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Structure and Coherence: Telling the Story of the Journey to Algebra William McCallum Illustrative Mathematics UnboundEd, Ft. Lauderdale, July 2016 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Memorizing a Poem Sea Fever I must go down to the seas again, to the lonely sea and the sky, And all I ask is a tall ship and a star to steer her by; And the wheel’s kick and the wind’s song and the white sail’s shaking, And a grey mist on the sea’s face, and a grey dawn breaking. I must go down to the seas again, for the call of the running tide Is a wild call and a clear call that may not be denied; And all I ask is a windy day with the white clouds flying, And the flung spray and the blown spume, and the sea-gulls crying. I must go down to the seas again, to the vagrant gypsy life, To the gull’s way and the whale’s way where the wind’s like a whetted knife; And all I ask is a merry yarn from a laughing fellow-rover, And quiet sleep and a sweet dream when the long trick’s over. John Masefield William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Memorizing a Poem Sea Fever I must go down to the seas again, to the lonely sea and the sky, And all I ask is a tall ship and a star to steer her by; And the wheel’s kick and the wind’s song and the white sail’s shaking, And a grey mist on the sea’s face, and a grey dawn breaking. I must go down to the seas again, for the call of the running tide Is a wild call and a clear call that may not be denied; And all I ask is a windy day with the white clouds flying, And the flung spray and the blown spume, and the sea-gulls crying. I must go down to the seas again, to the vagrant gypsy life, To the gull’s way and the whale’s way where the wind’s like a whetted knife; And all I ask is a merry yarn from a laughing fellow-rover, And quiet sleep and a sweet dream when the long trick’s over. John Masefield To memorize this poem, you use the structure, and the fact that each verse tells a story about the same thing: the sea. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra The Common Core was built on progressions William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra The pathway to algebra Opera&ons*and*Algebraic* Thinking* Expressions* →* and* Equa&ons* Number*and*Opera&ons— Base*Ten* →* 1" 2" 3" 4" Algebra* The*Number* →* System* Number*and* Opera&ons— Frac&ons* K" →* →* 5" William McCallum Illustrative Mathematics 6" 7" 8" High"School" Structure and Coherence: Telling the Story of the Journey to Algebra Kindergarten: Understanding place value Kindergarten: Understanding Place Value Kindergartners arrange teen numbers andsome some Kindergartners arrange teen numbersinto into10 10 ones ones and more more ones,ones, in preparation forfor viewing unit in preparation viewing10 10ones ones as as aa new new unit calledcalled a ten ainten Grade 1 1. in Grade Children)place)small)objects)into)105 frames)to)show)the)ten)as)two)rows)of) five)and)the)extra)ones)within)the)next) 105frame. Layered)place)value)cards)help) children)see)the)10)``hiding'')inside)any) teen)number.) William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1: Making a ten 9 ` 6 “ 9 ` p1 ` 5q “ p9 ` 1q ` 5 “ 10 ` 5 “ 15. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1: Making a ten 9 ` 6 “ 9 ` p1 ` 5q “ p9 ` 1q ` 5 “ 10 ` 5 “ 15. 2.OA.2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1: Making a ten 9 ` 6 “ 9 ` p1 ` 5q “ p9 ` 1q ` 5 “ 10 ` 5 “ 15. 2.OA.2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. How do students come to “know from memory?” William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1: Making a ten 9 ` 6 “ 9 ` p1 ` 5q 9 ` 7 “ 16 “ p9 ` 1q ` 5 9 ` 8 “ 17 “ 10 ` 5 “ 15. .. . 2.OA.2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. How do students come to “know from memory?” William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1: Making a ten 9 ` 6 “ 9 ` p1 ` 5q 9 ` 7 “ 16 “ p9 ` 1q ` 5 9 ` 8 “ 17 “ 10 ` 5 .. . “ 15. 19 ` 6 “ 25 2.OA.2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. 29 ` 6 “ 35 .. . How do students come to “know from memory?” William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1: Making a ten 9 ` 6 “ 9 ` p1 ` 5q 9 ` 7 “ 16 “ p9 ` 1q ` 5 9 ` 8 “ 17 “ 10 ` 5 .. . “ 15. 19 ` 6 “ 25 2.OA.2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. 29 ` 6 “ 35 .. . 36 ` 29 “ 65 How do students come to “know from memory?” William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1:toThe Kindergarten Gradeconnection 1: The connectionbetween between addition addition and subtraction and subtraction • Lucy has 3 apples. Julie has 5 apples. How many more apples does Julie have than Lucy? • Lucy has 3 apples. Julie has 5 apples. How many fewer apples does Lucy have than Julie? • If x + 2 = 5, then x = 5 - 2. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 1–2: Addition using place value and the properties of operations order, any grouping property • Students might start by counting on by 10s, then by 1s. • (40 + 6) + (30 + 7) = (40 + 6 + 30) + 7 • They move towards the standard algorithm by adding tens and ones separately. • 40 + 6 + 30 + 7 = 40 + 30 + 6 + 7. • 4x + 6 + 3x + 7 = 4x + 3x + 6 + 7 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra 0 0 3 1 0 1 3 2 3 2 3 3 4 3 3 1 5 3 William McCallum Illustrative Mathematics 6 3 4 2 7 3 8 3 5 The number line marked off in thirds 3 4 9 3 10 11 12 3 3 3 3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com. The number line 3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com. Grade 3: Extending from whole numbers to fractions 6 etc. Structure and Coherence: Telling the Story of the Journey to Algebra A note on the number line When community college students were asked to mark the approximate locations of ´0.7 and 1 38 on a number line, only 21% were able to locate both correctly.1 1 Cathy Kessel, citing What Community College Developmental Mathematics Students Understand About Mathematics Stigler, Givvin, and Thompson William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Using the number line to see that 0 1 5 segments put end to end 7 4 ` 5 5 William McCallum Illustrative Mathematics 5 3 “ 1 3 5 3 “ ` 1 3 1 3 ` 2 ` 1 3 1 3 ` ` 1 3 1 3 ` 3 ` 1 3 ` 1 3 3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com. Grade 4: Extending operations with whole numbers to fractions Segment of length 13 4 1 3 “ Structure and Coherence: Telling the Story of the Journey to Algebra Using the number line to see that 0 1 5 segments put end to end 7 4 ` 5 5 “ “ “ 5 3 7`4 . 5 William McCallum Illustrative Mathematics “ 7 hkkkkikkkkj 1 3 5 3 “ ` 1 3 1 3 ` 2 ` 1 3 1 3 ` ` 1 3 1 3 ` 3 ` 1 3 ` 1 3 3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com. Grade 4: Extending operations with whole numbers to fractions Segment of length 13 4 1 3 4 hkkkkikkkkj 1 1 1 1 ` ¨¨¨ ` ` ¨¨¨ 5 5 5 5 7`4 hkkkkkkkkikkkkkkkkj 1 ` 1 ` ¨¨¨ ` 1 5 Structure and Coherence: Telling the Story of the Journey to Algebra Grade 5: Connection between division and fractions Why is 0 5 “ 5 ˜ 3? 3 1 2 William McCallum Illustrative Mathematics 3 4 5 Structure and Coherence: Telling the Story of the Journey to Algebra Grade 5: Connection between division and fractions Why is 5 “ 5 ˜ 3? 3 0 1 2 3 4 5 0 1 2 3 4 5 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 5: Connection between division and fractions Why is 5 “ 5 ˜ 3? 3 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 5: Connection between division and fractions Why is 5 “ 5 ˜ 3? 3 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 6: Ratios and Equivalent Ratios William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 7: From Ratios to Proportional Relationships William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Grade 7–8: From Proportional Relationships to Linear Functions William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra 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 ¨ when you go 3 units to the right, you go up 3 ¨ when you go 4 units to the right, you go up 4 ¨ when you go x units to the right, you go up x ¨ 2 5 2 5 2 5 2 5 units. units. units. units. starting from p0, 0q, to get to a point px , y q on the graph, go x units to the right, so go up x ¨ 25 units. therefore y “ x ¨ 2 5 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra High School: Seeing Structure in Expressions A-SSE.1. Interpret expressions that represent a quantity in terms of its context. a Interpret parts of an expression, such as terms, factors, and coefficients. b Interpret complicated expressions by viewing one or more of their parts as a single entity. A-SSE.2. Use the structure of an expression to identify ways to rewrite it. A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept y “ mx ` b y “ y1 ` mpx ´ x1 q y “ y1 ` William McCallum Illustrative Mathematics y2 ´ y1 px ´ x1 q x2 ´ x1 Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept y “ mx ` b y “ y1 ` mpx ´ x1 q y “ y1 ` y2 ´ y1 px ´ x1 q x2 ´ x1 To remember these formulas you use the structure, and the fact that they are all about the same thing: the slope. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept y “ mpx ´ 0q ` b y “ y1 ` mpx ´ x1 q y “ y1 ` y2 ´ y1 px ´ x1 q x2 ´ x1 To remember these formulas you use the structure, and the fact that they are all about the same thing: the slope. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept slope “ William McCallum Illustrative Mathematics rise run Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept px , y q mx rise “ slope ¨ run p0, b q x William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept px , y q mpx ´ x1 q rise “ slope ¨ run px1 , y1 q x ´ x1 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept px , y q slope “ rise run y 2 ´y 1 x2 ´x1 px px2 , y2 q px1 , y1 q ´ x1 q rise “ slope ¨ run x ´ x1 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Slope as a Unifying Concept slope “ William McCallum Illustrative Mathematics rise run Structure and Coherence: Telling the Story of the Journey to Algebra Thinking Ahead: Strategic Manipulation (A-SSE.3) After a container of ice-cream has been sitting in a room for t minutes, its temperature in degrees Fahrenheit is a ´ b2´t ` b , where a and b are positive constants. Write this expression in a form that shows that the temperature is always greater than a. shows that the temperature is always less than a ` b. https://www.illustrativemathematics.org/ content-standards/HSA/SSE/B/3/tasks/551 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Thinking Ahead: Strategic Manipulation (A-SSE.3) After a container of ice-cream has been sitting in a room for t minutes, its temperature in degrees Fahrenheit is a ´ b2´t ` b , where a and b are positive constants. Write this expression in a form that shows that the temperature is always greater than a. a ` b p1 ´ 2´t q shows that the temperature is always less than a ` b. https://www.illustrativemathematics.org/ content-standards/HSA/SSE/B/3/tasks/551 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Thinking Ahead: Strategic Manipulation (A-SSE.3) After a container of ice-cream has been sitting in a room for t minutes, its temperature in degrees Fahrenheit is a ´ b2´t ` b , where a and b are positive constants. Write this expression in a form that shows that the temperature is always greater than a. a ` b p1 ´ 2´t q shows that the temperature is always less than a ` b. a ` b ´ b2´t https://www.illustrativemathematics.org/ content-standards/HSA/SSE/B/3/tasks/551 William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Interpreting and Building Functions F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F-BF.1. Write a function that describes a relationship between two quantities. a Determine an explicit expression, a recursive process, or steps for calculation from a context. b Combine standard function types using arithmetic operations. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra The Menagerie of Function Types Is your zoo like this? William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra The Menagerie of Function Types Or like this? William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Building a Saturating Exponential Suppose a can of cold soda is left in a warm room on a summer day. The graph shows the temperature of the soda as it gradually increased. The function that describes the temperature, F, of the soda (in degrees Fahrenheit) after t minutes can be expressed by F pt q “ C ´ Re ´kt , for some positive values of C, R, and k . Use the graph to estimate C. Use the graph to estimate R. What was the approximate room temperature? What was the initial temperature of the soda when placed in the room? William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra Better to be behind but on the pathway rather than lost in the wood. William McCallum Illustrative Mathematics Structure and Coherence: Telling the Story of the Journey to Algebra