Progressions for the Common Core StateStandards in Mathematics (draft)ccopyThe Common Core Standards Writing Team

2 July 2013

Suggested citationCommon Core Standards Writing Team (2013 March1) Progressions for the Common Core State Standardsin Mathematics (draft) High School Algebra TucsonAZ Institute for Mathematics and Education Univer-sity of ArizonaFor updates and more information about theProgressions see httpimematharizonaeduprogressionsFor discussion of the Progressions and related top-ics see the Tools for the Common Core blog httpcommoncoretoolsme

Draft 7022013 comment at commoncoretoolswordpresscom

High School AlgebraOverviewTwo Grades 6ndash8 domains are important in preparing students forAlgebra in high school The Number System prepares students tosee all numbers as part of a unified system and become fluent infinding and using the properties of operations to find the values ofnumerical expressions that include those numbers The standardsof the Expressions and Equations domain ask students to extendtheir use of these properties to linear equations and expressionswith letters These extend uses of the properties of operations inearlier grades in Grades 3ndash5 Number and OperationsmdashFractionsin Kndash5 Operations and Algebraic Thinking and Kndash5 Number andOperations in Base Ten

The Algebra category in high school is very closely allied withthe Functions category

bull An expression in one variable can be viewed as defining afunction the act of evaluating the expression at a given inputis the act of producing the functionrsquos output at that input

bull An equation in two variables can sometimes be viewed asdefining a function if one of the variables is designated asthe input variable and the other as the output variable and ifthere is just one output for each input For example this is thecase if the equation is in the form y (expression in x) or ifit can be put into that form by solving for y

bull The notion of equivalent expressions can be understood interms of functions if two expressions are equivalent they de-fine the same function

The study of algebra occupies a large part of a studentrsquos high school careerand this document does not treat in detail all of the material studied Rather itgives some general guidance about ways to treat the material and ways to tie ittogether It notes key connections among standards points out cognitive difficultiesand pedagogical solutions and gives more detail on particularly knotty areas of themathematics

The high school standards specify the mathematics that all students should studyin order to be college and career ready Additional material corresponding to (+)standards mathematics that students should learn in order to take advanced coursessuch as calculus advanced statistics or discrete mathematics is indicated by plussigns in the left margin

Draft 7022013 comment at commoncoretoolswordpresscom

3

bull The solutions to an equation in one variable can be under-stood as the input values which yield the same output in thetwo functions defined by the expressions on each side of theequation This insight allows for the method of finding ap-proximate solutions by graphing the functions defined by eachside and finding the points where the graphs intersect

Because of these connections some curricula take a functions-basedapproach to teaching algebra in which functions are introducedearly and used as a unifying theme for algebra Other approachesintroduce functions later after extensive work with expressions andequations The separation of algebra and functions in the Stan-dards is not intended to indicate a preference between these twoapproaches It is however intended to specify the difference asmathematical concepts between expressions and equations on theone hand and functions on the other Students often enter college-level mathematics courses apparently conflating all three of theseFor example when asked to factor a quadratic expression a stu-dent might instead find the solutions of the corresponding quadraticequation Or another student might attempt to simplify the expres-sion sin xx by cancelling the x rsquos

The algebra standards are fertile ground for the Standards forMathematical Practice Two in particular that stand out are MP7ldquoLook for and make use of structurerdquo and MP8 ldquoLook for and expressregularity in repeated reasoningrdquo Students are expected to seehow the structure of an algebraic expression reveals properties ofthe function it defines They are expected to move from repeatedreasoning with pairs of points on a line to writing equations invarious forms for the line rather than memorizing all those formsseparately In this way the Algebra standards provide focus in away different from the Kndash8 standards Rather than focusing on afew topics students in high school focus on a few seed ideas thatlead to many different techniques

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4

Seeing Structure in ExpressionsStudents have been seeing expressions since Kindergarten start-ing with arithmetic expressions in Grades Kndash5 and moving on toalgebraic expressions in Grades 6ndash8 The middle grades standardsin Expression and Equations build a ramp from arithmetic expres-sions in elementary school to more sophisticated work with alge-braic expressions in high school As the complexity of expressions

Animal Populations3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

Suppose P and Q give the sizes of two different animalpopulations where Q iexcl P In 1ndash4 say which of the given pair ofexpressions is larger Briefly explain your reasoning in terms ofthe two populations

1 P Q and 2P2

PP Q and

P Q2

3 pQ Pq2 and Q P24 P 50t and Q 50t

Task from Illustrative Mathematics For solutions and discussionsee httpwwwillustrativemathematicsorgillustrations436

increases students continue to see them as being built out of basicoperations they see expressions as sums of terms and products offactorsA-SSE1a

A-SSE1a Interpret expressions that represent a quantity interms of its context

a Interpret parts of an expression such as terms factorsand coefficients

For example in ldquoAnimal Populationsrdquo in the margin studentscompare P Q and 2P by seeing 2P as P P They distinguishbetween pQ Pq2 and Q P2 by seeing the first as a quotientwhere the numerator is a difference and the second as a differencewhere the second term is a quotient This last example also illus-trates how students are able to see complicated expressions as builtup out of simpler onesA-SSE1b As another example students can see

A-SSE1b Interpret expressions that represent a quantity interms of its context

b Interpret complicated expressions by viewing one or moreof their parts as a single entity

the expression 5px1q2 as a sum of a constant and a square andthen see that inside the square term is the expression x 1 Thefirst way of seeing tells them that it is always greater than or equalto 5 since a square is always greater than or equal to 0 the secondway of seeing tells them that the square term is zero when x 1Putting these together they can see that this expression attains itsminimum value 5 when x 1 The margin lists other tasks fromthe Illustrative Mathematics project (illustrativemathematicsorg) forA-SSE1

In elementary grades the repertoire of operations for building

Illustrations of interpreting the structure of expressions3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

The following tasks can be found by going tohttpillustrativemathematicsorgillustrations and searching forA-SSE

bull Delivery Trucks

bull Kitchen Floor Tiles

bull Increasing or Decreasing Variation 1

bull Mixing Candies

bull Mixing Fertilizer

bull Quadrupling Leads to Halving

bull The Bank Account

bull The Physics Professor

bull Throwing Horseshoes

bull Animal Populations

bull Equivalent Expressions

bull Sum of Even and Odd

expressions is limited to the operations of arithmetic addition sub-traction multiplication and division Later it is augmented by expo-nentiation first with whole numbers in Grades 5 and 6 then withintegers in Grade 8 By the time they finish high school studentshave expanded that repertoire to include radicals and trigonometricexpressions along with a wider variety of exponential expressions

For example students in physics classes might be expected tosee the expression

L0c1

v2

c2 which arises in the theory of special relativity as the product of theconstant L0 and a term that is 1 when v 0 and 0 when v cmdashand furthermore they might be expected to see it without having togo through a laborious process of written or electronic evaluationThis involves combining the large-scale structure of the expressionmdasha product of L0 and another termmdashwith the structure of internalcomponents such as v2c2

Seeing structure in expressions entails a dynamic view of analgebraic expression in which potential rearrangements and ma-nipulations are ever presentA-SSE2 An important skill for college A-SSE2 Use the structure of an expression to identify ways to

rewrite it

Draft 7022013 comment at commoncoretoolswordpresscom

5

readiness is the ability to try possible manipulations mentally with-out having to carry them out and to see which ones might be fruitfuland which not For example a student who can see

p2n 1qnpn 1q

6

as a polynomial in n with leading coefficient 13n3 has an advantage

when it comes to calculus a student who can mentally see theequivalence R1R2

R1 R2

11R1 1R2

without a laborious pencil and paper calculation is better equippedfor a course in electrical engineering

The Standards avoid talking about simplification because it isoften not clear what the simplest form of an expression is and even

Which form is ldquosimplerrdquo3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

After a container of ice cream has been sitting in a room for tminutes its temperature in degrees Fahrenheit is

a b 2t bwhere a and b are positive constants

Write this expression in a form that

1 Shows that the temperature is always less than a b

2 Shows that the temperature is never less than a

Commentary The form a b b 2t for the temperatureshows that it is a b minus a positive number so always lessthan a b On the other hand the form a bp1 2tq revealsthat the temperature is never less than a because it is a plus apositive number

ldquoIce Creamrdquo task from Illustrative Mathematics For solutions anddiscussion seehttpwwwillustrativemathematicsorgillustrations551

in cases where that is clear it is not obvious that the simplest form isdesirable for a given purpose The Standards emphasize purposefultransformation of expressions into equivalent forms that are suitablefor the purpose at hand as illustrated in the marginA-SSE3

A-SSE3 Choose and produce an equivalent form of an expres-sion to reveal and explain properties of the quantity representedby the expression

For example there are three commonly used forms for a quadraticexpression

bull Standard form eg x2 2x 3

bull Factored form eg px 1qpx 3q

bull Vertex form (a square plus or minus a constant) eg px1q24Rather than memorize the names of these forms students need togain experience with them and their different uses The traditionalemphasis on simplification as an automatic procedure might leadstudents to automatically convert the second two forms to the firstrather than convert an expression to a form that is useful in a givencontextA-SSE3ab This can lead to time-consuming detours in alge- a Factor a quadratic expression to reveal the zeros of the

function it defines

b Complete the square in a quadratic expression to revealthe maximum or minimum value of the function it defines

braic work such as solving px 1qpx 3q 0 by first expandingand then applying the quadratic formula

The introduction of rational exponents and systematic practicewith the properties of exponents in high school widen the field of op-erations for manipulating expressionsA-SSE3c For example students c Use the properties of exponents to transform expressions

for exponential functionsin later algebra courses who study exponential functions seePp1 r

12q12n as P p1 r

12q12n

in order to understand formulas for compound interest

Illustrations of writing expressions in equivalent forms3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

The following tasks can be found by going tohttpillustrativemathematicsorgillustrations and searching forA-SSE

bull Ice Cream

bull Increasing or Decreasing Variation 2

bull Profit of a Company

bull Seeing Dots

Much of the ability to see and use structure in transformingexpressions comes from learning to recognize certain fundamentalsituations that afford particular techniques One such technique isinternal cancellation as in the expansion

pa bqpa bq a2 b2Draft 7022013 comment at commoncoretoolswordpresscom

6

An impressive example of this ispx 1qpxn1 xn2 x 1q xn 1

in which all the terms cancel except the end terms This identityis the foundation for the formula for the sum of a finite geometricseriesA-SSE4 A-SSE4 Derive the formula for the sum of a finite geometric se-

ries (when the common ratio is not 1) and use the formula tosolve problems

Draft 7022013 comment at commoncoretoolswordpresscom

7

Arithmetic with Polynomials and Rational ExpressionsThe development of polynomials and rational expressions in highschool parallels the development of numbers in elementary andmiddle grades In elementary school students might initially seeexpressions for the same numbers 8 3 and 11 or 3

4 and 075 asreferring to different entities 8 3 might be seen as describing acalculation and 11 is its answer 3

4 is a fraction and 075 is a deci-mal They come to understand that these different expressions aredifferent names for the same numbers that properties of operationsallow numbers to be written in different but equivalent forms andthat all of these numbers can be represented as points on the num-ber line In middle grades they come to see numbers as forming aunified system the number system still represented by points onthe number line The whole numbers expand to the integersmdashwithextensions of addition subtraction multiplication and division andtheir properties Fractions expand to the rational numbersmdashand thefour operations and their properties are extended

A similar evolution takes place in algebra At first algebraic ex-pressions are simply numbers in which one or more letters are usedto stand for a number which is either unspecified or unknown Stu-dents learn to use the properties of operations to write expressionsin different but equivalent forms At some point they see equiva-lent expressions particularly polynomial and rational expressionsas naming some underlying thingA-APR1 There are at least two

A-APR1 Understand that polynomials form a system analogousto the integers namely they are closed under the operations ofaddition subtraction and multiplication add subtract and multi-ply polynomialsways this can go If the function concept is developed before or

concurrently with the study of polynomials then a polynomial canbe identified with the function it defines In this way x2 2x 3px1qpx3q and px1q24 are all the same polynomial becausethey all define the same function Another approach is to thinkof polynomials as elements of a formal number system in whichyou introduce the ldquonumberrdquo x and see what numbers you can writedown with it In this approach x2 2x 3 px 1qpx 3q andpx 1q2 4 are all the same polynomial because the properties ofoperations allow each to be transformed into the others Each ap-proach has its advantages and disadvantages the former approachis more common Whichever is chosen and whether or not the choiceis explicitly stated a curricular implementation should nonethelessbe constructed to be consistent with the choice that has been made

Either way polynomials and rational expressions come to forma system in which they can be added subtracted multiplied anddividedA-APR7 Polynomials are analogous to the integers rational

A-APR7(+) Understand that rational expressions form a systemanalogous to the rational numbers closed under addition sub-traction multiplication and division by a nonzero rational expres-sion add subtract multiply and divide rational expressionsexpressions are analogous to the rational numbers

Polynomials form a rich ground for mathematical explorationsthat reveal relationships in the system of integersA-APR4 For exam- A-APR4 Prove polynomial identities and use them to describe

numerical relationshipsple students can explore the sequence of squares1 4 9 16 25 36

and notice that the differences between themmdash3 5 7 9 11mdashareDraft 7022013 comment at commoncoretoolswordpresscom

8

consecutive odd integers This mystery is explained by the polyno-mial identity

pn 1q2 n2 2n 1A more complex identity

px2 y2q2 px2 y2q2 p2xyq2allows students to generate Pythagorean triples For example tak-ing x 2 and y 1 in this identity yields 52 32 42

A particularly important polynomial identity treated in advanced+courses is the Binomial TheoremA-APR5

A-APR5(+) Know and apply the Binomial Theorem for the ex-pansion of px yqn in powers of x and y for a positive integer nwhere x and y are any numbers with coefficients determined forexample by Pascalrsquos Triangle1

+

px yqn xn n1

xn1y

n2

xn2y2

n3

xn3y3 yn

for a positive integer n The binomial coefficients can be obtained+using Pascalrsquos triangle+

n 0 1n 1 1 1n 2 1 2 1n 3 1 3 3 1n 4 1 4 6 4 1

+

in which each entry is the sum of the two above Understanding+why this rule follows algebraically from+

px yqpx yqn1 px yqnis excellent exercise in abstract reasoning (MP2) and in expressing+regularity in repeated reasoning (MP8)+

Polynomials as functions Viewing polynomials as functions leadsto explorations of a different nature Polynomial functions are onthe one hand very elementary in that unlike trigonometric and ex-ponential functions they are built up out of the basic operations ofarithmetic On the other hand they turn out to be amazingly flexi-ble and can be used to approximate more advanced functions suchas trigonometric and exponential functions Experience with con-structing polynomial functions satisfying given conditions is usefulpreparation not only for calculus (where students learn more aboutapproximating functions) but for understanding the mathematics be-hind curve-fitting methods used in applications to statistics and com-puter graphics

A simple step in this direction is to construct polynomial functionswith specified zerosA-APR3 This is the first step in a progression

A-APR3 Identify zeros of polynomials when suitable factoriza-tions are available and use the zeros to construct a rough graphof the function defined by the polynomialwhich can lead as an extension topic to constructing polynomial

functions whose graphs pass through any specified set of points inthe plane

Draft 7022013 comment at commoncoretoolswordpresscom

9

Polynomials as analogues of integers The analogy between poly-nomials and integers carries over to the idea of division with remain-der Just as in Grade 4 students find quotients and remainders ofintegers4NBT6 in high school they find quotients and remainders of

4NBT6Find whole-number quotients and remainders with up tofour-digit dividends and one-digit divisors using strategies basedon place value the properties of operations andor the relation-ship between multiplication and division Illustrate and explain thecalculation by using equations rectangular arrays andor areamodels

polynomialsA-APR6 The method of polynomial long division is anal-A-APR6 Rewrite simple rational expressions in different formswrite apxq

bpxq in the form qpxq rpxqbpxq where apxq bpxq qpxq and

rpxq are polynomials with the degree of rpxq less than the degreeof bpxq using inspection long division or for the more compli-cated examples a computer algebra system

ogous to and simpler than the method of integer long divisionA particularly important application of polynomial division is the

case where a polynomial ppxq is divided by a linear factor of theform x a for a real number a In this case the remainder is thevalue ppaq of the polynomial at x aA-APR2 It is a pity to see this

A-APR2 Know and apply the Remainder Theorem For a poly-nomial ppxq and a number a the remainder on division by x ais ppaq so ppaq 0 if and only if px aq is a factor of ppxq

topic reduced to ldquosynthetic divisionrdquo which reduced the method toa matter of carrying numbers between registers something easilydone by a computer while obscuring the reasoning that makes theresult evident It is important to regard the Remainder Theorem asa theorem not a technique

A consequence of the Remainder Theorem is to establish theequivalence between linear factors and zeros that is the basis ofmuch work with polynomials in high school the fact that ppaq 0if and only if x a is a factor of ppxq It is easy to see if x a is afactor then ppaq 0 But the Remainder Theorem tells us that wecan write

ppxq px aqqpxq ppaq for some polynomial qpxqIn particular if ppaq 0 then ppxq px aqqpxq so x a is afactor of ppxq

Draft 7022013 comment at commoncoretoolswordpresscom

10

Creating EquationsStudents have been seeing and writing equations since elementarygradesKOA1 1OA1 with mostly linear equations in middle grades At

KOA1Represent addition and subtraction with objects fingersmental images drawings2 sounds (eg claps) acting out situa-tions verbal explanations expressions or equations

1OA1Use addition and subtraction within 20 to solve word prob-lems involving situations of adding to taking from putting to-gether taking apart and comparing with unknowns in all posi-tions eg by using objects drawings and equations with a sym-bol for the unknown number to represent the problem3

first glance it might seem that the progression from middle grades tohigh school is fairly straightforward the repertoire of functions thatis acquired during high school allows students to create more com-plex equations including equations arising from linear and quadraticexpressions and simple rational and exponential expressionsA-CED1

A-CED1 Create equations and inequalities in one variable anduse them to solve problems

students are no longer limited largely to linear equations in mod-eling relationships between quantities with equations in two varia-blesA-CED2 and students start to work with inequalities and systems

A-CED2 Create equations in two or more variables to representrelationships between quantities graph equations on coordinateaxes with labels and scales

of equationsA-CED3

A-CED3 Represent constraints by equations or inequalities andby systems of equations andor inequalities and interpret solu-tions as viable or nonviable options in a modeling context

Two developments in high school complicate this picture Firststudents in high school start using parameters in their equations torepresent whole classes of equationsF-LE5 or to represent situations

F-LE5 Interpret the parameters in a linear or exponential func-tion in terms of a context

where the equation is to be adjusted to fit databull

bull As noted in the Standards

Analytic modeling seeks to explain data on thebasis of deeper theoretical ideas albeit with pa-rameters that are empirically based for exampleexponential growth of bacterial colonies (until cut-off mechanisms such as pollution or starvation in-tervene) follows from a constant reproduction rateFunctions are an important tool for analyzing suchproblems (p 73)

Second modeling becomes a major objective in high school Twoof the standards just cited refer to ldquosolving problemsrdquo and ldquointerpret-ing solutions in a modeling contextrdquo And all the standards in theCreating Equations group carry a modeling star denoting their con-nection with the Modeling category in high school This connotesnot only an increase in the complexity of the equations studied butan upgrade of the studentrsquos ability in every part of the modelingcycle shown in the margin

The modeling cycle 3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

Problem Formulate Validate Report

Compute Interpret

Variables parameters and constants Confusion about these termsplagues high school algebra Here we try to set some rules for us-ing them These rules are not purely mathematical indeed from astrictly mathematical point of view there is no need for them at allHowever users of equations by referring to letters as ldquovariablesrdquoldquoparametersrdquo or ldquoconstantsrdquo can indicate how they intend to use theequations This usage can be helpful if it is consistent

In elementary and middle grades students solve problems withan unknown quantity might use a symbol to stand for that quantityand might call the symbol an unknown1OA2 In Grade 6 students 1OA2Solve word problems that call for addition of three whole

numbers whose sum is less than or equal to 20 eg by usingobjects drawings and equations with a symbol for the unknownnumber to represent the problem

begin to use variables systematically6EE6 They work with equa-

6EE6Use variables to represent numbers and write expressionswhen solving a real-world or mathematical problem understandthat a variable can represent an unknown number or dependingon the purpose at hand any number in a specified set

tions in one variable such as p 005p 10 or equations in twovariables such as d 5 5t relating two varying quantitiesbull In

bull Some writers prefer to retain the term ldquounknownrdquo for the firstsituation and the word ldquovariablerdquo for the second This is not theusage adopted in the Standards

each case apart from the variables the numbers in the equation aregiven explicitly The latter use presages the use of varibles to definefunctions

In high school things get more complicated For example stu-dents consider the general equation for a non-vertical line y mx b Here they are expected to understand that m and b arefixed for any given straight line and that by varying m and b weobtain a whole family of straight lines In this situation m and b arecalled parameters Of course in an episode of mathematical workthe perspective could change students might end up solving equa-Draft 7022013 comment at commoncoretoolswordpresscom

11

tions for m and b Judging whether to explicitly indicate thismdashldquonowwe will regard the parameters as variablesrdquomdashor whether to ignoreit and just go ahead and solve for the parameters is a matter ofpedagogical judgement

Sometimes an equation like y mx b is used not to workwith a parameterized family of equations but to consider the generalform of an equation and prove something about it For example youmight want take two points px1 y1q and px2 y2q on the graph ofy mx b and show that the slope of the line they determine ism In this situation you might refer to m and b as constants ratherthan as parameters

Finally there are situations where an equation is used to de-scribe the relationship between a number of different quantities towhich none of these terms applyA-CED4 For example Ohmrsquos Law A-CED4 Rearrange formulas to highlight a quantity of interest

using the same reasoning as in solving equationsV IR relates the voltage current and resistance of an electricalcircuit An equation used in this way is sometimes called a formulaIt is perhaps best to avoid using the terms ldquovariablerdquo ldquoparameterrdquoor ldquoconstantrdquo when working with this formula because there aresix different ways it can be viewed as defining one quantity as afunction of the other with a third held constant

Different curricular implementations of the Standards might nav-igate these terminological shoals in different ways (that might in-clude trying to avoid them entirely)Modeling with equations Consider the Formulate node in the mod-eling cycle In elementary school students formulate equations to

The modeling cycle 3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

Problem Formulate Validate Report

Compute Interpret

solve word problems They begin with situations that can be rep-resented by ldquosituation equationsrdquo that are also ldquosolution equationsrdquoThese situations and their equations have two important character-istics First the actions in the situations can be straightforwardlyrepresented by operations For example the action of putting to-gether is readily represented by addition (eg ldquoThere were 2 bun-nies and 3 more came how many were thererdquo) but representingan additive comparison (ldquoThere were 2 bunnies more came Thenthere were 5 How many more bunnies camerdquo) requires a moresophisticated understanding of addition Second the equations leaddirectly to a solution eg they are of the form 2 3 l with theunknown isolated on one side of the equation rather than 2l 5or 5l 2 More comprehensive understanding of the operations(eg understanding subtraction as finding an unknown addend) al-lows students to transform the latter types of situation equationsinto solution equations first for addition and subtraction equationsthen for multiplication and division equations

In high school there is again a difference between directly repre-senting the situation and finding a solution For example in solving

Selina bought a shirt on sale that was 20 less thanthe original price The original price was $5 more than

Draft 7022013 comment at commoncoretoolswordpresscom

12

the sale price What was the original price Explain orshow work

students might let p be the original price in dollars and then expressthe sale price in terms of p in two different ways and set them equalOn the one hand the sale price is 20 less than the original priceand so equal to p 02p On the other hand it is $5 less than theoriginal price and so equal to p 5 Thus they want to solve theequation

p 02p p 5In this task the formulation of the equation tracks the text of theproblem fairly closely but requires more than a direct representationof ldquoThe original price was $5 more than the sale pricerdquo To obtain anexpression for the sale price this sentence needs to be reinterpretedas ldquothe sale price is $5 less than the original pricerdquo Because thewords ldquolessrdquo and ldquomorerdquo have often traditionally been the subject ofschemes for guessing the operation required in a problem withoutreading it this shift is significant and prepares students to readmore difficult and realistic task statements

Indeed in a high school modeling problem there might be sig-nificantly different ways of going about a problem depending on thechoices made and students must be much more strategic in formu-lating the model

For example students enter high school understanding a solutionof an equation as a number that satisfies the equation6EE6 rather

6EE6Use variables to represent numbers and write expressionswhen solving a real-world or mathematical problem understandthat a variable can represent an unknown number or dependingon the purpose at hand any number in a specified setthan as the outcome of an accepted series of manipulations for a

given type of equation Such an understanding is a first step inallowing students to represent a solution as an unknown numberand to describe its properties in terms of that representation

Formulating an equation by checking a solution3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

Mary drives from Boston to Washington and she travels at anaverage rate of 60 mph on the way down and 50 mph on the wayback If the total trip takes 18 1

3 hours how far is Boston fromWashington

Commentary How can we tell whether or not a specific numberof miles s is a solution for this problem Building on theunderstanding of rate time and distance developed in Grades 7and 8 students can check a proposed solution s eg 500miles They know that the time required to drive down is 500

60

hours and to drive back is 50050 hours If 500 miles is a solution

the total time 50060 500

50 should be 18 13 hours This is not the

case How would we go about checking another proposedsolution say 450 miles Now the time required to drive down is45060 hours and to drive back is 450

50 hoursFormulating these repeated calculations be formulated in termsof s rather than a specific number (MP8) leads to the equations60 s

50 18 13

Task and discussion adapted from Cuoco 2008 ldquoIntroducingExtensible Tools in High School Algebrardquo in Algebra andAlgebraic Thinking in School Mathematics National Council ofTeachers of Mathematics

The Compute node of the modeling cycle is dealt with in the nextsection on solving equations

The Interpret node also becomes more complex Equations inhigh school are also more likely to contain parameters than equa-tions in earlier grades and so interpreting a solution to an equationmight involve more than consideration of a numerical value but con-sideration of how the solution behaves as the parameters are varied

The Validate node of the modeling cycle pulls together manyof the standards for mathematical practice including the modelingstandard itself (ldquoModel with mathematicsrdquo MP4)

Draft 7022013 comment at commoncoretoolswordpresscom

13

Reasoning with Equations and InequalitiesEquations in one variable A naked equation such as x2 4without any surrounding text is merely a sentence fragment neithertrue nor false since it contains a variable x about which nothing issaid A written sequence of steps to solve an equation such as inthe margin is code for a narrative line of reasoning using words like

Fragments of reasoning3Grade3ThemeaningoffractionsInGrades1and2studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares2G3In2G3PartitioncirclesandrectanglesintotwothreeorfourequalsharesdescribethesharesusingthewordshalvesthirdshalfofathirdofetcanddescribethewholeastwohalvesthreethirdsfourfourthsRecognizethatequalsharesofidenticalwholesneednothavethesameshapeGrade3theystarttodeveloptheideaofafractionmoreformallybuildingontheideaofpartitioningawholeintoequalpartsThewholecanbeacollectionofobjectsashapesuchasacircleorrect-anglealinesegmentoranyfiniteentitysusceptibletosubdivisionandmeasurementThewholeasacollectionofobjectsIfthewholeisacollectionof4bunniesthenonebunnyis14ofthewholeand3bunniesis34ofthewholeGrade3studentsstartwtihunitfractions(fractionswithnumer-ator1)Theseareformedbydividingawholeintoequalpartsandtakingonepartegifawholeisdividedinto4equalpartstheneachpartis14ofthewholeand4copiesofthatpartmakethewholeNextstudentsbuildfractionsfromunitfractionsseeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14rsquostogether3NF1Anyfractioncanbereadthiswayandinparticular3NF1Understandafraction1asthequantityformedby1partwhenawholeispartitionedintoequalpartsunderstandafrac-tionasthequantityformedbypartsofsize1thereisnoneedtointroducetheconceptsofldquoproperfractionandldquoimproperfractioninitially53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalpartsTwoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6)bullSpecifyingthewholeTheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedareaIftheleftsquareisthewholeitrepresentsthefraction32iftheentirerectangleisthewholeitrepresents34bullExplainingwhatismeantbyldquoequalpartsrdquoInitiallystudentscanuseanintuitivenotionofcongruence(ldquosamesizeandsameshaperdquo)toexplainwhythepartsareequalegwhentheydivideasquareintofourequalsquaresorfourequalrectanglesArearepresentationsof14IneachrepresentationthesquareisthewholeThetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshapeandsothesameareaInthethreesquaresontherighttheshadedareais14ofthewholeareaeventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsizeStudentscometounderstandamoreprecisemeaningforldquoequalpartsrdquoasldquopartswithequalmeasurementrdquoForexamplewhenarulerisdividedintohalvesorquartersofaninchtheyseethateachsubdivisionhasthesamelengthInareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3)Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractionsinthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbersjustaseverywholenumberisobtainedbycombiningasufficientnumberof1severyfractionisobtainedbycombiningasufficientnumberofunitfractionsThenumberlineOnthenumberlinethewholeistheunitintervalthatistheintervalfrom0to1measuredbylengthIteratingthiswholetotherightmarksoffthewholenumberssothattheintervalsbetweenconsecutivewholenumbersfrom0to11to22to3etcareallofthesamelengthasshownStudentsmightthinkofthenumberlineasaninfiniterulerThenumberline0123456etcToconstructaunitfractiononthenumberlineeg13studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13Theylocatethenumber13onthenumberDraft5292011commentatcommoncoretoolswordpresscom

x2 4

x2 4 0

px 2qpx 2q 0

x 22

This sequence of equations is short-hand for a line of reasoning

If x is a number whose square is 4 thenx2 4 0 Since x2 4 px 2qpx 2q for allnumbers x it follows that px 2qpx 2q 0 Soeither x 2 0 in which case x 2 orx 2 0 in which case x 2

More might be said a justification of the last step for exampleor a check that 2 and 2 actually do satisfy the equation whichhas not been proved by this line of reasoning

ldquoifrdquo ldquothenrdquo ldquofor allrdquo and ldquothere existsrdquo In the process of learning tosolve equations students learn certain standard ldquoifndashthenrdquo moves forexample ldquoif x y then x2 y2rdquo The danger in learning algebrais that students emerge with nothing but the moves which may makeit difficult to detect incorrect or made-up moves later on Thus thefirst requirement in the standards in this domain is that studentsunderstand that solving equations is a process of reasoningA-REI1

A-REI1 Explain each step in solving a simple equation as follow-ing from the equality of numbers asserted at the previous stepstarting from the assumption that the original equation has a so-lution Construct a viable argument to justify a solution method

This does not necessarily mean that they always write out the fulltext part of the advantage of algebraic notation is its compactnessOnce students know what the code stands for they can start writingin code Thus eventually students might go from x2 4 to x 2without intermediate steps4

Understanding solving equations as a process of reasoning de-mystifies ldquoextraneousrdquo solutions that can arise under certain solutionproceduresA-REI2 The reasoning begins from the assumption that x A-REI2 Solve simple rational and radical equations in one vari-

able and give examples showing how extraneous solutions mayariseis a number that satisfies the equation and ends with a list of pos-

sibilities for x But not all the steps are necessarily reversible andso it is not necessarily true that every number in the list satisfiesthe equation For example it is true that if x 2 then x2 4 Butit is not true that if x2 4 then x 2 (it might be that x 2)Squaring both sides of an equation is a typical example of an irre-versible step another is multiplying both sides of the equation by aquantity that might be zero

With an understanding of solving equations as a reasoning pro-cess students can organize the various methods for solving differenttypes of equations into a coherent picture For example solvinglinear equations involves only steps that are reversible (adding aconstant to both sides multiplying both sides by a non-zero con-stant transforming an expression on one side into an equivalentexpression) Therefore solving linear equations does not produce ex-traneous solutionsA-REI3 The process of completing the square also A-REI3 Solve linear equations and inequalities in one variable

including equations with coefficients represented by lettersinvolves only this same list of steps and so converts any quadraticequation into an equivalent equation of the form px pq2 q thathas exactly the same solutionsA-REI4a The latter equation is easy

A-REI4a Solve quadratic equations in one variablea Use the method of completing the square to transform any

quadratic equation in x into an equation of the form px pq2 q that has the same solutions Derive the quadraticformula from this form

to solve by the reasoning explained aboveThis example sets up a theme that reoccurs throughout algebra

finding ways of transforming equations into certain standard formsthat have the same solutions For example an exponential equationof the form c dkx constant can be transformed into one of the form

4It should be noted however that calling this action ldquotaking the square root ofboth sidesrdquo is dangerous because it suggests the erroneous statement 4 2

Draft 7022013 comment at commoncoretoolswordpresscom

14

bx a the solution to which is (by definition) a logarithm Studentsobtain such solutions for specific casesF-LE4 and those intending

F-LE4 For exponential models express as a logarithm the solu-tion to abct d where a c and d are numbers and the base bis 2 10 or e evaluate the logarithm using technologystudy of advanced mathematics understand these solutions in terms

of the inverse relationship between exponents and logarithmsF-BF5 F-BF5(+) Understand the inverse relationship between expo-nents and logarithms and use this relationship to solve problemsinvolving logarithms and exponents

It is traditional for students to spend a lot of time on various tech-niques of solving quadratic equations which are often presented asif they are completely unrelated (factoring completing the squarethe quadratic formula) In fact as we have seen the key step in com-pleting the square involves at its heart factoring And the quadraticformula is nothing more than an encapsulation of the method ofcompleting the square expressing the actions repeated in solving acollection of quadratic equations with numerical coefficients with asingle formula (MP8) Rather than long drills on techniques of dubi-ous value students with an understanding of the underlying reason-ing behind all these methods are opportunistic in their applicationchoosing the method that best suits the situation at handA-REI4b b Solve quadratic equations by inspection (eg for x2

49) taking square roots completing the square thequadratic formula and factoring as appropriate to the ini-tial form of the equation Recognize when the quadraticformula gives complex solutions and write them as abifor real numbers a and b

Systems of equations Student work with solving systems of equa-tions starts the same way as work with solving equations in onevariable with an understanding of the reasoning behind the varioustechniquesA-REI5 An important step is realizing that a solution to a A-REI5 Prove that given a system of two equations in two vari-

ables replacing one equation by the sum of that equation and amultiple of the other produces a system with the same solutions

system of equations must be a solution all of the equations in thesystem simultaneously Then the process of adding one equation toanother is understood as ldquoif the two sides of one equation are equaland the two sides of another equation are equal then the sum of theleft sides of the two equations is equal to the sum of the right sidesrdquoSince this reasoning applies equally to subtraction the process ofadding one equation to another is reversible and therefore leads toan equivalent system of equations

Understanding these points for the particular case of two equa-tions in two variables is preparation for more general situationsSuch systems also have the advantage that a good graphical visu-alization is available a pair px yq satisfies two equations in twovariables if it is on both their graphs and therefore an intersectionpoint of the graphsA-REI6 A-REI6 Solve systems of linear equations exactly and approxi-

mately (eg with graphs) focusing on pairs of linear equations intwo variablesAnother important method of solving systems is the method of

substitution Again this can be understood in terms of simultaneityif px yq satisfies two equations simultaneously then the expressionfor y in terms of x obtained from the first equation should form atrue statement when substituted into the second equation Since alinear equation can always be solved for one of the variables in itthis is a good method when just one of the equations in a system islinearA-REI7 A-REI7 Solve a simple system consisting of a linear equation

and a quadratic equation in two variables algebraically and graph-icallyIn more advanced courses students see systems of linear equa-+

tions in many variables as single matrix equations in vector vari-+ablesA-REI8 A-REI9 A-REI8(+) Represent a system of linear equations as a single

matrix equation in a vector variable

A-REI9(+) Find the inverse of a matrix if it exists and use it tosolve systems of linear equations (using technology for matricesof dimension 3 3 or greater)

+

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15

Visualizing solutions graphically Just as the algebraic work withequations can be reduced to a series of algebraic moves unsup-ported by reasoning so can the graphical visualization of solu-tions The simple idea that an equation f pxq gpxq can be solved(approximately) by graphing y f pxq and y gpxq and findingthe intersection points involves a number of pieces of conceptualunderstandingA-REI11 This seemingly simple method often treated

A-REI11 Explain why the x-coordinates of the points where thegraphs of the equations y fpxq and y gpxq intersect are thesolutions of the equation fpxq gpxq find the solutions approx-imately eg using technology to graph the functions make ta-bles of values or find successive approximations Include caseswhere fpxq andor gpxq are linear polynomial rational absolutevalue exponential and logarithmic functions

as obvious involves the rather sophisticated move of reversing thereduction of an equation in two variables to an equation in onevariable Rather it seeks to convert an equation in one variablef pxq gpxq to a system of equations in two variables y f pxq andy gpxq by introducing a second variable y and setting it equal toeach side of the equation If x is a solution to the original equationthen f pxq and gpxq are equal and thus px yq is a solution to thenew system This reasoning is often tremendously compressed andpresented as obvious graphically in fact following it graphically ina specific example can be instructive

Fundamental to all of this is a simple understanding of what agraph of an equation in two variables meansA-REI10 A-REI10 Understand that the graph of an equation in two vari-

ables is the set of all its solutions plotted in the coordinate planeoften forming a curve (which could be a line)

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