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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 Download • www.onetwoinfinity.ca • Kentuckyfractions