Lesso 16 Mathematics Assessmet Project Formative Assessmet Lesso Materials Iterpretig Algebraic Expressios MARS Shell Ceter Uiversity of Nottigham & UC Berkeley Beta Versio If you ecouter errors or other issues i this versio, please sed details to the MAP team c/o map.feedback@mathshell.org. 011 MARS Uiversity of Nottigham
Iterpretig Algebraic Expressios Teacher Guide Beta Versio 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). Commo Core State Stadards This lesso ivolves mathematical cotet i the stadards from across the grades, with emphasis o: A-SSE: Iterpret the structure of expressios. A-APR: Rewrite ratioal expressios. This lesso ivolves a rage of mathematical practices, with emphasis o:. Reaso abstractly ad quatitatively. 7. Look for ad make use of structure. Itroductio 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 ad 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 idividual studet will eed two copies of the assessmet task Iterpretig Expressios ad a mii-whiteboard, 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 eeded Approximately te miutes for the assessmet task, a oe-hour lesso ad te miutes i a follow-up lesso (or for homework). All timigs are approximate. Exact timigs will deped o the eeds of the class. 011 MARS Uiversity of Nottigham 1
Iterpretig Algebraic Expressios Teacher Guide Beta Versio Before the lesso 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. 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 studets are allowed to aswer the questios without assistace, as far as possible. 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. Iterpretig Expressios 1. Write algebraic expressios for each of the followig: a. Multiply by 5 the add 4. b. Add 4 to the multiply your aswer by 5. c. Add 4 to the divide your aswer by 5. d. Multiply by the multiply your aswer by 3. e. Multiply by 3 the square your aswer.. Imagie you are a teacher. Decide whether the followig work is right or wrog. If you see a wrog aswer, the: a. Cross it out ad replace it with a correct aswer. b. Explai what is wrog usig words or diagrams. ( + 3) = + 3 10 5 5 (5) = 5 = 1 ( + 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 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 table o page 3. 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. 011 MARS Uiversity of Nottigham
Iterpretig Algebraic Expressios Teacher Guide Beta Versio 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 3 1 3 3 3 3 011 MARS Uiversity of Nottigham 3
Iterpretig Algebraic Expressios Teacher Guide Beta Versio Suggested lesso outlie 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, 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 two to the multiply by four. ( + 4) matches Add four to the multiply by two.] 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: Teacher: Pupil: Three add divided by two. Oh, but i the secod oe you are dividig it all by two. So ca you try readig the first oe agai, so it souds differet from the secod oe? Three add [pause] divided by two [said quickly]. Or divided by two, the add three. 011 MARS Uiversity of Nottigham 4
Iterpretig Algebraic Expressios Teacher Guide Beta Versio 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 +1 + 36 E10 3 + W9 Square, the add six W10 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 six to, the multiply by two ad W10: Square, the multiply by ie. 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. 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. 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? T1 T3 T5 T7 1 3 4 As 14 16 18 0 1 3 4 As 10 15 1 3 4 As 81 100 1 3 4 As 4 5 Card Set C: Tables T T4 T6 T8 1 3 4 As 81 144 1 3 4 As 3 7 48 1 3 4 As 10 1 14 1 3 4 As 6.5 7 7.5 8 011 MARS Uiversity of Nottigham 5
Iterpretig Algebraic Expressios Teacher Guide Beta Versio 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? Ask studets to idetify groups of expressios that are equivalet ad explai their reasoig. For example, E1 is to E9, ad E4 is equivalet to E5. equivalet to E10, E8 is equivalet Pleary 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. 3( x + 4) Draw me a area that shows this expressio: Write me a differet expressio that gives the same area. 011 MARS Uiversity of Nottigham 6
Iterpretig Algebraic Expressios Teacher Guide Beta Versio 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: Write me a differet expressio that gives the same area. w + 6 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. 011 MARS Uiversity of Nottigham 7
Iterpretig Algebraic Expressios Teacher Guide Beta Versio Solutios 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 011 MARS Uiversity of Nottigham 8
Iterpretig Algebraic Expressios Studet Materials Beta Versio Iterpretig Expressios 1. Write algebraic expressios for each of the followig: a. Multiply by 5 the add 4. b. Add 4 to the multiply your aswer by 5. c. Add 4 to the divide your aswer by 5. d. Multiply by the multiply your aswer by 3. e. Multiply by 3 the square your aswer.. Imagie you are a teacher. Decide whether the followig work is correct or icorrect. If you see a error: a. Cross it out ad replace it with a correct aswer. b. Explai the error usig words or diagrams. ( + 3) = + 3 10 5 5 = 1 (5) = 5 ( + 3) = + 3 = + 9 011 MARS Uiversity of Nottigham S-1
Iterpretig Algebraic Expressios Studet Materials Beta Versio Card Set A: Expressios E1 + 6 E 3 E3 +1 E4 + 6 E5 ( + 3) E6 + 6 E7 (3) E8 ( + 6) E9 E10 +1 + 36 3+ E11 + 6 E1 + 6 E13 E14 011 MARS Uiversity of Nottigham S-
Iterpretig Algebraic Expressios Studet Materials Beta Versio Card Set B: Words W1 W Multiply by two, the add six. 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 011 MARS Uiversity of Nottigham S-3
Iterpretig Algebraic Expressios Studet Materials Beta Versio 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 8 011 MARS Uiversity of Nottigham S-4
Iterpretig Algebraic Expressios Studet Materials Beta Versio Card Set D: Areas A1 A 6 3 A3 A4 A5 A6 1 6 1 1 A7 6 A8 6 1 6 011 MARS Uiversity of Nottigham S-5
Matchig Expressios ad Words 4( +) 5&.(/0.1##1#(3%)#(*+,#6""#$%&'4# ( + 4) ""#$%&'#(%#)#(*+,#-&.(/0.1#1#(3%4# 4 + ""#(3%#(%#)#(*+,#-&.(/0.1#1#$%&'4# Beta Versio 011 MARS Uiversity of Nottigham Projector resources: 1