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ConnectingStandards
FAMSJanuary2017
TerriL.Sebring
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1
Expressions
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NumericExpressions
MAFS.5.OA.1.1
Useparentheses,brackets,orbracesinnumericalexpressions,and
evaluateexpressionswiththesesymbols.
MAFS.5.OA.1.2
Writesimpleexpressionsthatrecordcalculationswithnumbers,and
interpretnumericalexpressionswithoutevaluatingthem.
MAFS.6.EE.1.1
Writeandevaluatenumericalexpressionsinvolvingwhole-number
exponents.
MAFS.8.EE.1.1
Knowandapplythepropertiesofintegerexponentstogenerate
equivalentnumericalexpressions.
Expressions
Connectsnumericexpressionswithalgebraicexpressions.
MAFS.6.EE.1.2
Write,read,andevaluateexpressionsinwhichlettersstandfor
numbers.
a. Writeexpressionsthatrecordoperationswithnumbersandwith
lettersstandingfornumbers.
b. Identifypartsofanexpressionusingmathematicalterms(sum,
term,product,factor,quotient,coefficient);viewoneormoreparts
ofanexpressionasasingleentity.
c. Evaluateexpressionsatspecificvaluesoftheirvariables.Include
expressionsthatarisefromformulasusedinreal-worldproblems.
Performarithmeticoperations,includingthoseinvolvingwholenumberexponents,intheconventionalorderwhenthereareno
parenthesestospecifyaparticularorder(OrderofOperations).
AlgebraicExpressions
MAFS.6.EE.1.3
Applythepropertiesofoperationstogenerateequivalentexpressions.
MAFS.6.EE.2.6
Usevariablestorepresentnumbersandwriteexpressionswhensolving
areal-worldormathematicalproblem;understandthatavariablecan
representanunknownnumber,or,dependingonthepurposeathand,
anynumberinaspecifiedset.
AlgebraicExpressions
MAFS.7.EE.1.1
Applypropertiesofoperationsasstrategiestoadd,subtract,factor,and
expandlinearexpressionswithrationalcoefficients.
MAFS.7.EE.1.2
Understandthatrewritinganexpressionindifferentformsinaproblem
contextcanshedlightontheproblemandhowthequantitiesinitare
related.
GroupDiscussion
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Mathematicalexpressionsexpresscalculationswith
numbers.Anexpressioncanconsistofjustasinglenumber.
AsstudentsprogressthroughMAFS,expressionsgrowfrom
numerictoalgebraic.
Thestandardsalsobuildastudent’sunderstanding
ofequations.Howcanyouhelpteachersunderstand
theimportanceofteachingtheirstudentsthedistinct
differencebetweenexpressionsandequationsas
wellashowanequationisarelationshipbetween
twoexpressions?
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InGradesK–5studentshavebeenwritingnumericalequationsand
simpleequationsinvolvingoneoperationwithavariable.InGrade
6theystartthesystematicstudyofequationsandinequalitiesand
methodsofsolvingthem.Solvingisaprocessofreasoningtofind
thenumberswhichmakeanequationtrue,whichcaninclude
checkingifagivennumberisasolution.
Althoughtheprocessofreasoningwilleventuallyleadtostandard
methodsforsolvingequations,studentsshouldstudyexamples
wherelookingforstructurepaysoff,suchasin4x+ 3x= 3x+ 20,
wheretheycanseethat4xmustbe20tomakethetwosidesequal.
ProgressionsfortheCommonCoreStateStandardsinMathematics
MAFS.6.EE.2.5
Understandsolvinganequationorinequalityasaprocessofanswering
aquestion:whichvaluesfromaspecifiedset,ifany,maketheequation
orinequalitytrue?Usesubstitutiontodeterminewhetheragiven
numberinaspecifiedsetmakesanequationorinequalitytrue.
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Equations
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Equations
Writing numeric equations begins asearly asKindergarten.
MAFS.K.OA.1.1
Representadditionandsubtractionwithobjects,fingers,mentalimages,
drawings,sounds(e.g.,claps),actingoutsituations,verbalexplanations,
expressions,orequations.
MAFS.5.NF.2.6
Solverealworldproblems involving multiplication of fractions and
mixed numbers,e.g.,by using visualfraction models or equations to
represent the problem.
ProgressionofSolvingEquations
MAFS.6.EE.2.7
Solvereal-worldandmathematicalproblemsbywritingandsolving
equationsoftheformx+p=qandpx =qforcasesinwhichp,qandx
areallnon-negativerationalnumbers.
MAFS.7.EE.2.4a
Usevariablestorepresentquantitiesinareal-worldormathematical
problem,andconstructsimpleequationsandinequalitiestosolve
problemsbyreasoningaboutthequantities.
Solvewordproblemsleadingtoequationsoftheformpx +q=rand
p(x+q)=r,wherep,q,andrarespecificrationalnumbers.Solve
equationsoftheseformsfluently.Compareanalgebraicsolutiontoan
arithmeticsolution,identifyingthesequenceoftheoperationsusedin
eachapproach
ProgressionofSolvingEquations
MAFS.8.EE.3.7Solvelinearequationsinonevariable.
a. Giveexamplesoflinearequationsinonevariablewithonesolution,
infinitelymanysolutions,ornosolutions.Showwhichofthese
possibilitiesisthecasebysuccessivelytransformingthegiven
equationintosimplerforms,untilanequivalentequationofthe
formx=a,a=a,ora=bresults(whereaandbaredifferent
numbers).
b. Solvelinearequationswithrationalnumbercoefficients,including
equationswhosesolutionsrequireexpandingexpressionsusingthe
distributivepropertyandcollectingliketerms.
DoAlgebra1andAlgebra2teachersknowhowsolving
equationshaveprogressedthroughthestandards?
MAFS.8.EE.3.7b
Solvelinearequationswithrationalnumbercoefficients,including
equationswhosesolutionsrequireexpandingexpressionsusingthe
distributivepropertyandcollectingliketerms.
WhatshouldtherigorbeforAlgebra1andAlgebra2?
REI.1.1(shared) Explaineachstepinsolvingasimpleequationas
followingfromtheequalityofnumbersassertedatthepreviousstep,
startingfromtheassumptionthattheoriginalequationhasasolution.
Constructaviableargumenttojustifyasolutionmethod.
MathematicalContext
REI.2.3Solvelinearequationsandinequalitiesinonevariable,
includingequationswithcoefficientsrepresentedbyletters.
MathematicalContext
GroupDiscussion
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Kindergartenthrough7thgradelaysthefoundationforastudent’s
understandingtheconceptofafunction.Beforeastudentlearnsthe
term“function”he/shegainsexperiencewithfunctions.Thenotionofa
functionisintroducedinGrade8.Linearfunctionsareamajorfocus,but
studentsarealsoexpectedtogiveexamplesoffunctionsthatarenot
linear.
Elementary:Howarethe“pattern”standards
(4.OA.3.5and5.OA.2.3)layingthefoundationforthe
conceptofafunction?
Middle:Howis7.RP.1.2layingthefoundationfor
theconceptofafunction?
High:Howcouldspiralingbackto“patterns”and
proportionalrelationshipsaswellas8.F.1.1and
8.F.1.3helpstudentsdistinguishwhenanequation
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couldalsobewrittenusingfunctionnotation?
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MAFS.4.OA.3.5
Generateanumberorshapepatternthatfollowsagivenrule.Identify
apparentfeaturesofthepatternthatwerenotexplicitintheruleitself.
MAFS.5.OA.2.3
Generatetwonumericalpatternsusingtwogivenrules.Identify
apparentrelationshipsbetweencorrespondingterms.Formordered
pairsconsistingofcorrespondingtermsfromthetwopatterns,and
graphtheorderedpairsonacoordinateplane.
MAFS.7.RP.1.2
Recognizeandrepresentproportionalrelationshipsbetweenquantities.
a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingfor
equivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthe
graphisastraightlinethroughtheorigin.
b. Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,
andverbaldescriptionsofproportionalrelationships.
c. Representproportionalrelationshipsbyequations.Forexample,iftotalcosttisproportional
tothenumbernofitemspurchasedataconstantpricep,therelationshipbetweenthetotal
costandthenumberofitemscanbeexpressedast=pn.
d. Explainwhatapoint(x,y)onthegraphofaproportionalrelationshipmeansintermsofthe
situation,withspecialattentiontothepoints(0,0)and(1,r)whereristheunitrate.
MAFS.8.F.1.1
Understandthatafunctionisarulethatassignstoeachinputexactly
oneoutput.Thegraphofafunctionisthesetoforderedpairsconsisting
ofaninputandthecorrespondingoutput.
MAFS.8.F.1.3
Interprettheequationy=mx+basdefiningalinearfunction,whose
graphisastraightline;giveexamplesoffunctionsthatarenotlinear.
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TheAlgebracategoryinhighschoolisveryclosely
allignedwiththeFunctionscategory.
Theseparationofalgebraandfunctionsisnot
intendedtoindicateapreferencebetweenan
equation-basedorafunction-basedapproach.Itis,
however,intendedtospecifythedifferenceas
mathematicalconceptsbetweenexpressionsand
equationsontheonehandandfunctionsonthe
other.Studentsoftenentercollege-level
mathematicscoursesapparentlyconflatingallthree
ofthese.
ProgressionsDocumentsfortheCommonCoreMathStandards
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Expressions,EquationsandFunctions
A-SSE.2.3(shared) Chooseandproduceanequivalentformofanexpression
torevealandexplainpropertiesofthequantityrepresentedbytheexpression.
Real-WorldContext
A-REI.2.4a(shared) Solvequadraticequationsinonevariable.
a. Usethemethodofcompletingthesquaretotransformanyquadratic
equationinx intoanequationoftheform(x– p)2 =qthathasthesame
solutions.Derivethequadraticformulafromthisform.
MathematicalContext
F-IF.3.8a(shared)Writeafunctiondefinedbyanexpressionindifferentbut
equivalentformstorevealandexplaindifferentpropertiesofthefunction.
a. Usetheprocessoffactoringandcompletingthesquareinaquadratic
functiontoshowzeros,extremevalues,andsymmetryofthegraph,and
interprettheseintermsofacontext.
MathematicalorReal-WorldContext
CreateEquationsandFunctions
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CreateEquations
MAFS.5.NF.2.6
Solverealworldproblemsinvolvingmultiplicationoffractionsand
mixednumbers,e.g.,byusingvisualfractionmodelsorequationsto
representtheproblem.Several5th gradestandardsrequirestudentsto
writenumericequations.
MAFS.6.EE.2.7
Solvereal-worldandmathematicalproblemsbywritingandsolving
equationsoftheformx+p=qandpx =qforcasesinwhichp,qandx
areallnon-negativerationalnumbers.
MAFS.7.EE.2.4
Usevariablestorepresentquantitiesinareal-worldormathematical
problem,andconstructsimpleequationsandinequalitiestosolve
problemsbyreasoningaboutthequantities.
MAFS.8.F.2.4
Constructafunctiontomodelalinearrelationshipbetweentwo
quantities.Determinetherateofchangeandinitialvalueofthefunction
fromadescriptionofarelationshiporfromtwo(x,y)values,including
readingthesefromatableorfromagraph.Interprettherateofchange
andinitialvalueofalinearfunctionintermsofthesituationitmodels,
andintermsofitsgraphoratableofvalues.
Thingstonote:
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Thestandarddoesnotmentionacertainformoftheline.
Thestandarddoesnotstatethatstudentwillchangefromoneformoftheline
toanother.
Equivalenceisathemethroughoutthestandards.
Rigorshouldincreaseasstudentsmovefrom8th gradetoAlgebra1aswellasto
Algebra2.
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CED.1.1(shared)Createequationsandinequalitiesinonevariableanduse
themtosolveproblems.
Real-WorldContext
CED.1.2(shared)Createequationsintwoormorevariablestorepresent
relationshipsbetweenquantities;graphequationsoncoordinateaxeswith
Real-WorldContext
labelsandscales.
LE.1.2 Constructlinearandexponentialfunctions,includingarithmeticand
geometricsequences,givenagraph,adescriptionofarelationship,ortwo
input-outputpairs(includingreadingthesefromatable). Real-WorldContext
BF.1.1a (shared)Writeafunctionthatdescribesarelationshipbetweentwo
quantities.a.Determineanexplicitexpression,arecursiveprocess,or
Real-WorldContext
stepsforcalculationfromacontext.
ID.2.6a Representdataontwoquantitativevariablesonascatterplot,and
describehowthevariablesarerelated.a.Fitafunctiontothedata;use
functionsfittedtodatatosolveproblemsinthecontextofthedata.Usegiven
functionsorchooseafunctionsuggestedbythecontext.Emphasizelinearand
exponentialmodels.
Real-WorldContext
Solvingequations
Algebra1andAlgebra2
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REI.2.4(shared)Solvequadraticequationsinonevariable.
a. Usethemethodofcompletingthesquaretotransformany
quadraticequationinx intoanequationoftheform(x – p)2 =q thathas
thesamesolutions.Derivethequadraticformulafromthisform.
b.Solvequadraticequationsbyinspection(e.g.,forx2 =49),taking
squareroots,completingthesquare,thequadraticformula,and
factoring,asappropriatetotheinitialformoftheequation.Recognize
whenthequadraticformulagivescomplexsolutionsandwritethemas
a ± biforrealnumbersa andb.
MathematicalContext
MAFS.8.EE.3.8
Analyzeandsolvepairsofsimultaneouslinearequations.
a.Understandthatsolutionstoasystemoftwolinearequationsintwo
variablescorrespondtopointsofintersectionoftheirgraphs,becausepoints
ofintersectionsatisfybothequationssimultaneously.
b.Solvesystemsoftwolinearequationsintwovariablesalgebraically,and
estimatesolutionsbygraphingtheequations.Solvesimplecasesbyinspection.
Forexample,3x+2y=5and3x+2y=6havenosolutionbecause3x+2y
cannotsimultaneouslybe5and6.
c.Solverealworldandmathematicalproblemsleadingtotwolinearequations
intwovariables.Forexample,givencoordinatesfortwopairsofpoints,
determinewhetherthelinethroughthefirstpairofpointsintersectstheline
throughthesecondpair.
REI.3.6(shared)Solvesystemsoflinearequationsexactlyand
approximately(e.g.,withgraphs),focusingonpairsoflinearequations
intwovariables.
MathematicalorReal-WorldContext
REI.4.11(shared)Explainwhythex-coordinatesofthepointswherethe
graphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsof
theequationf(x)=g(x);findthesolutionsapproximately(e.g.,using
technologytographthefunctions,maketablesofvalues,orfind
successiveapproximations).Includecaseswheref(x)and/org(x)are
linear,polynomial,rational,absolutevalue,exponential,and
logarithmicfunctions.
Thisstandardisnotaboutsolving.
MathematicalorReal-WorldContext
Equivalentforms
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Equivalence
InK-5,studentsproduceequivalentnumeric
expressionsusingpropertiesofoperationsasthey
workwithwholenumbers,decimalsandfractions.
Equivalencecontinuesasstudentsworkwith
algebraicexpressionsinmiddleschool.Thestudentis
alsointroducedtounderstandingequivalencewithin
areal-worldcontext.
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EquivalentExpressions
MAFS.6.EE.1.3
Applythepropertiesofoperationstogenerateequivalentexpressions
MAFS.6.EE.1.4
Identifywhentwoexpressionsareequivalent(i.e.,whenthetwo
expressionsnamethesamenumberregardlessofwhichvalueis
substitutedintothem).
MAFS.7.EE.1.1
Applypropertiesofoperationsasstrategiestoadd,subtract,factor,and
expandlinearexpressionswithrationalcoefficients.
MAFS.7.EE.1.2
Understandthatrewritinganexpressionindifferentformsinaproblem
contextcanshedlightontheproblemandhowthequantitiesinitare
related.
SSE.1.2(shared)Usethestructureofanexpressiontoidentifywaysto
rewriteit.Forexample,seex4 – y4 as(x2)2 – (y2)2,thusrecognizingitasa
differenceofsquaresthatcanbefactoredas(x2 – y2)(x2 +y2).
MathematicalorReal-WorldContext
SSE.2.3 (shared)Chooseandproduceanequivalentformofan
expressiontorevealandexplainpropertiesofthequantityrepresented
bytheexpression.
a.Factoraquadraticexpressiontorevealthezerosofthe
functionitdefines.
b.Completethesquareinaquadraticexpressiontorevealthe
maximumorminimumvalueofthefunctionitdefines. Real-WorldContext
IF.3.8(shared)Writeafunctiondefinedbyanexpressionindifferentbut
equivalentformstorevealandexplaindifferentpropertiesofthe
function.
a.Usetheprocessoffactoringandcompletingthesquareina
quadraticfunctiontoshowzeros,extremevalues,and
symmetryofthegraph,andinterprettheseintermsofa
context.
Real-WorldContext
SSE.2.3 (shared)Chooseandproduceanequivalentformofan
expressiontorevealandexplainpropertiesofthequantityrepresented
bytheexpression.
c.Usethepropertiesofexponents totransformexpressionsfor
exponentialfunctions.Forexample,theexpression1.15tcan
berewrittenas(1.151/12)12t ≈(1.012)12ttorevealthe
approximateequivalentmonthlyinterestrateiftheannual
rateis15%.
Real-WorldContext
IF.3.8(shared) Writeafunctiondefinedbyanexpressionindifferent
butequivalentformstorevealandexplaindifferentpropertiesofthe
function.
b.Usethepropertiesofexponents tointerpretexpressionsfor
exponentialfunctions.Forexample,identifypercentrateof
changeinfunctionssuchasy =(1.02)t ,y =(0.97)t ,y =(1.01)12t,andy
=(1.2)t/10 andclassifythemasrepresentingexponentialgrowthor
decay.
MathematicalorReal-WorldContext
DomainandConstraints
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CED.1.3(shared)Representconstraintsbyequationsorinequalitiesand
bysystemsofequationsand/orinequalities,andinterpretsolutionsas
viableornonviableoptionsinamodelingcontext.Forexample,
representinequalitiesdescribingnutritionalandcostconstraintson
combinationsofdifferentfoods. Real-WorldContext
IF.2.5(shared) Relatethedomainofafunctiontoitsgraphand,where
applicable,tothequantitativerelationshipitdescribes.Forexample,if
thefunctionh(n)givesthenumberofperson-hoursittakestoassemble
nenginesinafactory,thenthe positiveintegerswouldbean
appropriatedomainforthefunction. Real-WorldContext
KeyFeatures
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MAFS.8.F.1.2
Comparepropertiesoftwofunctionseachrepresentedinadifferent
way(algebraically,graphically,numericallyintables,orbyverbal
descriptions).
IF.2.4(shared) Forafunctionthatmodelsarelationshipbetween
twoquantities,interpretkeyfeaturesofgraphsandtablesintermsof
thequantitiesandsketchgraphsshowingkeyfeaturesgivenaverbal
descriptionoftherelationship.Keyfeaturesinclude:intercepts;
intervalswherethefunctionisincreasing,decreasing,positive,or
negative;relativemaximumsandminimums;symmetries;endbehavior;
Real-WorldContext
andperiodicity.
IF.3.9(shared) Comparepropertiesoftwofunctionseachrepresented
inadifferentway(algebraically,graphically,numericallyintables,orby
verbaldescriptions).Forexample,givenagraphofonequadratic
functionandanalgebraicexpressionforanother,saywhichhasthe
largermaximum.
MathematicalorReal-WorldContext
Interpreting
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MAFS.8.F.2.4
Constructafunctiontomodelalinearrelationshipbetweentwo
quantities.Determinetherateofchangeandinitialvalueofthefunction
fromadescriptionofarelationshiporfromtwo(x,y)values,including
readingthesefromatableorfromagraph.Interprettherateofchange
andinitialvalueofalinearfunctionintermsofthesituationitmodels,
andintermsofitsgraphoratableofvalues.
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SSE.1.1(shared)Interpretexpressionsthatrepresentaquantityintermsofits
context.
a.Interpretpartsofanexpression,suchasterms,factors,andcoefficients.
b.Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasa
singleentity.Forexample,interpretP(1+r)n astheproductofPandafactornot
Real-WorldContext
dependingonP.
IF.2.6(shared)Calculateandinterprettheaveragerateofchangeofa
function(presentedsymbolicallyorasatable)overaspecifiedinterval.
Real-WorldContext
Estimatetherateofchangefromagraph.
ID.3.7 Interprettheslope(rateofchange)andtheintercept(constantterm)
Real-WorldContext
ofalinearmodelinthecontextofthedata.
LE.2.5(shared)Interprettheparametersinalinearorexponentialfunctionin
Real-WorldContext
termsofacontext.
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CommonStandards
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A-APR.1.1
A-APR.2.3
A-CED.1.1
A-CED.1.2
A-CED.1.3
A-CED.1.4
A-REI.1.1
A-REI.2.4
A-REI.3.6
A-REI.4.11
A-SSE.1.1
A-SSE.1.2
F-BF.1.1
F-BF.2.3
F-IF.2.4
F-IF.2.5
F-IF.2.6
F-IF.3.7a,b,c,d,ande
F-IF.3.8
F-IF.3.9
F-LE.2.5
N-RN.1.1
N-RN.1.2
RealWorldContext
A-SSE.2.3
RealWorldorMathematicalContext
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