgaizig Outcomes for Two Idepedet Evets may be provided as a template. 64 Grade 7 Mathematics: Support Documet for Teachers

425 c) Determie the theoretical probability of a outcome for the experimet. d) Coduct the experimet usig maipulatives or computer applets or software programs. e) Determie the experimetal probability for the outcome. f) Compare the experimetal probability to the theoretical probability. Iclude descriptive statemets ad umerical statemets i the comparisos. Propose a explaatio for variatios i the results. 2. Have studets preset their ivestigatios to each other, or post the ivestigatios for studets to view. Variatios: Assig studets two idepedet evets to use for the experimet. Have all studets work o the same ivestigatio, ad compile their results for comparig the theoretical ad experimetal probability. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the theoretical probability of a outcome for a experimet ivolvig two idepedet evets. r Coduct a probability experimet for a outcome ivolvig two idepedet evets, with or without techology, to compare the experimetal probability to the theoretical probability. Statistics ad Probability 65

426 Suggestios for Istructio Solve a probability problem ivolvig two idepedet evets. Materials: demostratio board display board Orgaizatio: Whole class, pairs Procedure: 1. Idetify games of chace that ivolve two idepedet evets, such as the followig: Rock, Paper, Scissors (played with two people) games that ivolve rollig umber cubes (e.g., Lucky Seve) games that ivolve drawig marbles from a bag (e.g., Player A draws a marble from a bag cotaiig a give assortmet of marbles, ad returs the marble to the bag. Player B the draws a marble. If the colours match, Player B scores a poit; if they do t match, Player A scores the poit.) 2. Ask whether a game is fair, ad how oe judges whether or ot it is fair. 3. As a class, ivestigate a game of chace, such as Rock, Paper, Scissors, to determie whether or ot the game is fair. 4. Have idividuals or pairs of studets determie whether or ot other games of chace are fair. Have them provide evidece for their coclusios. 5. Provide opportuities for studets to play the games. 6. Use studets results to create a display etitled Would You Play This Game? Variatios: Have pairs of studets play a game based o the probability of two idepedet evets occurrig. After a certai time has passed, ask studets who thiks the game is fair. Challege studets to support their opiio usig what they kow about probability. Have studets desig a fair game based o the probability of two idepedet evets occurrig. Have them provide evidece that the game is fair. Host a Game Fair i which studets itroduce ad play their games with others. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a probability problem ivolvig two idepedet evets. 66 Grade 7 Mathematics: Support Documet for Teachers

427 Suggestios for Istructio Solve a probability problem ivolvig two idepedet evets. Materials: BLM.7.SP.6.2: Probability Problems Ivolvig Two Idepedet Evets file cards (optioal) Orgaizatio: Idividual, pairs Procedure: 1. Preset studets with probability problems ivolvig two idepedet evets, such as those o BLM.7.SP.6.2: Probability Problems Ivolvig Two Idepedet Evets. 2. Ask studets to complete the problems idepedetly ad the compare their resposes with those of a classmate, resolvig ay discrepacies. Variatios: Ask studets to desig their ow probability problems ad solutios. Have studets cosolidate problems as a assigmet sheet ad a aswer key, ad share questio sheets with others. Studets compare their resposes to the aswer key, ad discuss ay discrepacies with the authors. Place studet-created probability problems ad solutios o large file cards. Studets ca pick a card or two, ad complete the problems as part of a learig activity cetre or as a Exit Slip. Play olie probability games requirig players to solve probability problems ivolvig both simple ad idepedet evets. Sample Website: For a sample probability game, refer to the followig website: Math-Play.com. Probability Game < Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a probability problem ivolvig two idepedet evets. Statistics ad Probability 67

428 N o t e s 68 Grade 7 Mathematics: Support Documet for Teachers

429 G r a d e 7 M a t h e m a t i c s Appedix: Models for Computig Decimal Numbers

430

431 A p p e d i x : M o d e l s f o r C o m p u t i g D e c i m a l N u m b e r s This appedix focuses o demostratig computatio with decimal umbers usig base-10 blocks ad umber lies. Base-10 Blocks Base-10 blocks ca be used to represet the operatios of additio, subtractio, multiplicatio, ad divisio of decimal umbers. The model is based o viewig decimals as fractioal equivalets. A fractio is viewed as a whole, cut ito equivalet pieces. The whole is divided ito teths to represet or 0.1. The whole is divided ito hudredths to represet or Studets must have a fluet uderstadig of the umeric values of the model. If they lack this uderstadig, their attetio will be focused o tryig to make the model istead of learig to work with decimal umbers. Have studets physically separate the blocks, or cut paper grids, to help them attach meaigful values to the represetatios. Sped time amig various combiatios of blocks ad creatig represetatios of various decimal umbers to develop fluecy. Base-10 grid paper serves as a two-dimesioal represetatio of the base-10 blocks. (See BLM : Base-Te Grid Paper.) Note: It is importat that studets work flexibly with various represetatios to develop their uderstadig of operatios with decimal umbers, rather tha memorizig the steps without uderstadig their meaig. Represetatios* If the flat represets a whole, its value is 1 its dimesios are 1 uit by 1 uit it has a area of 1 uit 2 If the rod represets oe-teth of a whole, its value is 0.1 its dimesios are 1 uit by 0.1 uit it has a area of 0.1 uits 2 * For the purposes of this resource, models are represeted without all idividual blocks draw. Appedix: Models for Multiplyig Decimal Numbers 3

432 If the small cube represets oe-hudredth of a whole, its value is 0.01 its dimesios are 1 uit by 0.01 uit it has a area of 0.01 uits 2 Arragig the Blocks as Arrays or Area Models 0.2 uits 2 requires 2 teths rods (or 20 hudredths cubes) it may be arraged as or uits 2 requires 2 flats ad 2 rods (or 22 rods or 220 cubes) It is importat to establish a covetio of keepig the blocks orgaized, as it will help with developig future represetatios. This model is arraged as Grade 7 Mathematics: Support Documet for Teachers

433 Modellig Additio If you thik it would be helpful, have studets work o place value mats. Model : Select the blocks to represet (1 flat, 4 rods, 6 small cubes) Select the blocks to represet (4 rods, 5 small cubes) Combie the blocks together. Group similar blocks where possible. Te blocks of oe value are exchaged for oe block of the ext larger value. The ew quatity represets Appedix: Models for Multiplyig Decimal Numbers 5

434 Modellig Subtractio If you thik it would be beeficial, have studets work o place value mats. Model : Select the blocks to represet (2 flats, 3 rods, 6 small cubes) If possible, remove blocks that represet (8 rods, 5 small cubes) If there are isufficiet similar blocks to allow removal of the required amout, exchage oe block of the ext largest value for 10 of the required blocks. 6 Grade 7 Mathematics: Support Documet for Teachers

435 The ew quatity represets Modellig Multiplicatio Base-10 blocks ca be used to represet both the active uderstadig of multiplicatio as a specific umber of groups of a specific size, or the o-active array represetatio of a quatity. Whe the blocks are arraged as a rectagle, the rectagle may be rotated, ad the quatity does ot chage. This is a verificatio of the commutative property of multiplicatio. The orietatio of the array has o affect o the result, but i some places a covetio has bee established of represetig the first umber horizotally ad the secod vertically. The array also serves as a model for area. It represets the area covered by a rectagle with a legth of the multiplicad ad a width of the multiplier, or vice versa. Model usig active uderstadig: Uderstad the statemet as 2 groups of 0.4. Select blocks to represet 1 group of 4 teths. Select a secod group of 4 teths. Combie the two groups. The ew quatity represets 0.8. Appedix: Models for Multiplyig Decimal Numbers 7

436 Model usig o-active uderstadig: Arrage the blocks ito a rectagle, with oe dimesio havig a liear measure of 2 uits ad the other dimesio havig a liear measure of 0.4. The arragemet requires 8 rods ad represets 0.8. Rotatig this model represets the commutative property of multiplicatio. Modellig Divisio Model usig active uderstadig: Uderstad the statemet as either how may groups of 2 are i 1.4 or how may will be i each group if 1.4 is shared betwee 2 groups. Select base-10 blocks to represet 1.4. Divide the blocks ito 2 groups. The quotiet is 0.7. Model usig o-active uderstadig: Represet the divisor (1.4) as a rectagle with the legth of oe side equal to the divided (2). The legth of the other side will be the quotiet. 8 Grade 7 Mathematics: Support Documet for Teachers

437 Number Lies Modellig Divisio 8 4 = 2 Begi at 8, ad jump back to 0, puttig 4 steps i each jump. It takes 2 jumps to reach = 20 As 0.4 is less tha 0.5 of a jump, there are more tha 2 jumps of 0.4 i each 1, so there are more tha 16 jumps i 8. This is a good place to discuss estimatig strategies ad the cocept that the result of a umber divided by less tha 1 will always be larger tha the origial umber = 2 Chage the values o the lie to teths. Jumpig back by 0.4 requires 2 jumps = 0.2 Take the 0.8 spaces ad divide them ito 4 groups. Each group cotais 0.2. Appedix: Models for Multiplyig Decimal Numbers 9

438 N o t e s 10 Grade 7 Mathematics: Support Documet for Teachers

439 G r a d e 7 M a t h e m a t i c s Bibliography

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441 B i b l i o g r a p h y Earl, Lora M., Steve Katz, ad Wester ad Norther Caadia Protocol for Collaboratio i Educatio. Rethikig Classroom Assessmet with Purpose i Mid: Assessmet for Learig, Assessmet as Learig, Assessmet of Learig. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at < Gravett, Emily. The Rabbit Problem. Lodo, UK: Macmilla Childre s Books, MacPherso, Eric D., ad Leo A. Rousseau. A Nomological Network for Elemetary School Mathematics. 6th ed. Wiipeg, MB: The Uiversity of Maitoba, Maitoba Educatio. Grade 5 Mathematics: Support Documet for Teachers. Wiipeg, MB: Maitoba Educatio, Available olie at < math/support_gr5/idex.html>.. Grade 8 Mathematics: Support Documet for Teachers. Wiipeg, MB: Maitoba Educatio, Available olie at < support_gr8/idex.html>. Maitoba Educatio ad Traiig. Grades 5 to 8 Mathematics: A Foudatio for Implemetatio. Wiipeg, MB: Maitoba Educatio ad Traiig, Success for All Learers: A Hadbook o Differetiatig Istructio: A Resource for Kidergarte to Seior 4. Wiipeg, MB: Maitoba Educatio ad Traiig, Maitoba Educatio, Citizeship ad Youth. Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at < framework_9-12/>.. Kidergarte to Grade 8 Mathematics Glossary: Support Documet for Teachers. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at < Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at < Middle Years Assessmet: Grade 7 Mathematics: Support Documet for Teachers: Eglish Program. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, Available olie at < McCallum, A. Rabbits Rabbits Everywhere: A Fiboaci Tale. Illus. Gideo Kedall. Watertow, MA: Charlesbridge Publishig Ic., Neuschwader, Cidy. Sir Cumferece ad the Isle of Immeter: A Math Adveture. Illus. Waye Geeha. Watertow, MA: Charlesbridge Publishig Ic., Sachar, Louis. More Sideways Arithmetic from Wayside School. New York, NY: Scholastic Ic., Bibliography 3

442 . Sideways Arithmetic from Wayside School. New York, NY: Scholastic Ic., Small, Maria. Makig Math Meaigful to Caadia Studets, K 8. Toroto, ON: Nelso Educatio Ltd., Timmos, Daiel L., Catherie W. Johso, ad Soya M. McCook. Fudametals of Algebraic Modelig: A Itroductio to Mathematical Modelig with Algebra ad Statistics. 5th ed. Belmot, CA: Brooks/Cole Cegage Learig, Va de Walle, Joh A., ad Sadra Folk. Elemetary ad Middle School Mathematics: Teachig Developmetally. 2d Caadia ed. Toroto, ON: Pearso Educatio Caada, Va de Walle, Joh A., ad LouA H. Lovi. Teachig Studet-Cetered Mathematics Grades 5 8. Bosto, MA: Pearso Educatio, Ic., Wester ad Norther Caadia Protocol (WNCP). The Commo Curriculum Framework for K 9 Mathematics. Edmoto, AB: Govermets of Alberta, British Columbia, Maitoba, Northwest Territories, Nuavut Territory, Saskatchewa, ad Yuko Territory, May Available olie at < mathematics/ccf.aspx>. Websites Ay websites refereced i this documet are subject to chage. Educators are advised to preview ad evaluate websites ad olie resources before recommedig them for studet use. Barile, Margherita. Eye of Horus Fractios. MathWorld A Wolfram Web Resource, created by Eric W. Weisstei. < (12 Ja. 2012). BBC. Shape, Space ad Measures. KS2 Bitesize < ks2bitesize/maths/shape_space/trasformatio/play.shtml> (19 July 2011). Bogomoly, A. Area of Triagle. Iteractive Mathematics Miscellay ad Puzzles < (20 Dec. 2011). Crossword Puzzle Games. Create a Crossword Puzzle < (28 Dec. 2011). Cut the Kot. Area of Triagle. Geometry Articles, Theorems, Problems < (19 July 2011). FuBased Learig. Medium Versio of Graph Mole. Graphig < (19 July 2011). GreatScott.com. Eye of Horus Fractios. Hieroglyphs < (12 Ja. 2012). 4 Grade 7 Mathematics: Support Documet for Teachers

443 icoachmath.com. Circle Graphs Solved Examples. Iowa Math Framework < Circle-GraphsStatistics_cstlqvXEMXDXMAFMXEDGFFXBMDKGXD.html> (19 July 2011). Lawrece, Sezaa. Eye of Horus Fractios. Maths Is Good for You! < (12 Ja. 2012). Maitoba Educatio. Middle Years Activities ad Games. Mathematics. < (19 July 2011). The Math Drexel. Aciet Greek Maps. Chameleo Graphig < (19 July 2011). MathIsFu.com. Symmetry Artist. Geometry < symmetry-artist.html> (19 July 2011). Math Ope Referece. Agles < (23 Nov. 2011).. Area of a Triagle. Triagles < (24 Nov. 2011).. Cetral Agle. Circles < (19 July 2011).. Circles < (23 Nov. 2011). Math-Play.com. The Coordiate Plae. Coordiate Plae Game. < Coordiate%20Plae%20Game/Coordiate%20Plae%20Game.html> (19 Nov. 2011).. Probability Game < (19 July 2011). Math Playgroud. Circle Graphs. Math Maipulatives < (19 July 2011). Maths Olie. Trasformatio Golf. < (19 July 2011). Moore, David S., ad George P. McCabe. Statistical Applets. Itroductio to the Practice of Statistics. 4th ed. New York, NY: W. H. Freema, Available o the W. H. Freema website at < html> (15 Dec. 2011). North Cato City Schools. Explorig Pi. Excel Activities for the Classroom. < (19 July 2011). Plottig Coordiates.com. CoordiArt News < (19 July 2011). Bibliography 5

444 Reed, Jim The Coordiate Plae. Grade 7: The Learig Equatio Math < (19 July 2011). Scheider, Michael S. Geometry of the North Rose Widow of Chartres Cathedral. Costructig the Uiverse. < Chartres%20Widow.html> (19 July 2011). Scweb4free.com. Circle Graphs Game < (19 July 2011). Shodor. Experimetal Probability. Iteractive < (19 July 2011).. Trasmographer. Iteractive < activities/trasmographer/> (19 July 2011). Stapel, Elizabeth. Purplemath < (19 July 2011). Statistics Caada. Data ad Results. Cesus at School. 29 July < (19 Dec. 2011).. Learig Resources. 19 Aug < (19 Dec. 2011). Stich, Mike. Parallelogram. 26 May flickr. < (19 July 2011). U.S. Departmet of Educatio. Istitute of Educatio Scieces, Natioal Ceter for Educatio Statistics. Create a Graph. Kids Zoe, Learig with NCES. < (19 July 2011). Utah State Uiversity. Data Aalysis ad Probability. Natioal Library of Virtual Maipulatives < (19 July 2011).. Natioal Library of Virtual Maipulatives < (19 July 2011).. Number ad Operatios (Grades 3 5). Natioal Library of Virtual Maipulatives < (19 July 2011). Wolfram Demostratios Projects. Symbol Rotatio Patters. Cotributed by Daielle Nogle. Based o a program by Chris Carlso < (19 July 2011). World-Mysteries.com. Fiboacci Numbers i Nature ad the Golde Ratio. Sciece Mysteries < (19 July 2011). 6 Grade 7 Mathematics: Support Documet for Teachers

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