© ICCAMS 2013page 1

FOR TRIAL ONLY

Multiplicative Reasoning: Lesson 2A

Westgate Close

Students estimate the length of Westgate Close, in metres and feet (Task A).They then estimate and calculate two distances along Westgate Close (Task B and Task C).

Task A Task B Task C

Outline of the lesson1. TaskA:estimatethelengthofWestgateClose,inmetresandfeet.

• Askforsomequickestimates.• Askstudentstoworkonthetaskinsmallgroups.• Brieflydiscusstheirmethodsandresults.

2. TaskB:changea72mdistanceonamaptofeet.• Askstudentstoworkontheestimateinsmallgroups.• Brieflydiscusstheirmethodsandresults.• Askstudentstoworkonthecalculationinsmallgroups.• Discusstheirmethodsandresults.

3. TaskB:change72mtofeetusingaratiotable.• RepresenttheinformationinTaskBinaratiotable.

Discusswaysofoperatingonthenumberstofindtherequiredvalueinfeet.

4. TaskC:changea33mdistanceonamaptofeet.• Askforsomequickestimates.• RepresenttheinformationinTaskCinaratiotable.

Askstudentstoworkonthecalculationinsmallgroups,usingthemap(DNL)and/ortheratiotable.

• Discusstheirmethodsandresults.

5. Possibleextension:reviseTaskC.• Askstudentstochangethegivendistance,33m,

toadifferentdistance,tomakethetaskeasierorharder.

6. Possibleextension,TaskD:assessstudents’work.• AskstudentstoevaluatesomeresponsestoTaskB.

SummaryInthislessonwelookatmultiplicationintermsofscaling,andmodelthiswiththeDoubleNumberLine(DNL).Weusethecontextofamap,withscalesshowingdistancesinmetresandfeet,tointroducetheDNLmodel.AtthesametimeweusetheDNLmodeltoconvertmetrestofeet,andwerelatethistotheuseofratiotables.

Westgate Close Task A

Look at the scale on the map.Use the scale to estimate the length of Westgate Close i. in metres ii. in feet.

20 m

50 ft

Westgate Close Task B

Here is another map of Westgate Close and Roman Rd.

Ag’s house is 15 m, or 50 ft, along Westgate Close.

Bo’s house is 72 m along Westgate Close, at the very end of the road.

i. Estimate Bo’s distance in feet.

ii. Calculate Bo’s distance in feet (try to find several ways of calculating the answer).

Westgate Close

0R

oman

Rd

0

m

ft

15

50

Ag

72

?

Bo

Westgate Close Task C

Here is another map of Westgate Close and Roman Rd.

Co’s house is 6 m, or 20 ft, along Westgate Close.

Di’s house is 33 m along Westgate Close.

i. Estimate Di’s distance in feet.

ii. Calculate Di’s distance in feet (try to find several ways of calculating the answer).

Westgate Close

0

Rom

an R

d

0

6

20

33

?

m

ft

Di Co

6

20

33

15

50

72

Note: These materials are the subject of ongoing research and are made available on request to teachers as draft trial materials only. PleasesendfeedbacktoJeremy.Hodgen@kcl.ac.ukorDietmar.Kuchemann@kcl.ac.uk

© ICCAMS 2013page 2

FOR TRIAL ONLY

Multiplicative Reasoning: Lesson 2A Westgate Close (continued)

Overview

Students’ mathematical experiencesStudents

• estimatelengths

• talkaboutratioproblemsandhowtosolvethem

• usetheDNLandratiotablestomodelproblemsinvolvingratio.

Adapting the lessonTheextensionactivityinStage5asksstudentstoreviseTaskCbychoosingnumbersthatwouldmakeiteasierorharder.Thisisausefulwayforstudentstoreflectonthenatureofataskandmethodsofsolution,anditprovidesinsightfortheteacherintothenatureofstudents’understanding.Youmightwanttoadoptthisstrategyyourself,earlierinthelesson,ifthegivennumbersprovetobetooeasyorhard.Youmightalso,hereorinalaterlesson,wanttodiscusshowtheDNLandratiotablecanbeusedtomodelandsolvetheitemsintheMini Ratio Test.

Assessment and feedbackTheMini Ratio Test,inparticularstudents’responsestotheDollarsconversionitem,shouldhelpyouanticipatethekindsofmethodsthatwillariseinthelesson-egaddition,ratedaddition,scaling,theunitarymethod.Findspaceinthelessontomakesuchmethodsexplicitbutdon’tatthisstagefeelthatyouhavetotrytoresolveallthemisconceptionsthatmightcomeup.TaskDgivesstudentsanopportunitytoreflectondifferentmethodsandideas.Itinvolvesassessingworkdonebyotherstudentsandthusprovidespracticeinpeer-andself-assessment.Youmightalsowanttogivestudentssimilartaskstotheonesinthelessonbutinvolvingadifferentratiocontext-arecipe,forexample.Dostudentsfindthiseasierorharder?

Mathematical ideasInthislessonweuseamaptointroducestudentstotheDoubleNumberLine(DNL),andweusethisandratiotablestoexploreconversionsbetweenmeasurementsinvolvingmetresandfeet.

ThetasksinthislessonaresimilartotheDollarsconversionitemontheMini Ratio TestStarter,andtherelationsbetweenthegivennumbers(eg15×3⅓=50)areatasimilarlevelofcomplexity.Thissuggeststhatsomestudentsmightwellconstruethesituationasadditive–egtheymightconcludethat72misthesameas107ft,because50is35morethan15,and72plus35is107.ThislessonisdesignedtohelpstudentsseethatthesituationismultiplicativeandintroducesthemtoamultiplicativemodelintheformoftheDNL.

Key questionsHowdidyouworkthatout?We’vegottwoanswersandanexplanationforboth.Howcanwedecidebetweenthem?

© ICCAMS 2013page 3

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Multiplicative Reasoning: Lesson 2A Westgate Close (continued)

Outline of the lesson (annotated)

Thiscanbedonequiteeffectivelybyeye,bymarkingoffdistancesofapproximately15manddistancesofapproximately50ft.

Again,thiscanbedonebyeye,orbyusingtheideathatthereare‘50ftforevery15m’,andthatthat5×15(or15+15+15+15+15)isalittleover72:sothedistanceinfeetwillbeabitlessthan5×50(or50+50+50+50+50).

Itislikelythatsomestudentswillusethe‘additionstrategy’here,andthusgetananswerof107ftratherthan240ft(byarguing“15+57=72,so50+57=107”,or“15+35=50,so72+35=107”).Asktheclasstocritiquethis.Forexample,• Howdoesthisanswercomparetotheearlierestimates?• Howdoesitfitwiththeideaof‘50ftforevery15m’?• Howmayftwouldthemethodgivefor16m,or1m?

Trytomakelinksbetweenstudents’useofthedoublenumberlineandstudents’useoftheratiotable.

1. TaskA:estimatethelengthofWestgateClose,inmetresandfeet.• Askforsomequickestimates.• Askstudentstoworkonthetaskinsmallgroups.• Brieflydiscusstheirmethodsandresults.

2. TaskB:changea72mdistanceonamaptofeet.• Askstudentstoworkontheestimateinsmallgroups.• Brieflydiscusstheirmethodsandresults.

• Askstudentstoworkonthecalculationinsmallgroups.

• Discusstheirmethodsandresults.

3. TaskB:change72mtofeetusingaratiotable.• RepresenttheinformationinTaskBinaratiotable.

Discusswaysofoperatingonthenumberstofindtherequiredvalueinfeet.

4. TaskC:changea33mdistanceonamaptofeet.• Askforsomequickestimates.• RepresenttheinformationinTaskCinaratiotable.

Askstudentstoworkonthecalculationinsmallgroups,usingthemap(DNL)and/ortheratiotable.

• Discusstheirmethodsandresults.

5. Possibleextension:reviseTaskC.• Askstudentstochangethegivendistance,33m,toa

differentdistance,tomakethetaskeasierorharder.

6. Possibleextension,TaskD:assessstudents’work.• AskstudentstoevaluatesomeresponsestoTaskB.

Itispossibletofindthedesirednumberoffeet(240)byusingthemultiplier×3.̇3,whichmaps(thenumberof)metresonto(thenumberof)feet.Thus15×3.̇3=50,and72×3.̇3=240.

Itisalsopossibletousethemultiplier×4.8,whichmaps15monto72m,andthusmaps50ftonto50ft×4.8=240ft.Challengestudentstofindthesemultipliers,iftheydon’tarisespontaneously.

However,firstallowstudentstoreportontheirownideas.Thesemightincludesuccessfuladditivestrategies,eg‘ratedadition’:15+15+15+15+15–1/5of15=72,sosimilarly50+50+50+50+50–1/5of50=240.Ortheymightincludetheuseofintermediatevalues,egusing÷5tomap(15m,50ft)onto(3m,10ft)andthenusing×24tomapthisonto(72m,240ft).

15

×4.8

×4.8

50

72

15

×24

×24

÷5

÷5

50

3

10

72

15×3.3 ×3.3

. .

50

72

Askstudentstoevaluatetheseresponses(seeTaskD).Thiscouldbesetforhomework.

© ICCAMS 2013page 4

FOR TRIAL ONLY

Scaling in the context of conversions and enlargementThoughWestgateCloseinvolvesdistancesonamap,itdoesnotinvolvegeometricenlargement.Wearesimplyconvertingadistanceonamapmeasuredinoneunittothesamedistancemeasuredinanotherunit.Wearenotscalingadistanceonamaptotheequivalentdistanceinreallifeorontoamapdrawntoadifferentscale.

ThustheWestgateClosetasksareprobablyclosertotheconversionitems(Dollars)ontheMini Ratio Testthantotheenlargementitems(CurlyL).Assuch,theyareprobablylessdemandingthantheCurlyKtaskshownbelow,wherethescalarandfunctionalmultipliersarefractional(×1½and×11⁄8,respectively),aswiththeWestgateClosetasks.TheitemisfromtheCSMSRatiotest(Hart,1981),andwasansweredcorrectlybyjust14%ofarepresentativesample(N=309)ofYear8studentsin1976,andbyasimilarproportion(12%)ofarepresentativesample(N=754)ofYear8studentsin2008.

Onepossiblereasonwhythesuccessrateonthisitemissolowisthattheadditionstrategyresponseof13isquiteclosetothecorrectanswer,13.5.ThisisnotsoforWestgateClose.Theresponsebelow(foranearlyversionofTaskB)wasgivenbyastudentfromalowattainingYear8group.Hehasnotmadeanyattempttorelatemetresandfeetbuthisestimateof220ftforthelengthofWestgateCloseisneartothecorrectdistance(240ft)andisverydifferentfromthedistance(107ft)producedbytheadditionstrategy.Thusstudentswhousetheadditionstraegyherehaveastrongreasontothinkagain.

TheworkbelowisbyastudentfromaveryhighattainingYear8class.Shefirstcameupwiththeadditionstrategyanswer107(from72+35andperhaps50+57),but

Multiplicative Reasoning: Lesson 2A Westgate Close (continued)

Background

subsequentlyarrivedatthecorrectanswer,240,probablyinthelightofawhole-classdiscussion.

Note:Inthislessonweareusingtheconversion“15mcorrespondsto50feet”.Thisisnotexact-theactualrelationismorecomplex.

Ratio tablesAtablecanbeveryusefulfororganisinginformationinaprobleminvolvingratio,ieforshowinghowthevalueoneistryingtofindcorrespondstothegiveninformation.Italsoprovidesausefulmeansforreflectingonandrecordingtheoperationsthatonemightperformtofindthemissingvalue.RatiotablesareparticularlyeasytocreatefortheWestgateClosetasks,sincewecanuseasimilarlayouttothewaythenumbersarepresentedinthegivenschematicmaps.

Additionstrategy

Ratedaddit-ion

Useofarate(“10forevery3”)

Unitarymethod

Functionalmultiplier

Scalar multiplier

15+35 +35

50

72 15

+57

+57

50

72

3÷5

÷5

10

15 15+15+15+15+15 – 3 = 72

50 50+50+50+50+50–10 =

72

15

÷15

÷15

50 3.3.1 72

×72

×72

15

÷5

÷5

50 10

3 72

×24

×24

15

×4.8

×4.8

50

7215×3.3 ×3.3

. .

50

72

© ICCAMS 2013page 5

FOR TRIAL ONLY

Multiplicative Reasoning: Lesson 2A Tasks A, B, C, D (from the file MR-2A-tasks-ABCD.pdf)