数字与运算集合 A11：乘以和除以小数

GRADE5SUPPLEMENT

SetA11Number&Operations:Multiplying&DividingDecimals

Includes

★Activity1:MultiplyingbyPowersofTenA11.1

★Activity2:DividingbyPowersofTenA11.7

★Activity3:UsingDecimalstoCalculateSalePricesA11.15

★Activity4:MultiplyingDecimalsA11.21

★Activity5:BuildingaDeck,UsingPartialProducts&ArraysforDecimalMultiplicationA11.29

★Activity6:MultiplyingDecimals,More/LessA11.35

★Activity7:DividingDecimalswithMoney&MenusA11.45

★Activity8:UsingModels&StrategiestoDividewithDecimalsA11.53

★IndependentWorksheet1:ThinkingaboutTenths,Hundredths&ThousandthsA11.59

★IndependentWorksheet2:VeryLarge&VerySmallNumbersinContextA11.61

★IndependentWorksheet3:Multiplying&DividingbyPowersofTenA11.63

★IndependentWorksheet4:UsingLandmarkFractions&PercentstoMultiplybyDecimalsA11.65

★IndependentWorksheet5:MultiplyingTwoDecimalNumbersA11.67

★IndependentWorksheet6:Comparing&MultiplyingFractions&DecimalsA11.69

★IndependentWorksheet7:OlympicSwimmersA11.71

★IndependentWorksheet8:OlympicTrackStarA11.73

Skills&Concepts

★roundnumberstothenearest0.1,0.01,and0.001

★multiplyanddividebypowersof10,including0.01,0.1,1,10,100,and1,000

★multiplywholenumbersanddecimalnumbersbydecimalnumbers★applyfraction,decimal,andpercentequivalenciestosolveproblems

★describetheeffectofplacevaluewhenmultiplyingwholenumbersanddecimals★multiplydecimalusingconcretemodels,drawingandstrategies

★estimatesolutionstoarithmeticproblemsinordertoassessreasonablenessofresults

★dividedecimalsusingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations.

★relatethestrategytoawrittenmethodandexplainthereasoningused.

CJalem,

TheMATHLEARNINGCENTER

egOn

palliShed

P201503

BridgesinMathematicsGrade5Supplement

SetA11Number&Operations:Multiplying&DividingDecimals

TheMathLearningCenter,POBox12929,Salem,Oregon97309.Tel.1800575–8130.

©2013byTheMathLearningCenterAllrightsreserved.

PreparedforpublicationonMacintoshDesktopPublishingsystem.PrintedintheUnitedStatesofAmerica.

P201503

TheMathLearningCentergrantspermissiontoclassroomteacherstoreproduceblacklinemastersinappropriatequantitiesfortheirclassroomuse.

BridgesinMathematicsisastandards-basedK–5curriculumthatprovidesauniqueblendofconceptdevelopmentandskillspracticeinthecontextofproblemsolving.Itincorpo-ratestheNumberCorner,acollectionofdailyskill-buildingactivitiesforstudents.

TheMathLearningCenterisanonprofitorganizationservingtheeducationcommunity.

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

visitusat.

SetA11Number&Operations:Multiplying&DividingDecimals

SetA11★Activity1

ACTIVITY

MultiplyingbyPowersofTen

Overview

Studentscompleteastringofcalculationswithfractionsanddecimalsandthendiscusstherelationshipsamongthosecalculationstobuildgreatercomputationalfluencyandastrongernumbersensewithdecimals.Thentheyexplorewhathappens,andwhy,whentheymultiplybypowersof10(0.01,0.1,1,10,etc.).

Skills&Concepts

★multiplybypowersof10,including0.01,0.1,1,10,100,and1,000

★describetheeffectofplacevaluewhenmultiplyingwholenumbersanddecimalsby0.01,0.1,1,10,100,and1,000

★applyfractionanddecimalequivalenciestosolveproblems

You’llneed

★PatternsinMultiplyingbyPowersofTen(pagesA11.4

andA11.5,run1copyfordisplay,plusaclassset)

★MultiplyingbyPowersofTenPractice(pageA11.6,run

1copyfordisplay,plusaclassset)

★basetenpiecesforeachpairofstudents,plusasetfordisplay

★GreatWallofBaseTensavedfromUnitSix

AdvancePreparationTrytofindsomecopiesof

BridgesStudentBookpages160and161,Fraction&Deci-malEquivalents,whichstudentscompletedinUnitSix,

Session10.YoumightalsofillinDisplayMaster6.10,Frac-tion&DecimalEquivalencies,whichyouusedinSession12.Bothoftheseresourcesmayjogstudents’memoryofthefractionequivalentsofcommondecimalsinsteps1and3below.

InstructionsforMultiplyingbyPowersofTen

1.Explaintostudentsthatthey’regoingtobemultiplyingdecimalnumbersinthenextfewdaysandthatthey’llbeginwithpowersof10,like0.1,10,and100.Writethefollowingproblemsoneatatime

wherestudentscanseethem(answersincludedinparenthesesforyourreference).Askstudentsto

workinpairsforaminuteortwotosolveoneproblematatime,andthenhavestudentssharetheiran-swersandstrategiesasawholegroup.

•1⁄2×10(5)

•0.5×10(5)

•1⁄4×10(2.5)

•0.25×10(2.5)

•0.75×10(7.5)

Whentheyhavesolvedallfiveproblems,askstudentstodiscusstherelationshipstheynoticedamongtheproblems.Studentsarelikelytonoteequivalenciesbetween1⁄2and0.5,andbetween1⁄4and0.25.Theymayalsohavenoticedthattheycouldhalvehalfof10tofindone-fourthof10,andthatthree-

fourths(0.75)isthreetimesone-fourth.Theymightalsonoticethatwhenmultiplyingadecimalnum-berby10,youmovethedecimalpointoneplacetotheright(e.g.,0.25×10=2.5).

©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.1

SetA11Number&Operations:Multiplying&DividingDecimalsActvity1MultiplyingbyPowersofTen(cont.)

Describingtherelationshipsamongtheproblemsshouldhelpstudentsbegintodevelopefficientstrate-giesforcomputingwithdecimalnumbers.Studentswillsolvesimilarsetsofproblemsatthebeginningofeachactivityinthisset.

2.PlacePatternsinMultiplyingbyPowersofTenondisplayandgiveeachstudentacopy.Reviewthesheetwiththeclass.Discussthesampleequationsineachtableandhavestudentsconnecttheelementsofeachequationtotheproblemsituation.Alsobesurestudentsrememberhowtowriteeachdecimal

(0.01and0.1)asafraction.InvitethemtorefertoBridgesStudentBookpages160and161,FractionandDecimalEquivalents,orafilledincopyofDisplayMaster6.10,FractionandDecimalEquivalencies,ifyouwereabletoretrievetheseresourcesfromUnitSix.

SetA11Number&Operations:Mutpying&DivdingDecmasBlacklneRun1copyfordspay,plusaclassset

NAMEDATEPatternsinMultiplyingbyPowersofTen,page1of2

1aThepostofficesellsone-centstamps.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquantitiesofone-centstamps.

NumberofStamps

DecimalEquation

FractionEquation

TotalCost

1stamp

1×0.01=0.01

1×1/100=1/100

$0.01

2stamps

2×0.01=0.02

2×2/100=2/100

$0.02

10stamps

20stamps

45stamps

321stamps

404stamps

bWhatdoyounoticeaboutmultiplyingby0.01?

3.Givestudentstimetocompletethesheetinpairs.Thenreconvenetheclassasawholegroupand

openthediscussionbyaskingwhattheynoticedaboutmultiplyingby0.01,0.1,and10.Discusseach

multiplieroneatatime,andencouragestudentstoexplainwhythepatternstheysee(e.g.,“Whenyoumultiplyby0.01,thedecimalpointmovestwoplacestotheleft”)makesense.Encouragestudentstore-fertotheGreatWallofBaseTenandtousethebasetenpiecestoexplainthepatternstheysee.Remem-berthatwhenmodelingdecimals,thematrepresents1,thestrip0.10,andtheunit0.01.

JosieIsawwhenyoumultiplyanumberby0.01,likeinthefirstproblem,youcanjustmovethedecimalpointtwoplacestotheleftlikethis.Itworkseverytime.

45×0.01=0.45

45.0becomes0.45

TeacherWhydoesitwork?CanyouusetheGreatWallofBaseTenorthesebasetenpiecestoexplain?

JosieWell,45timesone-hundredthis45hundredths.40hundredthsisthesameasfour-tenths.

That’sthe.4partoftheanswer.And5hundredthsisjust5hundredths.Soit’slikeeachpartofthefirstnumbergetsahundredtimessmaller:40becomesfour-tenthsand5becomesfive-hundredths.Oryoucouldjustthink45hundredths,really.That’sahundredtimessmallerthan45.

A11.2•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter

SetA11Number&Operations:Multiplying&DividingDecimalsActvity1MultiplyingbyPowersofTen(cont.)

40hundredthsis4tenths

5hundredthsisjust5hundredths

4.Afterstudentshavediscussedthepatternsthatemergedwhenmultiplyingby0.01,0.1,and10,give

eachstudentacopyofMultiplyingbyPowersofTenPractice.Explainthatthey’llcompleteitindepen-dently,andthenselectacoupleofproblemsfromthesheettodotogetherbeforeaskingstudentstoworkontheirown.

Extensions

•Ifstudentsfinishearly,askthemtoturntheirpapersoverandwriteproblemsforeachotherinthisform:

45×=0.04545×=4,50045×=4.5

Thentheycantradepapersandfillinthemissingpowersof10ineachequation.

•Clarifytheterm“poweroften”usingtheGreatWallofBaseTen,andintroduceexponentnotation.Apoweroftenisanumberresultingfrommultiplying10byitselfanynumberoftimes.Weuseexpo-nentstoshowhowmanytimesanumber,inthiscase10,ismultipliedbyitself.Anegativeexponentindicatesanumberlessthan1(afractionoradecimal).

1000=103100=10210=1011=1000.1=10–10.01=10–2

©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.3

SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.

NAMEDATE

PatternsinMultiplyingbyPowersofTen,page1of2

1aThepostofficesellsone-centstamps.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquantitiesofone-centstamps.

NumberofStamps

DecimalEquation

FractionEquation

TotalCost

1stamp

1×0.01=0.01

1×

$0.01

2stamps

2×0.01=0.02

2×

$0.02

10stamps

20stamps

45stamps

321stamps

404stamps

bWhatdoyounoticeaboutmultiplyingby0.01?

2aAmeliafeedsherpetlizardcrickets.Thepetstoresellscricketsfortencentseach.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferent

quantitiesofcrickets.

NumberofCrickets

DecimalEquation

FractionEquation

TotalCost

1cricket

1×0.10=0.10

1×=

$0.10

2crickets

2×0.10=0.20

2×=

$0.20

10crickets

20crickets

(Continuedonnextpage.)

A11.4•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter

SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.

NAMEDATE

PatternsinMultiplyingbyPowersofTen,page2of2(cont.)

2a(cont.)

NumberofCrickets

DecimalEquation

FractionEquation

TotalCost

45crickets

321crickets

404crickets

bWhatdoyounoticeaboutmultiplyingby0.10?

3aAlfonso’scompanysellsT-shirtstosoccerteams.EachT-shirtcoststendol-lars.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferent

quantitiesofT-shirts.

NumberofShirts

Equation

TotalCost

1shirt

1×10=10

$10

2shirts

2×10=20

$20

10shirts

20shirts

45shirts

321shirts

404shirts

bWhatdoyounoticeaboutmultiplyingby10?

©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.5

SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.

NAMEDATE

MultiplyingbyPowersofTenPractice

Completethefollowingequations.

106×0.01=

47×0.01=

3×0.01=

0.6×0.01=10×0.01=

0.32×0.01=

0.1×0.01=

452×0.1=

302×0.1=

64×0.1=

0.9×0.1=

0.57×0.1=

0.04×0.1=

0.1×0.1=

360×10=

23×10=

4×10=

0.7×10=

0.54×10=

0.01×10=

0.32×100=

4.3×100=

4×100=

45×100=

309×100=

0.1×100=

0.17×1,000=603×1,000=

0.34×1,000=0.01×1,000=

9.6×1,000=

A11.6•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter

SetA11Number&Operations:Multiplying&DividingDecimals

SetA11★Activity2

ACTIVITY

DividingbyPowersofTen

Overview

Studentscompleteastringofcalculationswithfractionsanddecimalsandthendiscusstherelationshipsamongthosecalculationstobuildgreatercomputationalfluencyandastrongernumbersensewithdecimals.Thentheyexplorewhathappens,andwhy,whentheydivideby

powersof10(0.01,0.1,1,10,etc.).

Skills&Concepts

★dividebypowersof10,including0.01,0.1,1,10,100,and1,000

★describetheeffectofplacevaluewhendividing

wholenumbersanddecimalsby0.01,0.1,1,10,100,and1,000

★applyfractionanddecimalequivalenciestosolveproblems

You’llneed

★PatternsinDividingbyPowersofTen(pagesA11.10–A11.12,run1copyfordisplay,plusaclassset)

★DividingbyPowersofTenPractice(pageA11.13,run1copyfordisplay,plusaclassset)

★basetenpiecesforeachpairofstudents,plusasetfordisplay

★GreatWallofBaseTensavedfromUnitSix

InstructionsforDividingbyPowersofTen

1.Writethefollowingproblemsoneatatimewherestudentscanseethem(answersincludedinparen-thesesforyourreference).Askstudentstoworkinpairsforaminuteortwotosolveoneproblematatime,andthenhavestudentssharetheiranswersandstrategiesasawholegroup.

•10×0.1(1)

•10×0.6(6)

•600×0.01(6)

•600×0.04(24)

•40×0.8(32)

2.Whentheyhavesolvedallfiveproblems,askstudentstodiscusstherelationshipstheynoticed

amongtheproblems.Studentsarelikelytonotethatmultiplyingby0.1islikedividingby10,justas

multiplyingby0.01islikedividingby100.Withthisinmind,theycansolve600×0.04,forexample,inthefollowingway:600÷100=6and6×4=24.

3.Nowexplaintostudentsthattodaythey’regoingtobedividingbypowersof10,like0.1,10,and100.

PlacePatternsinDividingbyPowersofTenondisplayandgiveeachstudentacopy.Reviewthesheetwiththeclass.Discussthesampleequationsineachtableandhavestudentsconnecttheelementsofeachequa-tiontotheproblemsituation.Alsobesurestudentsrememberhowtowriteeachdecimalasafraction.

©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.7

SetA11Number&Operations:Multplying&DvidingDecimalsRun1copyfordisplay,plusaclasssetPatternsinDividingbyPowersofTen,page2of3

2bFilloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquantitiesofcrickets.

TotalCost

DecimalEquation

FractionEquation

NumberofCrickets

$0.10

0.10÷0.10=1

1⁄10÷1⁄10=1

1cricket

$0.20

0.20÷0.10=2

2⁄10÷1⁄10=2

2crickets

$1.00

$2.00

$3.30

$5.20

cWhatdoyounoticeaboutdividingby0.10?

3aThepostofficesellsone-centstamps.Ifyouspent$2.08,howmanyone-centstampscouldyoubuy?

bFilloutthetablebelowtoshowhowmanystampsyoucouldbuywithdiffer-entamountsofmoney.

TotalCost

DecimalEquation

FractionEquation

NumberofStamps

$0.01

0.01÷0.01=1

1⁄100÷1⁄100=1

1stamp

$0.02

0.02÷0.01=2

2⁄100÷1⁄100=2

2stamps

$0.10

$0.40

(Continuedonnextpage.)

SetA11Number&Operations:Multiplying&DividingDecimalsActivity2DividingbyPowersofTen(cont.)

SetA11Number&Operatons:Multiplyng&DividingDecimasBlacklineRun1copyfordisplay,plusaclassset

NAMEDATE

PatternsinDividingbyPowersofTen,page1of3

1aAlfonso’scompanysellsT-shirtstosoccerteams.EachT-shirtcoststendol-lars.Ifyouspent$1030,howmanyshirtscouldyoubuy?

bFilloutthetablebelowtoshowhowmanyT-shirtsyoucouldbuywithdiffer-entamountsofmoney.

TotalCost

Equation

NumberofShirts

$10

10÷10=1

1

$20

20÷10=2

2

$100

$200

$450

$3210

$1020

cWhatdoyounoticeaboutdividingby10?

2aAmeliafeedsherpetlizardcrickets.Thepetstoresellscricketsfortencentseach.IfAmeliaspent$1.30oncricketslastweek,howmanycricketsdidshebuy?

(Continuedonnextpage.)

Asyoureviewthesheet,discusshowtowritethenumbersthataregreaterthan1asafraction.Inthiscase,studentswillprobablyfinditmostusefultowritethemasimproperfractions.Forexample,theywould

write2.47as247⁄100inthefirsttable.Thiswillprobablymakedividingby1⁄100moresensibletothem.

4.Nowaskstudentstocompletethesheetinpairs.Encouragethemtousethebasetenpiecestothinkabouttheproblemsifthathelps.Thenreconvenetheclassasawholegroupandopenthediscussionbyaskingwhattheynoticedaboutdividingby0.01,0.1,and10.Discusseachdivisoroneatatime,anden-couragestudentstoexplainwhythepatternstheyseemakesense.(e.g.,“Whenyoudivideby0.01,thedecimalpointmovestwoplacestotheright.That’swhathappenswhenyoumultiplyby100too!”)In-

vitestudentstorefertotheGreatWallofBaseTenandtousethebasetenpiecestoexplainthepatternstheysee.Rememberthatwhenmodelingdecimals,thematrepresents1,thestrip0.10,andtheunit0.01.

SydneyWhenyoudividebyadecimalnumber,it’slikemultiplyingbythereversewholenumber,soyoumovethedecimalpointthatmanyplacestotheright.

TeacherPleaseusethebasetenpiecestoshowuswhatyoumeanandwhythisistrue.

SydneyWell,thinkaboutthesestrips.Theyshow40.Soifyoudivideby0.1,it’slikeasking,howmanytenthsin40?Thereare10tenthsineachlittleunitand40unitsaltogether,soyougo10×40=400.So40÷0.1=400.400islike40withthedecimaloneplacetotheright.

A11.8•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter

SetA11Number&Operations:Multiplying&DividingDecimalsActivity2DividingbyPowersofTen(cont.)

Thereare10tenthsineachlittleunit,and40

unitsaltogether.10×40=400.

Students’verbalexplanationswillvaryconsiderablyintheirclarity,soencouragethemtoshowtheirthinkingwithbasetenpiecesandequations.Thiswillallowyoutogetaclearersenseofwhattheyun-derstandandwillmaketheirexplanationsmorecomprehensibletootherstudents.

5.Afterstudentshavediscussedthepatternsthatemergedwhendividingby0.01,0.1,and10,giveeachstu-dentacopyofDividingbyPowersofTenPractice.Explainthatthey’llcompleteitindependently,andthenselectacoupleofproblemsfromthesheettodotogetherbeforeaskingstudentstoworkontheirown.

Extensions

•Ifstudentsfinishearly,askthemtoturntheirpapersoverandwriteproblemsforeachotherinthisform:

45÷=0.04545÷=45045÷=4.5

Thentheycantradepapersandfillinthemissingpowersof10ineachequation.

•Youmightalsoconsideraskingthemtowritetheirproblemsinthisform:

45÷10=45×45÷0.10=45×45÷0.01=45×

•Helpstudentsunderstandpowersof10inagraphicway.TheMolecularExpressionswebsite(see

URLbelow)featuresaphotographicdisplaycalledSecretWorlds:TheUniverseWithinthatillustratespowersof10startingwiththeMilkyWay,10millionlightyears(1020meters)fromEarth.AseriesofphotosmovecloserandclosertoEarth,decreasingindistancebyapowerof10eachtime,untilyoureachatalloaktreeseenatadistanceof1meter(100meter).Thephotosdon’tstopthere,however.Thepowersof10gonegativeastheseriesmovesinthemicroscopicworldofanoakleaf,andfinallyintoasubatomicuniverseofelectronsandprotons.

/primer/java/scienceopticsu/powersof10/

INDEPENDENTWORKSHEET

UseSetA11IndependentWorksheets1–3onpagesA11.59–A11.64toprovidestudentswithmorepracticemultiplyinganddividingbypowersoften,aswellasroundinganddoingcalculationswithdecimals.

©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.9

SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.

NAMEDATE

PatternsinDividingbyPowersofTen,page1of3

1Alfonso’scompanysellsT-shirtstosoccerteams.EachT-shirtcoststendollars.aIfyouspent$1030,howmanyshirtscouldyoubuy?

bFilloutthetablebelowtoshowhowmanyT-shirtsyoucouldbuywithdiffer-entamountsofmoney.

TotalCost

Equation

NumberofShirts

$10

10÷10=1

1

$20

20÷10=2

2

$100

$200

$450

$3210

$1020

cWhatdoyounoticeaboutdividingby10?

2Ameliafeedsherpetlizardcrickets.Thepetstoresellscricketsfortencentseach.aIfAmeliaspent$1.30oncricketslastweek,howmanycricketsdidshebuy?

(Continuedonnextpage.)

A11.10•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter

SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.

NAMEDATE

PatternsinDividingbyPowersofTen,page2of3(cont.)

bFilloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquanti-tiesofcrickets.

TotalCost

DecimalEquation

FractionEquation

NumberofCrickets

$0.10

0.10÷0.10=1

÷=1

1cricket

$0.20

0.20÷0.10=2

÷=2

2crickets

$1.00

$2.00

$3.30

$5.20

cWhatdoyounoticeaboutdividingby0.10?

3Thepostofficesellsone-centstamps.

aIfyouspent$2.08,howmanyone-centstampscouldyoubuy?

bFilloutthetablebelowtoshowhowmanystampsyoucouldbuywithdifferentamountsofmoney.

TotalCost

DecimalEquation

FractionEquation

NumberofStamps

$0.01

0.01÷0.01=1

÷=1

1stamp

$0.02

0.02÷0.01=2

÷=2

2stamps

$0.10

$0.40

(Continuedonnextpage.)

©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.11

SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.

NAMEDATE

PatternsinDividingbyPowersofTen,page3of3(cont.)

3bCont.

TotalCost

DecimalEquation

FractionEquation

NumberofStamps

$0.86

$2.47

$3.05

cWhatdoyounoticeaboutdividingby0.01?

A11.12•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter

SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plus