CONCEPT DEVELOPMENT Mathematics Assessmet Project CLASSROOM CHALLENGES A Formative Assessmet Lesso Iterpretig Algebraic Expressios Mathematics Assessmet Resource Service Uiversity of Nottigham & UC Berkeley For more details, visit: http://map.mathshell.org 015 MARS, Shell Ceter, Uiversity of Nottigham May be reproduced, umodified, for o-commercial purposes uder the Creative Commos licese detailed at http://creativecommos.org/liceses/by-c-d/3.0/ - all other rights reserved
Iterpretig Algebraic Expressios MATHEMATICAL GOALS This lesso uit is iteded to help you assess how well studets are able to traslate betwee words, symbols, tables, ad area represetatios of algebraic expressios. It will help you to idetify ad support studets who have difficulty: Recogizig the order of algebraic operatios. Recogizig equivalet expressios. Uderstadig the distributive laws of multiplicatio ad divisio over additio (expasio of paretheses). COMMON CORE STATE STANDARDS This lesso relates to the followig Stadards for Mathematical Cotet i the Commo Core State Stadards for Mathematics: A-SSE: Iterpret the structure of expressios. This lesso also relates to the followig Stadards for Mathematical Practice i the Commo Core State Stadards for Mathematics, with a particular emphasis o Practices 1,, ad 7: 1. Make sese of problems ad persevere i solvig them.. Reaso abstractly ad quatitatively. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Atted to precisio. 7. Look for ad make use of structure. 8. Look for ad express regularity i repeated reasoig. INTRODUCTION The lesso uit is structured i the followig way: Before the lesso, studets work idividually o a assessmet task desiged to reveal their curret uderstadig ad difficulties. You the review their work ad formulate questios for studets to aswer, to help them improve their solutios. Durig the lesso, studets work i pairs or threes to traslate betwee word, symbol, table of values, ad area represetatios of expressios. I a whole-class discussio, studets fid differet represetatios of expressios ad explai their aswers. Fially, studets retur to their origial assessmet task ad try to improve their ow resposes. MATERIALS REQUIRED Each studet will eed two copies of Iterpretig Expressios, a mii-whiteboard, pe, ad eraser. Each pair of studets will eed glue, a felt-tipped pe, a large sheet of poster paper, ad cut-up copies of Card Set A: Expressios, Card Set B: Words, Card Set C: Tables, ad Card Set D: Areas. Note that the blak cards are part of the activity. If you thik you will eed to cotiue with the activities ito a secod lesso, provide evelopes ad paper clips for storig matched cards betwee lessos. TIME NEEDED 10 miutes for the assessmet task, a 90-miute lesso (or two 50-miute lessos), ad 10 miutes i a follow-up lesso. All timigs are approximate ad will deped o the eeds of the class. Teacher guide Iterpretig Algebraic Expressios T-1
BEFORE THE LESSON Assessmet task: Iterpretig Expressios (10 miutes) Have studets do this task i class or for homework a day or more before the formative assessmet lesso. This will give you a opportuity to assess the work ad to fid out the kids of difficulties studets have with it. You should the be able to target your help more effectively i the subsequet lesso. Give each studet a copy of Iterpretig Expressios. I wat you to sped te miutes workig idividually o this task. Iterpretig Expressios 1. Write algebraic expressios for each of the followig: a. Multiply by 5 the add 4. b. Add 4 to the multiply by 5. c. Add 4 to the divide by 5. d. Multiply by the multiply by 3. e. Multiply by 3 the square the result.. The equatios below were created by studets who were asked to write equivalet expressios o either side of the equals sig. Imagie you are a teacher. Your job is to decide whether their work is right or wrog. If you see a equatio that is false, the: a. Cross out the expressio o the right ad replace it with a expressio that is equivalet to the oe o the left. b. Explai what is wrog, usig words or diagrams. Do t worry too much if you ca t uderstad or do everythig. There will be a lesso [tomorrow] with a similar task that will help you improve. It is importat that, as far as possible, studets are allowed to aswer the questios without assistace. If studets are strugglig to get started, ask them questios that help them to uderstad what is required, but do ot do the task for them. ( + 3) = + 3 10 " 5 = " 1 5 (5) = 5 ( + 3) = + 3 = + 9 Assessig studets resposes Collect studets resposes to the task. Make some otes about what their work reveals about their curret levels of uderstadig. The purpose of doig this is to forewar you of the difficulties studets will experiece durig the lesso itself, so that you may prepare carefully. We suggest that you do ot score studets papers. The research shows that this will be couterproductive, as it will ecourage studets to compare their scores ad distract their attetio from what they ca do to improve their mathematics. Istead, help studets to make further progress by summarizig their difficulties as a list of questios. Some suggestios for these are give i the Commo issues table o the ext page. We suggest that you make a list of your ow questios, based o your studets work, usig the ideas o the followig page. We recommed you either: write oe or two questios o each studet s work, or give each studet a prited versio of your list of questios ad highlight the questios relevat to each studet. If you do ot have time to do this, you could select a few questios that will be of help to the majority of studets ad write these o the board whe you retur the work to the studets i the follow-up lesso. Teacher guide Iterpretig Algebraic Expressios T-
Commo issues: Writes expressios left to right, showig little uderstadig of the order of operatios implied by the symbolic represetatio For example: The studet writes: Q1a. 5 + 4 (ot icorrect). Q1b. 4 + 5. Q1c. 4 + 5. Q1d. 3. Does ot costruct paretheses correctly or expads them icorrectly For example: The studet writes: Q1b. 4 + 5 istead of 5( + 4). Q1c. 4 + 5 istead of 4 + 5. Or: The studet couts: Q. ( + 3) = + 3 as correct. Q. (5) = 5 as correct. Q. ( + 3) = + 3 as correct. Idetifies errors but does ot give explaatios For example: The studet corrects the first, third, ad fourth statemets, but o explaatio or diagram is used to explai why they are icorrect (Q). Suggested questios ad prompts: Ca you write aswers to the followig? 4 + 1 5 4 + 5 4 + 3 5 Check your aswers with your (scietific) calculator. How is your calculator workig these out? So what does 4 + 5 mea? Is this the same as Q1b? Which oe of the followig is the odd oe out: Thik of a umber, add 3, ad the multiply your aswer by. Thik of a umber, multiply it by, ad the add 3. Thik of a umber, multiply it by, ad the add 6. Why? How would you write expressios for these areas? 3 1 3 3 3 3 3 Ca you do this i differet ways? Teacher guide Iterpretig Algebraic Expressios T-3
SUGGESTED LESSON OUTLINE Iteractive whole-class itroductio (10 miutes) Give each studet a mii-whiteboard, pe, ad eraser ad hold a short questio ad aswer sessio. If studets show ay icorrect aswers, write the correct aswer o the board ad discuss ay problems. O your mii-whiteboards, show me a algebraic expressio that meas: Multiply by 4 ad the add 3 to your aswer. 4 + 3 Add 3 to ad the multiply your aswer by 4. 4(3+ ) Add 5 to ad the divide your aswer by 3. + 5 3 Multiply by ad the multiply your aswer by 5. 5 Multiply by 5 ad the square your aswer. (5) Collaborative activity 1: matchig expressios ad words (0 miutes) The first activity is desiged to help studets iterpret symbols ad realize that the way the symbols are writte defies the order of operatios. Orgaize studets ito groups of two or three. Display Slide P-1 of the projector resource: Matchig Expressios ad Words 4( +) 5&.(/0.1#!#1#(3%)#(*+,#6""#$%&'4# ( + 4)!""#$%&'#(%#!)#(*+,#-&.(/0.1#1#(3%4# 4 +!""#(3%#(%#!)#(*+,#-&.(/0.1#1#$%&'4# Note that oe of the algebraic expressios does ot have a match i words. This is deliberate! It is to help you explai the task to studets. Model the activity briefly for studets, usig the examples o the projector resource: I am goig to give each group two sets of cards, oe with expressios writte i algebra ad the other with words. Take turs to choose a expressio ad fid the words that match it. [4( + ) matches Add to the multiply by 4 ; ( + 4) matches Add 4 to the multiply by.] Whe you are workig i groups, you should place these cards side by side o the table ad explai how you kow that they match. If you caot fid a matchig card, the you should write your ow usig the blak cards provided. [4 + does ot match ay of the word cards show o Slide P-1. The word card Multiply by two, the add four does ot match ay of the expressios.] Give each small group of studets a cut-up copy of Card Set A: Expressios ad Card Set B: Words: Teacher guide Iterpretig Algebraic Expressios T-4
Card Set A: Expressios Card Set B: Words E1 + 6 E 3 W1 Multiply by two, the add six. W Multiply by three, the square the aswer. E3 +1 E4 + 6 W3 Add six to the multiply by two. W4 Add six to the divide by two. E5 ( + 3) E6 + 6 W5 Add three to the multiply by two. W6 Add six to the square the aswer. E7 (3) E8 ( + 6) W7 Multiply by two the add twelve. W8 Divide by two the add six. E9 +1 + 36 E10 3 + W9 Square, the add six W10 Square, the multiply by ie E11 + 6 E1 + 6 W11 W1 E13 E14 W13 W14 Support studets i makig matches ad explaiig their decisios. As they do this, ecourage them to speak the algebraic expressios out loud. Studets may ot be used to talkig algebra ad may ot kow how to say what is writte, or may do so i ways that create ambiguities. For example, the followig coversatio betwee a teacher ad studet is fairly typical: Teacher: Tell me i words what this oe says. [Teacher writes: 3+.] Studet: Three add divided by two. Teacher: How would you read this oe the? [Teacher writes: (3+ ).] Studet: Three add divided by two. Oh, but i the secod oe you are dividig it all by two. Teacher: So ca you try readig the first oe agai, so it souds differet from the secod oe? Studet: Three add [pause] divided by two [said quickly]. Or divided by two, the add three. Studets will eed to make word cards to match E10: 3+ ad E1: + 6. They will also eed to make expressio cards to match W3: Add 6 to, the multiply by ad W10: Square, the multiply by 9. Some studets may otice that some expressios are equivalet, for example ( + 3) ad + 6. You do ot eed to commet o this ow as whe Card Set C: Tables is give out, studets will be able to otice this for themselves. Teacher guide Iterpretig Algebraic Expressios T-5
Collaborative activity : matchig expressios, words, ad tables (0 miutes) Give each small group of studets a cut-up copy of Card Set C: Tables: Card Set C: Tables T1 T 1 3 4 As 14 16 18 0 1 3 4 As 81 144 T3 T4 1 3 4 As 10 15 1 3 4 As 3 7 48 T5 T6 1 3 4 As 81 100 1 3 4 As 10 1 14 T7 T8 1 3 4 As 4 5 1 3 4 As 6.5 7 7.5 8 Card Set C: Tables will make studets substitute umbers ito the expressios ad will alert them to the fact that differet expressios are equivalet. Ask studets to match these ew cards to the two card sets they have bee workig o. Some tables have umbers missig ad studets will eed to write these i. Ecourage studets to use strategies for matchig. There are shortcuts that will help to miimize the work. For example, some may otice that: Sice ( + 3) is a eve umber, we ca just look at tables with eve umbers i them. Sice (3) is a square umber, we ca look for tables with oly square umbers i them. Studets will otice that there are fewer tables tha expressios. This is because some tables match more tha oe expressio. Allow studets time to discover this for themselves. As they do so, ecourage them to test that they match for all. This is the begiig of a geeralizatio. Do ( + 3) ad + 6 always give the same aswer whe = 1,, 3, 4, 5? What about whe = 346, or whe = -3, or whe = 0.45? Check o your calculator. Ca you explai how you ca be sure? This last questio is a importat oe ad will be followed up i the ext part of the lesso. Extedig the lesso over two days It is importat ot to rush the learig. At about this poit, some lessos ru out of time. If this happes, ask pupils to stack their cards i order, so that matchig cards are grouped together ad faste them with a paper clip. Ask studets to write their ames o a evelope ad store the matched cards i it. These evelopes ca the be reissued at the start of ext lesso. Teacher guide Iterpretig Algebraic Expressios T-6
Collaborative activity 3: matchig expressios, words, tables, ad areas (0 miutes) Give each small group of studets a cut-up copy of the Card Set D: Areas, a large sheet of paper, a felt-tipped pe, ad a glue stick. Card Set D: Areas A1 A 6 3!! A3 A4! A5! A6 1 6 1! A7 1! A8 6 6 1 6!! The Card Set D: Areas will help studets to uderstad why the differet expressios match the same tables of umbers. Studet materials Iterpretig Algebraic Expressios 014 MARS, Shell Ceter, Uiversity of Nottigham S-5 Each of these cards shows a area. I wat you to match these area cards to the cards already o the table. Whe you reach agreemet, paste dow your fial arragemet of cards oto the large sheet of paper, creatig a poster. Next to each group of cards write dow why the areas show that differet expressios are equivalet. The posters studets produce will eed to be displayed i the fial whole-class discussio. They may look somethig like this: Teacher guide Iterpretig Algebraic Expressios T-7
As studets match the cards, ecourage them to explai ad write dow why particular pairs of cards go together. Why does this area correspod to + 1 + 36? 6 6 Show me where is i this diagram. Where is 1? Where is the 36 part of the diagram? Now show me why it also shows ( + 6). Where is the + 6? Ask studets to idetify groups of expressios that are equivalet ad explai their reasoig. For example, E1 is equivalet to E10, E8 is equivalet to E9, ad E4 is equivalet to E5. Whole-class discussio (0 miutes) Hold a whole-class iteractive discussio to review what has bee leared over this lesso. Ask each group of studets to justify, usig their poster, why two expressios are equivalet. The use mii-whiteboards ad questioig to begi to geeralize the learig: Draw me a area that shows this expressio: 3(x + 4) Write me a differet expressio that gives the same area. Draw me a area that shows this expressio: (4y) Write me a differet expressio that gives the same area. Draw me a area that shows this expressio: (z + 5) Write me a differet expressio that gives the same area. Draw me a area that shows this expressio: w + 6 Write me a differet expressio that gives the same area. Follow-up lesso: improvig idividual solutios to the assessmet task (10 miutes) Retur studets work o the assessmet task Iterpretig Expressios, alog with a fresh copy of the task sheet. If you have ot added questios to idividual pieces of work, write your list of questios o the board. Studets should select from this list oly those questios they thik are appropriate to their ow work. Read through the solutio you wrote [yesterday] ad the questios (o the board/writte o your script). Aswer the questios ad the thikig about what you leared this lesso, write a ew solutio to see if you ca improve your work. Some teachers give this as a homework task. Teacher guide Iterpretig Algebraic Expressios T-8
SOLUTIONS Assessmet task: Iterpretig Expressios 1a. 5 + 4. 1b. 5( + 4). 1c. + 4 5. 1d. 3. 1e. (3).. ( + 3) + 3, ( + 3) = + 6. 10 5 5 Lesso task = 1is correct. (5) 5, (5) = 5. ( + 3) + 3, ( + 3) = + 6 + 9 ( + 3 does however equal + 9). This table is for coveiece oly: it is helpful ot to refer to cards by these letters i class, but rather to the cotet of the cards. Expressios Words Tables Areas E1 W4 T7 A5 E10 W13 (Blak) Divide by the add 3 E W11 (Blak) Square the multiply by 3 T4 A3 E3 W3 T1 A1 E13 (Blak) ( + 6) W7 E4 W1 T6 A E5 W5 E6 W8 T8 A6 E7 W T A4 E14 (Blak) 9 W10 E8 W6 T5 A7 E9 W14 (Blak) Square, add 1 multiplied by, add 36 E11 W9 T3 A8 E1 W1 (Blak) Square the add 6 squared Teacher guide Iterpretig Algebraic Expressios T-9
Iterpretig Expressios 1. Write algebraic expressios for each of the followig: a. Multiply by 5 the add 4. b. Add 4 to the multiply by 5. c. Add 4 to the divide by 5. d. Multiply by the multiply by 3. e. Multiply by 3 the square the result.. The equatios below were created by studets who were asked to write equivalet expressios o either side of the equals sig. Imagie you are a teacher. Your job is to decide whether their work is right or wrog. If you see a equatio that is false, the: a. Cross out the expressio o the right ad replace it with a expressio that is equivalet to the oe o the left. b. Explai what is wrog, usig words or diagrams. ( + 3) = + 3 10 " 5 5 = " 1 (5) = 5 ( + 3) = + 3 = + 9 Studet materials Iterpretig Algebraic Expressios S-1 015 MARS, Shell Ceter, Uiversity of Nottigham
Card Set A: Expressios E1 + 6 E 3 E3 +1 E4 + 6 E5 ( + 3) E6 + 6 E7 (3) E8 ( + 6) E9 +1 + 36 E10 3+ E11 + 6 E1 + 6 E13 E14 Studet materials Iterpretig Algebraic Expressios S- 015 MARS, Shell Ceter, Uiversity of Nottigham
Card Set B: Words W1 Multiply by two, the add six. W Multiply by three, the square the aswer. W3 Add six to the multiply by two. W4 Add six to the divide by two. W5 Add three to the multiply by two. W6 Add six to the square the aswer. W7 Multiply by two the add twelve. W8 Divide by two the add six. W9 Square, the add six W10 Square, the multiply by ie W11 W1 W13 W14 Studet materials Iterpretig Algebraic Expressios S-3 015 MARS, Shell Ceter, Uiversity of Nottigham
Card Set C: Tables T1 T 1 3 4 As 14 16 18 0 1 3 4 As 81 144 T3 T4 1 3 4 As 10 15 1 3 4 As 3 7 48 T5 T6 1 3 4 As 81 100 1 3 4 As 10 1 14 T7 T8 1 3 4 As 4 5 1 3 4 As 6.5 7 7.5 8 Studet materials Iterpretig Algebraic Expressios S-4 015 MARS, Shell Ceter, Uiversity of Nottigham
Card Set D: Areas A1 A 6 3 A3 A4 A5 A6 1 6 1 1! A7 6 A8 6 1 6 Studet materials Iterpretig Algebraic Expressios S-5 015 MARS, Shell Ceter, Uiversity of Nottigham
Matchig Expressios ad Words 4( +) Multiply by two, the add four. ( + 4) Add four to, the multiply by two. 4 + Add two to, the multiply by four. Projector Resources Iterpretig Algebraic Expresssios P-1
Mathematics Assessmet Project Classroom Challeges These materials were desiged ad developed by the Shell Ceter Team at the Ceter for Research i Mathematical Educatio Uiversity of Nottigham, Eglad: Malcolm Swa, Nichola Clarke, Clare Dawso, Sheila Evas, Coli Foster, ad Marie Joubert with Hugh Burkhardt, Rita Crust, Ady Noyes, ad Daiel Pead We are grateful to the may teachers ad studets, i the UK ad the US, who took part i the classroom trials that played a critical role i developig these materials The classroom observatio teams i the US were led by David Foster, Mary Bouck, ad Diae Schaefer This project was coceived ad directed for The Mathematics Assessmet Resource Service (MARS) by Ala Schoefeld at the Uiversity of Califoria, Berkeley, ad Hugh Burkhardt, Daiel Pead, ad Malcolm Swa at the Uiversity of Nottigham Thaks also to Mat Crosier, Ae Floyde, Michael Gala, Judith Mills, Nick Orchard, ad Alvaro Villaueva who cotributed to the desig ad productio of these materials This developmet would ot have bee possible without the support of Bill & Melida Gates Foudatio We are particularly grateful to Caria Wog, Melissa Chabra, ad Jamie McKee The full collectio of Mathematics Assessmet Project materials is available from http://map.mathshell.org 015 MARS, Shell Ceter, Uiversity of Nottigham This material may be reproduced ad distributed, without modificatio, for o-commercial purposes, uder the Creative Commos Licese detailed at http://creativecommos.org/liceses/by-c-d/3.0/ All other rights reserved. Please cotact map.ifo@mathshell.org if this licese does ot meet your eeds.