Cocept Developmet Lessos How ca I help studets develop a deeper uderstadig of Mathematics? HANDOUTS FOR TEACHERS Cotets 1 Assessmet tasks... 2 2 Sample studet work... 5 3 Sample follow- up questios... 8 4 Geeralizatios commoly made by studets... 11 5 Priciples to discuss... 12 6 Structure of the Cocept lessos... 13 7 Some geres of activity used i the Cocept Lessos... 14 8 Classifyig mathematical objects... 15 9 Iterpretig multiple represetatios... 16 10 Evaluatig mathematical statemets... 20 11 Studets modifyig a give problem... 21 12 Studets creatig problems for each other... 22 2012 MARS, Shell Cetre, Uiversity of Nottigham
Hadout 1: Assessmet tasks Assessmet Task: Distace time graphs Hadouts for Teachers Cocept Developmet Lessos H-2
Assessmet Task: Percet chages Hadouts for Teachers Cocept Developmet Lessos H-3
Assessmet Task: Iterpretig Expressios 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. 2. 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. 2( + 3) = 2 + 3 10! 5 5 = 2! 1 (5) 2 = 5 2 ( + 3) 2 = 2 + 3 2 = 2 + 9 Hadouts for Teachers Cocept Developmet Lessos H-4
Hadout 2: Sample studet work Iterpretig a distace time graph Every morig Tom walks alog a straight road from his home to a bus stop, a distace of 160 meters. The graph shows his jourey o oe particular day. 1. Describe what may have happeed. You should iclude details like how fast he walked. Distace from home i meters. Hadouts for Teachers Cocept Developmet Lessos H-5
Percet chages 1. Maria sees a dress i a sale. The dress is ormally priced at $56.99. The ticket says that there is 45% off. She wats to use her calculator to work out how much the dress will cost. It does ot have a percet butto. Which keys must she press o her calculator? Write dow the keys i the correct order. (You do ot have to do the calculatio.) ( ) 2. I a sale, the prices i a shop were all decreased by 20%. After the sale they were all icreased by 25%. What was the overall effect o the shop prices? Explai how you kow. George's respose Jurge's respose Hadouts for Teachers Cocept Developmet Lessos H-6
Iterpretig expressios Britey's respose Hadouts for Teachers Cocept Developmet Lessos H-7
Hadout 3: Sample follow-up questios Distace-time graphs: Commo issues Commo issues: Studet iterprets the graph as a picture For example: The studet assumes that as the graph goes up ad dow, Tom's path is goig up ad dow. Or: The studet assumes that a straight lie o a graph meas that the motio is alog a straight path. Or: The studet thiks the egative slope meas Tom has take a detour. Suggested questios ad prompts: If a perso walked i a circle aroud their home, what would the graph look like? If a perso walked at a steady speed up ad dow a hill, directly away from home, what would the graph look like? I each sectio of his jourey, is Tom's speed steady or is it chagig? How do you kow? How ca you figure out Tom's speed i each sectio of the jourey? Studet iterprets graph as speed time The studet has iterpreted a positive slope as speedig up ad a egative slope as slowig dow. Studet fails to metio distace or time For example: The studet has ot metioed how far away from home Tom has traveled at the ed of each sectio. Or: The studet has ot metioed the time for each sectio of the jourey. Studet fails to calculate ad represet speed For example: The studet has ot worked out the speed of some/all sectios of the jourey. Or: The studet has writte the speed for a sectio as the distace covered i the time take, such as 20 meters i 10 secods. Studet misiterprets the scale For example: Whe workig out the distace the studet has icorrectly iterpreted the vertical scale as goig up i tes rather tha tweties. If a perso walked for a mile at a steady speed, away from home, the tured roud ad walked back home at the same steady speed, what would the graph look like? How does the distace chage durig the secod sectio of Tom's jourey? What does this mea? How does the distace chage durig the last sectio of Tom's jourey? What does this mea? How ca you tell if Tom is travelig away from or towards home? Ca you provide more iformatio about how far Tom has traveled durig differet sectios of his jourey? Ca you provide more iformatio about how much time Tom takes durig differet sectios of his jourey? Ca you provide iformatio about Tom's speed for all sectios of his jourey? Ca you write his speed as meters per secod? What is the scale o the vertical axis? Studet adds little explaatio as to why the graph is or is ot realistic What is the total distace Tom covers? Is this realistic for the time take? Why?/Why ot? Is Tom's fastest speed realistic? Is Tom's slowest speed realistic? Why?/Why ot? Hadouts for Teachers Cocept Developmet Lessos H-8
Percet chages: Commo issues Commo issues: Studet makes the icorrect assumptio that a percetage icrease meas the calculatio must iclude a additio For example: 40.85 + 0.6 or 40.85 + 1.6. (Q1.) A sigle multiplicatio by 1.06 is eough. Studet makes the icorrect assumptio that a percetage decrease meas the calculatio must iclude a subtractio For example: 56.99! 0.45 or 56.99! 1.45. (Q2.) A sigle multiplicatio by 0.55 is eough. Studet coverts the percetage to a decimal icorrectly For example: 40.85 " 0.6. (Q1.) Studet uses iefficiet method For example: First the studet calculates 1%, the multiplies by 6 to fid 6%, ad the adds this aswer o: (40.85 100) " 6 + 40.85. (Q1.) Or: 56.99 " 0.45 = ANS, the 56.99! ANS (Q2.) A sigle multiplicatio is eough. Studet is uable to calculate percetage chage For example: 450! 350 = 100% (Q3.) Or: The differece is calculated, the the studet does ot kow how to proceed or he/she divides by 450. (Q3.) The calculatio (450! 350) 350 " 100 is correct. Studet subtracts percetages For example: 25! 20 = 5%. (Q4.) Because we are combiig multipliers: 0.8 " 1.25 = 1, there is o overall chage i prices. Studet fails to use brackets i the calculatio For example: 450 350 350 " 100. (Q4.) Studet misiterprets what eeds to be icluded the aswer For example: The aswer is just operator symbols. Suggested questios ad prompts: Does your aswer make sese? Ca you check that it is correct? Compared to last year 50% more people atteded the festival. What does this mea? Describe i words how you ca work out how may people atteded the festival this year. Give me a example. Ca you express the icrease as a sigle multiplicatio? Does your aswer make sese? Ca you check that it is correct? I a sale, a item is marked 50% off. What does this mea? Describe i words how you calculate the price of a item i the sale. Give me a example. Ca you express the decrease as a sigle multiplicatio? How ca you write 50% as a decimal? How ca you write 5% as a decimal? Ca you thik of a method that reduces the umber of calculator key presses? How ca you show your calculatio with just oe step? Are you calculatig the percetage chage to the amout $350 or to the amout $450? If the price of a t-shirt icreased by $6, describe i words how you could calculate the percetage chage. Give me a example. Use the same method i Q3. Make up the price of a item ad check to see if your aswer is correct. I your problem, what operatio will the calculator carry out first? If you just etered these symbols ito your calculator would you get the correct aswer? Hadouts for Teachers Cocept Developmet Lessos H-9
Iterpretig expressios: Commo issues 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 Q2 Q2 Q2 Writes 4 + 5 istead of 4 + 5. 2( + 3) = 2 + 3 is couted as correct. (5) 2 = 5 2 is couted as correct. ( + 3) 2 = 2 + 3 2 is couted as correct. Studet idetifies errors but does ot give explaatios. I questio 2, 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 + 2! 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 2. Thik of a umber, multiply it by 2, ad the add 3. Thik of a umber, multiply it by 2, ad the add 6. How would you write dow expressios for these areas? Ca you do this i differet ways? 3 3 1 2 2 3 3 3 3 Hadouts for Teachers Cocept Developmet Lessos H-10
Hadout 4: Geeralizatios commoly made by studets What other examples ca you add to this list? Ca you thik of ay miscoceptios you have had at some time? How were these overcome? 0.567 > 0.85 The more digits a umber has, the larger is its value. 3 6 = 2 You always divide the larger umber by the smaller oe. 0.4>0.62 The fewer the umber of digits after the decimal poit, the larger is its value. It's like fractios. 5.62 x 0.65 > 5.62 Multiplicatio always makes umbers bigger. 1 gallo costs $5.60; 4.2 gallos cost $5.60 x 4.2; 0.22 gallos cost $5.60 0.22 If you chage the umbers i a questio, you chage the operatio you have to do. B A C C B A Area of rectagle Area of triagle If you dissect a shape ad rearrage the pieces, you chage the area. A B C Agle A is greatest. Agle C is greatest. The size of a agle is related to the size of the arc or the legth of the arms of the agle. If x+4 < 10, the x = 5. Letters represet particular umbers. 3 + 4 = 7 + 2 = 9 + 5 = 14 Equals' meas 'makes'. I three rolls of a die, it is harder to get 6,6,6 tha 2,4,6. Special outcomes are less likely tha more represetative outcomes. Hadouts for Teachers Cocept Developmet Lessos H-11
Hadout 5: Priciples to discuss These priciples are backed up by research evidece. Discuss the implicatios for your ow teachig. Teachig approaches that ecourage the exploratio of miscoceptios through discussios result i deeper, loger-term learig tha approaches that try to avoid mistakes by explaiig the right way to see thigs from the start. It is helpful if discussios focus o kow difficulties. Rather tha posig log lists of questios, it is better to focus o a challegig task ad ecourage a variety of iterpretatios to emerge, so that studets ca compare ad evaluate their ideas. Questios ca be juxtaposed i ways that create a tesio (sometimes called a cogitive coflict ) that eeds resolvig. Cotradictios arisig from coflictig methods or opiios create awareess that somethig eeds to be leared. For example, askig studets to say how much medicie is i each of the followig syriges may result i aswers such as 1.3ml, 1.12ml ad 1.6ml. But these quatities are all the same! This provides a start for a useful discussio o the deary ature of decimal otatio. Syrige A 0 1 2 ml Syrige B 0 1 2 ml Syrige C 0 1 2 ml Activities should provide opportuities for meaigful feedback. This does ot mea providig summative iformatio, such as the umber of correct or icorrect aswers. More helpful feedback is provided whe studets compare results obtaied from alterative methods util they realize why they get differet aswers. Sessios iclude time for whole group discussio i which ew ideas ad cocepts are allowed to emerge. This requires sesitivity so that studets are ecouraged to share tetative ideas i a o-threateig eviromet. Opportuities should be provided for studets to cosolidate what has bee leared through the applicatio of the ewly costructed cocept. Hadouts for Teachers Cocept Developmet Lessos H-12
Hadout 6: Structure of the Cocept lessos Broadly speakig, each Cocept Formative Assessmet Lesso is structured i the followig way, with some variatio, depedig o the topic ad task: (Before the lesso) Studets complete a assessmet task idividually This assessmet task is desiged to clarify studets existig uderstadigs of the cocepts uder study. The teacher assesses a sample of these ad plas appropriate questios that will move studet thikig forward. These questios are the itroduced i the lesso at appropriate poits. Whole class itroductio Each lesso begis with the teacher presetig a problem for class discussio. The aim here is to itrigue studets, provoke discussio ad/or model reasoig, Collaborative work o a substatial activity At this poit the mai activity is itroduced. This activity is desiged to be a rich, collaborative learig experiece. It is both accessible ad challegig; havig multiple etry poits ad multiple solutio paths. It is usually doe with shared resources ad is preseted o a poster. Four types of activity are commoly used as show i Hadout 6. Studets are ivolved i: o classifyig mathematical objects & challegig defiitios o iterpretig multiple represetatios o evaluatig cojectures ad assertios o modifyig situatios & explorig their structure These will be explored more fully later i this module. It is ot ecessary for every studet to complete the activity. Rather we hope that studets will come to uderstad the cocepts more clearly. Studets share their thikig with the whole class Studets ow share some of their learig with other studets. It is through explaiig that studets begi to clarify their ow thikig. The teacher may the ask further questios to provoke deeper reflectio. Studets revisit the assessmet task Fially, studets are asked to look agai at their origial aswers to the assessmet task. They are either asked to improve their resposes or are asked to complete a similar task. This helps both the teacher ad the studet to realize what has bee leared from the lesso. Hadouts for Teachers Cocept Developmet Lessos H-13
Hadout 7: Some geres of activity used i the Cocept lessos The mai activities i the cocept lessos are built aroud the followig four geres. Each of these types of activity is desiged to provoke studets to reaso i differet ways; to recogize properties, to defie, to represet, to challege cojectures ad miscoceptios, to recogize deeper structures i problems. 1. Classifyig mathematical objects Mathematics is full of coceptual objects such as umbers, shapes, ad fuctios. I this type of activity, studets examie objects carefully, ad classify them accordig to their differet attributes. Studets have to select a object, discrimiate betwee that object ad other similar objects (what is the same ad what is differet?) ad create ad use categories to build defiitios. This type of activity is therefore powerful i helpig studets uderstad differet mathematical terms ad symbols, ad the process by which they are developed. 2. Iterpretig multiple represetatios Mathematical cocepts have may represetatios; words, diagrams, algebraic symbols, tables, graphs ad so forth. These activities allow differet represetatios to be shared, iterpreted, compared ad grouped i ways that allow studets to costruct meaigs ad liks betwee the uderlyig cocepts. 3. Evaluatig mathematical statemets These activities offer studets a umber of mathematical statemets or geeralizatios. These statemets may typically arise from studet miscoceptios, for example: The square root of a umber is smaller tha the umber. Studets are asked to decide o their validity ad give explaatios for their decisios. Explaatios usually ivolve geeratig examples ad couterexamples to support or refute the statemets. I additio, studets may be ivited to add coditios or otherwise revise the statemets so that they become always true. 4. Explorig the structure of problems I this type of activity, studets are give the task of devisig their ow mathematical problems. They try to devise problems that are both challegig ad that they kow they ca solve correctly. Studets first solve their ow problems ad the challege other studets to solve them. Durig this process, they offer support ad act as teachers whe the problem solver becomes stuck. Creatig ad solvig problems may also be used to illustrate doig ad udoig processes i mathematics. For example, oe studet might draw a circle ad calculate its area. This studet is the asked to pass the result to a eighbor, who must ow try to recostruct the circle from the give area. Both studets the collaborate to see where mistakes have arise. Hadouts for Teachers Cocept Developmet Lessos H-14
Hadout 8: Classifyig mathematical objects Similarities ad differeces Show studets three objects. "Which is the odd oe out?" "Describe properties that two share that the third does ot." "Now choose a differet object from the three ad justify it as the odd oe out." (a) (b) (c) Properties ad defiitios Show studets a object. "Look at this object ad write dow all its properties." "Does ay sigle property costitute a defiitio of the object? If ot, what other object has that property?" "Which pairs of properties costitute a defiitio ad which pairs do ot?" Creatig ad testig a defiitio Ask studets to write dow the defiitio of a polygo, or some other mathematical word. Which of these is a polygo accordig to your defiitio? "Exchage defiitios ad try to improve them." Show studets a collectio of objects. "Use your defiitio to sort the objects." "Now improve your defiitios." Classifyig usig a two-way table Give studets a two-way table to sort a collectio of objects. No rotatioal symmetry Rotatioal symmetry "Create your ow objects ad add these to the table." "Try to justify why particular etries are impossible to fill." Classify the objects accordig to your ow categories. Hide your category headigs. Ca your parter idetify the headigs from the way you have sorted the objects? No lies of symmetry Oe or two lies of symmetry More tha two lies of symmetry Hadouts for Teachers Cocept Developmet Lessos H-15
Hadout 9: Iterpretig multiple represetatios Each group of studets is give a set of cards. They are ivited to sort the cards ito sets, so that each set of cards have equivalet meaig. As they do this, they have to explai how they kow that cards are equivalet. They also costruct for themselves ay cards that are missig. The cards are desiged to force studets to discrimiate betwee commoly cofused represetatios. Card Set A: Algebra expressios E1 + 6 2 E2 3 2 E3 2 +12 E4 2 + 6 E5 2( + 3) E6 2 + 6 E7 (3) 2 E8 ( + 6) 2 E9 2 +12 + 36 E10 3+ 2 E11 2 + 6 E12 2 + 6 2 E13 E14 Hadouts for Teachers Cocept Developmet Lessos H-16
Card Set B: Verbal descriptios W1 W2 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 W12 W13 W14 Hadouts for Teachers Cocept Developmet Lessos H-17
Card Set C: Tables T1 T2 1 2 3 4 As 14 16 18 20 1 2 3 4 As 81 144 T3 T4 1 2 3 4 As 10 15 22 1 2 3 4 As 3 27 48 T5 T6 1 2 3 4 As 81 100 1 2 3 4 As 10 12 14 T7 T8 1 2 3 4 As 4 5 1 2 3 4 As 6.5 7 7.5 8 Hadouts for Teachers Cocept Developmet Lessos H-18
Card Set D: Areas A1 A2 6 3 2 2! A3 A4!! A5 A6 1 2 6 12 1 2! A7 6 A8 6 1 6 Swa, M. (2008), A Desiger Speaks: Desigig a Multiple Represetatio Learig Experiece i Secodary Algebra. Educatioal Desiger: Joural of the Iteratioal Society for Desig ad Developmet i Educatio, 1(1), article 3. Hadouts for Teachers Cocept Developmet Lessos H-19
Hadout 10: Evaluatig mathematical statemets Each group of studets is give a set of statemets o cards. Usually these statemets are related i some way. They have to decide whether they are always, sometimes or ever true. If they thik it is always or ever true, the they must try to explai how they ca be sure. If they thik it is sometimes true, they must defie exactly whe it is true ad whe it is ot. Pay rise Max gets a pay rise of 30%. Jim gets a pay rise of 25%. So Max gets the bigger pay rise. Sale I a sale, every price was reduced by 25%. After the sale every price was icreased by 25%. So prices wet back to where they started. Area ad perimeter Whe you cut a piece off a shape you reduce its area ad perimeter. Right agles A petago has fewer right agles tha a rectagle. Birthdays I a class of te studets, the probability of two studets beig bor o the same day of the week is oe. Lottery I a lottery, the six umbers 3, 12, 26, 37, 44, 45 are more likely to come up tha the six umbers 1, 2, 3, 4, 5, 6. Bigger fractios If you add the same umber to the top ad bottom of a fractio, the fractio gets bigger i value. Smaller fractios If you divide the top ad bottom of a fractio by the same umber, the fractio gets smaller i value. Square roots The square root of a umber is less tha or equal to the umber Series If the limit of the sequece of terms i a ifiite series is zero, the the sum of the series is zero. Hadouts for Teachers Cocept Developmet Lessos H-20
Hadout 11: Studets modifyig a give problem Here is a typical word problem from a textbook. The cadles problem A studet wats to ear some moey by makig ad sellig cadles. Suppose that she ca make 60 cadles from a $50 kit, ad that these will each be sold for $4. How much profit will she make? After aswerig such a questio, we might explore its structure ad attempt some geeralizatios. First remove all the umbers from the problem: The cost of buyig the kit: (This icludes the molds, wax ad wicks.) k $ 50 The umber of cadles that ca be made with the kit: 60 cadles s The price at which he sells each cadle: $ 4 per cadle p Total profit made if all the cadles are sold: $ 190 Now we ca ask the followig, first usig umerical values, the usig variables: 1. How did we calculate the profit p usig the give values of k,, ad s? Would your method chage if the values of k,, ad s were differet? 2. Write i the profit ad erase oe of the other values: the sellig price of each cadle, s. How ca you figure out the value of s from the remaiig values of k, ad p? Repeat, but ow erase the value of a differet variable ad say how it may be recostructed from the remaiig values. 3. Suppose you did t kow either of the values of ad p, but you kew the remaiig values. How will the profit deped o the umber of cadles made? Plot a graph. Repeat for other pairs of variables. 4. Write dow four geeral formulas showig the relatioships betwee the variables. p =.. s =.. =.. k =.. Hadouts for Teachers Cocept Developmet Lessos H-21
Hadout 12: Studets creatig problems for each other Ask studets to work i pairs. Each creates a problem for the other to solve. Doig: The problem poser geerates a equatio step-bystep, startig with, say, x = 4 ad doig the same to both sides draws a rectagle ad calculates its area ad perimeter. Udoig: The problem solver solves the resultig equatio: 10x + 9 " 7 = "0.875 8 tries to draw a rectagle with the give area ad perimeter. writes dow a equatio of the form y=mx+c ad plots a graph. tries to fid a equatio that fits the resultig graph. expads a algebraic expressio such as (x+3)(x-2) factorizes the resultig expressio: x 2 + x - 6 writes dow a polyomial ad differetiates it itegrates the resultig fuctio writes dow five umbers ad fids their mea, media, rage tries to fid five umbers with the give mea, media ad rage. Hadouts for Teachers Cocept Developmet Lessos H-22