Grade 7 Mathematics. Support Document for Teachers

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1 Grade 7 Mathematics Support Documet for Teachers

2

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

28

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

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