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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