grade 3, module 5: fractions as number on the number line lessons 2016 v2.pdf · grade 3, module 5:...
TRANSCRIPT
©2016Zearn,Inc.Portionsofthiswork,ZearnMath,arederivativeofEurekaandlicensedbyGreatMinds.
©2016GreatMinds.Allrightsreserved.
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
NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:
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
ConceptDevelopment(35minutes)
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
ConceptDevelopment(35minutes)
Materials: (S)ProblemSet,seeadditionalitemsforstationslistedbelow
Exploration:Studentsworkatstationstorepresentagivenfractionalunitusingavarietyofmaterials.Designatethefollowingstationsforgroupsof3students(morethan3notsuggested).
StationA:Halves StationE:SixthsStationB:Fourths StationF:NinthsStationC:Eighths StationG:FifthsStationD:Thirds StationH:Tenths
Lessons
5
NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:
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;!#
NOTESONMULTIPLEMEANSOFREPRESENTATION:
Whileintroducingthenewterms—unitform,fractionform,andunitfraction—checkforstudentunderstanding.Englishlanguagelearnersmaychoosetodiscussdefinitionsofthesetermsintheirfirstlanguagewiththeteacherortheirpeers.
1third;!$
TopicB:UnitFractionsandtheirRelationtotheWholeInTopicB,studentscompareunitfractionsandlearntobuildnon-unitfractionswithunitfractionsasbasicbuildingblocks.Thisparallelstheunderstandingthatthenumber1isthebasicbuildingblockofwholenumbers.
LESSON5
ConceptDevelopment(25minutes)
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
NOTESONMULTIPLEMEANSOFENGAGEMENT:
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.)
NOTESONMULTIPLEMEANSOFENGAGEMENT:
Reviewpersonalgoalswithstudents.
Forexample,ifstudentsworkingbelowgradelevelchosetosolveonewordproblem(perlesson)lastweek,encouragethemtoworktowardcompletingtwowordproblemsbytheendofthisweek.
MP.6
Lessons
8
LESSON7
ConceptDevelopment(28minutes)
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
ConceptDevelopment(28minutes)
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
NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:
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.
NOTESONMULTIPLEMEANSOFENGAGEMENT:
Forstudentsworkingbelowgradelevel,respectfullyfacilitateself-assessmentofpersonalgoals.Guidestudentstoreflectuponquestionssuchas,“WhichfractionskillsamIgoodat?WhatwouldIliketobebetterat?Whatismyplantoimprove?”Celebrateimprovement.
Lessons
10
TopicC:ComparingUnitFractionsandSpecifyingtheWholeInTopicC,studentspracticecomparingunitfractionswithfractionstrips,specifyingthewholeandlabelingfractionsinrelationtothenumberofequalpartsinthatwhole.
LESSON10
ConceptDevelopment(32minutes)
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
ConceptDevelopment(32minutes)
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.
NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:
ThispartneractivitybenefitsEnglish
languagelearnersasitincludesrepeateduseofmathlanguageinareliablestructure(e.g.,“__isgreaterthan__”).ItalsoofferstheEnglishlanguagelearneranopportunitytodiscussthemathwithapeer,whichmaybemorecomfortablethanspeakinginfrontoftheclassortotheteacher.
MP.6
Lessons
12
NOTESONMULTIPLEMEANSOFENGAGEMENT:
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
NOTESONMULTIPLEMEANSOFENGAGEMENT:
Organizestudentsworkingbelowgrade
levelatthestationswiththeeasierfractionalunitsandstudentsworkingabovegradelevelatthestationswiththemostchallengingfractionalunits.
LESSON12
ConceptDevelopment(32minutes)
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.)
NOTESONMULTIPLEMEANSOFREPRESENTATION:
GiveEnglishlanguagelearnersalittle
moretimetorespond,eitherinwritingorintheirfirstlanguage.
Lessons
14
NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:
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
ConceptDevelopment(33minutes)
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
&$,!$,#$,$$?àArefractionsnumbers?
T: Someofyoureadthewholenumbers,andothersreadwholenumbersandfractions.Fractionsarenumbers.Let’sreadthenumbersfromleasttogreatest,andlet’ssay0thirdsand3thirdsfornowratherthanzeroandone.
MP.7
Lessons
16
NOTESONMULTIPLEMEANSOFENGAGEMENT:
Thislessongraduallyleadsstudents
fromtheconcretelevel(fractionstrips)tothepictoriallevel(numberlines).
S: (Readnumbers,&$ ,
!$,#$,$$.)
T: Let’sreadagainandthistimesayzeroand1ratherthan0thirdsand3thirds.
S: (Readnumbers,0, !$,#$,1.)
Part3:Drawnumberbondstocorrespondwiththenumberlines.
Oncestudentshavebecomeexcellentatmakingandlabelingfractionsonnumberlinesusingstripstomeasure,havethemdrawnumberbondstocorrespond.Usequestioningwhilecirculatingtohelpthemseesimilaritiesanddifferencesbetweenthebonds,fractionstrips,andfractionsonthenumberline.Guidestudentstorecognizethatplacingfractionsonthenumberlineisanalogoustoplacingwholenumbersonthenumberline.Ifpreferred,thefollowingsuggestionscanbeused:
§ Whatdoboththenumberbondandnumberlineshow?
§ Whichmodelbestshowshowbigtheunitfractionisinrelationtothewhole?Explainhow.
§ Howdoyournumberlineshelpyoumakenumberbonds?
LESSON15
ConceptDevelopment(33minutes)
Materials: (S)Personalwhiteboard
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
ConceptDevelopment(31minutes)
Materials: (S)Personalwhiteboard
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.
NOTESONMULTIPLEMEANSOFENGAGEMENT:
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
ConceptDevelopment(32minutes)
Materials: (S)Personalwhiteboard
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?
NOTESONMULTIPLEMEANSOFENGAGEMENT:
Studentsworkingabovegradelevel
maysolvequicklyusingmentalmath.Pushstudentstonoticeandarticulatepatternsandrelationships.Astheyworkinpairstopartitionnumberlines,havestudentsmakeandanalyzetheirpredictions.
NOTESONMULTIPLEMEANSOFENGAGEMENT:
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
ConceptDevelopment(28minutes)
Materials: (S)Personalwhiteboard
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
NOTESONMULTIPLEMEANSOFENGAGEMENT:
Askstudentsworkingabovegradelevel
thismoreopen-endedquestion:“Howmanyhalvesareonthenumberline?”
NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:
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
ConceptDevelopment(33minutes)
Materials: (T)Linkingcubesin2colors(S)Thirds(Template),redcrayon,scissors,gluestick,andblankpaper
UselinkingcubestocreateModel1,asshowntotheright.
T: Thewholeisallofthecubes.Whispertoyourpartnerthefractionofcubesthatareblue.
S: (Whisper!".)
UselinkingcubestocreateModel2,asshowntotheright.
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
UselinkingcubestocreateModel3,asshowntotheright.
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
ConceptDevelopment(32minutes)
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
NOTESONMULTIPLEMEANSOFENGAGEMENT:
ChallengestudentsworkingabovegradeleveltowritemorethantwoequivalentfractionsontheProblemSet.Astheybegintogenerateequivalenciesmentallyandrapidly,guidestudentstoarticulatethepatternanditsrule.
Lessons
25
halves fourths
thirds sixths
1whole
1whole
12
12
12
12
14
Image1
LESSON24
ConceptDevelopment(33minutes)
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
NOTESONMULTIPLEMEANSOFENGAGEMENT:
Studentsworkingbelowgradelevelmayappreciatetangiblyprovingthat2halvesisthesameas4fourths.Encouragestudentstoplacethe(paper)fourthsontopofthehalvestoshowequivalency.
12
12
13 1
313
Image2
NOTESONMULTIPLEMEANSOFENGAGEMENT:
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
ConceptDevelopment(33minutes)
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.
ConceptDevelopment(30minutes)
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
NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:
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
ConceptDevelopment(30minutes)
Materials: (S)Personalwhiteboard,3wholes(Lesson25Template1)
Seatstudentsinpairsfacingeachotherinalargecirclearoundtheroom.3wholesshouldbeintheirpersonalwhiteboards.
T: Today,we’llonlyusethefirstrectangle.Atmysignal,drawand
shadeafractionlessthan!#,andlabelitbelowtherectangle.
(Signal.)
MP.2
NOTESONMULTIPLEMEANSOFENGAGEMENT:
ExtendPage1oftheProblemSetforstudentsworkingabovegradelevelsotheycanusetheirknowledgeofequivalencies.Say,“If2thirdsisgreaterthan2fifths,useequivalentfractionstonamethesamecomparison.Forexample,4sixthsisgreaterthan2fifths.”
NOTESONMULTIPLEMEANSOFENGAGEMENT:
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
T: Whisperingtoyourpartner,sayasentencecomparingthefractionsusingthewordsgreaterthan,lessthan,orequalto.
S:$+islessthan
$*.
T: Now,writethecomparisonasanumbersentencewiththecorrectsymbolbetweenthefractions.
S: (Write$+<
$*.)
Dootherexamples,ifnecessary,usingavarietyofshapesandunits.
T: Draw2numberlinesonyourboard,andlabeltheendpoints0and1.
S: (Drawandlabel.)
T: Partitionthefirstnumberlineintoeighthsandthesecondoneintotenths.
S: (Partition.)
T: Onthefirstnumberline,label,,.
S: (Label.)
T: Onthesecondnumberline,label2copiesof*!&.
S: (Label.)
T: Whisperingtoyourpartner,sayasentencecomparingthefractionsusingthewordsgreaterthan,lessthan,orequalto.
S: Wait,they’rethesame!,,isequalto
!&!&.
T: Howdoyouknow?
S: Becausetheyhavethesamepointonthenumberline.Thatmeansthey’reequivalent.
T: Now,writethecomparisonasanumbersentencewiththecorrectsymbolbetweenthefractions.
S: (Write,,=
!&!&.)
Dootherexampleswiththenumberline.Insubsequentexamplesthatusesmallerunitsorunitsthatarefartherapart,movetousingasinglenumberline.
Lesson30–OmittedfromIndependentZearnTime
ConceptDevelopment(30minutes)
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.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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.
© 2015 Great Minds. eureka-math.orgG3-M5-TE-1.3.0-06.2015
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.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
thirds
© 2015 Great Minds. eureka-math.orgG3-M5-TE-1.3.0-06.2015
Lesson 24: Express whole numbers as fractions and recognize equivalence with different units.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 24 Template NYS COMMON CORE MATHEMATICS CURRICULUM 3 5
halv
es
third
s
four
ths
1
who
le
sixth
s
fraction pieces
© 2015 Great Minds. eureka-math.orgG3-M5-TE-1.3.0-06.2015
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.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
3 wholes
© 2015 Great Minds. eureka-math.orgG3-M5-TE-1.3.0-06.2015
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.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6 wholes
Mod
el 1
M
odel
3
Mod
el 2
© 2015 Great Minds. eureka-math.orgG3-M5-TE-1.3.0-06.2015
Lesson 30 Template NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 30: Partition various wholes precisely into equal parts using a number line method.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
lined paper
© 2015 Great Minds. eureka-math.orgG3-M5-TE-1.3.0-06.2015