From the Ark of History to the Arc of Reasoning William McCallum The University of Arizona & Illustrative Mathematics JMM, 2016
Themes Decluttering Attending to historically unattended leaps Distinguishing between objects, uses of those objects, and ways of viewing them (why not just call a ratio an ordered pair of numbers?)
0 0 1 0 1 2 2 4 1 5 6 4 2 7 8 5 The number line marked off in thirds 4 9 10 11 12 GradeThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.In2.G.Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Gradetheystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeandbunniesis4ofthewhole.Gradestudentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-atorof4assayingthat4iswhatyougetbyputtingofthe14 stogether..nf.1anyfractioncanbereadthisway,andinparticular.nf.1understandafraction1 asthequantityformedby1partwhenawholeispartitionedinto equalparts;understandafrac-tion asthequantityformedby partsofsize1.thereisnoneedtointroducetheconceptsof properfraction"and improperfraction"initially;5iswhatonegetsbycombining5partstogetherwhenthewholeisdividedintoequalparts.twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(mp6): Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction2;iftheentirerectangleisthewhole,itrepresents4. Explainingwhatismeantby equalparts. Initially,studentscanuseanintuitivenotionofcongruence( samesizeandsameshape )toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.arearepresentationsof14ineachrepresentationthesquareisthewhole.thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.studentscometounderstandamoreprecisemeaningfor equalparts as partswithequalmeasurement. Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline012456etc.Toconstructaunitfractiononthenumberline,e.g.1,studentsdividetheunitintervalintointervalsofequallengthandrecognizethateachhaslength1.Theylocatethenumber1onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com. The number line GradeThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.In2.G.Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Gradetheystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeandbunniesis4ofthewhole.Gradestudentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-atorof4assayingthat4iswhatyougetbyputtingofthe14 stogether..nf.1anyfractioncanbereadthisway,andinparticular.nf.1understandafraction1 asthequantityformedby1partwhenawholeispartitionedinto equalparts;understandafrac-tion asthequantityformedby partsofsize1.thereisnoneedtointroducetheconceptsof properfraction"and improperfraction"initially;5iswhatonegetsbycombining5partstogetherwhenthewholeisdividedintoequalparts.twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(mp6): Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction2;iftheentirerectangleisthewhole,itrepresents4. Explainingwhatismeantby equalparts. Initially,studentscanuseanintuitivenotionofcongruence( samesizeandsameshape )toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.arearepresentationsof14ineachrepresentationthesquareisthewhole.thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.studentscometounderstandamoreprecisemeaningfor equalparts as partswithequalmeasurement. Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline012456etc.Toconstructaunitfractiononthenumberline,e.g.1,studentsdividetheunitintervalintointervalsofequallengthandrecognizethateachhaslength1.Theylocatethenumber1onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com. From whole numbers to fractions 6 etc.
Connection between division and fractions Why is 5 5? 0 1 2 4 5 0 1 2 4 5 0 1 2 4 5
From Fractions to Ratios
From Equivalent Ratios to Proportional Relationships cups grape cups peach 5 2 10 4 15 6 20 8 25 10 cups grape cups peach 5 2 10 4 15 6 20 8 25 10
From Proportional Relationships to Linear Functions
MP8: Look for and express regularity in repeated reasoning Moving from the table and the graph to the equation for each 1 unit you move to the right, move up 2 5 of a unit. when you go 2 units to the right, you go up 2 2 when you go units to the right, you go up 2 when you go 4 units to the right, you go up 4 2 when you go x units to the right, you go up x 2 starting from p0, 0q, to get to a point px, yq on the graph, go x units to the right, so go up x 2 therefore y x 2 5