Grade 3, Module 1: Properties of Multiplication and ...

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Grade3,Module1:

PropertiesofMultiplicationandDivisionandSolvingProblemswithUnitsof2-5and10

Mission:MultiplyandDivideFriendlyNumbers

Fluency

TableofContents

FLUENCY...........................................................................................................................2-14

TopicA:MultiplicationandtheMeaningoftheFactors............................................................2

TopicB:DivisionasanUnknownFactorProblem......................................................................4

TopicC:MultiplicationUsingUnitsof2and3................................................................................6

TopicD:DivisionUsingUnitsof2and3.....................................................................................9

TopicE:MultiplicationandDivisionUsingUnitsof4...............................................................11

TopicF:DistributivePropertyandProblemSolvingUsingUnitsof2-5and10........................12

FLUENCY

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Thinkoffluencyactivityashavingthreegoals:

1. Maintenance:Stayingsharponpreviouslylearnedskills

2. Preparation:Targetedpracticeforthecurrentlesson

3. Anticipation:Skillsthatensurethatstudentswillbereadyforthein-depthworkofupcoming

lessons

a. Exampleofanticipatoryfluency:Studentsmustbesecureinworkingwithfractionswith

thesamedenominatorbeforetheycanbeexpectedtoworkwithfractionswithunlike

denominators.

TopicA:MultiplicationandtheMeaningofFactorsInthistopic,studentsuseavarietyoffactorssincetheselessonsemphasizeunderstandingtheconceptofmultiplyingratherthanfindingtotals.LESSON1GroupCounting(10minutes)Note:Basicskip-countingskillsfromGrade2shiftfocusinthisGrade3activity.Group-countinglaysafoundationforinterpretingmultiplicationasrepeatedaddition.Whenstudentscountgroupsinthisactivity,theyaddandsubtractgroupsoftwowhencountingupanddown.

T:Let’scountto20forwardandbackward.Watchmyfingerstoknowwhethertocountupordown.Aclosedhandmeansstop.(Showsignalsasyouexplain.)T:(Rhythmicallypointupuntilachangeisdesired.Showaclosedhandthenpointdown.)S:1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0.T:Let’scountto20forwardandbackwardagain.Thistimewhispereveryothernumber.Saytheothernumbersinaregularvoice.S:(Studentswhisperthenspeakeveryothernumberto20forwardandbackward.)T:Let’scountto20forwardandbackwardagain.Thistime,humeveryothernumberinsteadofwhispering.Asyouhum,thinkofthenumber.S:(Hum),2,(hum),4,(hum),6,etc.T:Let’scountto20forwardandbackwardagain.Thistime,thinkeveryothernumberinsteadofhumming.S:(Think),2,(think),4,(think),6,etc.T:Whatdidwejustcountby?Turnandtalktoyourpartner.S:Twos.T:Let’scountbytwos.(Directstudentstocountforwardtoandbackwardfrom20,changingdirectionsattimes.)

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LESSON2GroupCounting(3minutes)Note:Basicskip-countingskillsfromGrade2shiftfocusinthisGrade3activity.Group-countinglaysafoundationforinterpretingmultiplicationasrepeatedaddition.Whenstudentscountgroupsinthisactivity,theyaddandsubtractgroupsofthreewhencountingupanddown.

T:Let’scountto18forwardandbackward.Iwantyoutowhisper,whisper,andthenspeaknumbers.T:Watchmyfingerstoknowwhethertocountupordown.Aclosedhandmeansstop.(Showsignalsasyouexplain.)T:(Rhythmicallypointupuntilachangeisdesired.Showaclosedhandthenpointdown.)S:(Whisper1,whisper2,speak3,etc.)T:Let’scountto18forwardandbackwardagain.Thistime,thinkeverynumberinsteadofwhispering.S:(Think),(think),3,(think),(think),6,(think),(think),9,etc.T:Whatdidwejustcountby?Turnandtalktoyourpartner.S:Threes.T:Let’scountbythrees.(Directstudentstocountforwardandbackwardto18,periodicallychangingdirections.Emphasizethe9to12transition.)

AddEqualGroups(3minutes)Materials:(S)PersonalwhiteboardsNote:ThisactivityreviewsLesson1.Studentsdirectlyrelaterepeatedadditiontomultiplication.Theyinterpretproductsasthenumberofequalgroupstimesthenumberofobjectsineachgroup.

T:(Projectapicturearraywith3groupsof2circled.)Howmanygroupsarecircled?S:3.T:Howmanyareineachgroup?S:2.T:Writethisasanadditionsentence.S:(Write2+2+2=6.)T:Writeamultiplicationsentencefor3twosequals6.S:(Write3×2=6.)

Continuewithpossiblesequence:3groupsof5,5groupsof10,and3groupsof4.LESSON3GroupCounting(3minutes)Note:Basicskip-countingskillsfromGrade2shiftfocusinthisGrade3activity.Group-countingreviewsinterpretingmultiplicationasrepeatedaddition.CountingbytwosandthreesinthisactivityanticipatesworkwiththosefactorsinTopicB.

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T:Let’scountbytwos.(Directstudentstocountforwardandbackwardto20,periodicallychangingdirections.)T:Let’scountbythrees.(Directstudentstocountforwardandbackwardto21,periodicallychangingdirections.Emphasizethe9to12and18to21transitions.)

AddtoMultiply(3minutes)Materials:(S)PersonalwhiteboardsNote:ThisactivityreviewsLesson2.Studentsdirectlyrelaterepeatedadditiontomultiplication.Theyinterpretproductsusingthearray.

T:(Projectapicturearraywith3groupsof5circled.)Howmanygroupsarecircled?S:3.T:Howmanyareineachgroup?S:5.T:Writeitasanadditionsentence.S:(Write5+5+5=15.)T:Writeamultiplicationsentencerepresenting3fivesequals15.S:3×5=15.

Continuewithpossiblesequence:3groupsof10,3groupsof4,and7groupsof2.TopicB:DivisionasanUnknownFactorProblemThestudyoffactorslinksTopicsAandB.TopicBextendsthestudytodivision.Studentscontinuetouseavarietyoffactorsinthistopicastheemphasisintheselessonsrestsonconceptuallyunderstandingdivisionandlearningtointerpretproblemsbywritingdivisionequations.LESSON4GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbytwosandthreesinthisactivityanticipatesworkwiththosefactorsinthislesson.

T:Let’scountbytwos.(Directstudentstocountforwardandbackwardto20,periodicallychangingdirections,e.g.,2,4,6,8,10,8,10,12,10,12,14,16,18,20,18,20,18,16,14,12,10,12,10,8,10,8,6,4,2,0.)T:Let’scountbythrees.(Directstudentstocountforwardandbackwardto24,periodicallychangingdirections.Emphasizethe9to12and18to21transitions,e.g.,3,6,9,12,9,12,9,12,15,18,21,18,21,18,21,24,21,18,21,18,15,12,15,12,9,12,9,6,3,0.)

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ArrayMultiplication(2minutes)Materials:(S)Personalwhiteboards.Note:ThisactivityreviewsTopicA’sobjectives.Studentsdirectlyrelaterepeatedadditiontomultiplication.Theyinterpretproductsusingthearray.

T:(Projectapicturearraywith3groupsof2circled.)Saytherepeatedadditionsentence.S:2+2+2=6.T:(Write3×____=____.)Onyourpersonalboards,completethemultiplicationsentence.S:(Write3×2=6.)

Continuewithpossiblesequence:4groupsof10,3groupsof4,7groupsof3,and8groupsof2.LESSON5GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.CountingbytwosandthreesinthisactivitysupportsworkwiththosefactorsinTopicB.

T:Let’scountbytwos.(Directstudentstocountforwardandbackwardto20,emphasizingthe8to10,10to12,and18to20transitions.)T:Let’scountbythrees.(Directstudentstocountforwardandbackwardto27,changingdirections.Emphasizethe9to12and18to21transitions.)

Recordthecount-bythreeupto24touselaterinthelesson.DivideEqualGroups(5minutes)Materials:(S)Personalwhiteboards.Note:Studentsdirectlyrelaterepeatedadditiontodivision.Theyinterpretthenumberofgroupsastheunknownindivision.Thisactivityanticipatesthelessonobjective.

T:(Projectanarraywith2groupsof5circled.)Howmanygroupsarecircled?S:2.T:Howmanyareineachgroup?S:5.T:Saythetotalasarepeatedadditionsentence.S:5+5=10.T:Writeadivisionsentencefor10dividedinto2equalgroups.S:(Write10÷2=5.)

Continuewithpossiblesequence4groupsof2,3groupsof4and2groupsof6.

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LESSON6GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.CountingbytwosandthreesinthisactivitysupportsworkwiththosefactorsinTopicB.

T:Let’scountbytwos.(Directstudentstocountforwardandbackwardto20,emphasizingthe8to10,10to12,and18to20transitions.)T:Let’scountbythrees.(Directstudentstocountforwardandbackwardto30,periodicallychangingdirections.Emphasizethe9to12,18to21,and27to30transitions.)

DivideEqualGroups(5minutes)Materials:(S)PersonalwhiteboardsNote:Studentsdirectlyrelaterepeatedadditiontodivision.Theyinterprettheunknownindivision.ThisactivitybridgesLessons5and6.

T:(Projectanarraywith3groupsof5circled.)Saythetotalasarepeatedadditionsentence.S:5+5+5=15.T:Writeadivisionsentencefor15dividedinto3equalgroups.S:(Write15÷3=5.)

Continuewithpossiblesequence:5groupsof3,4groupsof3,3groupsof4,9groupsof2,and2groupsof9.Alternatebetweenstudentswritingdivisionsentenceswherethequotientrepresentseitherthenumberofobjectsinagrouporthenumberofgroups.TopicC:MultiplicationUsingUnitsof2and3InTopicC,studentsbeginbuildingfluencywithfactsof2and3usingthearraymodelandfamiliarskip-countingstrategies.LESSON7GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.CountingbytwosandthreesinthisactivityreviewsworkwiththosefactorsfromTopicC.

T:Let’scountbytwos.(Directstudentstocountforwardandbackwardto20,emphasizingthe8to10,10to12,and18to20transitions.)T:Let’scountbythrees.(Directstudentstocountforwardandbackwardto30,periodicallychangingdirections.Emphasizethe9to12,18to21,and27to30transitions.)

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DivideEqualGroups(5minutes)Materials:(S)PersonalwhiteboardsNote:Studentsdirectlyrelaterepeatedadditiontodivision.Theyinterprettheunknownindivision.ThisactivityreviewsLesson6.

T:(Projectapicturearraywith2groupsof4circled.)Saythetotalasarepeatedadditionsentence.S:4+4=8.T:Writeadivisionsentencefor8dividedinto2equalgroups.S:(Write8÷2=4.)T:Belowthatdivisionsentencewriteadivisionsentencedividing8into4equalgroups.S:(Write8÷4=2.)

Continuewithpossiblesequence:5groupsof3,3groupsof4,and6groupsof2.LESSON8GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbytwos,threesandfoursinthisactivitysupportsworkwithunitsof2and3inthistopic,andanticipatesworkusingunitsof4inTopicE.

T:Let’scountbytwosto20.Whisperthenspeakthenumbers.T:Let’scountbytwosto20again.Thistime,humthefirstnumberandthenspeak.Asyouhum,thinkofthenumber.T:Let’scountbytwosto20.Thistime,insteadofhumming,thinkeveryothernumber.T:Whatdidwejustcountby?S:Fours.T:Let’scountbyfours.(Directstudentstocountforwardandbackwardto20,periodicallychangingdirections.)T:Let’scountbythrees.(Directstudentstocountforwardandbackwardto30,periodicallychangingdirections.Emphasizethe9to12,18to21,and27to30transitions.)

CommutativeMultiplying(3minutes)Materials: (S)PersonalwhiteboardNote:Practicingthisconcept,whichwastaughtinLesson7,helpsstudentsbuildconfidenceandautomaticity.

T: (Projecta3×2array.)Howmanygroupsof2doyousee?S: 3groupsof2.T: Writetwodifferentmultiplicationsentencesforthearray.S: (Write3×2=6and2×3=6.)

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Continuewiththefollowingpossiblesequence:3by5and4by3.T: (Write4×2=2×___.)Onyourboard,fillintheblank.S: (Write4×2=2×4.)

Repeattheprocessfor9×5=5×___and3×6=6×___.LESSON9GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbythreesandfoursinthisactivitysupportsworkwithunitsof3inthistopicandanticipatesworkusingunitsof4inTopicE.

T: Let’scountbythrees.(Directstudentstocountforwardandbackwardto30,emphasizingthe

transitionfrom18to21.)T: Let’scountbyfours.(Directstudentstocountforwardandbackwardto24,emphasizingthe

16to20transition.)FormsofMultiplication(4minutes)Materials: (S)PersonalwhiteboardNote:Studentsdirectlyrelaterepeatedadditiontomultiplicationinpreparationforusingthedistributivepropertyinthislesson.

T: (Projecta3×5array.)Representthisarrayasarepeatedadditionsentenceusing5asthesize

ofthegroups.S: (Write5+5+5=15.)T: (Projecta3×4array.Write____fours=____.)Completetheequationonyourpersonal

whiteboard.S: (Write3fours=12.)T: (Projecta7×2array.)Writetwomultiplicationsentencesfor7groupsof2.S: (Write7×2=14and2×7=14.)T: (Projecta6×3array.Write18=6×____.)Completetheequationonyourpersonalwhite

board.S: (Write18=6×3.)T: (Projecta5×3array.Write5threes=___.)Completetheequationonyourpersonalwhite

board.S: (Write5threes=15.)T: (Addonemoregroupof3tothearray.Write5threes+1three=____threes=____ones.)

Completetheequationonyourpersonalwhiteboard.S: (Write5threes+1three=6threes=18ones.)

FLUENCY

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LESSON10GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbythreesandfoursinthisactivitysupportsworkwithunitsof3inthistopicandanticipatesworkusingunitsof4inTopicE.

T: Let’scountbythrees.(Directstudentstocountforwardandbackwardto30,emphasizingthe

transitionfrom18to21.)T: Let’scountbyfours.(Directstudentstocountforwardandbackwardto24,emphasizingthe

16to20transition.)

TopicD:DivisionUsingUnitsof2and3InTopicD,studentssolvetwotypesofdivisionsituations—partitive(groupsizeunknown)andmeasurement(numberofgroupsunknown)—usingfactorsof2and3.LESSON11GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbytwosandfoursinthisactivityreviewsmultiplicationwithunitsof2fromTopicCandanticipatesusingunitsof4inTopicE.

T: Let’scountbytwos.(Directstudentstocountforwardandbackwardto20.)T: Let’scountbyfours.(Directstudentstocountforwardandbackwardto36,emphasizingthe

20to24and28to32transitions.)

LESSON12GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbytwosandfoursinthisactivityreviewsmultiplicationwithunitsof2fromTopicCandanticipatesusingunitsof4inTopicE.

T: Let’scountbytwos.(Directstudentstocountforwardandbackwardto20.)T: Let’scountbyfours.(Directstudentstocountforwardandbackwardto36,emphasizingthe

20to24and28to32transitions.)

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Divide(2minutes)Materials: (S)PersonalwhiteboardNote:Thisactivitybuildsfluencywithmultiplicationanddivision.Itworkstowardthegoalofstudentsknowingfrommemoryallproductsoftwoone-digitnumbersandreviewstheobjectiveofLesson11.

T: (Projecta2by4arrayofobjects.)Drawanarraytomatchmypicture.S: (Draw2by4array.)T: Skip-countbytwostofindhowmanytotalobjectsthereare.(Pointasstudentscount.)S: 2,4,6,8.T: Howmanygroupsof2arethere?S: 4.T: Saythetotalasamultiplicationsentencestartingwiththenumberofgroups.S: 4×2=8.T: (Write4×2=8.Belowit,write8÷4=__.)Fillintheblanktomakeatruedivisionsentence.

Then,divideyourarrayinto4equalgroupstofindtheanswer.S: (Drawlinesseparatingthearrayinto4groupsof2,andwrite8÷4=2.)T: Erasethelinesthatdividedthearray.S: (Eraselines.)T: Show8÷4bymakinggroupsof4.S: (Circle2groupsof4.)

Repeatprocessforthefollowingpossiblesequence:9÷3,12÷2,and12÷3.LESSON13GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbythreesandfoursinthisactivityreviewsmultiplicationwithunitsof3fromTopicCandanticipatesusingunitsof4inTopicE.

T: Let’scountbythrees.(Directstudentstocountforwardandbackwardto30.)T: Let’scountbyfours.(Directstudentstocountforwardandbackwardto40,emphasizingthe

20to2428to32,and36to40transitions.)Divide(2minutes)Materials: (S)PersonalwhiteboardNote:Thisactivitybuildsfluencywithmultiplicationanddivision.Itworkstowardstudentsknowingfrommemoryallproductsoftwoone-digitnumbers.

T: (Write2×3=___.)Saythemultiplicationsentence.S: 2×3=6.T: (Write2×3=6.Directlybelowit,write___÷3=2.)Onyourpersonalwhiteboard,writethe

equationandfillintheblank.

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S: (Write6÷3=2.)Repeattheprocessforthefollowingpossiblesequence:3×3,5×3,and9×3.TopicE:MultiplicationandDivisionUsingUnitsof4TopicEbeginsbyintroducingstudentstomultiplicationby4throughskip-countingobjectsinarraymodelsinLesson14.LESSON14ReadTapeDiagrams(3minutes)Materials:(S)PersonalwhiteboardsNote:Studentspracticereadingthedifferencebetweenthevalueoftheunit(thesizeofthegroups)andthenumberofunits.Theactivityanticipatesusingthetapediagramasamodelforcommutativity.

T:(Projectatapediagrampartitionedinto5equalunits,drawing2starsinthefirstunit.)Whatisthevalueofeachunit?S:2stars.T:Howmanyunitsarethere?Lesson14:Date:S:5units.T:Writeamultiplicationsentenceforthistapediagram.S:(Write5×2=10.)

Repeattheprocess,alternatingbetweenfindingthenumberofgroupsandthesizeofthegroupsfor4×3=12,8÷4=2,and15÷3=5.LESSON15GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbytwosandthreesinthisactivityreviewsmultiplicationwithunitsof2and3fromTopicsCandD.

T: Let’scountbytwos.(Directstudentstocountforwardandbackwardto20.)T: Let’scountbythrees.(Directstudentstocountforwardandbackwardto30.)

LESSON16GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbytwosandthreesinthisactivityreviewsmultiplicationwithunitsof2and3fromTopicsCandD.

T: Let’scountbytwos.(Directstudentstocountforwardandbackwardto20.)

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T: Let’scountbythrees.(Directstudentstocountforwardandbackwardto30.Whisperthenumbersbetweenthreesandspeakeachthreeoutloud.Forexample,whisper1,whisper2,say3,whisper4,whisper5,say6,andsoon.)

ReadTapeDiagrams(3minutes)Materials:(S)PersonalwhiteboardsNote:Studentspracticereadingthedifferencebetweenthevalueoftheunit(thesizeofthegroups)andthenumberofunits.Theactivityreviewsusingthetapediagramasamodelforcommutativity.

T:(Projectatapediagrampartitionedinto2equalunits.Draw8starsineachunitandbracketthetotalwithaquestionmark.)Saytheadditionsentence.S:8+8=16.T:Saythemultiplicationsentencestartingwiththenumberofgroups.S:2x8=16.T:Drawthetapediagramandlabelunitswithnumbersinsteadofstars.Labelthemissingtotal.Beneaththediagram,writeamultiplicationsentence.S:(Drawatapediagramwith8writteninsidebothunitsand16writtenasthetotal.Beneaththediagram,theywrite2x8=16.)

Repeatprocessfor3x7and4x6.LESSON17Noadditionalfluencyactivitiesforthislesson.TopicF:DistributivePropertyandProblemSolvingUsingUnitsof2-5and10TopicFintroducesthefactors5and10,familiarfromskip-countinginGrade2.StudentsapplythemultiplicationanddivisionstrategiestheyhaveusedtomixedpracticewithallofthefactorsincludedinModule1.LESSON18Noadditionalfluencyactivitiesforthislesson.LESSON19GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbythrees,fours,fives,andsixesinthisactivityreviewsmultiplicationwithunitsof3,4,and5andanticipatesmultiplicationwithunitsof6inModule3.

T: Let’scountbyfives.(Directstudentstocountforwardandbackwardto50.)

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T: Let’scountbyfours.(Directstudentstocountforwardandbackwardto40.)T: Let’scountbythrees.(Directstudentstocountforwardandbackwardto30.)T: Let’scountbysixes.(Directstudentstocountforwardandbackwardto36,emphasizingthe24

to30transition.)CommutativeMultiplying(3minutes)Note:Thisactivityreviewsthecommutativityofmultiplication,learnedinLessons7,8,and15.

T:(Write3x2=___.)Saythemultiplicationsentence.

S:3x2=6. T:Flipit.

S:2x3=6.Repeatprocessfor5x2,5x3,3x4,2x8,and3x7.DecomposeandMultiply(4minutes)Materials: (S)PersonalwhiteboardNote:Thisactivityanticipatesmultiplicationusingunitsof6,7,8,and9bydecomposinglargerfactsintosmallerknownfacts.Itreviewsthebreakapartanddistributestrategy.

T: (Write7×4=___.)Rewritetheequationinunitform.S: (Write7fours=___.)T: (Write7fours=(5fours)+(___fours)=___.)7

foursisthesameas5foursandhowmanyfours?S: 2fours.T: (Write(5fours)+(2fours)=___.Belowit,write20

+___=___.)Fillintheblanks.S: (Write20+8=28.)T: 7×4equals?S: 28!

Repeatforthefollowingpossiblesequence:8×3,9×2,and6×4.Changetheunknownsthatstudentsneedtofillin.LESSON20Noadditionalfluencyactivitiesforthislesson.LESSON21GroupCounting(3minutes)Note:Groupcountingreviewsinterpretingmultiplicationasrepeatedaddition.Countingbythrees,fours,andsixesinthisactivityreviewsmultiplicationwithunitsof3and4andanticipatesmultiplicationwithunitsof6inModule3.

7×4=___7fours=(5fours)+(___fours)=___ (5fours)+(2fours)=___ 20+___=___

SampleTeacherboard

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T: Let’scountbythrees.(Directstudentstocountforwardandbackwardto30.)T: Let’scountbyfours,think/talkforwardandbackward.(Directstudentstocountforwardand

backwardto40.Think1,2,3;say4.Think5,6,7;say8,etc.)T: Let’scountbysixes.(Directstudentstocountforwardandbackwardto48,emphasizingthe24

to30and36to42transitions.)CommutativeMultiplying(3minutes)Note:Thisactivityreviewsthecommutativityofmultiplication,learnedinLessons7,8,and15.

T: (Write4×2=___.)Saythemultiplicationsentence.S: 4×2=8.T: Flipit.S: 2×4=8.

Repeattheprocessfor5×3,9×2,4×3,2×7,and3×8.