uncomplicating fractions - one, two, infinity…€¦ · • meaning of fractions ... issue of...

UncomplicatingFractions

MarianSmallLexington,KYJuly,2016

Let’sstartbydoing!

• Useyourpatternblockstocreateadesignthatis½yellow.• Let’sseewhatwenotice.

Halfinwhatway?• Halfinvolume?• Halfinareaofthetop?

Halfinwhatway?• Halfinnumberofblocks?

Whatwouldhappenif…

• Youduplicatedyourdesign?

Arethereothernames?

• Whatotherfractionnamecouldsomeonelegitimatelyuseforyouryellowpart?

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.usingwordsliketwothirdsinsteadof2/3?

SettingtheStage

• Istherevalueinpartitioningshapesotherthanrectanglesandcirclesintheearlygrades?

SettingtheStage

• Istherevalueinworkingongeoboardsorwithsquaretilessothatpartitionscanbebothpartitionsofwholesandpartitionsofsetssimultaneously?

Forexample

• Use16tilesandmakeasquare.• Howcouldyoudivideitupsothatitishalfred?

SettingtheStage

• IstherevalueinclarifyingthatafractionisALWAYSrelativetoawhole?• Youcanshow2,butyoucan’tshowonehalfunlessyouknowwhatthewholeisORyouassumethewholeisthenumber1.

Whatfractionsdoyousee?

SettingtheStage

• Whatdowemeanwhenwesaythepartshavetobeequal?• Itcouldbearea,orvolume,ornumber,ormass,etc.,andnotnecessarilyallofthem.

SettingtheStage

• StudentsneedtoseeNON-examplesaswellasexamples.• Createordrawanon-exampleofonefourth.

Whatmight3/4looklike?

• Drawsomepicturesorusematerialstoshow3/4asmanywaysasyoucan.

Partofalength

Partofaregion

Partofavolume

Partofacapacity

Partofadurationoftime

Partofaweight

Anymarbleontherighthas¾oftheweightofamarbleontheleft.

Whyisthat?

Partofaset

Anumber

• Itis3jumpsof¼from0.

01/43/41

Adivision• If4peopleshare3things,eachgets3/4ofoneofthosethings.

Adivision• Ifthedistancefrom0to3issplitinto4equalparts,thelengthofeachpartis3/4.

0 1 2 3

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’tyousee2/3withoutseeingatleastoneotherfraction?

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?

Whycanwemultiplynumeratoranddenominatorbythesameamount?• Notbecausewearemultiplyingby1(whichislearnedlater)• Becauseweareturningeachsectioninton timesasmanysections.• Orbecausewearemergingn sectionsintoone.

Forexample

• 3/4=6/8

Buttheimportantissue

• Thereason3/5=6/10isNOTbecauseyoumultipliedbothnumeratoranddenominatorby2.• Itisbecause6is3/5of10.

Isthislegitimate?

• Is1½/3=½?

Introducingequivalenceusingpartofset

• Trickyforkidstobelievethat2/3=4/6basedonthisvisual.

Somewhatbetter

Equivalenceonnumberline

• Howcouldyouuseanumberlinetoshowwhy4/6=6/9?

01

Equivalencebasedondivision

• Whyis2÷ 3thesameas4÷ 6?

• (If3peopleshare2,then6peoplewouldbesharing4.)

Mixednumber/improperfractionequivalence

• Noticethatweoftenrelyonthedivisionmeaningoffractionstoexplainwhy,e.g.8/3=22/3.• Howcouldweuseanumberline?• Apartofwholemodel?

Comparingfractions

• Wecannotcomparefractionsthatarepartsofwholesunlesstheyarefractionsofthesameorequalsizedwholes.• Forexample,2/3of3islessthan1/3of9.

Usingmodels

• Fractiontowerisexcellentforthis.

Usingmodels

• Tocompare,e.g.2/3and4/5,studentscoulduseobjectswherethirdsandfifthsareeasytosee,e.g.a3x 5gridORtherightCuisenairerods,e.g.

3x 5grid

2/3vs 4/5

Wewantstudentstorealizethat

• itiseasytocompare2/8and3/8because….• itiseasytocompare2/8and2/9because…• itiseasytocompare1/10and5/6because…

Youmightaskstudents--

• Writedowntwofractionsthatmightlookcomplicated,butarereallyeasytocompare.

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

• Askstudentstoplacealloftheseonanumberlinethatgoesfrom0to4

1/2,2½,1¾,15/12

Somestudentsthink

• Thatwhenthenumeratoranddenominatorareclosertogether,thefractionisgreater,e.g.4/5>3/6• Whatdoyouthink?

Infact…

• 4/7<5/6BUT• 1/3<95/100

Didyounotice…

• Thatnumeratorsanddenominatorsofequivalentfractionsarepredictablyfarapart.• Forexample,createanequivalentfractionfor2/5wherethenumeratoranddenominatorare27apart.

Danger

• Somekidsthinkthatthisshowsthat1/5=1/6.

• Howwillyouhandlethis?

Didyouknow….

• Choosetwofractions.• Createanewfractionbyaddingthenumeratorsandthedenominatorstogetthenewnumeratoranddenominator.• Howdoesthatfractioncomparetotheoriginaltwo?

Addingandsubtracting

• Onewaytointroduceadditionoffractionsistorealizethatanyfractionotherthanaunitfractionisanaddition.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– 1/5onthetowerbylookingforafractionthatexactlyfitsthespacebetweenthetwo.

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- 1¾2(– ¼)=1¾

Subtractingmixednumbers

3½- 1¾

Startat1¾.Add¼togetto2andthenanother1½togetto3½.

Multiplyingafractionbyawholenumber

• Whyis4/5=4x 1/5?• Howare6x 2/3and2/3x 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 1½2½

1½

Whendowemultiply?

• Partofapart• Scaling• Calculatingarea

Dividingfractions

• Unitfractionsbyawhole,e.g.ifIcut1/3into4equalpieces,howbigiseachpiece?• 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,howfarinanhour?

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.• Findasmanyfractionpairsasyoucansothatthesmalleronefitsintothebiggeroneexactly2½times.

Morex and÷ questions

• Useyourfractiontower.• Findfractionsoftheform2/[].Multiplyeachoneby1½.(Thatmeanstake1½ofthem.)• Whatfractionsdoyouget?

Morex and÷ questions

• Youmultiplytwofractions.• Theproductisalmostasmuchasoneofthefractionsandmorethanoneofthefractions.• Whatmightyouhavebeenmultiplying?

Morex and÷ questions

• Youdividetwofractions.• Thequotientisjustabitlessthan1.• Whatcouldthefractionsbe?

Yourquestions

• Aretherestillissuesyouwishtoraise?

Whataboutgaps?

• Tofillingaps,re-teachingthesamewayisprobablynotthewaytogo.• Usingadifferentstrategyseemsmorevaluable.

Aresource

Uncomplicating Fractions

Download

• www.onetwoinfinity.ca• Kentuckyfractions