Download Uncomplicating Fractions

Uncomplicating
Fractions
MarianSmall
Lexington,KY
July,2016
Let’sstartbydoing!
• Useyourpatternblockstocreateadesignthatis½yellow.
• Let’sseewhatwenotice.
Halfinwhatway?
• Halfinvolume?
• Halfinareaofthe
top?
Halfinwhatway?
• Halfinnumberof
blocks?
Whatwouldhappenif…
• Youduplicatedyourdesign?
Arethereothernames?
• Whatotherfractionnamecouldsomeonelegitimatelyuseforyour
yellowpart?
Howabout…
• Adesignthatis2/3redand1/3green?
Somequestions
• Whataboutyourdesignwas2/3red?
• Whatotherfractionmightdescribetheredpartofyourdesign?
Somequestions
• Didyouhaveanycolorsbesidesredandgreen- whyorwhynot?
• Whatifyouduplicatedyourdesign?
Nowmakeadesign…
• 1/5blueand1/2green
Nowmakeadesign…
• 1/5blueand1/2green
Agendatoday
• SettingtheStage
• Meaningoffractions
• Equivalence
• Comparison
• Addingandsubtracting
• Multiplyinganddividing
• Closinggaps
SettingtheStage
• Istherevalueinintroducinghalves,quarters,thirds,etc.usingwords
liketwothirdsinsteadof2/3?
SettingtheStage
• Istherevalueinpartitioningshapesotherthanrectanglesandcircles
intheearlygrades?
SettingtheStage
• Istherevalueinworkingongeoboardsorwithsquaretilessothat
partitionscanbebothpartitionsofwholesandpartitionsofsets
simultaneously?
Forexample
• Use16tilesandmakeasquare.
• Howcouldyoudivideitupsothatitishalfred?
SettingtheStage
• IstherevalueinclarifyingthatafractionisALWAYSrelativetoa
whole?
• Youcanshow2,butyoucan’tshowonehalfunlessyouknowwhat
thewholeisORyouassumethewholeisthenumber1.
Whatfractionsdoyousee?
SettingtheStage
• Whatdowemeanwhenwesaythepartshavetobeequal?
• Itcouldbearea,orvolume,ornumber,ormass,etc.,andnot
necessarilyallofthem.
SettingtheStage
• StudentsneedtoseeNON-examplesaswellasexamples.
• Createordrawanon-exampleofonefourth.
Whatmight3/4looklike?
• Drawsomepicturesorusematerialstoshow3/4asmanywaysas
youcan.
Partofalength
Partofaregion
Partofavolume
Partofacapacity
Partofadurationoftime
Partofaweight
Anymarbleontherighthas
¾oftheweightofamarble
ontheleft.
Whyisthat?
Partofaset
Anumber
• Itis3jumpsof¼from0.
01/43/41
Adivision
• If4peopleshare3things,
eachgets3/4ofoneofthose
things.
Adivision
• Ifthedistancefrom0to3issplit
into4equalparts,thelengthof
eachpartis3/4.
0123
3/4
Acomparison
• 3is3/4of4.
• 4is11/3(4/3)of3.
Acomparison
• Justlikefor3/6
• 3is3/6of6.
• 6is6/3of3.
Ascalefactor
• 3/4canbeusedasawaytodescribehowanobject’slengthchanges,
e.g.
becomes
Relatingmeanings
• Youalreadysawhowsomearerelated.
Whyis2/3of12equalto8?
Whatabout2/3of13?
Whatdoyoumeanwhen…
• Yousaythepartshavetobeequal?
• Is1/3red?
Whatdoyoumeanwhen…
• Yousaythepartshavetobeequal?
• Is1/3gray?
Whyistherealwaysmorethanonefraction?
• Whenafractionispartofawhole,whycan’tyousee2/3without
seeingatleastoneotherfraction?
Forexample
• Isthis4/3or4/6?
Issueofimproperfractions
• Easiestwithnumbermeaningordivision
• Let’slookatpartofawhole
Issueofimproperfractions
• Withpartofwhole,reallyneedtofocusonwhatthewholeis
Evenwithpartsofwholes….
vs.
Evenwithpartsofwholes
• Usingrectanglesvs.circles
• Why1/8sareeasierthan1/3sand1/5s
Importanceofestimating
• Studentsshouldestimatefractionamounts,e.g.
• Aboutwhatfractionoftherectangleisred?
Importanceofestimating
• Aboutwhatfractionofthekidsintheschoolareinthirdgrade?
Shouldstudentscreatetheirownmodels
• Yes,but…..nottocompareclosefractionsortoaddorsubtract.
Evenwithpartsofwholes
• Givingapartandaskingforawhole.
• Ifthistriangleis1/3,whatdoesthewholelooklike?
• Whatifit’s2/3?
Ifit’s1/3
Ifit’s2/3
Relationshipsbetweenfractions
• Whatfractionrepresentseachpartoftheareaofthiswhole?
Whatmakesfractionsequal?
• Usethefractiontowertofindequivalentsto
• 1/2
• 2/3
• 4/5
• Whatdoyounotice?
Usingpatternblocksforequivalence
• Whatequivalentfractionscanyoushowusingpatternblocks?
Whycanwemultiplynumeratorand
denominatorbythesameamount?
• Notbecausewearemultiplyingby1(whichislearnedlater)
• Becauseweareturningeachsectioninton timesasmanysections.
• Orbecausewearemergingn sectionsintoone.
Forexample
• 3/4=6/8
Buttheimportantissue
• Thereason3/5=6/10isNOTbecauseyoumultipliedbothnumerator
anddenominatorby2.
• Itisbecause6is3/5of10.
Isthislegitimate?
• Is1½/3=½?
Introducingequivalenceusingpartofset
• Trickyforkidstobelievethat2/3=4/6basedonthisvisual.
Somewhatbetter
Equivalenceonnumberline
• Howcouldyouuseanumberlinetoshowwhy4/6=6/9?
0
1
Equivalencebasedondivision
• Whyis2÷ 3thesameas4÷ 6?
• (If3peopleshare2,then6peoplewouldbesharing4.)
Mixednumber/improperfractionequivalence
• Noticethatweoftenrelyonthedivisionmeaningoffractionsto
explainwhy,e.g.8/3=22/3.
• Howcouldweuseanumberline?
• Apartofwholemodel?
Comparingfractions
• Wecannotcomparefractionsthatarepartsofwholesunlesstheyare
fractionsofthesameorequalsizedwholes.
• Forexample,2/3of3islessthan1/3of9.
Usingmodels
• Fractiontowerisexcellentforthis.
Usingmodels
• Tocompare,e.g.2/3and4/5,studentscoulduseobjectswherethirds
andfifthsareeasytosee,e.g.a3x 5gridORtherightCuisenaire
rods,e.g.
3x 5grid
2/3vs 4/5
Wewantstudentstorealizethat
• itiseasytocompare2/8and3/8because….
• itiseasytocompare2/8and2/9because…
• itiseasytocompare1/10and5/6because…
Youmightaskstudents-• Writedowntwofractionsthatmightlookcomplicated,butarereally
easytocompare.
So…
• Howwouldyoudecidewhichofeachpairisgreater?
• 2/3vs.4/7
• 3/8vs.12/30
• 3/8vs 8/15
Somestudentsthink
• Thatifafractionhasagreaternumeratorandagreaterdenominator,
itisagreaterfraction.
• Forexample,5/6>4/5since5ismorethan4and6ismorethan5.
• Doyouagree?Disagree?
Forexample
• 5/7>3/5but
• 6/5<5/4and
• 9/10<12/11
Mixingupimproper,mixedandproper
• Askstudentstoplacealloftheseonanumberlinethatgoesfrom0
to4
1/2,2½,1¾,15/12
Somestudentsthink
• Thatwhenthenumeratoranddenominatorareclosertogether,the
fractionisgreater,e.g.4/5>3/6
• Whatdoyouthink?
Infact…
• 4/7<5/6BUT
• 1/3<95/100
Didyounotice…
• Thatnumeratorsanddenominatorsofequivalentfractionsare
predictablyfarapart.
• Forexample,createanequivalentfractionfor2/5wherethe
numeratoranddenominatorare27apart.
Danger
• Somekidsthinkthatthisshowsthat1/5=1/6.
• Howwillyouhandlethis?
Didyouknow….
• Choosetwofractions.
• Createanewfractionbyaddingthenumeratorsandthe
denominatorstogetthenewnumeratoranddenominator.
• Howdoesthatfractioncomparetotheoriginaltwo?
Addingandsubtracting
• Onewaytointroduceadditionoffractionsistorealizethatany
fractionotherthanaunitfractionisanaddition.
e.g.3/5=1/5+1/5+1/5
Addingandsubtracting
• Wecancolordifferentlytoshowthat5/8=3/8+2/8or½+1/8.
Addingandsubtracting
• Sowhyisitthat2/5+1/5=3/5,andnot3/10?
Usingaclock
• 1/2+1/3
• ½hour=30min
• 1/3hour=20min
• Total=50minutes,
Or50/60ofanhour
Usingaclock
• 2/3– 1/5
• 2/3hour=40min
• 1/5hour=12min
• difference=28minutes,
Or28/60ofanhour(also7/15)
Usingaclock
• 4/3– 1/2
• 4/3hours=60+20min
• ½hour=30min
• 80– 30min=50min=5/6hour
Usingameterstick
• 2/5– 1/4
• 2/5of100cm=40cm
• 1/4of100cm=25cm
• Differenceis15cm,or
• 15/100m (or3/20)
Usinganynumber
• 2/3+3/7
• Iwilluse21.
• 2/3of21is14
• 3/7of21is9
• 14+9=23
• Theansweris23/21(or12/21)
Usingagrid
• 3/5+1/3
Usingagrid
• 3/5+1/3
Usingagrid
• 4/5– 1/3
Usingagrid
• 4/5– 1/3
Usingagrid
• 4/5– 1/3
Fractionstrips
• 3/5+1/3
Fractionstrips
• 3/5– 1/3
Subtractionoffractions
• Youcansubtract2/3– 1/5onthetowerbylookingforafractionthat
exactlyfitsthespacebetweenthetwo.
Subtractioncontexts
• Shouldinvolvetakeaway,butalsocomparisonormissingaddend,
e.g.
Subtractioncontexts
• Ihad1½c offlour.Iused2/3cup.Howmuchisleft?
• Iused1½c offlourand2/3c sugar.Howmuchmoreflour?
• Ineed1½c offlourandhave2/3c.HowmuchmoredoIneed?
Howdoyouknowthat…
• 2/3+5/6isbetween1and2?
• 9/10+1/4isalmost1¼?
• 2/3– 1/5ismorethan1/3?
Some+and– problems
• Yousubtracttwofractions.
• Theresultisveryclosetothefractionyousubtracted.
• Whatcouldthetwofractionsbe?
Some+and– problems
• Youaddtwofractionslessthan1.
• Theresultisjustabitmorethan1¼.
• Whatcouldthefractionsbe?
Some+and– problems
• Youaddtwofractionsandthenyousubtractthem.
• Thesumisabout1/3morethanthedifference.
• Whatcouldthefractionsbe?
Some+and– problems
• Youaddtwofractionsandthesumhasadenominatorof8.
• Whataresomepossibilitiesforthefractionsyouadded?
Mixednumbers
• Whymightitbebettertoleavemixednumbersasmixednumbers,
notimproperfractions,toaddandsubtractthem?
Subtractingmixednumbers
• Didyouknow:
3½ =2+1+1/2
- 1¾
– 1–3/4
1+¼+½
Subtractingmixednumbers
• Didyouknow:
3½
- 1¾
2(– ¼)=1¾
Subtractingmixednumbers
3½
- 1¾
Startat1¾.
Add¼togetto2andthenanother1½togetto3½.
Multiplyingafractionbyawholenumber
• Whyis4/5=4x 1/5?
• Howare6x 2/3and
2/3x 6different?
Multiplyingfractions
• Thinkof2/3x 3/4as2/3of3/4.
• Howdoesthismodelshowit?
Usingagrid
• 2/3x 2/5
Usingfractionstrips
• 2/3of2/5
Scaling
• 2/3x 3/5meansstartwithalength.
• Take3/5ofit.
• Thentake2/3ofthat3/5.
• Comparethefinaltotheoriginal.
Multiplyingmixednumbers
2½x 1½
2½
1
½
Whendowemultiply?
• Partofapart
• Scaling
• Calculatingarea
Dividingfractions
• Unitfractionsbyawhole,e.g.ifIcut1/3into4equalpieces,howbig
iseachpiece?
• Fractiontowerisideal.
• Anotherwaytodescribe1/3÷ 4is--- howmuchof4is1/3?
Howmuchofa4is1/3?
0
1
4
Dividingfractions
• Awholebyaunitfraction.
• IfIwanttoknowhowmany1/3sfitin4,i.e.4÷ 1/3.
• IfIwantaunitrate,e.g.IfIcanjog4kmin1/3ofanhour,howfarin
anhour?
Dividingfractions
• Awholebyaunitfraction.
• Whatmodelwouldyouuse?
Dividingfractionsmoregenerally(grade6)
• Usingpatternblocks
• 5/6÷ 2/3askshowmanysetsof2/3fitin5/6
Usinggrids
• 3/5÷ 1/3askshowmany1/3sfitin3/5.
• 1/3is5squaresbelow,so
thereare14/5 (9/5)setsof5squaresin3/5.
Usingfractionstrips
• Noticethat1/3fitsinto½1½times.
Mult/divproblems
• Useyourfractiontower.
• Findasmanyfractionpairsasyoucansothatthesmalleronefitsinto
thebiggeroneexactly2½times.
Morex and÷ questions
• Useyourfractiontower.
• Findfractionsoftheform2/[].Multiplyeachoneby1½.(Thatmeans
take1½ofthem.)
• Whatfractionsdoyouget?
Morex and÷ questions
• Youmultiplytwofractions.
• Theproductisalmostasmuchasoneofthefractionsandmorethan
oneofthefractions.
• Whatmightyouhavebeenmultiplying?
Morex and÷ questions
• Youdividetwofractions.
• Thequotientisjustabitlessthan1.
• Whatcouldthefractionsbe?
Yourquestions
• Aretherestillissuesyouwishtoraise?
Whataboutgaps?
• Tofillingaps,re-teachingthesamewayisprobablynotthewaytogo.
• Usingadifferentstrategyseemsmorevaluable.
Aresource
Uncomplicating Fractions
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• www.onetwoinfinity.ca
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