Download Keynote: Structure and Coherence: Telling the Story of the Journey

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
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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
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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