sample lesson study research lesson

LessonResearchProposalfor5thGradeMultiplicationofaDecimalNumberbyaWholeNumber

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LessonDate:Instructor:Lessonplandevelopedby:

1. TitleoftheLesson:MultiplicationofaDecimalNumberbyaWholeNumber 2. BriefdescriptionofthelessonStudentswillusepreviousunderstandingsofmultiplyingwholedigitnumbers,estimationandplacevaluetocalculate0.8x6.

3. ResearchThemeTeachscholarstoconstructviableargumentsandcritiquethereasoningofothersthroughnote-taking,boardwork,andstudentdiscourse.Teachscholarstomakesenseofproblemsandpersevereinsolvingthembyteachingmathematicsthroughproblemsolving.

4. UnitGoals Studentswillunderstandthemeaningofmultiplyinganddividingadecimalbyawholenumber.Theywillunderstandthesecalculationsbyrelatingthemtowhattheyalreadyknowaboutmultiplyinganddividingwholenumbersandtheirunderstandingofplacevalue.Specifically,thatthevalueofdigitsincreasebypowersoftenwhenmovingtotheleft,anddecreasebypowersoftenwhenmovingtotheright.Studentswillunderstandthemeaningsofthecalculationsofdecimalnumbersxwholenumbers,decimalnumbers÷wholenumbers,andwholenumbers÷wholenumberswhenthequotientisadecimalnumber.Additionally,studentswillunderstandhowtocarryoutsuchcalculations,andtheywillfurthertheirunderstandingoftheplacevaluestructureofdecimalnumbers.Studentswillbeabletouseconcreteobjects,diagrams,andmathsentencestofindsolutions,communicatetheirideastoothers,andsummarizehowtodothecalculations.Studentswillbeabletodocalculationsofdecimalnumbersxwholenumbers,decimalnumbers÷wholenumbers,andwholenumbers÷wholenumberswhenthequotientisadecimalnumber.

5. LessonGoalsStudentswillunderstandthemeaningofmultiplicationofadecimalbyawholenumberbyrelatingthemtowhattheyhavealreadylearnedaboutwholenumbermultiplicationandaboutplacevalue,specificallytheirpreviousexperiencemultiplyingwholenumbersanddecimalsbytenandrecognizingthatthedigitsdonotchangewhenmultipliedbyten,thedigitshiftsoneplacetoleft,increasingit’svaluetentimes.


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6. RelationshipoftheUnittotheStandards

PreviousGrade CurrentGrade NextGrade

4.NBT.1–Recognizethatinamulti-digitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.Forexample,recognizethat700÷70=10byapplyingconceptsofplacevalueanddivision.

4.NBT.3–Useplacevalueunderstandingtoroundmulti-digitwholenumberstoanyplace.

5.NBT.A.1-Recognizethatinamulti-digitnumber,adigitinoneplacerepresents10timesasmuchasitrepresentsintheplacetoitsrightand1/10ofwhatitrepresentsintheplacetoitsleft.5.NBT.B.7-Add,subtract,multiply,anddividedecimalstohundredths,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.

6.NS.2–Fluentlydividemulti-digitnumbersusingthestandardalgorithm.

6.NS.3–Fluentlyadd,subtract,multiply,anddividemulti-digitdecimalsusingthestandardalgorithmforeachoperation.

7. BackgroundandRationaleInpreviousyears,multiplyingdecimalshasbeenintroducedafterstudentshavedemonstratedmasteryofwholedigitmultiplication.Additionally,studentshavebeenmostsuccessfulwhentheyhavepracticewithestimatingreasonablesolutionsandfocusingonplacevalueduringoperations.Thiscurrentclasshasdoneexceptionallywellwiththeprecedingtopics:numbertheory,wholenumbermultiplication,estimationandplacevalue.Itistheteam’sopinionbasedonpreviousexperiencethatstudentscanbeleftwithonlyaproceduralunderstandingofdecimalmultiplicationifcertainaspectsarenotemphasizedduringlessons.Thistendstohappenwhenbothteacherandstudentsfocusonthestepsortherule.Duringtheunittheteamwillregularlystressestimationandplacevaluebeforeandduringalloperations.Thegoalistounderstandthemechanicsofthedecimalmultiplicationalgorithmandhaveaconceptualunderstandingforit.Theconceptualunderstandingforthedecimalmultiplicationalgorithmwillbegroundedintheuseofestimationandvisualizationwithadualnumberline.Thedualnumberlineservesasatoolthatlaysthefoundationforunderstandingthemultiplicativerelationshipofproportions.Bothstrategieswereusedduringwholedigitmultiplication.Thedualnumberlineandestimationwillcontinuetobeusedasstudentsstudyfractionmultiplicationanddivisionofdecimalsandfractionslaterintheyear.

8. ResearchandKyozaikenkyu

TheCommonCoreStateStandardsforMathematicsemphasizesthatinfifthgradeinstructionaltimeshouldfocusonthreecriticalareasincluding“extendingdivisionto2-digitdivisors,integratingdecimalfractionsintotheplacevaluesystem,developingunderstandingofoperationswithdecimalstohundredths,anddevelopingfluencywithwholenumberanddecimaloperations.”TheCCSS-Mgoesontospecifythatstudentsinfifthgradeshould“usetherelationshipbetweenfinitedecimalsandwholenumbers(i.e.,afinitedecimalmultipliedbyanappropriatepoweroftenisawholenumber),tounderstandandexplainwhytheproceduresformultiplyinganddividingfractionsmakessense”AccordingtotheProgressionsfortheCommonCoreStateStandardsinMathematicswrittenbytheCommonCoreStandardsWritingTeam,


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becauseoftheuniformityofthestructureofthebase-tensystem,studentsusethesameplacevalueunderstandingformultiplyinganddividingdecimalsthattheyuseformultiplyinganddividingandwholenumbers.TheProgressiondocumentsuggeststhatbeforestudentsconsiderdecimalmultiplicationmoregenerally,theystudythe“effectofmultiplyingby0.1andby0.01toexplainwhytheproductistenorahundredtimesassmallasthemultiplicand.”Achallengefortheteamistodesignalessonthatnotonlyhelpsstudentstodevelopthisdeepunderstandingofmultiplyingwithdecimals,butalsoaddressesandundevelopedunderstandingofplacevaluethatmayexistforsomestudents.Inresearchingthistopic,theteamlookedatVandeWalle’stext,ElementaryandMiddleSchoolMathematics:TeachingDevelopmentally.OurstudyofVandeWalle’stextondevelopmentallyappropriatemethodsforteachingmathematicsechoedtheCommonCoreWritingTeam.VandeWallestressestheimportanceofstudentshavingaclearunderstandingoftheunderlyingfoundationofthebasetensystem.Thatthereexistsa“10-to-1relationshipbetweenthevalueofanytwoadjacentpositions”.Inregardsspecificallytomultiplyingdecimals,VandeWallearguesthatspecificrulesfordecimalcomputationarenotreallynecessaryifcomputationisbuiltonafirmunderstandingofplacevalue.VandeWallewarnsthatbyfocusingonroteapplicationofrulesstudentsloseoutonapproachesthatemphasizeopportunitiestounderstandthemeaningandeffectsofoperationsandaremorepronetomisapplyprocedures.VandeWallealsoarguesthatstudentsshouldbecomeadeptatestimatingdecimalcomputationbeforetheyareabletomasterpaperandpencilcomputation.Hesuggeststhatinaproblemsuchas3.7times4thatstudentsuseestimationstrategiestodiscusstherangeofreasonableanswers.Couldtheanswerbelessthan12?Coulditbemorethat16?Thisallowsstudentstothensolvetheanswerandjudgeitsreasonablenessbasedontheestimationdiscussedpreviously.AfterlookingattheCCSS,theCCSSProgressionDocument,andVandeWalle’sbookonteachingmathematicsdevelopmentally,theteamnextlookedatTokyoShoseki’sMathematicsInternational(MI)tolookathowmultiplicationwastaughtandhowplacevaluewastaughttodeveloplessonsthatwouldleaduptothelaunchofthisunitonmultiplyingdecimals.OneimportantthingwenoticedthatdiffersfromCommonCoreisthatMIteachesmultiplicationasmultiplicandxmultiplier=product.Forinstance,inourresearchlessonthestorycontextof6cartonsofjuiceeachcontaining0.8Lofjuicewouldbewrittenas0.8x6=4.8andinterpretedas0.8Lofjuicemultipliedby6cartonsequals4.8L.TheteamalsolookedcarefullyathowMItreatsplacevalue.InthefirstunitofMIgrade5,placevalueisdiscussedinproblemsolvinglessonsinwhichdecimalnumbersaremultipliedby10and100.Studentslearnthatwhenadecimalnumberismultipliedbyapoweroftenthedigitsintheproductstaythesameasthedigitsinthemultiplicand,theysimplymovetothelefthowevermanypowersoftenitwasmultipliedby.Ifmultipliedbyten,thedigitsmovetothenextlargerplacevalue.Ifmultipliedby100thedigitsmovetwoplacestotheleftbecause100is10x10.Byfocusingonbothwholenumbersanddecimals,studentslearnthattheunderlyingstructureofthebasetenplacevaluesystem,thateachadjacentplacevaluepositionhasa10to1relationship,appliestodecimalsaswellaswholenumbers.Thisisintendedtohelpstudentsavoidthinkingaboutoperationswithdecimalsassomeseparatebranchofmath,andratheracontinuationofwhattheyhavealreadylearnedaboutwholenumberoperations.ThesetwoimportanttopicsinMI,multiplicationofwholenumbersandunderstandingofplacevalue,wereimportantindesigningthelessonsthatleaduptotheresearchlesson,whichisthefirstlessoninaunitonmultiplyingdecimals,adaptedbytheteamfromthe4thgradeMIunit,“Let’sthinkaboutmultiplyinganddividingdecimalnumbers”


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9. UnitPlan

Let’sThinkaboutMultiplyingandDividingDecimalNumbersMathInternationalGrade4,Chapter15

Assessment

Lesson1ResearchLesson

ProblemWebought6cartonsofjuice.Eachcartonofjuicecontains0.8Lofjuice.Howmuchjuiceistherealtogether?AnticipatedResponses*Repeatedaddition*Traditionalalgorithmwithmisplaceddecimal*Estimationtodetermineplacementofdecimalpoint(.8iscloseto1,and6x1=6,sotheanswerisprobably4.8)*Multiplyasiftheywerewholenumbers,andthendividetheproductby10SummaryTheproductof0.8x6canbecalculatedbyfirstmaking0.810timesasmuch,thenbycalculating8x6,andthenbydividingtheproductby10.

ExitSlip:0.2x60.3x5

Lesson2 ProblemThereare7waterjugseachofwhichcanhold3.6Lofwater.Ifwefillthesejugswithwater,howmuchwaterwilltherebealtogether?AnticipatedResponses*36x7=252,so3.6x7=25.2Summary3.6x7canbecalculatedbyfirstmaking3.610timesasmuch,thenbycalculating36x7,andthenbydividingtheproductby10.AdditionalPracticeIXL5.I.3Multiplyadecimalbyaone-digitwholenumberMultiplicationScootMultiplyadecimalbyaone-digitwholenumber.

ExitSlip:1.2x43.7x4

Lesson3 PracticeLet’ssolvethesameproblemfromyesterdayusingthemultiplicationalgorithm:3.6x7

ExitSlip:1.2x43.7x4


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TeacherGuidance3.6x7

1. Writethetwonumbersbyliningupthedigitsontheright.Disregardthedecimalpoint.

2. Calculateasifyouweremultiplyingtwowholenumbers.

3. Writethedecimalpointfortheproductdirectlybeneaththedecimalpointinthetopnumber(themultiplicand).

SummaryTomultiplydecimalnumbers,firstfindanestimateforthesolution.Thenignorealldecimalsandmultiplyasiftheywerewholenumbers.Reasonwheretoplacethedecimalbasedofftheestimate.AdditionalPracticeIXL5.I.3Multiplyadecimalbyaone-digitwholenumber

Lesson4 PracticeExplainhow0.2x4,0.8x5,and7.5x4canbecalculatedusingthemultiplicationalgorithm.TeacherGuidanceLet’suseestimationtocheckhowreasonableouranswersare.SummaryTodaywepracticedusingestimationtomakesureouransweriscorrect,especiallywhentheproductendsinzero.

ExitSlip:0.3x28.5x6

Lesson5

ProblemExplainhowtocalculate1.8x34usingthemultiplicationalgorithm.AnticipatedResponses*Multiply18by34,thendividetheproductby10*Multiply1.8bytheonesplacefirst,andthenbythetensplaceTeacherGuidance*72and54aretheresultsofwhatcalculations?*Howshouldwedecidewheretowritethedecimalpointintheproduct?SummaryWethinkaboutplacevaluetodeterminewherewewritethedecimalpointintheproduct.

ExitSlip:Wearegoingtogive2.6mofribbontoeachof13people.Howmanymofribbondoweneed?


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AdditionalPracticeIXL5.I.4Multiplyadecimalbyamulti-digitnumber

Lesson6 ProblemOnemeterofanironbarweighs1.36kg.Howmuchwill7mofthisironbarweigh?AnticipatedResponses*Multiply136by7,thendividetheproductby100*Multiplyeachplacevalueby7TeacherGuidance*136ishowmanytimesasmuchas1.36?*Firstmake1.36100timesasmuch,thencalculate136x7usingthemultiplicationalgorithm.Finally,dividetheproductby100.*Explainhow1.36x7iscalculatedusingthemultiplicationalgorithm.SummaryTheproductof1.36x7canbecalculatedbyfirstmaking1.36100timesasmuch,thenbycalculating136x7,andthenbydividingtheproductby100.

ExitSlip:4.83x21.25x5

Lesson7 PracticeProblemExplainhowthemultiplicationalgorithmisusedinthecalculationsontheright:0.25x3=0.752.45x4=9.80

ExitSlip:0.16x44.28x5

10. LessonPlan

Steps,LearningActivitiesTeacher’sQuestionsandExpectedStudentReactions

TeacherSupport PointsofEvaluation

1.PosingtheTaskWebought6cartonsofjuice.Eachcartonofjuicecontains0.8Lofjuice.Howmuchjuiceistherealtogether?T:Talkwithyourtableanddecidewhatnumbersentencewewillsolvefortoday’slesson.Whataretheimportantnumbers?Howdoweknowwhatoperationtodo?

Teacherwillwritequestionontheboardandinstructstudentstowritethequestionintheirnotebooks.Doesitmatterifwewritetheproblemas0.8x6versus6x0.8?Whichpropertyofmathematicstellsustheyhave

Whatevidenceistherethatstudentsunderstandtheproblem?Weretheyabletohighlightkeyideas?


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ExpectedStudentResponses:

Ournumbersentenceis0.8x6Ournumbersentenceis6x0.8Ournumbersentenceis0.8+0.8+0.8+0.8+0.8+0.8

Writeonboard:0.8x6=?Writeonboard:Let’sthinkabouthowtomultiplyadecimalbyawholenumber.Offerdualnumberlineasahelpfultoolinsolvingtheproblem.Stressthatitsuseisoptional.Postonboard:

thesamevalue(CommutativePropertyofMultiplication)?Whatistherelationshipbetweenmultiplicationandaddition?

Dostudentsrecognizethat0.8x6representsthissituation?

2.AnticipatedStudentResponsesResponse1:00.81.62.43.244.8L

Response2:0.8+0.8+0.8+0.8+0.8+0.8_____4.8Response3:0.8x6____48

Encouragestudentswhoonlysolvewithrepeatedadditiontousedualnumberline.

Whatstrategiesarestudentsusingtosolvetheproblem?


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Myanswerneedstobecloseto6x1,because0.8isalmost1.4.8iscloserto6than48and0.48Response4:6x0.8=?(10timesmore)⃕⃕⃕⃕⃕⃕↓6x8=48(10timesless)↓↓6x0.8=4.8Response5:0.8means8of0.1,so6x0.8=(6x8)of0.1=48of0.1=4.8Possiblemisconceptions:

6x0.8=486x0.8=0.48

Askstudenttomakeaballparkestimatetocheckthereasonablenessofhisorheranswer.Oraskstudenttochecksolutionwithrepeatedaddition.

3.ComparingandDiscussingShare1Studentusesdualnumberlinetodeterminetotallitersoffluidfor2,3,4,5,and6cartonsofjuice.Share2Studentusesrepeatedadditionof0.8sixtimestoget4.8litersofjuice.Share3Studentmultiplies0.8x6similarlyto8x6=48.Heorsheplacesthedecimalinthecorrectplacebyreasoningthat0.8iscloseto1.Thereforemysolutionneedstobecloseto1x6or6.So4.8iscloserto6than48and0.48.Share4Studentsolves0.8x6=4.8byfirstmaking0.8tentimeslargerandcalculating8x6=48.Heorshethenmakes48tentimessmallertoget4.8liters.TeacherGuidanceForstudentswhodonotexplicitlyrefertomultiplyingby10anddividingby10asareasoningforwhywecanuse8

Teachershouldreferbacktotheoriginalequation0.8x6=?sostudentsunderstandthattheirstrategiesrelatetosolvinganewtypeofmultiplicationproblem.

Whenstudentsareexplainingtheirclassmates’solutionsintheirownwords,aretheirresponsesshowingevidenceofunderstandingoftheproblemandsolutions?


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x6=48tosolve0.8x6=4.8,ask:(1)Whatdoyoumultiply0.8bytoget8?(2)Whatdoyoumultiply48bytoget4.8?4.SumminguptheLessonTosolve0.8x6= ⃞,itishelpfultoknow8x6=48.8is10timesasmuchas0.8.`Therefore48is10timesasmuchas�.Reflection:Afterthesummary,studentswillwriteareflectionintheirnotebook.Theteacherpromptforthisreflectionwillbe“Usewhatyoulearnedtodaytosolvetheproblem0.9x5=?Explainhowyougotyouranswer”

Arestudentsabletocorrectlysolve0.9x5intheirreflection?Whichstrategiesdotheyusetosolveit?

11. EvaluationArestudentsabletomultiplyadecimalbyawholenumber?Howdidstudentsusewhattheyknowaboutplacevalueandwholenumbermultiplicationtomakesenseofthisnewkindofproblem?Whatevidencewastherethatthestudentnotetakingandteacherboardworkhelpedmakeideasvisibletoothers,andhelpedstudentstocommunicatetheirmathematicalthinkingtoothers?

12. BoardPlan

sample lesson study research lesson

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