grade 3, module 5: fractions as number on the number line lessons 2016 v2.pdf · grade 3, module 5:...

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1

Grade3,Module5:

FractionsasNumberontheNumberLineMission:FractionsasNumbers

Lessons

TableofContents

Lessons.............................................................................................................................2-41

TopicA:PartitioningaWholeintoEqualParts..............................................................................2

TopicB:UnitFractionsandtheirRealtiontotheWhole................................................................6

TopicC:ComparingUnitFractionsandSpecifyingtheWhole.....................................................10

TopicD:FractionsontheNumberLine........................................................................................15

TopicE:EquivalentFractions.......................................................................................................22

TopicF:Comparison,Order,andSizeofFractions.......................................................................29

ProblemSetsandTemplatesforLessons4,20,24,25,27,29,and30.......................................35

Lessons

2

NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:

Somestudentsmaybenefitfroma

reviewofhowtousearulertomeasure.Suggestthefollowingsteps:

1.Identifythe0markontheruler.

2.Lineupthe0markwiththeleftendofthepaperstrip.

3.Pushdownontherulerasyoumakeyourmark.

NOTESONMULTIPLEMEANSOFREPRESENTATION:

Reviewandpostfrequentlyused

vocabulary,suchas1fourth,accompaniedbyapictureof1fourth,

1outof4equalparts,and!".

TopicA:PartitioningaWholeintoEqualParts

TopicAopensModule5withstudentsactivelypartitioningdifferentmodelsofwholesintoequalparts(e.g.,concretemodels,fractionstrips,anddrawnpictorialareamodelsonpaper).Theyidentifyandcountequalpartsas1half,1fourth,1third,1sixth,and1eighthinunitformbeforeanintroductiontotheunitfraction1/b.

LESSON1

ConceptDevelopment(32minutes)

Materials: (T)1—clearplasticcupfullofcoloredwater,2—otheridenticalclearplasticcups(empty),2—12"×1"stripsofconstructionpaper(S)2—12"×1"stripsofconstructionpaper,12-inchruler

Note:StudentsshouldsavethefractionstripstheycreateduringthislessonforuseinfutureModule5lessons.

Part1:Partitionfractionstripsintoequalparts.

T: Measureyourpaperstripusinginches.Howlongisit?

S: 12inches.

T: Makeasmallmarkat6inchesatboththetopandbottomofthestrip.Connectthetwopointswithastraightline.

T: (Afterstudentsdoso.)HowmanyequalpartshaveIsplitthepaperintonow?

S: 2.

T: Thefractionalunitfor2equalpartsishalves.Whatfractionofthewholestripisoneoftheparts?

S: 1half.

T: Pointtothehalvesandcountthemwithme.(Pointtoeachhalfofthestripasstudentscount“onehalf,twohalves.”)Discusswithyourpartnerhowweknowthesepartsareequal.

S: WhenIfoldthestripalongtheline,thetwosidesmatchperfectly.àImeasuredandsawthateachpartwas6incheslong.àThewholestripis12incheslong.12dividedby2is6.à6times2or6plus6is12,sotheyareequalinlength.

Continuewithfourthsonthesamestrip.

Fourths:Repeatthesamequestionsaskedwhenmeasuringhalves.(Studentswhobenefitfromachallengecanthinkabouthowtofindeighthsaswell.)

T: Makeasmallmarkat3inchesand9inchesatthetopandbottomofyourstrip.Connectthetwopointswithastraightline.Howmanyequalpartsdoyouhavenow?

S: 4.

Lessons

3


ForEnglishlanguagelearnersandothers,sentenceframessupportEnglishlanguageacquisition.Studentsareabletoformcompletesentenceswhileprovidingdetailsaboutthefractiontheyareanalyzing.

Askstudentsworkingabovegradelevel

forapossiblemethodtopartitionthewholeintoninths(e.g.,afterpartitioningthirds).

T: Thefractionalunitfor4equalpartsisfourths.Countthefourths.

S: 1fourth,2fourths,3fourths,4fourths.

T: Discusswithyourpartnerhowyouknowthatthesepartsareequal.

Distributeasecondfractionstrip,andrepeattheprocesswiththirdsandsixths.

Thirds:Havethestudentsmarkpointsat4inchesand8inchesatthetopandbottomofanewstrip.Askthemtoidentifythefractionalunit.Askthemhowtheyknowthepartsareequal,andthenhavethemcounttheequalparts,“1third,2thirds,3thirds.”

Sixths:Havethestudentsmarkpointsat2inches,6inches,and10inches.Repeatthesameprocessaswithhalves,fourths,andthirds.Askstudentstothinkabouttherelationshipofthehalvestothefourthsandthethirdstothesixths.

Part2:Partitionawholeamountofliquidintoequalparts.

T: Justaswemeasuredawholestripofpaperwitharulertomakehalves,let’snowmeasurepreciselytomake2equalpartsofawholeamountofliquid.

Leadademonstrationusingthefollowingsteps(picturedtotheright).

1. Presenttwoidenticalglasses.Makeamarkabout1fourthofthewayupthecuptotheright.

2. Fillthecuptothatmark.

3. Pourthatamountofliquidintothecupontheleft,andmarkoffthetopofthatamountofliquid.

4. Repeattheprocess.Fillthecupontherighttothemarkagain,andpouritintothecupontheleft.

5. Markthetopoftheliquidinthecupontheleft.Thecupontheleftnowshowsthemarkingsforhalftheamountofwaterandthewholeamountofwater.

6. Havestudentsdiscusshowtheycanmakesurethemiddlemarkshowshalfoftheliquid.Comparethestripshowingawholepartitionedinto2equalpartsandtheliquidpartitionedinto2equalparts.Havestudentsdiscusshowtheyarethesameanddifferent.

LESSON2


Materials: (S)8paperstripssized4!"”×1”(verticallycutan8

!#”×11”

paperdownthemiddle),pencil,crayon

Note:StudentsshouldsavethefractionstripstheycreateduringthislessonforuseinfutureModule5lessons.

Havestudentstakeonestripandfoldittomakehalves.(Theymightfolditoneoftwoways.Thisiscorrect,butforthepurposeofthislesson,itisbesttofoldaspicturedbelow.)

Lessons

4

NOTESONMULTIPLEMEANSOFENGAGEMENT:

Organizestudentsworkingbelowgrade

levelatthestationswitheasierfractionalunitsandstudentsworkingabovegradelevelatstationswiththemostchallengingfractionalunits.Tocreateagreaterchallenge,makestationsforseventhsandtwelfths.

T: Howmanyequalpartsdoyouhaveinthewhole?

S: Two.

T: Whatfractionofthewholeis1part?

S: 1half.

T: Drawalinetoshowwhereyoufoldedyourpaper.Writethenameofthefractiononeachequalpart.

Usethefollowingsentenceframeswiththestudentschorally.

1. Thereare____________equalpartsinall.

2. 1equalpartiscalled____________________.

Studentsshouldfoldandlabelstripsshowingfourthsandeighthstostart,followedbythirdsandsixthsandfifthsandtenths.Somestudentsmaycreatemorestripsthanothers.

Whilecirculating,watchforstudentswhoarenotfoldinginequalparts.Encouragestudentstotryspecificstrategiesforfoldingequalparts.Awordwallwouldbehelpfultosupportthecorrectspellingofthefractionalunits,especiallyeighths.

Whenthestudentshavecreatedtheirfractionstrips,askaseriesofquestionssuchasthefollowing:

§ Lookatyoursetoffractionstrips.Imaginetheyare4piecesofdeliciouspasta.Raisethestripintheairthatbestshowshowtocut1pieceofpastaintoequalpartswithyourfork.

§ Lookatyourfractionstrips.Imaginetheyarelengthsofribbon.Raisethestripintheairthatbestshowshowtodividetheribboninto3equalparts.

§ Lookatyourfractionstrips.Imaginetheyarecandybars.Whichbestshowshowtoshareyourcandybarfairlywith1person?Whichshowshowtoshareyourhalffairlywith3people?

LESSON4


Materials: (S)ProblemSet,seeadditionalitemsforstationslistedbelow

Exploration:Studentsworkatstationstorepresentagivenfractionalunitusingavarietyofmaterials.Designatethefollowingstationsforgroupsof3students(morethan3notsuggested).

StationA:Halves StationE:SixthsStationB:Fourths StationF:NinthsStationC:Eighths StationG:FifthsStationD:Thirds StationH:Tenths

Lessons

5


Asstudentsmovearoundtheroomduringthemuseumwalk,havethemgentlypickupthematerialstoencouragebetteranalysis.Thisencouragesmoreconversation,too.

Equipeachstationwiththefollowingsuggestedmaterials:§ 1-meterlengthofyarn§ 1rectangularpieceofyellowconstructionpaper(1”×12”)§ 1pieceofbrownconstructionpaper(candybar)(2”×6”)§ 1squarepieceoforangeconstructionpaper(4”×4”)§ Alargecupcontainingawholeamountofwaterthatcorrespondstothedenominatorofthe

station’sfractionalunit(e.g.,thefourthsstationgetsawholeof4ouncesofwater)§ Anumberofsmall,clearplasticcupscorrespondingtothedenominatorofthestation’s

fractionalunit(e.g.,thefourthsstationgets4cups)§ A200-gramballofclayorplaydough(besuretohavepreciselythesameamountateach

station)Tohelpthestudentsstart,giveaslittledirectionaspossiblebutenoughdependingontheparticularclass.Itissuggestedthatstudentsworkwithoutscissorsorcutting.Paperandyarncanbefolded.Pencilcanbeusedonpapertodesignateequalpartsratherthanfolding.Belowaresomepossibledirectionsforstudents:

§ Youwillpartitioneachitemandmakeadisplayatyourstationaccordingtoyourfractionalunit.§ Eachitematyourstationrepresents1whole.Youmustuseallofeachwhole.(Forexample,if

showingthirds,alloftheclaymustbeused.)§ Useyourfractionalunittoshoweachwholepartitionedintoequalparts.§ Partitiontheclaybydividingitintosmallerequalpieces.(Possiblydothisbyformingtheclay

intoequal-sizedballs.Ifnecessary,demonstrate.)§ Partitionthewholeamountofwaterbyestimatingtopourequalamountsfromthelargecup

intoeachofthesmallercups.Thewaterineachsmallercuprepresentsanequalpartofthewhole.

Givethestudents15minutestocreatetheirdisplay.Next,conductamuseumwalkwheretheytourtheworkoftheotherstations.Beforethemuseumwalk,chartandreviewthefollowingpoints.Iftheanalysisdwindlesduringthetour,circulateandreferstudentsbacktothechart.StudentscompletetheirProblemSetsastheymovebetweenstations;theymayalsousetheirProblemSetsasaguide.

§ Identifythefractionalunit.§ Thinkabouthowtheunitsrelatetoeachotherat

thatstation.§ Comparetheyarntotheyellowstrip.§ Comparetheyellowstriptothebrownpaperor

candybar.§ Comparethewatertotheclay.

Thinkabouthowthatunitrelatestoyourownandtootherunits.

Lessons

6

1half;!#


Whileintroducingthenewterms—unitform,fractionform,andunitfraction—checkforstudentunderstanding.Englishlanguagelearnersmaychoosetodiscussdefinitionsofthesetermsintheirfirstlanguagewiththeteacherortheirpeers.

1third;!$

TopicB:UnitFractionsandtheirRelationtotheWholeInTopicB,studentscompareunitfractionsandlearntobuildnon-unitfractionswithunitfractionsasbasicbuildingblocks.Thisparallelstheunderstandingthatthenumber1isthebasicbuildingblockofwholenumbers.

LESSON5


Materials: (S)Personalwhiteboard

T: (Projectordrawacircle,asshownbelow.)Whisperthenameofthisshape.

S: Circle.

T: WatchasIpartitionthewhole.(Drawalinetopartitionthecircleinto2equalparts,asshown.)Howmanyequalpartsarethere?

S: 2equalparts.

T: What’sthenameofeachunit?

S: 1half.

T: (Shadeoneunit.)Whatfractionisshaded?

S: 1half.

T: Justlikeanynumber,wecanwriteonehalfinmanyways.Thisistheunitform.(Write1half

underthecircle.)Thisisthefractionform.(Write!#underthecircle.)Bothofthesereferto

thesamenumber,1outof2equalunits.Wecall1halfaunitfractionbecauseitnamesoneoftheequalparts.

T: (Projectordrawasquare,asshownbelow.)What’sthenameofthisshape?

S: It’sasquare.

Lessons

7


Studentsworkingabovegradelevelmay

enjoyidentifyingfractionswithanaddedchallengeofeachshaperepresentingafractionratherthanthewhole.Forexample,askthefollowing:

“Ifthesquareis1third,namethe

shadedregion”(e.g.,$!#or

!").

T: Drawitonyourpersonalwhiteboard.(Afterstudentsdrawthesquare.)Estimatetopartitionthesquareinto3equalparts.

S: (Partition.)

T: What’sthenameofeachunit?

S: 1third.

T: Shadeoneunit.Then,writethefractionfortheshadedamountinunitformandfractionformonyourboard.

S: (Shadeandwrite1thirdand!$.)

T: Talktoapartner:Isthenumberthatyouwrotetorepresenttheshadedpartaunitfraction?Whyorwhynot?

S: (Discuss.)

Continuetheprocesswithmoreshapesasneeded.Thefollowingsuggestedshapesincludeexamplesofbothshadedandnon-shadedunitfractions.Alterlanguageaccordingly.

T: (Projectordrawthefollowingimage.)Discusswithyourpartner:Doestheshapehaveequalparts?Howdoyouknow?

S: No.Thepartsarenotthesamesize.àThey’realsonotexactlythesameshape.àThepartsarenotequalbecausethebottompartsarelarger.Thelinesonthesidesleaninatthetop.

T: Mostagreethatthepartsarenotequal.Howcouldyoupartitiontheshapetomakethepartsequal?

S: Icancutitinto2equalparts.Youhavetocutitrightdownthemiddlegoingupanddown.Thelinesaren’tallthesamelengthlikeinasquare.

T: Turnandtalk:Ifthepartsarenotequal,canwecallthesefourths?Whyorwhynot?

S: (Discuss.)


Reviewpersonalgoalswithstudents.

Forexample,ifstudentsworkingbelowgradelevelchosetosolveonewordproblem(perlesson)lastweek,encouragethemtoworktowardcompletingtwowordproblemsbytheendofthisweek.

MP.6

Lessons

8

LESSON7


Materials: (T)1-literbeaker,water(S)Paper,scissors,

crayons,mathjournalShowabeakerofliquidhalffull.

T: Whisperthefractionofliquidthatyouseetoyourpartner.

S: 1half.T: Whataboutthepartthatisnotfull?Talktoyourpartner:Couldthatbeafraction,too?Why

orwhynot?S: No,becausethere’snothingthere.àIdisagree.It’sanotherpart.It’sjustnotfull.àIt’s

anotherhalf.Becausehalfisfullandhalfisempty.Twohalvesmakeonewhole.T: Eventhoughpartsmightnotbefullorshaded,theyarestillpartofthewhole.Let’sexplore

thisideasomemore.I’llgiveyou1sheetofpaper.Partitionitintoanyshapeyouchoose.Justbesureofthese3things:1. Thepartsmustbeequal.2. Therearenofewerthan5,andnomorethan20partsinall.3. Youusetheentiresheetofpaper.

S: (Partitionbyestimatingtofoldthepaperintoequalparts.)T: Now,useacrayontoshadeoneunit.S: (Shadeonepart.)T: Next,you’regoingtocutyourwholeintopartsbycuttingalongthelinesyoucreatedwhenyou

foldedthepaper.You’llreassembleyourpartsintoauniquepieceofartforourfractionmuseum.Asyoumakeyourart,makesurethatallpartsaretouchingbutnotontopoforundereachother.

S: (Cutalongthefoldsandreassemblepieces.)

T: Asyoutourourmuseumadmiringtheart,identifywhichunitfractiontheartistchoseandidentifythefractionrepresentingtheunshadedequalpartsoftheart.Writebothfractionsinyourjournalnexttoeachother.

S: (Walkaroundandcollectdata,whichwillbeusedintheDebriefportionofthelesson.)

LESSON9


Materials: (S)Personalwhiteboard,fractionstrips

T: Ibrought2orangesforlunchtoday.IcuteachoneintofourthssothatIcouldeatthemeasily.DrawapictureonyourpersonalwhiteboardtoshowhowIcutmy2oranges.

S: (Draw.)

T: If1orangerepresents1whole,howmanycopiesof1fourtharein1whole?

S: 4copies.

T: Then,whatisourunit?

S: Fourths.

T: Howmanycopiesof1fourthareintwowholeoranges?

NOTESONMATERIALS:

Ifabeakerisnotavailable,useaclear

containerthathasaconsistentdiameterfrombottomtotop,andmeasuretheamountofliquidtopreciselyshowthecontainerhalffull.

Lessons

9


TurnandTalkisanexcellentwayfor

EnglishlanguagelearnerstouseEnglishtodiscusstheirmaththinking.LetEnglishlanguagelearnerschoosethelanguagetheywishtousetodiscusstheirmathreasoning,particularlyiftheirEnglishlanguagefluencyislimited.

S: 8copies.

T: Let’scountthem.

S: 1fourth,2fourths,3fourths,…,8fourths.

T: Areyousureourunitisstillfourths?Talkwithyourpartner.

S: No,it’seighthsbecausethereare8pieces.àIdisagreebecausetheunitisfourthsineachorange.àRemember,eachorangeisawhole,sotheunitisfourths.2orangesaren’tthewhole!

T: IwassohungryIate1wholeorangeand1pieceofthesecondorange.ShadeinthepiecesIate.

S: (Shade.)

T: HowmanypiecesdidIeat?

S: 5pieces.

T: Andwhat’sourunit?

S: Fourths.

T: SowecansaythatIate5fourthsofanorangeforlunch.Let’scountthem.

S: 1fourth,2fourths,3fourths,4fourths,5fourths.

T: Onyourboard,worktogethertoshow5fourthsasanumberbondofunitfractions.

S: (Workwithapartnertodrawanumberbond.)

T: ComparethenumberofpiecesIateto1wholeorange.Whatdoyounotice?

S: Thenumberofpiecesislarger!àYouatemorepiecesthanthewhole.

T: Yes.Ifthenumberofpartsisgreaterthanthenumberofequalpartsinthewhole,thenyouknowthatthefractiondescribesmorethan1whole.

T: Workwithapartnertomakeanumberbondwith2parts.Onepartshouldshowthepiecesthatmakeupthewhole.Theotherpartshouldshowthepiecesthataremorethanthewhole.

S: (Workwithapartnertodrawanumberbond.)

Demonstrateagainusinganotherconcreteexample.Followbyworkingwithfractionstrips.Foldfractionstripssothatstudentshaveatleast2stripsrepresentinghalves,thirds,fourths,sixths,andeighths.Studentscanthenbuildandidentifyfractionsgreaterthan1withthesetsoffractionstrips.NotethatthesefractionstripsareusedagaininLesson10.Itmightbeagoodideatocollectthemorhavestudentsstoretheminasafeplace.


Forstudentsworkingbelowgradelevel,respectfullyfacilitateself-assessmentofpersonalgoals.Guidestudentstoreflectuponquestionssuchas,“WhichfractionskillsamIgoodat?WhatwouldIliketobebetterat?Whatismyplantoimprove?”Celebrateimprovement.

Lessons

10

TopicC:ComparingUnitFractionsandSpecifyingtheWholeInTopicC,studentspracticecomparingunitfractionswithfractionstrips,specifyingthewholeandlabelingfractionsinrelationtothenumberofequalpartsinthatwhole.

LESSON10


Materials: (S)Foldedfractionstrips(halves,thirds,fourths,sixths,andeighths)fromLesson9,personalwhiteboard,1setof<,>,=cardsperpair

T: Takeoutthefractionstripsyoufoldedyesterday.

S: (Takeoutstripsfoldedintohalves,thirds,fourths,sixths,andeighths.)

T: Lookatthedifferentunits.Takeaminutetoarrangethestripsinorderfromthelargesttothesmallestunit.

S: (Placethefractionstripsinorder:halves,thirds,fourths,sixths,andeighths.)

T: Turnandtalktoyourpartneraboutwhatyounotice.

S: Eighthsarethesmallesteventhoughthenumber8isthebiggest.àWhenthewholeisfoldedintomoreunits,eachunitissmaller.Ionlyfoldedoncetogethalves,andthey’rethebiggest.

T: Lookat1halfand1third.Whichunitfractionislarger?

S: 1half.

T: Explaintoyourpartnerhowyouknow.

S: Icanjustsee1halfislargeronthestrip.àWhenyousplititbetween2people,thepiecesarelargerthanifyousplititbetween3people.àTherearefewerpieces,sothepiecesarelarger.

Continuewithotherexamplesusingthefractionstripsasnecessary.

T: Whathappenswhenwearen’tusingfractionstrips?Whatifwe’retalkingaboutsomethinground,likeapizza?Is1halfstilllargerthan1third?Turnandtalktoyourpartneraboutwhyorwhynot.

S: I’mnotsure.àSharingapizzaamong3peopleisnotasgoodassharingitbetween2people.Ithinkpiecesthatarehalvesarestilllarger.àIagreebecausethenumberofpartsdoesn’tchangeeveniftheshapeofthewholechanges.

T: Let’smakeamodelandseewhathappens.Draw5circlesthatarethesamesizetorepresentpizzasonyourpersonalwhiteboard.

S: (Draw.)

T: Estimatetopartitionthefirstcircleintohalves.Labeltheunitfraction.

S: (Drawandlabel.)

T: Estimatetopartitionthesecondcircleintothirds.(Modelifnecessary.)Labeltheunitfraction.

S: (Drawandlabel.)

T: Themorewecut,what’shappeningtoourpieces?

S: They’regettingsmaller!

T: So,is1thirdstillsmallerthan1half?

S: Yes!

MP.2

Lessons

11

Myglass Mybrother’sglass

T: Partitionyourremainingcirclesintofourths,sixths,andeighths.Labeltheunitfractionineachone.

S: (Drawandlabel.)

T: Compareyourdrawingstoyourfractionstrips.Talktoapartner:Doyounoticethesamepatternaswithyourfractionstrips?

S: (Discuss.)

Continuewithotherrealworldexamplesifnecessary.

T: Let’scompareunitfractions.Foreachturn,youandyourpartnerwilleachchooseanysinglefractionstrip.Choosenow.

S: (Chooseastriptoplay.)

T: Now,compareunitfractionsbyfoldingtoshowonlytheunitfraction.Then,placetheappropriatesymbolcard(<,>,or=)onthetablebetweenyourstrips.

S: (Fold,compare,andplacesymbolcards.)

T: (Holdsymbolcardsfacedown.)Iwillfliponeofmysymbolcardstoseeiftheunitfractionthatisgreaterthanorlessthanwinsthisround.IfIflipequals,it’satie.(Flipacard.)

Continueatarapidpaceforafewrounds.

LESSON11


Materials: (T)2different-sizedclearplasticcups,foodcoloring,water(S)Personalwhiteboard

T: (Write1isthesameas1.)Showthumbsupifyouagree,thumbsdownifyoudisagree.

S: (Showthumbsuporthumbsdown.)

T: 1literofsodaand1canofsoda.(Drawpicturesorshowobjects.)Is1stillthesameas1?Turnandtalktoyourpartner.

S: Yes,they’restillthesameamount.àNo,aliterandacanaredifferent.àHowmanystaysthesame,butaliterislargerthanacan,sohowmuchineachisdifferent.

T: Howmanyandhowmuchareimportanttoourquestion.Inthiscase,whateachthingischangesit,too.Becausealiterislarger,ithasmoresodathanacan.Talktoapartner:Howdoesthischangeyourthinkingabout1isthesameas1?

S: Ifthethingislarger,thenithasmore.àEventhoughthenumberofthingsisthesame,whatitismightchangehowmuchofitthereis.àIfwhatitisandhowmuchitisaredifferent,then1and1aren’texactlythesame.

T: Asyoucompare1and1,Ihearyousaythatthesizeofthewholeandhowmuchisinitmatters.Thesameistruewhencomparingfractions.

T: Forbreakfastthismorning,mybrotherandIeachhadaglassofjuice.(Presentdifferent-sizedglassespartitionedintohalvesandfourths.)Whatfractionofmyglasshasjuice?

S: 1fourth.


ThispartneractivitybenefitsEnglish

languagelearnersasitincludesrepeateduseofmathlanguageinareliablestructure(e.g.,“__isgreaterthan__”).ItalsoofferstheEnglishlanguagelearneranopportunitytodiscussthemathwithapeer,whichmaybemorecomfortablethanspeakinginfrontoftheclassortotheteacher.

MP.6

Lessons

12


Manystudents,includingthoseworkingbelowgradelevel,maybenefitfromhavingpre-drawnwholesofthesameshapeandsize.

islessthan

!%islessthan

!$

T: Whatfractionofmybrother’sglasshasjuice?

S: 1half.

T: Whenthewholesarethesame,1halfisgreaterthan1fourth.Doesthispictureprovethat?Discussitwithyourpartner.

S: 1halfisalwayslargerthan1fourth.àItlookslikeyoumighthavedrunkmore,butthewholesaren’tthesame.àTheglassesaredifferentsizes—likethecanandtheliterofsoda.Wecan’treallycompare.

T: I’mhearingyousaythatwehavetoconsiderthesizeofthewholewhenwecomparefractions.

Tofurtherillustratethepoint,poureachglassofjuiceintocontainersthatarethesamesize.Itmaybehelpfultopurposefullyselectyourcontainerssothat1fourthofthelargeglassisthelargerquantity.

Totransitionintothepictorialworkwithwholesthatarethesame,offeranotherconcreteexample.Thistimeuserectangularshapedwholesthataredifferentinsize,suchasthoseshowntotheright.

T: Let’sseehowthecomparisonchangeswhenourwholesarethesame.Onyourboard,drawtworectanglesthatarethesamesize.Partitioneachintothirds.

S: (Drawandpartitionrectangles.)

T: Now,partitionthefirstrectangleintosixths.

S: (Partitionthefirstrectanglefromthirdstosixths.)

T: Shadetheunitfractionineachrectangle.Labelyourmodelsandusethewordsgreaterthanorlessthantocompare.

S: (Shade,label,andcomparemodels.)

T: Doesthispictureprovethat1sixthislessthan1third?Whyorwhynot?Discusswithyourpartner.

S: Yes,becausetheshapesarethesamesize.àOneisjustcutintomorepiecesthantheother.àWeknowthepiecesaresmalleriftherearemoreofthem,aslongasthewholeisthesame.

Demonstratewithmoreexamplesifnecessary,perhapsrotatingoneoftheshapessoitappearsdifferentbutdoesnotchangeinsize.

Lessons

13


Organizestudentsworkingbelowgrade

levelatthestationswiththeeasierfractionalunitsandstudentsworkingabovegradelevelatthestationswiththemostchallengingfractionalunits.

LESSON12


Materials: (S)UsesimilarmaterialstothoseusedinLesson4(atleast75copiesofeach),10-centimeterlengthofyarn,4”×1”rectangularpieceofyellowconstructionpaper,3”×1”brownpaper,1”×1”orangesquare,water,smallplasticcups,clay

Exploration:Designatethefollowingstationsforgroupsof3(morethan3notsuggested).

StationA:1halfand1fourthStationB:1halfand1thirdStationC:1thirdand1fourthStationD:1thirdand1sixthStationE:1fourthand1sixthStationF:1fourthand1eighthStationG:1fifthand1tenthStationH:1fifthand1sixth

Thestudentsrepresent1wholeusingthematerialsattheirstations.Notes:

§ Eachitematthestationrepresentstheindicatedunitfractions.

§ Studentsshow1wholecorrespondingtothegivenunitfraction.Eachstationincludes2objectsrepresentingunitfractions,andtherefore2differentwholeamounts.

§ Theentirequantityofeachitemmustbeusedasthefractionindicated.Forexample,ifshowing1thirdwiththeorangesquare,thewholemustuse3thirdsor3oftheorangesquares(picturedtotheright).

T: (Holdupthesamesizeballofclay—200g—fromLesson4.)Thispieceofclayrepresents1third.Whatmight1wholelooklike?Discusswithyourpartner.

S: (Discuss.)T: (Afterdiscussion,modelthewholeas3equallumpsof

clayweighing600g.)T: (Holdupa12-inchby1-inchyellowstrip.)Thisstrip

represents1fourth.Whatmight1wholelooklike?S: (Discuss.)T: (Afterdiscussion,modelthewholeusing4equalstripslaidend-to-endforalengthof48

inches.)T: (Showa12-ouncecupofwater.)Thewaterinthiscuprepresents1fifth.Whatmightthe

wholelooklike?Whatifthewaterrepresents1fourth?(Measurethe2quantitiesinto2separatecontainers.)


GiveEnglishlanguagelearnersalittle

moretimetorespond,eitherinwritingorintheirfirstlanguage.

Lessons

14


Themuseumwalkisarichopportunity

forstudentstopracticelanguage.Pairstudentsandgivethemsentenceframesorpromptstouseateachstationtohelpthemdiscusswhattheyseewiththeirpartner.

Givethestudents15minutestocreatetheirdisplay.Next,conductamuseumwalkwheretheytourtheworkoftheotherstations.Duringthetour,studentsshouldidentifythefractionsandthinkabouttheirrelationships.Usethefollowingpointstoguidethestudents:

§ Identifytheunitfraction.§ Thinkabouthowthewholeamountrelatestoyour

ownandtootherwholeamounts.§ Comparetheyarntotheyellowstrip.§ Comparetheyellowstriptothebrownpaper.

Lessons

15

TopicD:FractionsontheNumberLineStudentstransfertheirworktothenumberlineinTopicD.Theybeginbyusingtheintervalfrom0to1asthewhole.Continuingbeyondthefirstinterval,theypartition,place,count,andcomparefractionsonthenumberline.

LESSON14


Materials: (T)Boardspace,yardstick,largefractionstripformodeling(S)Fractionstrips,blankpaper,ruler

Part1:Measurealineoflength1whole.T: (Modelthestepsbelowasstudentsfollowalongontheirpersonalwhiteboards.)

1. Drawahorizontallinewithyourrulerthatisabitlongerthanoneofyourfractionstrips.

2. Placeawholefractionstripjustabovethelineyoudrew.

3. Makeasmallmarkonyourlinethatisevenwiththeleftendofyourstrip.

4. Labelthatmark0abovetheline.Thisiswherewestartmeasuringthelengthofthestrip.

5. Makeasmallmarkonyourlinethatisevenwiththerightendofyourstrip.

6. Labelthatmark1abovetheline.Ifwestartat0,the1tellsuswhenwe’vetravelled1wholelengthofthestrip.

Part2:Measurethefractions.T: (Modelthestepsbelowasstudentsfollowalongontheir

boards.)

1. Placeyourfractionstripwithhalvesabovetheline.

2. Makeamarkonthenumberlineattherightendof1half.Thisisthelengthof1halfofthefractionstrip.

3. Labelthatmark!#.Label0halvesand2halves.

4. Repeattheprocesstomeasureandlabelotherfractionalnumbersonanumberline.

T: Lookatyournumberlinewiththirds.Readthenumbersonthislinetoapartner.

S: 0,1.àIthinkit’s0,!$,#$,1.àWhatabout

&$,!$,#$,$$?àArefractionsnumbers?

T: Someofyoureadthewholenumbers,andothersreadwholenumbersandfractions.Fractionsarenumbers.Let’sreadthenumbersfromleasttogreatest,andlet’ssay0thirdsand3thirdsfornowratherthanzeroandone.

MP.7

Lessons

16


Thislessongraduallyleadsstudents

fromtheconcretelevel(fractionstrips)tothepictoriallevel(numberlines).

S: (Readnumbers,&$ ,

!$,#$,$$.)

T: Let’sreadagainandthistimesayzeroand1ratherthan0thirdsand3thirds.

S: (Readnumbers,0, !$,#$,1.)

Part3:Drawnumberbondstocorrespondwiththenumberlines.

Oncestudentshavebecomeexcellentatmakingandlabelingfractionsonnumberlinesusingstripstomeasure,havethemdrawnumberbondstocorrespond.Usequestioningwhilecirculatingtohelpthemseesimilaritiesanddifferencesbetweenthebonds,fractionstrips,andfractionsonthenumberline.Guidestudentstorecognizethatplacingfractionsonthenumberlineisanalogoustoplacingwholenumbersonthenumberline.Ifpreferred,thefollowingsuggestionscanbeused:

§ Whatdoboththenumberbondandnumberlineshow?

§ Whichmodelbestshowshowbigtheunitfractionisinrelationtothewhole?Explainhow.

§ Howdoyournumberlineshelpyoumakenumberbonds?

LESSON15



Problem1:Locatethepoint2thirdsonanumberline.

T: 2thirds.Howmanyequalpartsareinthewhole?

S: Three.

T: Howmanyofthoseequalpartshavebeencounted?

S: Two.

T: Countupto2thirds,startingat1third.

S: 1third,2thirds.

T: Drawa2-partnumberbondof1wholewith1partas2thirds.

S: (Drawanumberbond.)

T: Whatistheunknownpart?

S: 1third.

T: Drawanumberlinewithendpointsof0and1—with0thirdsand3thirds—tomatchyournumberbond.

S: (Drawanumberline,andlabeltheendpoints.)

T: Markoffyourthirdswithoutlabelingthefractions.

S: (Markthethirds.)

T: Slideyourfingeralongthelengthofthefirstpartofyournumberbond.Speakthefractionasyoudo.

Lessons

17

S: 2thirds(slidinguptothepoint2thirds).

T: Labelthatpointas2thirds.

S: (Label2thirds.)

T: Putyourfingerbackon2thirds.Slideandspeakthenextpart.

S: 1third.

T: Atwhatpointareyounow?

S: 3thirdsor1whole.

T: Ournumberbondiscomplete.

Problem2:Locatethepoint3fifthsonanumberline.

T: 3fifths.Howmanyequalpartsareinthewhole?

S: Five.

T: Howmanyofthoseequalpartshavebeencounted?

S: Three.

T: Countupto3fifths,startingat1fifth.

S: 1fifth,2fifths,3fifths.

T: Drawa2-partnumberbondof1wholewith1partas3fifths.

S: (Drawanumberbond.)

T: Whatistheunknownpart?

S: 2fifths.

T: Drawanumberlinewithendpointsof0and1—with0fifthsand5fifths—tomatchyournumberbond.

S: (Drawanumberline,andlabeltheendpoints.)

T: Markoffyourfifthswithoutlabelingthefractions.

S: (Markthefifths.)

T: Slideyourfingeralongthelengthofthefirstpartofyournumber.Speakthefractionasyoudo.

S: 3fifths(slidinguptothepoint3fifths).

T: Labelthatpointas3fifths.

S: (Label3fifths.)

T: Putyourfingerbackon3fifths.Slideandspeakthenextpart.

S: 2fifths.

T: Atwhatpointareyounow?

S: 5fifthsor1whole.

T: Ournumberbondiscomplete.

Repeattheprocesswithotherfractionssuchas3fourths,6eighths,2sixths,and1seventh.Releasethestudentstoworkindependentlyastheydemonstratetheirskillsandunderstanding.

Lessons

18

12

234

""

*"

%"

+"

,"

12

LESSON16



T: Drawanumberlineonyourboardwiththeendpoints1and2.Thelastfewdays,ourleftendpointwas0.Talktoapartner:Wherehas0gone?

S: Itdidn’tdisappear;itistotheleftofthe1.àThearrowonthenumberlinetellsusthattherearemorenumbers,butwejustdidn’tshowthem.

T: It’sasifwetookapictureofapieceofthenumberline,butthosemissingnumbersstillexist.Partitionyourwholeinto4equallengths.(Model.)

T: Ournumberlinedoesn’tstartat0,sowecan’tstartat0fourths.Howmanyfourthsarein1whole?

S: 4fourths.

T: Wewilllabel4fourthsatwholenumber1.Labeltherestofthefractionsupto2.Checkyourworkwithapartner.(Allowworktime.)Whatarethewholenumberfractions—thefractionsequalto1and2?

S: 4fourthsand8fourths.

T: Drawboxesaroundthosefractions.(Model.)

T: 4fourthsisthesamepointonthenumberlineas1.Wecallthatequivalence.Howmanyfourthswouldbeequivalentto,oratthesamepointas,2?

S: 8fourths.

T: Talktoapartner:Whatfractionisequivalentto,atthesamepointas,3?

S: (Afterdiscussion.)12fourths.

T: Drawanumberlinewiththeendpoints2and4.Whatwholenumberismissingfromthisnumberline?

S: Thenumber3.

T: Let’splacethenumber3.Itshouldbeequallyspacedbetween2and4.Drawthatin.(Model.)

T: Wewillpartitioneachwholenumberintervalinto3equallengths.Tellyourpartnerwhatyournumberlinewilllooklike.

S: (Discuss.)

T: Tolabelthenumberlinethatstartsat2,wehavetoknowhowmanythirdsareequivalentto2wholes.Discusswithyourpartnerhowtofindthenumberofthirdsin2wholes.

S: 3thirdsmade1whole.So,6unitsofthirdsmake2wholes.à6thirdsareequivalentto2wholes.

T: Fillintherestofyournumberline.


Ifgaugingthatstudentsworkingbelowgradelevelneedit,buildunderstandingwithpicturesorconcretematerials.Extendthenumberlinebackto0.Havestudentsshadeinfourthsastheycount.UsefractionstripsasinLesson14,ifneeded.

MP.7

Lessons

19

234

%$

+$

,$

-$

!&$

!!$

!#$

1234

Followwithanexampleusingendpoints3and6sostudentsplace2wholenumbersonthenumberline,andthenpartitionintohalves.

Closetheguidedpracticebyhavingstudentsworkinpairs.PartnerAnamesanumberlinewithendpointsbetween0and5andaunitfraction.Partnersbeginwithhalvesandthirds.Whentheyhavedemonstratedthattheyhavedone2numberlinescorrectly,theymaytryfourthsandfifths,etc.PartnerBdraws,andPartnerAassesses.Then,partnersswitchroles.

LESSON17



T: Drawanumberlinewithendpoints1and4.Labelthewholes.Partitioneachwholeintothirds.Labelallofthefractionsfrom1to4.

T: Afteryoulabeledyourwholenumbers,whatdidyouthinkabouttoplaceyourfractions?

S: Evenlyspacingthemarksbetweenwholenumberstomakethirds.àWritingthenumbersinorder:3thirds,4thirds,5thirds,etc.àStartingwith3thirdsbecausetheendpointwas1.

T: Whatdothefractionshaveincommon?Whatdoyounotice?

S: Allofthefractionsarethirds.àAllareequaltoorgreaterthan1whole.àThenumberof

thirdsthatnamewholenumberscountbythrees:1=3thirds,2=6thirds,3=9thirds.à$$,%$,

-$,and

!#$ areatthesamepointonthenumberlineas1,2,3,and4.Thosefractionsare

equivalenttowholenumbers.

T: Drawanumberlineonyourboardwithendpoints1and4.

T: (Write##,*#,+#,and

,#.)Lookatthesefractions.Whatdoyou

notice?


Studentsworkingabovegradelevel

maysolvequicklyusingmentalmath.Pushstudentstonoticeandarticulatepatternsandrelationships.Astheyworkinpairstopartitionnumberlines,havestudentsmakeandanalyzetheirpredictions.


Tohelpstudentsworkingbelowgradelevel,locateandlabelfractionsonthenumberline.Elicitanswersthatspecifythewholeandthefractionalunit.Say,“Pointtoandcountthewholeswithme.Howmanywholes?Intowhatfractionalunitarewepartitioningthewhole?Labelaswecountthefractions.”

Lessons

20

0!"

#"

$"1

S: Theyareallhalves.àTheyareallequaltoorgreaterthan1.àTheyareinorder,butsomearemissing.

T: Placethesefractionsonyournumberline.(Afterstudentsplacefractionsonthenumberline.)Comparewithyourpartner.Checkthatyournumberlinesarethesame.

Followasimilarsequencewiththefollowingpossiblesuggestions:

§ Numberlinewithendpoints1and4,markingfractionsinthirds

§ Numberlinewithendpoints2and5,markingfractionsinfifths

§ Numberlinewithendpoints4and6,markingfractionsinthirds

Closethelessonbyhavingpairsofstudentsgeneratecollectionsoffractionstoplaceonnumberlineswithspecifiedendpoints.Studentsmightthenexchangeproblems,challengingeachothertoplacefractionsonthenumberline.Studentsshouldreasonaloudabouthowthepartitionedfractionalunitischosenforeachnumberline.

LESSON19



T: Draw2same-sizedrectanglesonyourboard,andpartitionbothinto4equalparts.Shadeyourtoprectangletoshow1fourth,andshadethebottomtoshow3copiesof1fourth.

T: Comparethemodels.Whichshadedfractionislarger?Tellyourpartnerhowyouknow.

S: Iknow3fourthsislargerbecause3partsisgreaterthanjust1partofthesamesize.

T: Useyourrectanglestomeasureanddrawanumberlinefrom0to1.Partitionitintofourths.Labelthewholesandfractionsonyournumberline.

S: (Drawandlabelthenumberline.)

T: Talkwithyourpartnertocompare1fourthto3fourthsusingthenumberline.Howdoyouknowwhichisthelargerfraction?

S: 1fourthisashorterdistancefrom0,soitisthesmallerfraction.3fourthsisagreaterdistanceawayfrom0,soitisthelargerfraction.

T: Manyofyouarecomparingthefractionsbyseeingtheirdistancefrom0.You’reright;1unitisashorterdistancefrom0than3units.Ifweknowwhere0isonthenumberline,howcanithelpusfindthesmallerorlargerfraction?

S: Thesmallerfractionwillalwaysbetotheleftofthelargerfraction.

T: Howdoyouknow?

14

34


Askstudentsworkingabovegradelevel

thismoreopen-endedquestion:“Howmanyhalvesareonthenumberline?”


ForEnglishlanguagelearners,modelthedirectionsorusegesturestoclarifyEnglishlanguage(e.g.,extendbotharmstodemonstratelong).

GiveEnglishlanguagelearnersalittle

moretimetodiscusswithapartnertheirmaththinkinginEnglish.

Lessons

21

S: Becausethefartheryougototherightonthenumberline,thefartherthedistancefrom0.àThatmeansthefractiontotheleftisalwayssmaller.It’scloserto0.

T: ThinkbacktoourApplicationProblem.Whatwerewetryingtofind?Thelengthofthepagefromtheedgetoeachhole?Orwerewesimplyfindingthelocationofeachhole?

S: Thelocationofeachhole.

T: Rememberthepepperproblemfromyesterday?Whatwerewecomparing?Thelengthofthepeppersorthelocationofthepeppers?

S: Wewerelookingforthelengthofeachpepper.

T: Talktoapartner:Whatisthesameandwhatisdifferentaboutthewaywesolvedtheseproblems?

S: Inboth,weplacedfractionsonthenumberline.àTodothat,weactuallyhadtofindthedistanceofeachfrom0,too.àYes,butinThomas’s,weweremoreworriedaboutthepositionofeachfraction,sohe’dputtheholesintherightplaces.àAndinthepepperproblem,thedistancefrom0tothefractiontoldusthelengthofeachpepper,andthenwecomparedthat.

T: Howdodistanceandpositionrelatetoeachotherwhenwecomparefractionsonthenumberline?

S: Youusethedistancefrom0tofindthefraction’splacement.àOryouusetheplacementtofindthedistance.àSo,they’rebothpartofcomparing.Thepartyoufocusonjustdependsonwhatyou’retryingtofindout.

T: Relatethattoyourworkonthepepperandhole-punchproblems.

S: Sometimes,youfocusmoreonthedistance,likeinthepepperproblem,andsometimesyoufocusmoreontheposition,likeinThomas’sproblem.Itdependsonwhattheproblemisasking.

T: Tryandusebothwaysofthinkingaboutcomparingasyouworkthroughtheproblemsontoday’sProblemSet.

Lessons

22

Model2

Model1

TopicE:EquivalentFractionsInTopicE,theynoticethatsomefractionswithdifferentunitsareplacedatthe

exactsamepointonthenumberline,andthereforeareequal.Forexample,!#,#",$

%,and",areequivalentfractions.Studentsrecognizethatwholenumberscanbe

writtenasfractions.

LESSON20


Materials: (T)Linkingcubesin2colors(S)Thirds(Template),redcrayon,scissors,gluestick,andblankpaper

UselinkingcubestocreateModel1,asshowntotheright.

T: Thewholeisallofthecubes.Whispertoyourpartnerthefractionofcubesthatareblue.

S: (Whisper!".)


T: Again,thewholeisallofthecubes.Whispertoyourpartnerthefractionofcubesthatareblue.

S: (Whisper!".)

T: Discusswithyourpartnerwhetherthefractionofcubesthatareblueinthesemodelsisequal,eventhoughthemodelsarenotthesameshape.

S: Theydon’tlookthesame,sotheyaredifferent.

àIdisagree.Theyareequalbecausetheyareboth!"blue.

àTheyareequalbecausetheunitsarestillthesamesize,andthewholeshavethesamenumberofunits.Theyareinadifferentshape.

T: Ihearyounoticingthattheunitsmakeadifferentshapeinthesecondmodel.It’ssquareratherthanrectangular.Goodobservation.Takeanotherminutetonoticewhatissimilaraboutourmodels.

S: Theybothusethesamelinkingcubesasunits.àTheybothhavethesameamountofbluesandreds.àBothwholeshavethesamenumberofunits,andtheunitsarethesamesize.

T: Thesizeoftheunitsandthesizeofthewholedidn’tchange.

Thatmeans!"and

!"areequal,orwhatwecallequivalent

fractions,eventhoughtheshapesofourwholesaredifferent.

Ifnecessary,dootherexamplestodemonstratethepointmadewithModel2.

NOTESONVOCABULARY:

TheconceptofequivalentfractionswasfirstintroducedinLesson16inreferencetofractionsthatareatthesamepointonthenumberline.Inthislesson,thestudents’understandingofequivalentfractionsexpandstoincludepictorialmodels,wheretheequivalentfractionsnamethesamesize.Guidestudentstorecognizethedifferencesandsimilaritiesbetweenthesemethodsforfindingequivalentfractions.

Lessons

23

ThirdsTemplate

Model3

SampleStudentWork


T: Whyisn’tthefractionrepresentedbythebluecubesequaltotheotherfractionswemadewithcubes?

S: Thisfractionshows#"ofthecubesareblue.

T: Whenwearefindingequivalentfractions,theshapesofthewholescanbedifferent.However,equivalentfractionsmustdescribepartsofthewholethatarethesamesize.

EquivalentShapesCollageActivity

Studentsusethethirdstemplate,andfollowthedirectionsbelowtocreatevariousrepresentationsof2thirds.

Directionsforthisactivityareasfollows:

1. Colorthewhite1thirdred.

2. Cutouttherectangle.Cutitinto2–4smallershapes.

3. Reassembleallofthepiecesintoanewshapewithnooverlaps.

4. Gluethenewshapeontoablankpaper.

Invitestudentstolookattheirclassmates’workanddiscusstheequivalencerepresentedbytheseshapes.Eachofthe6shapespicturedtotherightisanexampleofpossiblestudentwork.

Theseshapesareequivalentbecausetheyallshow#$grey,

althoughclearlyindifferentshapes.

Lessons

24

LESSON23


Materials: (S)Indexcard(1perpair,describedbelow),sentencestrip(1perpair),chartpaper(1pergroup),markers,glue,mathjournal

Studentsworkinpairs.Eachpairreceivesonesentencestripandanindexcard.Theindexcarddesignatesendpointsonanumberlineandaunitwithwhichtopartition(examplesontheright).

Dividetheclasssoeachgroupiscomposedofpairs(eachgroupcontainsmorethanonepair).Createthefollowingindexcards,anddistributeonecardtoeachpairpergroup.

GroupA:Interval3–5,thirdsandsixths

GroupB:Interval1–3,sixthsandtwelfths

GroupC:Interval3–5,halvesandfourths

GroupD:Interval1–3,fourthsandeighths

GroupE:Interval4–6,sixthsandtwelfths

GroupF:Interval6–8,halvesandfourths

Note:Differentiatetheactivitybystrategicallyassigningjustrightintervalsandunitstopairsofstudents.

T: Withyourpartner,useyoursentencestriptomakeanumberlinewithyourgiveninterval.Then,estimatetopartitionintoyourgivenunitbyfoldingyoursentencestrip.Labeltheendpointsandfractions.Renamethewholes.

S: (Workinpairs.)

T: (Giveonepieceofchartpapertoamemberofeachlettergroup.)Now,standupandfindyourotherlettergroupmembers.Onceyou’vefoundthem,glueyournumberlinesinacolumnsothattheendsmatchuponyourchartpaper.Comparenumberlinestofindequivalentfractions.Recordallpossibleequivalentfractionsinyourmathjournals.

S: (Findlettergroupmembers,andgluefractionstripsontochartpaper.Lettergroupmembersdiscussandrecordequivalentfractions.)

T: (Hangeachchartpaperaroundtheroom.)Now,we’regoingtodoamuseumwalk.Asalettergroup,youwillvisittheothergroups’chartpapers.Onepersonineachgroupwillbetherecorder.Youcanswitchrecorderseachtimeyouvisitanewchartpaper.Yourjobwillbetofindandlistalloftheequivalentfractionsyouseeateachchartpaper.

S: (Gotoanotherlettergroup’schartpaperandbegin.)

T: (Rotategroupsbrisklysothat,atthebeginning,studentsdon’tfinishfindingallfractionsat1station.Aslettergroupsrotateandchartpapersfillup,challengegroupstocheckothers’worktoensurenofractionsaremissing.)

T: (Afterrotationiscomplete.)Gobacktoyourownchartpaperwithyourlettergroup.Takeyourmathjournals,andcheckyourfriends’work.Didtheynamethesameequivalentfractionsyoufound?

GroupA

Interval:3–5Unit:thirds

ExampleIndexCardsforGroupA

GroupA

Interval:3–5Unit:sixths


ChallengestudentsworkingabovegradeleveltowritemorethantwoequivalentfractionsontheProblemSet.Astheybegintogenerateequivalenciesmentallyandrapidly,guidestudentstoarticulatethepatternanditsrule.

Lessons

25

halves fourths

thirds sixths

1whole

1whole

12

12

12

12

14

Image1

LESSON24


Materials: (S)Fractionpieces(Template),scissors,envelope,personalwhiteboard,sentencestrip,crayons

Eachstudentstartswiththefractionpieces,anenvelope,andscissors.

T: Cutoutalloftherectanglesonthefractionpieces,andinitialeachrectanglesoyouknowwhichonesareyours.

S: (Cutandinitial.)

T: Placetherectanglethatsays1wholeonyourpersonalwhiteboard.Takeanotherrectangle.Howmanyhalvesmake1whole?Showbyfoldingandlabelingeachunitfraction.

S: (Foldthesecondrectangleinhalf,andlabel!#oneachofthe2parts.)

T: Now,cutonthefold.Drawcirclesaroundyourwholeandyourpartstomakeanumberbond.

S: (Drawanumberbondusingtheshapestorepresentwholesandparts.)

T: Inyourwhole,writeanequalitythatshowshowmanyhalvesareequalto1whole.Remember,theequalsignislikeabalance.Bothsideshavethesamevalue.

S: (Write1whole=##inthe1wholerectangle.)

T: Putyourhalvesinsideyourenvelope.

Followthesamesequenceforeachrectanglesothatstudentscutallpiecesindicated.Havestudentsupdatetheequalityontheir1wholerectangleeachtimetheycutanewpiece.Attheend,itshould

read:1whole=## =

$$ =

"" =

%%.Discusstheequalitywithstudentstoensurethattheyunderstandthe

meaningoftheequalsignandtheroleitplaysinthisnumbersentence.

ProjectorshowImage1,showntotheright.

T: Useyourpiecestomakethisnumberbondonyourboard.

S: (Makethenumberbond.)

T: Discusswithyourpartner:Isthisnumberbondtrue?Whyorwhynot?

S: No,becausethewholehasonly2pieces,butthereare4parts!àButfourthsarejusthalvescutin2.So,they’rethesamepieces,butsmallernow.à#"isequivalentto

!#.àSo,

## =

"",justlikewhatwewrotedownonour1wholerectangle.

T: Ihearsomeofyousayingthat##and

""bothequal

1whole.So,canwesaythatthisistrue?(ProjectorshowImage2,shownonthenextpage.)

MP.714

14

14

Lessons

26


Studentsworkingbelowgradelevelmayappreciatetangiblyprovingthat2halvesisthesameas4fourths.Encouragestudentstoplacethe(paper)fourthsontopofthehalvestoshowequivalency.

12

12

13 1

313

Image2


ForEnglishlanguagelearners,

demonstratethatwordscanhavemultiplemeanings.Here,cutmeanstodrawaline(orlines)thatdividestheunitintosmallerequalparts.

Studentsworkingbelowgradelevelmaybenefitfromrevisitingthediscussionofdoubling,tripling,halving,andcuttingunitfractionsaspresentedinLesson22.

S: No,becausethirdsaren’thalvescutin2.Theylookcompletelydifferent.àBut,whenweputourthirdstogetherandhalvestogether,theymakethesamewhole.àBefore,wefoundwith

ourpiecesthat1whole= ## =

$$ =

"".àThen,itmustbetrue!

Followthesamesequencewithavarietyofwholesandpartsuntilstudentsarecomfortablewiththisrepresentationofequivalence.

T: Now,let’splaceourdifferentunitsonthesamenumberline.Useyoursentencestriptorepresenttheintervalfrom0to1onanumberline.Marktheendpointswithyourpencilnow.

S: (Markendpoints0and1belowthenumberline.)

T: Goaheadandfoldyoursentencestriptopartitiononeunitatatimeintohalves,fourths,thirds,andthensixths.Labeleachfractionabovethenumberline.Asyoucount,besuretorename0andthewhole.Useadifferentcolorcrayontomarkandlabelthefractionforeachunit.

S: (Foldthesentencestripandfirstlabelhalves,thenfourths,thenthirds,andthensixthsindifferentcolors.Rename0and1intermsofeachnewunit.)

T: Youshouldhaveacrowdednumberline!Compareittoyourpartner’s.

S: (Compare.)

T: Beforetoday,we’vebeennoticingalotofequivalentfractionsbetweenwholesonthenumberline.Today,noticethefractionsyouwroteat0and1.Lookfirstatthefractionsfor0.Whatpatterndoyounotice?

S: Theyallhave0copiesoftheunit!àThetotalnumberofequalpartschanges.Itshowsyouwhatunityou’regoingtocountby.àSinceournumberlinestartsat0,thereis0ofthatunitinallofthefractions.

T: Eventhoughtheunitisdifferentineachofourfractionsat0,aretheyequivalent?Thinkbacktoourworkwithshapesearlier.

S: Wesawbeforethatfractionswithdifferentunitscanstillmakethesamewhole.Thistime,thewholeisjust0.

Followthesequencetostudythefractionswrittenat1.Forboth0and1,studentsshouldseethateverycolortheyusedispresent.

LESSON27


Materials: (S)3wholes(Lesson25Template1),personalwhiteboard,fractionstrips(3perstudent),mathjournal

Passout3wholes,andhavestudentsslipitintotheirpersonalwhiteboards.

T: Eachrectanglerepresents1whole.Estimatetopartitioneachrectangleintothirds.

MP.7

Lessons

27

3wholes(Lesson25Template1)

S: (Partition.)

T: Howcanwedoublethenumberofunitsinthesecondrectangle?

S: Wecuteachthirdin2.

T: Goaheadandpartition.

S: (Partition.)

T: What’sournewunit?

S: Sixths!

Repeatthisprocessforthethirdrectangle.Insteadofhavingstudentsdouble,havethemtripletheoriginalthirds.

T: Labelthefractionsineachmodel.

S: (Label.)

T: Whatisdifferentaboutthesemodels?

S: Theyallstartedasthirds,butthenwecutthemintodifferentparts.àThepartsaredifferentsizes.àYes,they’redifferentunits.

T: Whatisthesameaboutthesemodels?

S: Thewhole.

T: Talktoyourpartnerabouttherelationshipbetweenthenumberofpartsandthesizeofpartsineachmodel.

S: 3isthesmallestnumber,butthirdshavethebiggestsize.àAsIdrewmorelinestopartition,thesizeofthepartsgotsmaller.àThat’sbecausethewholeiscutintomorepieceswhenthereareninthsthanwhentherearethirds.

T: (Giveeachstudent3fractionstrips.)Foldall3fractionstripsintohalves.

S: (Fold.)

T: Foldyoursecondandthirdfractionstripstodoublethenumberofunits.

S: (Fold.)

T: What’sthenewunitonthesefractionstrips?

S: Fourths!

T: Foldyourthirdfractionstriptodoublethenumberofunitsagain.

S: (Fold.)

T: What’sthenewunitonyourthirdfractionstrip?

S: Eighths!

T: Comparethenumberofpartsandthesizeofthepartswiththenumberoftimesyoufoldedthestrip.Whathappenstothesizeofthepartswhenyoufoldthestripmoretimes?

S: ThemoreIfolded,thesmallerthepartsgot.àYeah,that’sbecauseyoufoldedthewholetomakemoreunits.

T: Openyourmathjournaltoanewpage,andglueyourstripsinacolumn,makingsuretheendslineup.Gluethemfromthelargestunittothesmallest.

S: (Glue.)

T: Useyourfractionstripstofindthefractionsequivalentto",.Shadethem.

MP.3

Lessons

28

S: (Shade", ,

#",and

!#.)

T: Talkwithyourpartner:Whatdoyounoticeaboutthesizeofpartsandnumberofpartsinequivalentfractions?

S: Youcanseethattherearemoreeighthsthanhalvesorfourthsshadedtocoverthesameamountofthestrip.àIt’sthesameasbeforethen.Asthenumberofpartsgetslarger,thesizeofthemgetssmaller.àThat’sbecausetheshadedareainequivalentfractionsdoesn’tchange,eventhoughthenumberofpartsgetslarger.

Ifnecessary,reinforcetheconceptwithotherexamplesusingthesefractionstrips.

T: (ShowImage1.)Let’spracticethisideaabitmoreonourpersonalwhiteboards.Drawmyshapeonyourboard.Theentirefigurerepresents1whole.

S: (Draw.)

T: Writetheshadedfraction.

S: (Write!".)

T: Talktoyourpartner:Howcanyoupartitionthisshapetomakeanequivalentfractionwithsmallerunits?

S: Wecancuteachsmallrectanglein2piecesfromtoptobottomtomakeeighths.àOrwecanmake2horizontalcutstomaketwelfths.

T: Useoneofthesestrategiesnow.(Circulateasstudentsworktoselectafewdifferentexamplestosharewiththeclass.)

S: (Partition.)

T: Let’slookatourclassmates’work.(Showexamplesof#, ,

$!# ,

"!%,etc.)Aswepartitionedwith

moreparts,whathappenstotheshadedareaandnumberofpartsneededtomakethemequivalent?

S: Thesizeofthepartsgetssmaller,butthenumberofthemgetslarger.

T: Eventhoughthepartschanged,didtheareacoveredbytheshadedregionchange?

S: No.

Considerhavingstudentspracticeindependently.Theshapetotherightismorechallengingbecausetrianglesaremoredifficulttomakeintoequalparts.

Image1

Lessons

29

TopicF:Comparison,Order,andSizeofFractionsTopicFconcludesthemodulewithcomparingfractionsthathavethesamenumerator.Astheycomparefractionsbyreasoningabouttheirsize,studentsunderstandthatfractionswiththesamenumeratorandalargerdenominatorareactuallysmallerpiecesofthewhole.TopicFleavesstudentswithanewmethodforpreciselypartitioninganumberlineintounitfractionsofanysizewithoutusingaruler.

Lesson28

ApplicationProblem(8minutes)

LaTonyahas2equal-sizedhotdogs.Shecutthefirstoneintothirdsatlunch.Later,shecutthesecondhotdogtomakedoublethenumberofpieces.DrawamodelofLaTonya’shotdogs.

a. Howmanypiecesisthesecondhotdogcutinto?

b. Ifshewantstoeat#$ofthesecondhotdog,howmany

piecesshouldsheeat?

Note:ThisproblemreviewstheconceptofequivalentfractionsfromTopicE.Encouragestudentstofindotherequivalentfractionsbasedontheirmodels.ThisproblemisusedintheConceptDevelopmenttoprovideacontextinwhichstudentscancomparefractionswiththesamenumerators.


Materials: (S)WorkfromApplicationProblem,personalwhiteboard

T: LookagainatyourmodelsofLaTonya’shotdogs.Let’schangetheproblemslightly.WhatifLaTonyaeats2piecesofeachhotdog?Figureoutwhatfractionofeachhotdogsheeats.

S: (Work.)Sheeats#$ofthefirstoneand

#%ofthesecondone.

T: DidLaTonyaeatthesameamountofthefirsthotdogandsecondhotdog?

S: (Usemodelsforhelp.)No.

T: Butsheate2piecesofeachhotdog.Whyistheamountsheatedifferent?

S: Thenumberofpiecessheateisthesame,butthesizeofeachpieceisdifferent.àJustlikewesawyesterday,themoreyoucutupawhole,thesmallerthepiecesget.àSo,eating2piecesofthirdsismorehotdogthan2piecesofsixths.

T: (Projectordrawthecirclesontheright.)Drawmypizzasonyourpersonalwhiteboard.

S: (Drawshapes.)

T: Estimatetopartitionbothpizzasintofourths.

MP.2

Lessons

30


Asstudentsplayacomparisongame,

facilitatepeer-to-peertalkforEnglishlanguagelearnerswithsentenceframes,suchasthefollowing:

§ “Ipartitionedinto____(fractionalunit).Ishaded___(numberof)____(fractionalunit).”

§ “Idrew___(fractionalunit),too.Ishaded___(numberof)____(fractionalunit).____islessthan____.”

3wholes(Lesson25Template1)

S: (Partition.)

T: Partitionthesecondpizzatodoublethenumberofunits.

S: (Partition.)

T: Whatunitsdowehave?

S: Fourthsandeighths.

T: Shadein3fourthsand3eighths.

S: (Shade.)

T: Whichshadedportionwouldyourathereat?Thefourthsoreighths?Why?

S: I’drathereatthefourthsbecauseit’swaymorepizza.àI’drathereattheeighthsbecauseI’mnotthathungry,andit’sless.

T: Butbothchoicesare3pieces.Aren’ttheyequivalent?

S: No.Youcanseefourthsarelarger.àWeknowbecausethemoretimesyoucutthewhole,thesmallerthepiecesget.àSo,eighthsaretinycomparedtofourths!àThenumberofpiecesweshadedisthesame,butthesizesofthepiecesaredifferent,sotheshadedamountsarenotequivalent.

Ifnecessary,continuewithotherexamplesvaryingthepictorialmodels.

T: Let’sworkinpairstoplayacomparisongame.PartnerA,drawawholeandshadeafractionofthewhole.Labeltheshadedpart.

S: (PartnerAdrawsandlabels.)

T: PartnerB,drawafractionthatislessthanPartnerA’sfraction.Usethesamewholeandsamenumberofshadedparts,butchooseadifferentfractionalunit.Labeltheshadedparts.

S: (PartnerBdrawsandlabels.)

T: PartnerA,checkyourfriend’sworktobesurethefractionislessthanyours.

S: (PartnerAchecksandhelpsmakeanycorrectionsnecessary.)

T: PartnerB,drawawhole,andshadeafraction.IwillsaylessthanorgreaterthanforPartnerAtodrawanotherfraction.

Playseveralrounds.

Lesson29


Materials: (S)Personalwhiteboard,3wholes(Lesson25Template1)

Seatstudentsinpairsfacingeachotherinalargecirclearoundtheroom.3wholesshouldbeintheirpersonalwhiteboards.

T: Today,we’llonlyusethefirstrectangle.Atmysignal,drawand

shadeafractionlessthan!#,andlabelitbelowtherectangle.

(Signal.)

MP.2


ExtendPage1oftheProblemSetforstudentsworkingabovegradelevelsotheycanusetheirknowledgeofequivalencies.Say,“If2thirdsisgreaterthan2fifths,useequivalentfractionstonamethesamecomparison.Forexample,4sixthsisgreaterthan2fifths.”


Givestudentsworkingbelowgradeleveltheoptionofrectangularpizzas(ratherthancircles)toeasethetaskofpartitioning.

Lessons

31

S: (Drawandlabel.)

T: Checkyourpartner’sworktomakesureit’slessthan!#.

S: (Check.)

T: Thisishowwe’regoingtoplayagametoday.Forthenextround,we’llseewhichpartnerisquickerbutstillaccurate.Assoonasyoufinishdrawing,raiseyourpersonalwhiteboard.Ifyouarequicker,thenyouarethewinneroftheround.Ifyouarethewinneroftheround,youwillstandup,andyourpartnerwillstayseated.Ifyouarestanding,youwillthenmovetopartnerwiththepersononyourright,whoisstillseated.Ready?Eraseyourboards.Atmy

signal,drawandlabelafractionthatisgreaterthan!#.(Signal.)

S: (Drawandlabel.)

Thestudentwhogoesaroundtheentirecircleandarrivesbackathisoriginalplacefasterthantheotherstudentswinsthegame.Thewinnercanalsobethestudentwhohasmovedthefurthestifittakestoolongtoplayallthewayaround.Movethegameatabriskpace.Useavarietyoffractions,andmixitupbetweengreaterthanandlessthansothatstudentsconstantlyneedtoupdatetheirdrawingsandfeelchallenged.Ifpreferred,mixitupbycallingoutequalto.

T: (Draworshowtheimagesontheright.)Drawmyshapesonyourboard.Makesuretheymatchinsizelikemine.

S: (Draw.)

T: Partitionbothshapesintosixths.

S: (Partition.)

T: Partitionthesecondshapetoshowdoublethenumberofunitsinthesamewhole.

S: (Partition.)

T: Whatfractionalunitsdowehave?

S: Sixthsandtwelfths.

T: Shadein4unitsofeachshape,andlabeltheshadedfractionbeloweachshape.

S: (Shadeandlabel.)

T: Whisperingtoyourpartner,sayasentencecomparingthefractionsusingthewordsgreaterthan,lessthan,orequalto.

S:"%isgreaterthan

"!#.

T: Now,writethecomparisonasanumbersentencewiththecorrectsymbolbetweenthefractions.

S: (Write"%>

"!#.)

T: (Draworshowtheimagesontheright.)Drawmyrectanglesonyourboard.Makesuretheymatchinsizelikemine.

S: (Draw.)

T: Partitionthefirstrectangleintoseventhsandthesecondoneintofifths.

S: (Partition.)

T: Shadein3unitsofeachrectangle,andlabeltheshadedfractionbeloweachrectangle.

S: (Shadeandlabel.)

Lessons

32


S:$+islessthan

$*.


S: (Write$+<

$*.)

Dootherexamples,ifnecessary,usingavarietyofshapesandunits.

T: Draw2numberlinesonyourboard,andlabeltheendpoints0and1.

S: (Drawandlabel.)

T: Partitionthefirstnumberlineintoeighthsandthesecondoneintotenths.

S: (Partition.)

T: Onthefirstnumberline,label,,.

S: (Label.)

T: Onthesecondnumberline,label2copiesof*!&.

S: (Label.)


S: Wait,they’rethesame!,,isequalto

!&!&.

T: Howdoyouknow?

S: Becausetheyhavethesamepointonthenumberline.Thatmeansthey’reequivalent.


S: (Write,,=

!&!&.)

Dootherexampleswiththenumberline.Insubsequentexamplesthatusesmallerunitsorunitsthatarefartherapart,movetousingasinglenumberline.

Lesson30–OmittedfromIndependentZearnTime


Materials: (S)9-inch×1-inchstripsofredconstructionpaper(atleast5perstudent),linedpaper(Template)orwide-rulednotebookpaper(severalpiecesperstudent),12-inchruler

Note:PleasereadthedirectionsfortheExitTicketbeforebeginning.

T: Thinkbackonourlessons.Talktoyourpartnerabouthowtopartitionanumberlineintothirds.

S: Drawtheline,andthenestimate3equalparts.àUseyourfoldedfractionstriptomeasure.

NOTESONMATERIALS:

Itishighlyrecommendedtotrytheactivitywiththepreparedmaterialsbeforepresentingittostudents.Evensmallvariationsinthewidthofspacesonwide-rulednotebookpaperorinthe9-inch×1-inchpaperstripsmayresultinadjustingthedirectionsslightlytoobtainthedesiredresult.

Lessons

33

àMeasurea3-inchlinewitharuler,andthenmarkoffeachinch.àOrona6-inchline,1markwouldbeateach2inches.àDon’tforgettomark0.àYes,youalwayshavetostartmeasuringfrom0.

T: Let’sexploreamethodtomarkoffanyfractionalunitpreciselywithouttheuseofaruler,justwithlinedpaper.

Step1:Drawanumberlineandmarkthe0endpoint.

T: (Givestudentsthelinedpaperornotebookpaper.)Turnyourpapersothemarginishorizontal.Drawanumberlineontopofthemargin.

T: Mark0onthepointwhereIdid.(Demonstrate.)Talktoyourpartner:Howcanweequallyandpreciselypartitionthisnumberlineintothirds?

S: Wecanusetheverticallines.àEachlinecanbeanequalpart.àWecancount2linesforeachthird.àOr3spacesor4tomakeanequalpart,justsolongaseachparthasthesamenumber.àOh,Isee;thisistheanswer.àButtheteachersaidanypieceofpaper.Ifwemakethirdsonthispaper,itwon’thelpusmakethirdsoneverypaper.

Step2:Measureequalunitsusingthepaper’slines.

T: Usethepaper’sverticallinestomeasure.Let’smakeeachpart5spaceslong.Labelthenumberlinefrom0to1using5spacesforeachthird.Discussinpairshowyouknowtheseareprecisethirds.

Step3:Extendtheequalpartstothetopofthenotebookpaperwithaline.

T: Drawverticallinesupfromyournumberlinetothetopofthepaperateachthird.(Holdup1redstripofpaper.)Talktoyourpartnerabouthowwemightusetheselinestopartitionthisredstripintothirds.

S: (Discuss.)

T: (Passout1redstriptoeachstudent.)Thechallengeistopartitiontheredstrippreciselyintothirds.Lettheleftendofthestripbe0.Therightendofthestripis1.

S: Thestripistoolong.àWecan’tcutit?àNo.Theteachersaidno.Howcanwedothis?(Circulateandlisten,butdon’tgiveananswer.)

Step4:Angletheredstripsothattheleftendtouchesthe0endpointontheoriginalnumberline.Therightendtouchesthelineat1.

Step5:Markoffequalunits,whichareindicatedbytheverticalextensionsofthepointsontheoriginalnumberline.

T: Doyourunitslookequal?

S: I’mnotsure.àTheylookequal.àIthinkthey’reequalbecauseweusedthespacesonthepapertomakeequalunitsofthirds.

T: Verifythattheyareequalwithyourruler.Measurethefulllengthoftheredstripininches.Measuretheequalparts.

S: (Measure.)

MP.6

Lessons

34

T: Imadethisstrip9incheslongjustsoyoucouldverifythatourmethodpartitionsprecisely.

Havestudentsthinkaboutwhythismethodworks.Havethemreviewtheprocessstepbystep.

Lesson 4: Represent and identify fractional parts of different wholes.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Lesson 4 Problem Set NYS COMMON CORE MATHEMATICS CURRICULUM

Name Date

1. Draw a picture of the yellow strip at 3 (or 4) different stations. Shade and label 1 fractional unit of each.

2. Draw a picture of the brown bar at 3 (or 4) different stations. Shade and label 1 fractional unit of each.

3. Draw a picture of the square at 3 (or 4) different stations. Shade and label 1 fractional unit of each.

© 2015 Great Minds. eureka-math.orgG3-M5-TE-1.3.0-06.2015

Lesson 4: Represent and identify fractional parts of different wholes.


Lesson 4 Problem Set NYS COMMON CORE MATHEMATICS CURRICULUM

4. Draw a picture of the clay at 3 (or 4) different stations. Shade and label 1 fractional unit of each.

5. Draw a picture of the water at 3 (or 4) different stations. Shade and label 1 fractional unit of each.

6. Extension: Draw a picture of the yarn at 3 (or 4) different stations.


Lesson 20 Template NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 20: Recognize and show that equivalent fractions have the same size, though not necessarily the same shape.


thirds


http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US




Lesson 24: Express whole numbers as fractions and recognize equivalence with different units.


Lesson 24 Template NYS COMMON CORE MATHEMATICS CURRICULUM 3 5

halv

es

third

s

four

ths

1

who

le

sixth

s

fraction pieces






Lesson 25 Template 1 NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 25: Express whole number fractions on the number line when the unit interval is 1.


3 wholes






Lesson 25 Template 2 NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 25: Express whole number fractions on the number line when the unit interval is 1.


6 wholes

Mod

el 1

M

odel

3

Mod

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Lesson 30 Template NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 30: Partition various wholes precisely into equal parts using a number line method.


lined paper






grade 3, module 5: fractions as number on the number line lessons 2016 v2.pdf · grade 3, module 5:...

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