gap closing - edugains€¦ · for example, 7 × 6 is 5 × 6 and 2 × 6. 9 × : to multiply by 9,...

Gap ClosinG

Multiplying and Dividing

Junior / Intermediate

Student Book

Module 5

Multiplying and Dividing

Diagnostic .........................................................................................4

Multiplication Fact Strategies ....................................................8

Multiplying by 1-digit Numbers ...............................................12

Multiplying Two 2-digit Numbers ............................................16

Relating Multiplication to Division .........................................21

Relating Division Calculations ................................................24

Dividing by 1-digit Numbers .....................................................29

4 © Marian Small, 2010 Multiplying and Dividing

Diagnostic

1. Howmuchmoreisthefirstanswerthanthesecond?

a) 4×5than3×5

b) 7×8than5×8

c) 6×9than5×9

d) 8×8than5×8

2.Writetheanswerbyfiguringitoutinyourhead.

a) 5×7 b) 4×3

c) 8×9 d) 4×7

e) 6×8 f) 7×9

g) 8×4 h) 3×9

i) 7×30 j) 6×60

k) 5×300 l) 4×600


Diagnostic (Continued)

3. Fillinthemissingvalues.

a) 5×14=5×___and5×4

b) 6×28=6×20and__×___

c) 7×312=7×___and7×10and__×__

4. Howmuchmoreisthefirstanswerthanthesecond?Youmaywritetheanswerasaproduct.

a) 23×48than20×48

b) 42×53than40×53

c) 70×23than69×23

d) 90×58than88×58

5. Showhowtocalculate23×48.

Youmayuseadiagramorblockstohelpyou.



6. Estimateeachanswer.

a) 8×58 b) 9×42

c) 3×812 d) 5×790

e) 51×32 f) 62×78

7. Howmuchmoreisthefirstanswerthanthesecond?Youmaywritetheanswerasaquotientifyouwish.

a) 35÷7than28÷7 b) 48÷6than30÷6

c) 72÷9than63÷9 d) 56÷8than40÷8

8.Writetheanswerbyfiguringitoutinyourhead.

a) 63÷9 b) 42÷7

c) 32÷8 d) 28÷4

e) 56÷8 f) 64÷8

g) 45÷5 h) 24÷6

i) 120÷3 j) 180÷9

k) 1800÷6 l) 4500÷9



9. Fillinthemissingvalues.

a) 420÷4=400÷4and_____÷4

b) 600÷4=400÷4and_____÷_____

c) 600÷5=_____÷_____and_____÷_____

10. Circlethevaluesthatarethesameas400÷8.

a) 400÷4addedto400÷4 b) 200÷8addedto200÷8

c) 400÷4thendividedby2 d) 400÷5addedto400÷3

11. Estimateeachquotient.

a) 97÷3 b) 450÷8

c) 125÷3 d) 2547÷8

12. Showhowtocalculate328÷8.

Youmayuseadiagramorbase-tenblockstohelpyou.


Multiplication Fact Strategies

LearningGoal• reasoningabouttherelationshipbetweenproductsofsingledigit

numbers.

Open Question

Ifweknowthat4×5=20,weknowthat4groupsof5thingsis20things.

Thisshouldhelpusfigureoutthat:

5×5=25sincethereisonemoregroupof5.

4×6=24sincethereisoneextraitemineachofthe4groups.

5×4=20sinceanarrayof4×5isanarrayof5×4ifyouturnitaround

Choosetwoofthefactsbelow.Foreachofthetwo,tellasmanyotherfactsasyoucanthatwouldhelpyoufigureouttheanswer.Howwouldeachonehelpyou?

3×6=18 4×6=24

3×4=12 5×6=30


Multiplication Fact Strategies (Continued)

Think Sheet

Therearestrategieswecanusetohelpusremembermultiplicationfacts(e.g.,4×6or8×9)

Amultiplicationtellsthetotalinanumberofequalgroups.Forexample,thepicturefor4×5isshownhere.The4tellshowmanygroupsandthe5tellshowmanyineachgroup.

4groupsof5

Wecanusearepeatedadditiontofigureoutthetotal.Forexample,for4×5,youcanaddfourfives:5+5+5+5.

Wecanrelateanewmultiplicationfacttoonewealreadyknow.

Theeasiestonestolearnare:

2× :Todoubleanumber,ormultiplyby2,weaddittoitself. 2×8=8+8

5× :Tomultiplyby5,wecanskipcountby5s. 8×5isthe8thnumberwesay:5,10,15,20,25,30,35,40,45

Thenwecanrelateotherfactstothesefacts.

4× :Tomultiplyby4,wecandoubleadouble.That’sbecause4groupsofsomethingistwiceasmuchastwogroups.



8× :Tomultiplyby8,wecandoubletheresultwhenwemultiplyby4.That’sbecause8groupsofsomethingistwiceasmanyas4groupsofsomething.

3× :Tomultiplyby3,wecanaddthenumbertoitsdouble.That’sbecause3groupsofsomethingis1groupofitandanother2groups.

6× :Tomultiplyby6,wecanmultiplyby3anddouble(6groupsistwiceasmanyas3groups)orwecanmultiplyby5andaddanothergroup(5groupsandanothergroup).

7× :Tomultiplyby7,wecanmultiplyby5andaddthedouble.Forexample,7×6is5×6and2×6.

9× :Tomultiplyby9,wecanmultiplyby10andsubtractthenumberwearemultiplyingby.That’sbecause9groupsofsomethingis1lessgroupthan10groupsofit.

1. Youknowthat3×4=12.Howcanthathelpyoufigureouteachofthesefacts?

a) 4×3

b) 4×4

c) 6×4

d) 3×8



2. Youknowthat3×5=15.Howcanthathelpyoufigureouteachofthesefacts?

a) 4×5

b) 6×5

c) 3×7

3.Whichmultiplicationfactsresultineachproduct?

a) 12

b) 24

c) 18

d) 30

4.Whycanyoualwaysmultiplyanumberby8bymultiplyingitby5,multiplyingitby2andthenaddingthosetwoanswerstothenumber?

5. Describe2strategiestofigureouteach:

a) 8×9

b) 7×8

c) 6×8

d) 6×7


Multiplying by 1-digit Numbers

LearningGoal

• representingaproductusingrepeatedadditionorusinganareamodel.

Open Question

6×53means6groupsof53.

Thinkofitas6groupsof50and6groupsof3.

50 3 50 3

50 3 50 3

50 3 50 3

Wemultiplya1-digitnumberbya3-digitnumber.Whenwearedone,thereisa5inthetensplace.

× = 5

Figureoutatleastfivepairsofnumberswemighthavemultiplied.


Multiplying by 1-digit Numbers (Continued)

Think Sheet

Wewanttoknowhowmanymuffinsarein6boxesof12muffins.

Wecouldmultiplytofigureitout.

6×12isthesameas6×10+6×2.

That’shelpfulsinceweprobablyknowthat6×10is6tens,or60.

Wejustadd6×2=12to60toget72muffins.

Wecouldusethesameideaiftherewere

60 1260

6boxesof22muffins.

Wecouldthinkof6×10+6×10+6×2,orwecoulduseamodeltoseethattherearetwosetsof6×10(or6×20)and6setsof2:

Thetotalvalueis132.

Wecanusethesameideatomultiply6×144whenwewanttoknowhowmanypeoplecansitin6partsofagym,ifeachparthas144chairs.

Since144=100+40+4,then6×144is6×100+6×40+6×4.



Wecanusediagrams,models,orotherstrategiestofigureoutthepartsthatwecanputtogethertogetthetotal.

Wecanestimateproductsbyestimatingthenumbersbeingmultiplied.

Forexample,5×352isabout5×3hundred.Thatis15hundred(1500).

5×679isabout5×7hundred.Thatis35hundred(3500).

1. a) Howmanypencilsarein4boxeswith18pencilsineachbox?

b) Howmanycupcakesarein8boxeswith24cupcakesineachbox?

c) Howmanydaysarein19weeks?

1 2 3 4 5

6 7 8 9 10 11 12

13 14 15 16 17 18 19

20 21 22 23 24 25 26

27 28 29 30 31

d) Howmanysidesarein49triangles?

2. Listtwoorthreeproductsyoucanaddtogethertofigureeachproduct:

a) 4×29

b) 7×56

c) 8×212

d) 9×573



3. Estimateeachproduct:

a) 8×212

b) 7×589

c) 3×421

d) 6×818

4. Lucysaidthattofigureout9×88,shestartsbyfiguringout10×88.

Howwillthathelpherfigureout9×88?

5. Zacharysaidthattofigureout6×95,youcould add(2×95)+(2×95)+(2×95)?Doyouagreeornot?Explain.

6. Putthedigitsintheboxesintheblanksinatleast3differentways.Predictwhichanswerwillbegreater.Testyourprediction.

4 6 3 8 ×

× ×


Multiplying Two 2-digit Numbers (Continued)

LearningGoal

• representingaproductusinganareamodelorrelatingtoasimplerproduct.

Open Question

16×23means16groupsof23.

Thinkofitas10groupsof23and6groupsof23.

That’sthesameas10groupsof20( )and10groupsof3( )and6groupsof3( )and6groupsof20( ).

• Multiplyeachofthefourpairsofnumbers:

37×39

38×38

19×76

33×41

Compareeachpairofnumberswitheachotherpairandtellonewaytheproducts(answers)arealike.



Think Sheet

Wewanttoknowhowmanymuffinstherearein16boxesof24muffins.

Therearemanywaystothinkabouttheproblem.

• Wecouldfigureouthowmanymuffinsarein8boxesanddoubleit,since16isdouble8. 8×24=192and2×192=384muffins. Since16×24iscloseto20×20=400,thismakessense.

• Wecouldfigureouthowmanymuffinsarein10boxesandhowmanymuffinsarein6boxesandaddthem.

Weknowthat10×24=240. Wealsoknowthat6×24=6×20+6×4,or144. Thetotalis384muffins.

• Wecouldalsomodelarectanglethatis16by24andfinditsarea.That’sbecausethereare16rowsof24.

Noticethatthetwobluesectionsareeach10×10.Wecouldcallthem100s. Noticethatthe40greensquaresareactually4columnsof10. Altogether,thereis200+40+120+24=384.



1.Matchthemodelwiththeproductitshows.

a) 12×35 _____

b) 24×27 _____

c) 53×48 _____

d) 18×83 _____

E:

F:

G:

H:



2.Matchtheproductswiththeirdescriptions.

a) 14×73 E: 10×30+10×7+4×30+4×7

b) 41×37 F: 10×30+10×4+7×30+7×4

c) 14×37 G: 10×70+10×3+4×70+4×3

d) 17×34 H: 40×30+40×7+1×30+1×7

3.Whichofthesearecloseto2000?Circleallthatmakesense.

a) 60×55

b) 70×29

c) 97×35

d) 49×37

e) 33×15

4. Figureoutthetotalnumberof:

a) pencilsin14boxeswith15pencilsineachbox.

b) cupcakesin24boxeswith24cupcakesineachbox.

c) studentsin13classeswith25studentsineachclass.

d) photoson32pageswith16photosonapage.



5. Lucysaidthattofigureout39×88,shestartsbyfiguringout40×88.

Howwillthathelpherfigureout39×88?

6. Tellwhy38×50hasthesameansweras76×25.

7. Putthedigitsintheboxesintheblanksinatleast3differentways.Predictwhichanswerwillbethegreatest.Testyourprediction.

4 6 3 8 ×

× ×


Relating Multiplication to Division

LearningGoal

• connectinganydivisionquestiontoarelatedmultiplication.

Open Question

Thequestion42÷6= isanotherwayofwriting6× =42.

Wewanttoknowwhatnumbertomultiply6bytoget42.

EachofthesepairsdescribeAandB:

1. Aisbetween40and60andBis6




• DivideAbyB.

• Tellwhathastobetrueaboutthequotient.

• Explainhowyouknow.

• ChoosetwoothervaluesforAandBandanswerthesamethreequestionsabove.


Relating Multiplication to Division (Continued)

Think Sheet

Wedividewhenwewanttofindouthowmanygroupsofacertainsizeareinanumber.

Thereare7daysinaweek.Ifwewanttoknowhowmanyfullweekstherearein35days,wedivide35by7.

Wecanfigureitoutdifferentways:

• Wecankeepsubtracting7stoseehowmany7sarein35.

35–7–7–7–7–7=0. Wecouldsubtractfive7s,so35÷7=5weeks.

• Wecouldfigureoutwhattomultiply7bytogetto35,inotherwords7× =35.

Thinkofmultiplicationfacts.Since7×5=35,then35÷7=5weeks.

Wealsodividetoshareequally.

• Suppose7peopleshare$35.Wecankeepsubtracting7suntilwegetto0.

Thatwouldtellustohowmanytimeseachpersoncanget$1.

• Wecanalsofigureoutwhattomultiply7bytoget35.

Ifthe7peoplehadshared$350(35$10bills),theanswerwouldbe10timesasmuch.Since7×5=35,7×50=350.

1. Describetwowaystosolveeachproblem.

a) Howmanyhandsarethere,ifyoucansee45fingersandthumbs?

b) Howmanyfullweeksaretherein56days?

c) Howmanypackagesof6applesdoyoubuy,ifyouwant54apples?


Relating Multiplication to Division (Continued)

2.Whatmultiplicationfactswouldhelpyoufigureeachout?

a) 36÷4

b) 32÷8

c) 630÷9

d) 490÷7

3. Youdivide2numbersandthequotient(answerindivision)is8. List5possiblepairsthatyoumighthavedivided.

4. Youdivide2numbersandthequotientis40.List5possiblepairsthatyoumighthavedivided.

5. Youfigureout480÷6.Howdoyouknowthatthequotientfor540÷2mustbeagreaternumber?


Relating Division Calculations

LearningGoal

• reasoningabouttherelationshipbetweenquotients.

Open Question

Ifyouknowthat20÷4=5,youknowthatthereare5groupsof4in20.

Therearealso4groupsof5in20.

Thisshouldhelpyoufigureoutthat:

• 24÷4=6sincethereisonemoregroupof4.

• 16÷4=4sincethereisonefewergroupof4.

• 40÷4=5+5sincethereare5groupsof4ineach20buttherearetwo20s.


Relating Division Calculations (Continued)

• 200÷4=50since200is20tens,sothereare4groupsof5tens(or50)in200.

Choosetwoofthedivisionquestionsbelow.

Foreachdivisionquestion,tell3or4divisionquestionsthatarerelatedandexplainhowtheyarerelated.

63÷9=7 28÷7=4

36÷9=4 56÷8=7



Think Sheet

Wecanfigureouttheanswertoadivisionquestionbyusinganotherdivisionquestionwithananswerweknow.

Forexample,supposewewanttoknowhowmanypaperclipseachpersongets,if4peopleshare64paperclipsequally.

• Onestrategyistobreakupwhatwearesharingintoparts:

Wecouldthink:Iknowthat40÷4=10. If4peopleshare40paperclips,theyeachget10. Since64=40+24,wewillshare40paperclipsandthentheother24. 40÷4=10and24÷4=6,so64÷4=10+6=16.

• Anotherstrategyistomultiplyordividebothnumbersbythesameamount.

Forexample,48÷8=24÷4.Wecandividebothnumbersby2.That’sbecauseif8peopleshare48,then4(halfasmanyas8)ofthemshare24(halfofthewhole48).

Wecanalwaystrytorelatethenewquestiontoquestionsthatwefindeasier.



1. Howmuchmoreisthefirstquotientthanthesecondone?Tellwhy.

a) 48÷4than44÷4

b) 30÷5than25÷5

c) 56÷8than40÷8

d) 78÷6than60÷6

2. Howwouldyoubreakupeachamounttomakeiteasytoworkwith thegroups?

a) 55tomakegroupsof5

b) 75tomakegroupsof3

c) 96tomakegroupsof6

d) 108tomakegroupsof4

3. Howwouldyoubreakupeachamounttomakeiteasiertoshare?

a) 68into4groups

b) 92into4groups

c) 84into7groups

d) 93into3groups



4. Alyssasaysthatyoucanfigureout72÷6bytakinghalfof72anddividingthathalfby3.Soshefiguresout72÷2andthendividestheanswerby3.DoyouagreewithAlyssa?Explain.

5.Whyisiteasiertocompare48÷8with40÷8thantocompare48÷8with48÷3?

6. Describethreedifferentstrategiesyoucouldusetocalculate96÷4.


Dividing by 1-digit Numbers

LearningGoal

• representingaquotientusingsharingandgroupingmodels.

Open Question

Select5differentnumbersbetween50and500.

Divideeachofthembyatleastthreedifferentwholenumbersbetween1and10.Trytogetasmanydifferentremaindersasyoucan.

• Whatdidyoudivideby?

• Whatareyourremainders?


Dividing by 1-digit Numbers (Continued)

Think Sheet

Wehavetoput86sandwichesintobagsof2sandwicheseach.

Wewanttoknowhowmanybagsweneed.

• Tofigureout86÷2, 86÷2= wemightthink:I’llputthefirst40into20bagsof2. ( ÷2= ) ThenI’llputthenext40into20bagsof2. ( ÷2= ) ThenI’llputthelast6into3bagsof2. ( ÷2= ) Iused20+20+3=43bags. ÷2=

• Wecouldalsothink:86=80+6 So86÷2=(80÷2)+(6÷2) =40+3,or43

Suppose2peopleweresharing$87.

• Wecouldmodel8tensand7onesusingbase-tenrodsandsmallcubes.

• Wecouldmake2equalpilesof4tensand3ones,witharemainderof1.

So$87÷2=$43R1(R means remainder). Eachshareis$43with$1left.Sinceitismoney,wecouldtradethe

$1forquartersandeachpersonwouldget2quartersor50cents.Eachperson’sshareis$43.50.

1.Completeeachquestionwithasingledigit.

a) 5÷3isabout20. b) 44÷7isabout30.

c) 71÷9isabout40. d) 02÷5isabout80.



2.Calculateeachquotient.Tellwhyyouranswerseemsreasonable.

a) 95÷5

b) 79÷6

c) 120÷8

d) 153÷6

3. Inwhatwayscanyouseat96peopleinequalrowsofchairs?

4. Youknowthat96÷3=32.

a) Howdoyouknowthat98÷3=32R2?

b) Howdoyouknowthat99÷3=33(andnot32R3)?

5. Usethedigits4,1,and2tomakefourdifferent3-digitnumbers(e.g.,412).

• Divideeachnumberby3.



• Whatdoyounoticeabouttheremainders?

• Doesthesamethinghappenifyouuseadifferentsetofthreedigitstostartwith?

gap closing - edugains€¦ · for example, 7 × 6 is 5 × 6 and 2 × 6. 9 × : to multiply by 9,...

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