CONCEPT DEVELOPMENT Mathematics Assessmet Project CLASSROOM CHALLENGES A Formative Assessmet Lesso Iterpretig Algebraic Expressios Mathematics Assessmet Resource Service Uiversity of Nottigham & UC Berkeley Beta Versio For more details, visit: http://map.mathshell.org 01 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 i: 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. A-APR: Rewrite ratioal expressios. This lesso also relates to the followig Stadards for Mathematical Practice i the Commo Core State Stadards for Mathematics:. Reaso abstractly ad quatitatively. 7. Look for ad make use of structure. INTRODUCTION The lesso uit is structured i the followig way: Before the lesso, studets work idividually o a assessmet task that is 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 the assessmet task Iterpretig Expressios ad a miiwhiteboard, pe, ad eraser. Each pair of studets will eed a copy of Card set A: Expressios, Card Set B: Words, Card Set C: Tables, Card Set D: Areas, a glue stick, a felt-tipped pe, ad a large sheet of paper or card for makig a poster. 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. There are some projector resources to help with istructios. The card sets should be cut up before the lesso. Note that the blak cards are part of the activity. TIME NEEDED 10 miutes for the assessmet task, a 1-hour lesso ad 10 miutes i a follow-up lesso (or for homework). 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 the 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 will the be able to target your help more effectively i the follow-up lesso. Give each studet a copy of Iterpretig Expressios. Itroduce the task briefly ad help studets to uderstad what they are beig asked to do. 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. ( + 3) = + 3 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. 10 " 5 = " 1 5 (5) = 5 ( + 3) = + 3 = + 9 If studets are strugglig to get started, ask them questios that help them uderstad what is required, but do ot do the task for them. 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 o the ext page. These have bee draw from commo difficulties observed i trials of this lesso uit. We suggest that you write your ow lists of questios, based o your ow studets work, usig the ideas i the Commo issues table o the ext page. You may choose to write questios o each studet s work. If you do ot have time to do this, you could write a few questios that will help the majority of studets. These ca the be displayed o the board at the ed of the lesso. Teacher guide Iterpretig Algebraic Expressios T-
Commo issues: Studet writes expressios left to right, showig little uderstadig of the order of operatios implied by the symbolic represetatio. For example: Q1a Writes 5 + 4 (ot icorrect). Q1b Writes 4 + 5. Q1c Writes 4 + 5. Q1d Writes 3. Studet does ot costruct paretheses correctly or expads them icorrectly. For example: Q1b Writes 4 + 5 istead of 5( + 4). Q1c Q Q Q Writes 4 + 5 istead of 4 + 5. ( + 3) = + 3 is couted as correct. (5) = 5 is couted as correct. ( + 3) = + 3 is couted as correct. Studet idetifies errors but does ot give explaatios. I questio, there are correctios to the first, third, ad fourth statemets, but o explaatio or diagram is used to explai why they are icorrect. Suggested questios ad prompts: Ca you write aswers to the followig? 4 + 1 5 4 + 5 4 + 3 5 Check your aswers with your 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 ad why? 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. How would you write dow expressios for these areas? Ca you do this i differet ways? 3 1 3 3 3 3 3 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. 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: matchig expressios ad words (15 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 the projector resource P-1, Matchig Expressios ad Words. Note that oe of the algebraic expressios o the slide 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 it i 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. Use the blak cards provided. [4+ does ot match ay of the word cards show o the slide. The word card Multiply by two, the add four does ot match ay of the expressios.] Give a copy of Card Set A: Expressios ad Card Set B: Words to each small group. Support studets i makig matches ad explaiig their decisios. As they do this, ecourage studets 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 pupil is fairly typical: Teacher: Tell me i words what this oe says. [Teacher writes: 3+.] Pupil: Three add divided by two. Teacher: How would you read this oe the? [Teacher writes: (3+ ).] Pupil: Three add divided by two. Oh, but i the secod oe you are dividig it all by two. Teacher guide Iterpretig Algebraic Expressios T-4
Teacher: So ca you try readig the first oe agai, so it souds differet from the secod oe? Pupil: Three add [pause] divided by two [said quickly]. Or divided by two, the add three. E1 E3 Card Set A: Expressios + 6 +1 E E4 3 + 6 W1 W3 Multiply by two, the add six. Add six to the multiply by two. Card Set B: Words W W4 Multiply by three, the square the aswer Add six to the divide by two E5 E7 ( + 3) (3) E6 E8 + 6 ( + 6) W5 W7 Add three to the multiply by two. Multiply by two the add twelve W6 W8 Add six to the square the aswer Divide by two the add six. E9 E10 W9 W10 +1 + 36 3 + Square, the add six Square, the multiply by ie W11 W1 E11 + 6 E1 + 6 W13 W14 E13 E14 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: whe the Card Set C: Tables is give out, studets will otice this for themselves. Collaborative activity: matchig expressios, words, ad tables (15 miutes) Card Set C: Tables will make studets substitute umbers ito the expressios ad will alert them to the fact that differet expressios are equivalet. T1 1 3 4 As 14 16 18 0 Card Set C: Tables T 1 3 4 As 81 144 Give each small group of studets a copy of Card Set C: Tables ad ask studets to match these to the card sets already o the table. Some tables have umbers missig: studets will eed to write these i. T3 T5 1 3 4 As 10 15 T4 T6 1 3 4 As 3 7 48 Ecourage studets to use strategies for matchig. There are shortcuts that will help to miimize the work. For example, some may otice that: T7 1 3 4 As 81 100 3 4 1 As 4 5 T8 1 3 4 As 10 1 14 3 4 1 As 6.5 7 7.5 8 Teacher guide Iterpretig Algebraic Expressios T-5
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. 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 be reissued i the ext lesso. Collaborative activity: matchig expressios, words, tables, ad areas (15 miutes) The Card Set D: Areas will help studets to uderstad why the differet expressios match the same tables of umbers. Give each small group of studets a copy of the Card Set D: Areas, a large sheet of paper, ad a glue stick. 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 oto the large sheet, creatig a poster. Next to each group write dow why the areas show that differet expressios are equivalet. These posters will be displayed i the fial class discussio. 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). Whereis the + 6? Teacher guide Iterpretig Algebraic Expressios T-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 (15 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. Review 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 chose to write a list of questios rather tha write questios o idividual papers, display your questios ow. Read through the solutio you wrote [yesterday] ad thik about what you leared this lesso. Write a ew solutio, bearig i mid what you ve leared, to see if you ca improve your work. If you are ruig out of time you could postpoe this activity util the ext lesso, or set it for homework. Teacher guide Iterpretig Algebraic Expressios T-7
SOLUTIONS 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 A5 E T4 A3 E3 W7 T1 A1 E4 W1 T6 A E5 W5 A E6 W8 T8 A6 E7 W T A4 E8 W6 T5 A7 E9 W6 A7 E10 T7 A5 E11 W9 T3 A8 E1 W3 W10 Teacher guide Iterpretig Algebraic Expressios T-8
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 01 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- 01 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 01 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 01 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 01 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 CHALLENGES This lesso was desiged ad developed by the Shell Ceter Team at the Uiversity of Nottigham Malcolm Swa, Nichola Clarke, Clare Dawso, Sheila Evas with Hugh Burkhardt, Rita Crust, Ady Noyes, ad Daiel Pead It was refied o the basis of reports from teams of observers led by David Foster, Mary Bouck, ad Diae Schaefer based o their observatio of trials i US classrooms alog with commets from teachers ad other users. This project was coceived ad directed for MARS: Mathematics Assessmet Resource Service by Ala Schoefeld, Hugh Burkhardt, Daiel Pead, ad Malcolm Swa ad based at the Uiversity of Califoria, Berkeley We are grateful to the may teachers, i the UK ad the US, who trialed earlier versios of these materials i their classrooms, to their studets, ad to Judith Mills, Carol Hill, ad Alvaro Villaueva who cotributed to the desig. 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 01 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.