Grade 4, Module 3: Multi-Digit Multiplication and Division

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Grade4,Module3:

Multi-DigitMultiplicationandDivisionMission:MultiplyandDivideBigNumbers

WordProblems

TableofContentsWORDPROBLEMS............................................................................................................2-18TopicA:MultiplicativeComparisonWordProblems..................................................................2

TopicB:Multiplicationby10,100,and1,000............................................................................3

TopicC:MultiplicationofuptoFourDigitsbySingle-DigitNumbers.........................................4

TopicD:MultiplicationWordProblems......................................................................................6

TopicE:DivisionofTensandOneswithSuccessiveRemainders………………………………………………….6

TopicF:ReasoningwithDivisibility..........................................................................................10

TopicG:DivisionofThousands,Hundreds,Tens,andOnes.....................................................12

TopicH:MultiplicationofTwo-DigitbyTwo-DigitNumbers....................................................16

TemplateforLesson35............................................................................................................18

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TopicA:MultiplicativeComparisonWordProblemsInTopicA,studentsinvestigateformulasforareaandperimeterandusetheirunderstandingoftheareaandperimeterformulastosolvemultiplicativecomparisonproblemswithafocusonproblemsusingareaandperimeterasacontext.LESSON1Thereisnoapplicationproblemforthislesson.LESSON2ApplicationProblem(6minutes)Tommy’sdadisteachinghimhowtomaketablesoutoftiles.Tommymakesasmalltablethatis3feetwideand4feetlong.Howmanysquare-foottilesdoesheneedtocoverthetopofthetable?Howmanyfeetofdecorativebordermaterialwillhisdadneedtocovertheedgesofthetable?Extension:Tommy’sdadismakingatable6feetwideand8feetlong.Whenbothtablesareplacedtogether,whatwilltheircombinedareabe?Note:ThisApplicationProblembuildsfrom3.MD.5,3.MD.6,and3.MD.8andbridgesbacktotheConceptDevelopmentofLesson1,duringwhichstudentsinvestigatedandusedtheformulasfortheareaandperimeterofrectangles.

LESSON3Thereisnoapplicationproblemforthislesson.

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NOTESONMULTIPLEMEANSOFENGAGEMENT:

DifferentiatethedifficultyoftheApplicationProblembyadjustingthenumbers.Extendforstudentsworkingabovegradelevelwiththeseopen-endedquestions:§ Howmanystudentswouldyou

predictattendthemiddleschool?Explainyourreasoning.

§ Ifthesewereestimatesofthenumberofstudents,whatmightbetheactualnumbers?

TopicB:Multiplicationby10,100,and1,000InTopicB,studentsexaminemultiplicationpatternswhenmultiplyingby10,100,and1,000.LESSON4ApplicationProblem(4minutes)Samanthareceivedanallowanceof$3everyweek.Bybabysitting,sheearnedanadditional$30everyweek.HowmuchmoneydidSamanthahaveinfourweeks,combiningherallowanceandherbabysitting?Note:Themultiplicationoftwo-digitmultiplesof10bysingle-digitnumbersisaGrade3standard(3.NBT.3).Thesecondstepofthisproblemrelatestotoday’sConceptDevelopment.Studentsmaysolveitonewayhereandmayfindasimplifyingstrategytosolveafterthelessonhasbeencompleted.LESSON5Thereisnoapplicationproblemforthislesson.LESSON6ApplicationProblem(5minutes)Thereare400childrenatParkElementarySchool.ParkHighSchoolhas4timesasmanystudents.

a. Howmanystudentsinallattendbothschools?b. LaneHighSchoolhas5timesasmanystudentsasPark

Elementary.HowmanymorestudentsattendLaneHighSchoolthanParkHighSchool?

Note:TheseproblemsareareviewofworkfromLesson5.

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NOTESONMULTIPLEMEANSOFENGAGEMENT:

ExtendtheApplicationProblemforstudentsabovegradelevelwithopen-endedquestions,suchasthefollowing:§ Whatmightbeanexplanationfor

thelargedifferenceinT-shirtsalesbetweenMondayandTuesday?

§ Basedonyourthoughts,whatmightbeastrategyforgeneratingthemostmoneyfromT-shirtsales?

§ GiventheincreaseinT-shirtssold,shouldtheteamincreaseordecreasethepriceoftheshirt?Explainyourreasoning.

TopicC:MultiplicationofuptoFourDigitsbySingle-DigitNumbersInTopicC,studentsdecomposenumbersintobasetenunitsinordertofindproductsofsingle-digitbymulti-digitnumbers.LESSON7ApplicationProblem(10minutes)ThebasketballteamissellingT-shirtsfor$9each.OnMonday,theysold4T-shirts.OnTuesday,theysold5timesasmanyT-shirtsasonMonday.HowmuchmoneydidtheteamearnaltogetheronMondayandTuesday?

Note:Thisisamulti-stepwordproblemreviewingmultiplyingbymultiplesof10fromLesson5,includingmultiplicativecomparison.LESSON8ApplicationProblem(8minutes)Andrebuysastamptomailaletter.Thestampcosts46cents.Andrealsomailsapackage.Thepostagetomailthepackagecosts5timesasmuchasthecostofthestamp.Howmuchdoesitcosttomailthepackageandletter?Note:ThisproblemisareviewofLesson7andincorporatesmultiplicativecomparison.Studentswhoexaminethetapediagramfindamorerapidsolutionistomultiplytofind6unitsof46cents.

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LESSON9ApplicationProblem(4minutes)Calculatethetotalamountofmilkinthreecartonsifeachcartoncontains236mLofmilk.

Note:ThisproblemisareviewofLesson8,practicingthree-digitbyone-digitmultiplication.LESSON10ApplicationProblem(5minutes)Theprincipalwantstobuy8pencilsforeverystudentatherschool.Ifthereare859students,howmanypencilsdoestheprincipalneedtobuy?

Note:ThisproblemisareviewofLesson9.Studentsmaysolveusingthealgorithmorpartialproducts.Bothareplacevaluestrategies.LESSON11ApplicationProblem(7minutes)

Writeanequationfortheareaofeachrectangle.Then,findthesumofthetwoareas.

4

8

30

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NOTESONMULTIPLEMEANSOFREPRESENTATION:

Englishlanguagelearnersandothersmaybenefitfromabriefexplanationofthetermacre.

Extension:Findafastermethodforfindingtheareaofthecombinedrectangles.

Note:ThisproblemisdesignedtobridgelearningfromTopicA,inwhichstudentssolvedforthearea,tothislesson,wheretheylearntomodelmultiplicationproblemsusingtheareamodel.Theplacementofthesmallrectangletotherightofthelargerrectangleisintentionalforshowingthetensandonesoftheareamodel.ItisrecommendedthatthisproblembepresentedimmediatelypriortotheConceptDevelopment.TopicD:MultiplicationWordProblemsInTopicD,studentsaregiventheopportunitytoapplytheirnewmultiplicationskillstosolvemulti-stepwordproblemsandmultiplicativecomparisonproblems.AttheconclusionofTopicD,studentsapplyalloftheiraddition,subtraction,andmultiplicationskillstosolvemulti-stepproblems.LESSON12Thereisnoapplicationproblemforthislesson.LESSON13Thereisnoapplicationproblemforthislesson.TopicE:DivisionofTensandOneswithSuccessiveRemaindersInTopicE,studentssynthesizetheirGrade3knowledgeofdivisiontypes(groupsizeunknownandnumberofgroupsunknown)withtheirnew,deeperunderstandingofplacevalue.Studentsfocusoninterpretingtheremainderwithindivisionproblemsinwordproblemsandwithinlongdivision.LESSON14ApplicationProblem(8minutes)Tylerplantedpotatoes,oats,andcorn.Heplanted23acresofpotatoes.Heplanted3timesasmanyacresofoatsaspotatoes,

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NOTESONMULTIPLEMEANSOFREPRESENTATION:

Modelingthearray(ratherthanthetapediagram)maygivestudentsaclearerpictureofthesolutiontotheApplicationProblem.Encouragestudentstousethelabelsphotoandpage,ifbeneficial.Discusshowtheequationinformsthesolution,yetthepicturerevealsthesolution.

andheplanted4timesasmanyacresofcornasoats.HowmanyacresdidTylerplantwithpotatoes,oats,andcorninall?Note:ThisApplicationProblemreviewstheobjectiveofLesson13:Usemultiplication,addition,orsubtractiontosolvemulti-stepwordproblems.ItprecedestheFluencyPracticeandConceptDevelopmentasareviewofmultiplicationskillspriortoworkwithdivisioninGrade4,whichstartsinthislesson.

LESSON15ApplicationProblem(5minutes)Chandraprinted38photostoputintoherscrapbook.Ifshecanfit4photosoneachpage,howmanypageswillsheuseforherphotos?Note:ThisApplicationProblemrelatestotheobjectiveofLesson14inthatstudentssolveadivisionwordproblemwitharemainder.Here,studentsinterprettheremaindertodeterminethetotalnumberofscrapbookpagesneeded.Thisanticipatesthelastprobleminthislesson.

LESSON16Thereisnoapplicationproblemforthislesson.

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LESSON17ApplicationProblem(5minutes)Audreyandhersisterfound9dimesand8pennies.Iftheysharethemoneyequally,howmuchmoneywilleachsisterget?

Note:ThisApplicationProblemreviewsdivisionofones.Sharing9dimesconnectstoProblems1and2oftoday’sConceptDevelopment,askingstudentstodecompose1tenfor10ones.LESSON18ApplicationProblem(7minutes)Malory'sfamilyisgoingtobuyoranges.TheGrandMarketsellsorangesat3poundsfor87cents.Howmuchdoes1poundoforangescostatGrandMarket?Note:ThisApplicationProblemreviewsdivisionwitharemainderinthetensfromLesson17.

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LESSON19ApplicationProblem(8minutes)Twofriendsstartabusinesswritingandsellingcomicbooks.After1month,theyhaveearned$38.Showhowtheycansharetheirearningsfairly,using$1,$5,$10,and$20bills.Note:StudentspracticedecomposingatenusinglongdivisionfromLesson17andwithamoneymodel.Otheracceptableanswersare1ten9ones,19ones,3fives4ones,or2fives9ones.

LESSON20ApplicationProblem(8minutes)Writeanexpressiontofindtheunknownlengthofeachrectangle.Then,findthesumofthetwounknownlengths.

Note:ThisApplicationProblemservesasanintroductiontotoday’sConceptDevelopment,inwhichstudentsfindthetotalunknownlengthofarectanglewithanareaof48,correspondingtoPart(a),and96,correspondingtoPart(b).

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LESSON21ApplicationProblem(8minutes)Arectanglehasanareaof36squareunitsandawidthof2units.Whatistheunknownsidelength?

Note:ThisApplicationProblemservesasanintroductiontoProblem1intheConceptDevelopment,inwhichstudentsfindthetotalunknownlengthofarectanglewithanareaof37andawidthof2.Intoday’sConceptDevelopment,studentsmoveontothecomplexityofusingtheareamodelwhenthereisaremainder.TopicF:ReasoningwithDivisibilityInTopicF,armedwithanunderstandingofremainders,studentsexplorefactors,multiples,andprimeandcompositenumberswithin100.Bydoingso,studentsgaininsightsintopatternsofdivisibility,evenusingtheirnewskillofdividingdouble-digitdividends.AllofthisworkpreparesthemforTopicG’sworkwithdividendsofuptofourdigits.LESSON22ApplicationProblem(5minutes)8×____=96.Findtheunknownsidelength,orfactor.Useanareamodeltosolvetheproblem.

Note:ThisApplicationProblemappliestheTopicEskillofdividingatwo-digitdividendusinganareamodelandservesasalead-intothislesson’sConceptDevelopmentbyusingareamodelstoillustratetheconceptoffactorpairs.

Method1: Method2:

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LESSON23ApplicationProblem(5minutes)Sashasaysthateverynumberinthetwentiesisacompositenumberbecause2iseven.Amandasaystherearetwoprimenumbersinthetwenties.Whoiscorrect?Howdoyouknow?Note:ThisApplicationProblembridgesLesson22’sworkwithusingdivisiontodetermineprimeandcompositenumberstothislesson’sobjectiveofusingdivisionpatternstodeterminefactorsofnumbers.LESSON24ApplicationProblem(5minutes)8cm×12cm=96squarecentimeters.Imaginearectanglewithanareaof96squarecentimetersandasidelengthof4centimeters.Whatisthelengthofitsunknownside?Howwillitlookwhencomparedtothe8centimeterby12centimeterrectangle?Drawandlabelbothrectangles.

Note:ThisApplicationProblemrelatesfindingfactors(Lessons22and23)tomultiples(Lesson24).Considerleadingstudentstovisualizethecolumnsof4or8squarecentimeters.Whencountingbythenumberofsquaresinthosecolumns,willthecountarriveexactlyat96?Whencountingbythenumberofsquaresinonerow,24or12,willthecountalsoarriveexactlyat96?(Considerusinggraphpapertodemonstrateforthosestudentswhowouldbenefitfrompictorialrepresentation.)Alsoconsidershowingstudentshowtheassociativepropertybeautifullyillustrateshowasthe8issplitintwo,the12doubles(picturedtotheright).LESSON25Thereisnoapplicationproblemforthislesson.

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TopicG:DivisionofThousands,Hundreds,Tens,andOnesTopicGextendsthepracticeofdivisionwiththree-andfour-digitdividendsusingplacevalueunderstanding.Placevaluediskssupportstudentsvisuallyastheydecomposeeachunitbeforedividing.Studentsthenpracticeusingthestandardalgorithmtorecordlongdivisionandmakeconnectionstotheareamodel.LESSON26ApplicationProblem(5minutes)Acoffeeshopuses8-ouncemugstomakeallofitscoffeedrinks.Inoneweek,theyserved30mugsofespresso,400lattes,and5,000mugsofcoffee.Howmanyouncesofcoffeedrinksdidtheymakeinthatoneweek?

Note:Byreviewingmultiplicationof10,100,and1,000,thisApplicationProblemleadsuptotoday’sConceptDevelopment,whichwillexploredivisionofmultiplesof10,100,and1,000.LESSON27ApplicationProblem(5minutes)Emmatakes57stickersfromhercollectionanddividesthemupequallybetween4ofherfriends.Howmanystickerswilleachfriendreceive?Emmaputstheremainingstickersbackinhercollection.HowmanystickerswillEmmareturntohercollection?

Note:ThisApplicationProblemreviewsworkwithtwo-digitdividendsfromLesson17.

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LESSON28ApplicationProblem(6minutes)Use846÷2towriteawordproblem.Then,drawanaccompanyingtapediagramandsolve.

Note:ThisApplicationProblemconnectstoLesson27’shalvingdiscussionintheStudentDebrief.Italsoreinforcestheuseofinverseoperationstocheckcalculations.ItusesthedivisionproblemfromthefluencyactivityDivideThree-DigitNumbers.Encouragestudentstorevisetheirwordproblemstousethewordhalf.LESSON29ApplicationProblem(5minutes)Janetuses4feetofribbontodecorateeachpillow.Theribboncomesin225-footrolls.Howmanypillowswillshebeabletodecoratewithonerollofribbon?Willtherebeanyribbonleftover?Note:ThisApplicationProblemreviewstheskillofdecomposingunitsinordertodivideandinterpretingaremainderwithinthecontextofawordproblemsothatthoseskillsmaybeappliedtotoday’sworkwithfour-digitdividends.

NOTESONMULTIPLEMEANSOFREPRESENTATION:

SimplifyandclarifytheApplicationProblemforEnglishlanguagelearnersandothers.Useimagesorillustrationstoexplainarollofribbon.Challengestudentsworkingabovegradeleveltomakepredictionsorestimatesbeforesolvingandtodetermineanddiscusswhichmodelandmethodismostefficient(forthem)tosolve225dividedby4.

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NOTESONMULTIPLEMEANSOFENGAGEMENT:

DifferentiatethedifficultyoftheApplicationProblembyadjustingthenumbers.

Extendforstudentsworkingabovegradelevelwiththesequestions:

§ Howorwhymighttheshirtsbesorted?

§ Wereyouabletopredictthatazerowouldbeinthequotient?How?

LESSON30ApplicationProblem(5minutes)Thestorewantedtoput1,455bottlesofjuiceintopacksof4.Howmanycompletepackscantheymake?Howmanymorebottlesdotheyneedtomakeanotherpack?Note:ThisproblemisareviewofLesson29,whichbridgesdividingwithremainderstothecurrentlesson.

LESSON31ApplicationProblem(5minutes)1,624shirtsneedtobesortedinto4equalgroups.Howmanyshirtswillbeineachgroup?

Note:ThisApplicationProblemisareviewofLesson30,practicingwithazerointhequotient.InProblem1oftheConceptDevelopment,studentsdiscusswhethertheunknowninthisproblemisthegroupsizeorthenumberofgroups.LESSON32ApplicationProblem(7minutes)Usethetapediagramtocreateadivisionwordproblemthatsolvesfortheunknown,thetotalnumberofthreesin4,194.Switchwordproblemswithapartnerandsolve.

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NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:

AllotextratimeandamomentfordiscussionforEnglishlanguagelearnersbeforetheywrite.Makeavailablealistofmeasurementunits(e.g.,dollars)andeverydayunits(e.g.,marbles).

Studentsworkingbelowgradelevelmaybenefitfromareview.Ask,“Whatisthewholeamount?4,194what?Howmanyareineachgroupaccordingtoourtapediagram?Howcanwegroupourunit?Inacan?Abasket?Ameasurement?Doweknowthenumberofgroups?Howcanwefindout?”

Note:ThisproblemextendsunderstandingfromLesson31aboutsolvingforanunknownnumberofgroups.ExtendthisproblemintheStudentDebriefusingadivisorof6,whichconnectstothislesson’sConceptDevelopment.LESSON33ApplicationProblem(5minutes)Writeanequationtofindtheunknownlengthofeachrectangle.Then,findthesumofthetwounknownlengths.

Note:ThisApplicationProblemservesasanintroductiontotoday’sConceptDevelopment,inwhichstudentsfindthetotalunknownlengthofarectanglewithanareaof672squaremeters.

They can make 1,398 sacks of flour.

72squarem

3m 600squarem 3m

The sum of the two unknown lengths is 224 meters.

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TopicH:MultiplicationofTwo-DigitbyTwo-DigitNumbersInTopicH,studentsmultiplytwo-digitbytwo-digitnumbers,usingtheirplacevalueunderstandingandunderstandingoftheareamodeltoempowerthemtomultiplybylargernumber.TopicHculminatesatthemostabstractlevelbyexplicitlyconnectingthepartialproductsappearingintheareamodeltothedistributivepropertyandrecordingthecalculationvertically.LESSON34ApplicationProblem(5minutes)Mr.Gogginsplanted10rowsofbeans,10rowsofsquash,10rowsoftomatoes,and10rowsofcucumbersinhisgarden.Heput22plantsineachrow.Drawanareamodel,labeleachpart,andthenwriteanexpressionthatrepresentsthetotalnumberofplantsinthegarden.Note:ThisApplicationProblembuildsonTopicB,wherestudentslearnedtomultiplybymultiplesof10,andTopicC,wherestudentslearnedtomultiplytwo-digitbyone-digitnumbersusinganareamodel.ThisApplicationProblemhelpsbridgetotoday’slessonasstudentslearntomultiplymultiplesof10bytwo-digitnumbers.LESSON35ApplicationProblem(6minutes)Materials: (S)Thousandsplacevaluechart(Lesson4Template)For30daysoutofonemonth,Katieexercisedfor25minutesaday.WhatisthetotalnumberofminutesthatKatieexercised?Solveusingaplacevaluechart.Note:ThisApplicationProblembuildsonthecontentofLesson34byusingaplacevaluecharttorepresentandthenmultiplyamultipleof10byatwo-digitnumber.Althoughsomestudentsmayeasilysolvethisproblemusingmentalmath,encouragethemtoseethatthemodelverifiestheirmentalmathskills.StudentscanusetheirmentalmathandplacevaluechartsolutiontoverifytheiranswerinProblem1oftheConceptDevelopment.LESSON36ApplicationProblem(6minutes)Mr.Gogginssetup30rowsofchairsinthegymnasium.Ifeachrowhad35chairs,howmanychairsdidMr.Gogginssetup?Drawanareamodeltorepresentandtohelpsolvethisproblem.Discusswithapartnerhowtheareamodelcanhelpyousolve30×35.

(4×10)×22

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Note:ThisApplicationProblembuildsonpriorlearningfromLesson35wherestudentsusedanareamodelandpartialproductstomultiplyatwo-digitmultipleof10byatwo-digitnumberusinganareamodel.ThisApplicationProblemalsohelpsbridgetotoday’slessoninthatstudentsapplypriorknowledgeoftheareamodelandpartialproductstorepresentandsolvetwo-digitbytwo-digitmultiplication.LESSON37ApplicationProblem(5minutes)Sylvie’steacherchallengedtheclasstodrawanareamodeltorepresenttheexpression24×56andthentosolveusingpartialproducts.Sylviesolvedtheexpressionasseentotheright.Isheranswercorrect?Whyorwhynot?Note:ThisApplicationProblembuildsonthecontentofLessons34,35,and36.Studentsnowhaveasolidfoundationuponwhichtobuildunderstandingoftwo-digitbytwo-digitmultiplication.Theymovefrompictorialrepresentationstoabstractrepresentations.ThisApplicationProblemguidessuchmovementandbuildstothecontentoftoday’slessonwherestudentsseehowalloftheworkthattheyhavedonefitstogetherandpreparesthemtosolveusingthestandardalgorithm(Lesson38).LESSON38ApplicationProblem(5minutes)Sandy’sgardenhas42plantsineachrow.Shehas2rowsofyellowcornand20rowsofwhitecorn.Drawanareamodel(representingtwopartialproducts)toshowhowmuchyellowcornandwhitecornhasbeenplantedinthegarden.Note:Thisproblemrevisitstheareamodelthatfocusesontwopartialproductsinpreparationforworkwiththestandardalgorithm.TheareamodelusedintheApplicationProblemisusedinProblem1oftheConceptDevelopment.

` Lesson 4 Template NYS COMMON CORE MATHEMATICS CURRICULUM 4 3

Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and 1,000 in arrays and numerically.

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thousands place value chart

thousands hundreds tens ones

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