學(xué)習(xí)使用 Rekenrek 進(jìn)行數(shù)學(xué)思考 補(bǔ)充活動(dòng)

therekenrek

LearningtoThink

Mathematically

withtheRekenrek

SupplementalActivities

JeffreyFrykholm,Ph.D.

LearningtoThinkMathematicallywiththeRekenrek,SupplementalActivitiesAcomplementaryresourcetotheLearningtoThinkMathematicallyseries

byJeffreyFrykholm,Ph.D.

PublishedbyTheMathLearningCenter

?2008TheMathLearningCenter.Allrightsreserved.

TheMathLearningCenter,POBox12929,Salem,Oregon97309.Tel1(800)575-8130

Originallypublishedin2008byCloudbreakPublishing,Inc.,Boulder,Colorado

TheMathLearningCentergrantspermissiontoreproduceandshareprintcopiesorelectroniccopiesofthematerialsinthispublicationforeducationalpurposes.Forusagequestions,pleasecontactTheMathLearningCenter.

TheMathLearningCentergrantspermissiontowriterstoquotepassagesandillustrations,withattribution,foracademicpublicationsorresearchpurposes.Suggestedattribution:

“LearningtoThinkMathematicallywiththeRekenrek,SupplementalActivities,”JeffreyFrykholm,2008.

TheMathLearningCenterisanonprofitorganizationservingtheeducationcommunity.

Ourmissionistoinspireandenableindividualstodiscoveranddeveloptheirmathematicalconfidenceandability.Weofferinnovativeandstandards-basedprofessionaldevelopment,curriculum,materials,andresourcestosupportlearningandteaching.

ISBN:978-1-60262-566-2

LearningtoThinkMathematicallywiththeRekenrek

SupplementalActivities

Authoredby

JeffreyFrykholm,Ph.D.

AcomplementaryresourcetotheLearningtoThinkMathematicallyseries

2

BookOverview

ThiscompilationofactivitieshasbeendevelopedasasupplementalresourcetotheThinkingMathematicallyseriesindependentlypublishedbyCloudbreakPublishing

().Theactivitiesinthebookareintendedtobuilduponthe

contextsandlearningobjectivesfoundintheseriesbookentitled:LearningtoThinkMathematicallywiththeRekenrek.

Departingsomewhatfromthefirstbook,theseactivitiesarelessaboutthemechanicsofusingtherekenrekitself,andmoreabouttheapplicationofthemodeltosolveproblemcontextsthatareimaginableandaccessibletostudents.Thepurposeofthisbook,then,istoprovide

teacherswithstudent-readyactivitiesthatcanbeprinted,projectedormodeledforimmediateuseintheclassroomsetting.

Theproblemshighlightcontexts(e.g.,icecreamcones,trains,andbusses)thatmotivatethekeymathematicalprinciplesthatarereadilyaccessedthroughtherekenrekmodel(e.g.,

cardinality,one-to-onerelationships,part-part-wholerelationships,subitizing,etc.).The

illustrationsintheproblemsmatchthestructureoftherekenrek,particularlyitsgroupingsof5,10,and20.Theproblemsthemselvesdonotfeaturearekenrekmodelperse;however,the

problemshavebeendesignedsuchthatteachersandstudentscanusearekenrektomodeltheproblems,discussthinkingstrategies,andmakeconnectionsbetweenkeymathematical

insightsthatariseintheflowofclassroomconversation.Therefore,whiletherekenrek

manipulativeisnotfeaturedintheproblemsexplicitly,theintentionofthissupplemental

resourceistoprovideteacherswithexercisesforwhichtherekenrekmodelcanbeusedveryeasilytorepresenttheproblem,modelthinkingandcontributetostudents’solutionstrategies.

IfyouarenewtotheRekenrekmanipulative,werecommendthatyouexploretheLearningtoThinkMathematicallywiththeRekenrekbook(

)

beforeusingtheseactivitieswithyourstudents,asitprovidesathoroughintroductionand

orientationtotherekenrekmanipulative.Afterusingtherekenrekasateachingtoolformanyyears,weareconvincedthatwithinashortperiodoftime,bothteachersandstudentsthatarenewtothemodelquicklygravitatetoit,andreaptherewards(towardthedevelopmentof

mathematicalunderstandinginyoungchildren)thatareinherentlyavailableinthemodel.

Werecommendthateachchildinyourclassroomhavehisorherownrekenrekmodelnotonly

fortheexercisesintheThinkingMathematicallyseries,butalsoasatoolthatwillbe

instrumentalinhelpingyourstudentsdevelopmathematicalreasoningstrategies,evenasyoungchildren.Thereisnomorepowerfulmathematicalmodelforyoungchildrenwhoareactivelydevelopingtheirmathematicalinsightsandintuitions–thefoundationuponwhich

theywillbasetheframeworkfortheirunderstandingsandexplorationsofmathematics

throughouttheirlives.Ifyoudonothaverekenrekmanipulativesforyourstudents,youmaypurchaseinexpensiverekenreksforyourchildrenonthewebsite.

3

HowtoTeachTheActivitiesinthisBook

Theproblemsetsinthisresourceareintendedtobestand-aloneactivities.Whiletheexercises

growincomplexityingeneralfromthestarttotheendofthebook,itisnotnecessarytobeginwiththefirstsetofproblems,andprogresssequentiallytothelast.Thereare14setsof

exercisesinthebook,clusteredinto3organizinglevelsofcomplexity.Thefirstsetdealswithnumbersintherageof0-5.Thesecondproblemsetcontainscontextsthatengagestudentswithnumberrelationshipsbetween0-10.Finally,thelastsetofproblemschallengesstudentstoextendtheirthinkingtonumberrelationshipsbetween0-20.

Youwillimmediatelyseetheconnectionbetweentheproblemillustrationsandtherekenrek

modelitself.Thefirst4activitiesaredesignedtohelpchildrendevelopcardinalityandone-to-onerelationshipsfornumbersbetween0-5.Atrayof5icecreamconesisusedtomotivatethemathematicalthinkingbehindtheproblems.Werecommendthatyouusearekenrekmodel

fortheseproblemsthatcontains5beadsofthesamecolor.(Unlikemostcommercialmodels,therekenreksthatwehavemadeavailableontheThinkingMathematicallywebsiteare

adaptable–onecaneasilychangethenumberofbeadsonthestringstomodelproblemsintherangeof0-20.)

Trayof5icecreamconesCorrespondingRekenrekModel

Thesecondsetofexercises(Activities5-8)focusonnumberrelationshipsintherangeof0-10.Again,werecommendthatstudentshavea10-beadrekenrekmodelreadilyavailableastheyengagetheseproblems.Thedesignoftheseproblems,aswellasthe10-beadrekenrekitself,areintendedtohelpchildrensubitize–i.e.,torecognizethenumberofitemspresentina

groupwithouthavingtocounteachindividualitem.Thisisacritical,andpowerful,strategy

thatyoungchildrenmustmasterearlyintheirmathematicspathway.Thecontextthatisusedtomotivatethesefundamentalideasintherangeof0-10isthatoftwotraincars,eachwith5passengerwindowsofadifferentcolor.

Twotraincars,fivewindowsineachcarCorrespondingRekenrekModel

4

Thefinalsetofproblems(Activities9-14)focusonnumbersintherangeof0-20.Muchlikethepreviousproblemsets,themathematicalcontentemphasizedintheseactivitiesincludes

subitizingongroupsof5and10,aswellaspart-part-wholerelationshipsto20.Studentshavenumerousopportunitiestoworkwithnumbercombinationsto20,aswellastoexplorethe

comparisonmodelofsubtraction.

Again,theimportanceoftherekenrekmodelinsolidifyingthesefundamentalmathematical

conceptscannotbeemphasizedenough.Ifchildrenbecomecomfortablewithnumbersandtheirrelationshiptoothersintherangeof0-20,theywillbeabletoextendtheirunderstandingandstrategyusetonumbersinamuchlargerrange.Thesuccesschildrenhavein

conceptualizing,modeling,andsolvingtheproblemsinthissetofactivitieswillbeanindicationofhowwelleachchildwillbeabletolaterengageandsucceedindevelopingstrongnumber

senseandcomputationalfluency.Theseproblemsmakeuseofadouble-deckerbusasthemotivatorfortheseessentialmathematicalconcepts.Useofthecorrespondingrekenrek

model(20beads,twostringsof10)isfundamentaltothedevelopmentoffluencyinthisrangeofnumbers.

Double-DeckerBus,2rowsof10(5+5)

CorrespondingRekenrekModel

Ourrecommendationfortheteachingoftheseproblemsisthatbothteacherandstudentmodeleachproblemwitharekenrektobeginthesolutionstrategyprocess.Recallthe

importanceofencouragingchildrennottocounteverypersoninthewindow…everybead.

Withtherightencouragementandpractice,studentsshouldbeganratherreadilytorecognizethegroupsof5and10thatareinherentintheproblems,andthenbuildfromthatrecognition.Sevenisseenas(5+2)…Seventeenmaybeseenas(10+5+2)…Studentsshouldbeabletoseegroupsof10ontherekenrekmodelbothhorizontally(5red,plus5yellow),orvertically(5red,

plus5red).Asyoumodeltheseproblems,takeeveryopportunitytoencouragechildrentosubitize–toavoidcountingeverybead(oricecreamcone,orpersoninthewindow,etc.)wheneverpossible.

Wehopethattheseactivitieswillprovideteacherswithpowerfulcontextsforlearning,and

thatchildrenwillenjoytheconfidencethatcomesfromasolidunderstandingofthesefundamentalnumberrelationships.

5

Rekenrek:SupplementalActivities

ContentFocus

ProblemSet1:Developingnumberrelationships,0-5

Theactivitiesinthissectionhelpstudentsformrelationshipsandunderstandingofnumbersintherangeof0-5.Thisincludescardinality,andbeginningpart-part-wholerelationships.

Studentsshouldbecomecomfortablewithnumbercombinationsuptofive.

Activity1:HowManyIceCreamCones?(one-to-onerelationships)Page3

Activity2:Make5:HowManyMore?(combinationsto5)Page4

Activity3:Howmanybehindthebox?(part-part-whole)Page5

Activity4:TheDogatemyIceCream!(part-part-whole)Page6

ProblemSet2:Developingnumberrelationships,0-10

Theactivitiesinthissectionhelpstudentsformunderstandingsofnumberrelationshipsand

combinationsintherangeof0-10.Theactivitiesinthissectionmakeuseoftwopassenger

traincarsthathave5windowseach.Byexploringhowmanypassengersareinthewindows,studentsexploremathematicalconceptsincludingcardinality,one-to-onerelationships,part-part-wholerelationships,andnumbercombinationsto10.Theprimarygoalintheseactivitiesistoencouragestudentstosubitize…torecognizeagroupof5or10objects(beads)withouthavingtocounteachbead.Forexample,thenumber7mightbethoughtofasthequantityoffive,andtwomore.

Activity5:HowManyontheTrain?(one-to-oneandcardinality)Page7

Activity6:HowManyMoretoMake10(combinationsto10)Page9

Activity7:Howmanyaresleeping?(part-part-whole)Page11

Activity8:Howmanyinthetunnel?(part-part-whole)Page12

ProblemSet3:Developingnumberrelationships,0-20

Theactivitiesinthissectionextendthepreviouslearningtoincludenumbersintherangeof0-20byusingadouble-deckerbusasthemotivatingcontext.Numbercombinationsto20are

highlighted,aswellaspart-part-wholerelationships,doubles,neardoubles,andthe

comparisonmodelofsubtraction.Theprocessofsubitizingis,again,avitalcomponentofthisproblemset.

Activity9:Howmanyonthebus?(one-to-oneandcardinality)Page14

Activity10:Howmanyseatsareleft?Make20.(combinationsto20)Page16

Activity11:Underthemud(part-part-whole)Page18

Activity12:Doubles(doublesfacts)Page20

Activity13:Neardoubles(doublesfacts+/-1)Page22

Activity14:Howmanymoreontheupperdeck?(comparisonmodel)Page24

6

Activity1:HowManyIceCreamCones?

TheScoopicecreamstoresellssinglescoopcones,withstrawberryicecream.

Thestorehasatraythatcanhold5conesatatime.

1.Howmanyicecreamconesinthetray?ANSWER

2.Howmanyicecreamconesinthetray?ANSWER

3.Howmanyicecreamconesinthetray?ANSWER

4.Howmanyicecreamconesinthetray?ANSWER

7

Activity2:Make5…HowManyMore?

5friendswanticecreamcones.

Howmanymoreicecreamconesmustbemadetohave5icecreamcones?

1.Howmanymoretomake5cones?ANSWER

2.Howmanymoretomake5cones?ANSWER

3.Howmanymoretomake5cones?ANSWER

4.Howmanymoretomake5cones?ANSWER

8

Activity3:HowManyareBehindtheBox?

Aworkerputsaboxonthecounterinfrontofthetrayof5cones.

Howmanyconesarebehindthebox?

1.Howmanyicecreamconesarebehindthebox?ANSWER

2.Howmanyicecreamconesarebehindthebox?ANSWER

3.Howmanyicecreamconesarebehindthebox?ANSWER

4.Howmanyicecreamconesarebehindthebox?ANSWER

9

Activity4:TheDogAtemyIceCream!

Crunch!

1.Howmanyconesdidthedogeat?ANSWER

Crunch!

2.Howmanyconesdidthedogeat?ANSWER

Crunch!

3.Howmanyconesdidthedogeat?ANSWER

Crunch!

4.Howmanyconesdidthedogeat?ANSWER

Crunch!

10

Activity5:HowManyontheTrain?

1.Howmanypeopleareonthetrain?ANSWER:

2.Howmanypeopleareonthetrain?ANSWER:

3.Howmanypeopleareonthetrain?ANSWER:

4.Howmanypeopleareonthetrain?ANSWER:

5.Howmanypeopleareonthetrain?ANSWER:

11

Activity5(Continued):HowManyontheTrain?

6.Howmanypeopleareonthetrain?ANSWER:

7.Howmanypeopleareonthetrain?ANSWER:

8.Howmanypeopleareonthetrain?ANSWER:

9.Howmanypeopleareonthetrain?ANSWER:

10.Howmanypeopleareonthetrain?ANSWER:

12

Activity6:HowManyMoretoMake10?

1.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

2.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

3.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

4.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

13

Activity6(Continued):HowManyMoretoMake10?

5.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

6.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

7.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

8.Howmanypeopleonthetrain?ANSWER

Howmanymoretomake10?ANSWER

14

Activity7:HowManyPeopleSleeping?

1.Howmanypeopleareawake?ANSWER

Howmanypeoplearesleeping?ANSWER

2.Howmanypeopleareawake?ANSWER

Howmanypeoplearesleeping?ANSWER

3.Howmanypeopleareawake?ANSWER

Howmanypeoplearesleeping?ANSWER

4.Howmanypeopleareawake?ANSWER

Howmanypeoplearesleeping?ANSWER

15

Activity8:HowManyPeopleintheTunnel?

Thetrainisfullwith10people.Howmanyarealreadyinthetunnel?

1.Howmanypeopleareinthetunnel?ANSWER:

2.Howmanypeopleareinthetunnel?ANSWER:

3.Howmanypeopleareinthetunnel?ANSWER

16

Activity8(Con’t):HowManyPeopleintheTunnel?

4.Howmanypeopleareinthetunnel?ANSWER

5.Howmanypeopleareinthetunnel?ANSWER

6.Howmanypeopleareinthetunnel?ANSWER

7.Modelwithyourownrekenrek:Sixpeopleonthetrainareinthetunnel.

Howmanyareoutsidethetunnel?ANSWER

17

Activity9:HowManyPeopleareontheDoubleDeckerBus?

TheDoubleDeckerBushas20seats–10upontop,and10downbelow.

1.Howmanypeopleareonthebus?ANSWER

2.Howmanypeopleareonthebus?ANSWER

3.Howmanypeopleareonthebus?ANSWER

4.Howmanypeopleareonthebus?ANSWER

18

Activity9(Con’t):HowManyPeopleareontheDoubleDeckerBus?

5.Howmanypeopleareonthebus?ANSWER

6.Howmanypeopleareonthebus?ANSWER

7.Howmanypeopleareonthebus?ANSWER

8.Howmanypeopleareonthebus?ANSWER

19

Activity10:HowManySeatsareLeft?Make20.

TheDoubleDeckerBushas20seats–howmanymorepeoplecangeton?

1.Howmanymorepeoplecangetonthebus?ANSWER

2.Howmanymorepeoplecangetonthebus?ANSWER

3.Howmanymorepeoplecangetonthebus?ANSWER

4.Howmanymorepeoplecangetonthebus?ANSWER

20

Activity10(Con’t):HowManySeatsareLeft?Make20.

5.Howmanymorepeoplecangetonthebus?ANSWER

6.Howmanymorepeoplecangetonthebus?ANSWER

7.Howmanymorepeoplecangetonthebus?ANSWER

8.Howmanymorepeoplecangetonthebus?ANSWER

21

Activity11:Howmanywindowsunderthemud?

Themudsplashedthebus.Howmanywindowsareunderthemud?

1.Howmanywindowsunderthemud?ANSWER

Splash!

2.Howmanywindowsunderthemud?ANSWER

Splash!

3.Howmanywindowsunderthemud?ANSWER

Splash!

Splash!

4.Howmanywindowsunderthemud?ANSWER

22

Activity11(Con’t):Howmanywindowsunderthemud?

5.Howmanywindowsunderthemud?ANSWER

Splash!

6.Howmanywindowsunderthemud?ANSWER

Splash!

7.Howmanywindowsunderthemud?ANSWER

Splash!

8.Howmanywindowsunderthemud?ANSWER

Splash!

23

Activity12:Doubles

1.Howmanypeopleareonthebus?ANSWER

2.Howmanypeopleareonthebus?ANSWER

3.Howmanypeopleareonthebus?ANSWER

4.Howmanypeopleareonthebus?ANSWER

24

Activity12(Con’t):Doubles

5.Howmanypeopleareonthebus?ANSWER

6.Howmanypeopleareonthebus?ANSWER

7.Howmanypeopleareonthebus?ANSWER

8.Howmanypeopleareonthebus?ANSWER

25

Activity13:NearDoubles

1.Howmanypeopleareonthebus?ANSWER

2.Howmanypeopleareonthebus?ANSWER

3.Howmanypeopleareonthebus?ANSWER

4.Howmanypeopleareonthebus?ANSWER

26

Activity13(Con’t):NearDoubles

5.Howmanypeopleareonthebus?ANSWER

6.Howmanypeopleareonthebus?ANSWER

7.Howmanypeopleareontheb