mathematics Standards for High School · mathematics Standards for High School ... 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents

Common Core State StandardS for matHematICSH

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mathematics Standards for High SchoolThehighschoolstandardsspecifythemathematicsthatallstudentsshould

studyinordertobecollegeandcareerready.Additionalmathematicsthat

studentsshouldlearninordertotakeadvancedcoursessuchascalculus,

advancedstatistics,ordiscretemathematicsisindicatedby(+),asinthis

example:

(+)Representcomplexnumbersonthecomplexplaneinrectangular

andpolarform(includingrealandimaginarynumbers).

Allstandardswithouta(+)symbolshouldbeinthecommonmathematics

curriculumforallcollegeandcareerreadystudents.Standardswitha(+)

symbolmayalsoappearincoursesintendedforallstudents.

Thehighschoolstandardsarelistedinconceptualcategories:

• NumberandQuantity

• Algebra

• Functions

• Modeling

• Geometry

• StatisticsandProbability

Conceptualcategoriesportrayacoherentviewofhighschool

mathematics;astudent’sworkwithfunctions,forexample,crossesa

numberoftraditionalcourseboundaries,potentiallyupthroughand

includingcalculus.

Modelingisbestinterpretednotasacollectionofisolatedtopicsbutin

relationtootherstandards.MakingmathematicalmodelsisaStandardfor

MathematicalPractice,andspecificmodelingstandardsappearthroughout

thehighschoolstandardsindicatedbyastarsymbol(★).Thestarsymbol

sometimesappearsontheheadingforagroupofstandards;inthatcase,it

shouldbeunderstoodtoapplytoallstandardsinthatgroup.


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mathematics | High School—number and QuantityNumbers and Number Systems.Duringtheyearsfromkindergartentoeighth

grade,studentsmustrepeatedlyextendtheirconceptionofnumber.Atfirst,

“number”means“countingnumber”:1,2,3...Soonafterthat,0isusedtorepresent

“none”andthewholenumbersareformedbythecountingnumberstogether

withzero.Thenextextensionisfractions.Atfirst,fractionsarebarelynumbers

andtiedstronglytopictorialrepresentations.Yetbythetimestudentsunderstand

divisionoffractions,theyhaveastrongconceptoffractionsasnumbersandhave

connectedthem,viatheirdecimalrepresentations,withthebase-tensystemused

torepresentthewholenumbers.Duringmiddleschool,fractionsareaugmentedby

negativefractionstoformtherationalnumbers.InGrade8,studentsextendthis

systemoncemore,augmentingtherationalnumberswiththeirrationalnumbers

toformtherealnumbers.Inhighschool,studentswillbeexposedtoyetanother

extensionofnumber,whentherealnumbersareaugmentedbytheimaginary

numberstoformthecomplexnumbers.

Witheachextensionofnumber,themeaningsofaddition,subtraction,

multiplication,anddivisionareextended.Ineachnewnumbersystem—integers,

rationalnumbers,realnumbers,andcomplexnumbers—thefouroperationsstay

thesameintwoimportantways:Theyhavethecommutative,associative,and

distributivepropertiesandtheirnewmeaningsareconsistentwiththeirprevious

meanings.

Extendingthepropertiesofwhole-numberexponentsleadstonewandproductive

notation.Forexample,propertiesofwhole-numberexponentssuggestthat(51/3)3

shouldbe5(1/3)3=51=5andthat51/3shouldbethecuberootof5.

Calculators,spreadsheets,andcomputeralgebrasystemscanprovidewaysfor

studentstobecomebetteracquaintedwiththesenewnumbersystemsandtheir

notation.Theycanbeusedtogeneratedatafornumericalexperiments,tohelp

understandtheworkingsofmatrix,vector,andcomplexnumberalgebra,andto

experimentwithnon-integerexponents.

Quantities.Inrealworldproblems,theanswersareusuallynotnumbersbut

quantities:numberswithunits,whichinvolvesmeasurement.Intheirworkin

measurementupthroughGrade8,studentsprimarilymeasurecommonlyused

attributessuchaslength,area,andvolume.Inhighschool,studentsencountera

widervarietyofunitsinmodeling,e.g.,acceleration,currencyconversions,derived

quantitiessuchasperson-hoursandheatingdegreedays,socialscienceratessuch

asper-capitaincome,andratesineverydaylifesuchaspointsscoredpergameor

battingaverages.Theyalsoencounternovelsituationsinwhichtheythemselves

mustconceivetheattributesofinterest.Forexample,tofindagoodmeasureof

overallhighwaysafety,theymightproposemeasuressuchasfatalitiesperyear,

fatalitiesperyearperdriver,orfatalitiespervehicle-miletraveled.Suchaconceptual

processissometimescalledquantification.Quantificationisimportantforscience,

aswhensurfaceareasuddenly“standsout”asanimportantvariableinevaporation.

Quantificationisalsoimportantforcompanies,whichmustconceptualizerelevant

attributesandcreateorchoosesuitablemeasuresforthem.


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The Real Number System

• extend the properties of exponents to rational exponents

• Use properties of rational and irrational numbers.

Quantities

• reason quantitatively and use units to solve problems

The Complex Number System

• Perform arithmetic operations with complex numbers

• represent complex numbers and their operations on the complex plane

• Use complex numbers in polynomial identities and equations

Vector and Matrix Quantities

• represent and model with vector quantities.

• Perform operations on vectors.

• Perform operations on matrices and use matrices in applications.

mathematical Practices

1. Makesenseofproblemsandpersevereinsolvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritiquethereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

number and Quantity overview


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the real number System n-rn

Extend the properties of exponents to rational exponents.

1. Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesofintegerexponentstothosevalues,allowingforanotationforradicalsintermsofrationalexponents.For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

2. Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthepropertiesofexponents.

Use properties of rational and irrational numbers.

3. Explainwhythesumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.

Quantities★ n-Q

Reason quantitatively and use units to solve problems.

1. Useunitsasawaytounderstandproblemsandtoguidethesolutionofmulti-stepproblems;chooseandinterpretunitsconsistentlyinformulas;chooseandinterpretthescaleandtheoriginingraphsanddatadisplays.

2. Defineappropriatequantitiesforthepurposeofdescriptivemodeling.

3. Choosealevelofaccuracyappropriatetolimitationsonmeasurementwhenreportingquantities.

the Complex number System n-Cn

Perform arithmetic operations with complex numbers.

1. Knowthereisacomplexnumberisuchthati2=–1,andeverycomplexnumberhastheforma +bi withaandbreal.

2. Usetherelationi2=–1andthecommutative,associative,anddistributivepropertiestoadd,subtract,andmultiplycomplexnumbers.

3. (+)Findtheconjugateofacomplexnumber;useconjugatestofindmoduliandquotientsofcomplexnumbers.

Represent complex numbers and their operations on the complex plane.

4. (+)Representcomplexnumbersonthecomplexplaneinrectangularandpolarform(includingrealandimaginarynumbers),andexplainwhytherectangularandpolarformsofagivencomplexnumberrepresentthesamenumber.

5. (+)Representaddition,subtraction,multiplication,andconjugationofcomplexnumbersgeometricallyonthecomplexplane;usepropertiesofthisrepresentationforcomputation.For example, (–1+√3i)3=8because(–1+√3i)has modulus2and argument120°.

6. (+)Calculatethedistancebetweennumbersinthecomplexplaneasthemodulusofthedifference,andthemidpointofasegmentastheaverageofthenumbersatitsendpoints.

Use complex numbers in polynomial identities and equations.

7. Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions.

8. (+)Extendpolynomialidentitiestothecomplexnumbers.For example, rewrite x2+4as(x+2i)(x–2i).

9. (+)KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.


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Vector and matrix Quantities n-Vm

Represent and model with vector quantities.

1. (+)Recognizevectorquantitiesashavingbothmagnitudeanddirection.Representvectorquantitiesbydirectedlinesegments,anduseappropriatesymbolsforvectorsandtheirmagnitudes(e.g.,v,|v|,||v||,v).

2. (+)Findthecomponentsofavectorbysubtractingthecoordinatesofaninitialpointfromthecoordinatesofaterminalpoint.

3. (+)Solveproblemsinvolvingvelocityandotherquantitiesthatcanberepresentedbyvectors.

Perform operations on vectors.

4. (+)Addandsubtractvectors.

a. Addvectorsend-to-end,component-wise,andbytheparallelogramrule.Understandthatthemagnitudeofasumoftwovectorsistypicallynotthesumofthemagnitudes.

b. Giventwovectorsinmagnitudeanddirectionform,determinethemagnitudeanddirectionoftheirsum.

c. Understandvectorsubtractionv–wasv+(–w),where–wistheadditiveinverseofw,withthesamemagnitudeaswandpointingintheoppositedirection.Representvectorsubtractiongraphicallybyconnectingthetipsintheappropriateorder,andperformvectorsubtractioncomponent-wise.

5. (+)Multiplyavectorbyascalar.

a. Representscalarmultiplicationgraphicallybyscalingvectorsandpossiblyreversingtheirdirection;performscalarmultiplicationcomponent-wise,e.g.,asc(v

x,v

y)=(cv

x,cv

y).

b. Computethemagnitudeofascalarmultiplecvusing||cv||=|c|v.Computethedirectionofcvknowingthatwhen|c|v≠0,thedirectionofcviseitheralongv(forc>0)oragainstv(forc<0).

Perform operations on matrices and use matrices in applications.

6. (+)Usematricestorepresentandmanipulatedata,e.g.,torepresentpayoffsorincidencerelationshipsinanetwork.

7. (+)Multiplymatricesbyscalarstoproducenewmatrices,e.g.,aswhenallofthepayoffsinagamearedoubled.

8. (+)Add,subtract,andmultiplymatricesofappropriatedimensions.

9. (+)Understandthat,unlikemultiplicationofnumbers,matrixmultiplicationforsquarematricesisnotacommutativeoperation,butstillsatisfiestheassociativeanddistributiveproperties.

10. (+)Understandthatthezeroandidentitymatricesplayaroleinmatrixadditionandmultiplicationsimilartotheroleof0and1intherealnumbers.Thedeterminantofasquarematrixisnonzeroifandonlyifthematrixhasamultiplicativeinverse.

11. (+)Multiplyavector(regardedasamatrixwithonecolumn)byamatrixofsuitabledimensionstoproduceanothervector.Workwithmatricesastransformationsofvectors.

12. (+)Workwith2×2matricesastransformationsoftheplane,andinterprettheabsolutevalueofthedeterminantintermsofarea.


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mathematics | High School—algebraExpressions.Anexpressionisarecordofacomputationwithnumbers,symbolsthatrepresentnumbers,arithmeticoperations,exponentiation,and,atmoreadvancedlevels,theoperationofevaluatingafunction.Conventionsabouttheuseofparenthesesandtheorderofoperationsassurethateachexpressionisunambiguous.Creatinganexpressionthatdescribesacomputationinvolvingageneralquantityrequirestheabilitytoexpressthecomputationingeneralterms,abstractingfromspecificinstances.

Readinganexpressionwithcomprehensioninvolvesanalysisofitsunderlyingstructure.Thismaysuggestadifferentbutequivalentwayofwritingtheexpressionthatexhibitssomedifferentaspectofitsmeaning.Forexample,p+0.05pcanbeinterpretedastheadditionofa5%taxtoapricep.Rewritingp+0.05pas1.05pshowsthataddingataxisthesameasmultiplyingthepricebyaconstantfactor.

Algebraicmanipulationsaregovernedbythepropertiesofoperationsandexponents,andtheconventionsofalgebraicnotation.Attimes,anexpressionistheresultofapplyingoperationstosimplerexpressions.Forexample,p+0.05pisthesumofthesimplerexpressionspand0.05p.Viewinganexpressionastheresultofoperationonsimplerexpressionscansometimesclarifyitsunderlyingstructure.

Aspreadsheetoracomputeralgebrasystem(CAS)canbeusedtoexperimentwithalgebraicexpressions,performcomplicatedalgebraicmanipulations,andunderstandhowalgebraicmanipulationsbehave.

Equations and inequalities.Anequationisastatementofequalitybetweentwoexpressions,oftenviewedasaquestionaskingforwhichvaluesofthevariablestheexpressionsoneithersideareinfactequal.Thesevaluesarethesolutionstotheequation.Anidentity,incontrast,istrueforallvaluesofthevariables;identitiesareoftendevelopedbyrewritinganexpressioninanequivalentform.

Thesolutionsofanequationinonevariableformasetofnumbers;thesolutionsofanequationintwovariablesformasetoforderedpairsofnumbers,whichcanbeplottedinthecoordinateplane.Twoormoreequationsand/orinequalitiesformasystem.Asolutionforsuchasystemmustsatisfyeveryequationandinequalityinthesystem.

Anequationcanoftenbesolvedbysuccessivelydeducingfromitoneormoresimplerequations.Forexample,onecanaddthesameconstanttobothsideswithoutchangingthesolutions,butsquaringbothsidesmightleadtoextraneoussolutions.Strategiccompetenceinsolvingincludeslookingaheadforproductivemanipulationsandanticipatingthenatureandnumberofsolutions.

Someequationshavenosolutionsinagivennumbersystem,buthaveasolutioninalargersystem.Forexample,thesolutionofx+1=0isaninteger,notawholenumber;thesolutionof2x+1=0isarationalnumber,notaninteger;thesolutionsofx2–2=0arerealnumbers,notrationalnumbers;andthesolutionsofx2+2=0arecomplexnumbers,notrealnumbers.

Thesamesolutiontechniquesusedtosolveequationscanbeusedtorearrangeformulas.Forexample,theformulafortheareaofatrapezoid,A=((b

1+b

2)/2)h,can

besolvedforhusingthesamedeductiveprocess.

Inequalitiescanbesolvedbyreasoningaboutthepropertiesofinequality.Many,butnotall,ofthepropertiesofequalitycontinuetoholdforinequalitiesandcanbeusefulinsolvingthem.

Connections to Functions and Modeling. Expressionscandefinefunctions,andequivalentexpressionsdefinethesamefunction.Askingwhentwofunctionshavethesamevalueforthesameinputleadstoanequation;graphingthetwofunctionsallowsforfindingapproximatesolutionsoftheequation.Convertingaverbaldescriptiontoanequation,inequality,orsystemoftheseisanessentialskillinmodeling.


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Seeing Structure in Expressions

• Interpret the structure of expressions

• Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Expressions

• Perform arithmetic operations on polynomials

• Understand the relationship between zeros and factors of polynomials

• Use polynomial identities to solve problems

• rewrite rational expressions

Creating Equations

• Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities

• Understand solving equations as a process of reasoning and explain the reasoning

• Solve equations and inequalities in one variable

• Solve systems of equations

• represent and solve equations and inequalities graphically










algebra overview


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Seeing Structure in expressions a-SSe

Interpret the structure of expressions

1. Interpretexpressionsthatrepresentaquantityintermsofitscontext.★

a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.

b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.For example, interpretP(1+r)nas the product of P and a factor not depending on P.

2. Usethestructureofanexpressiontoidentifywaystorewriteit.For example, see x4–y4as(x2)2–(y2)2,thus recognizing it as a difference of squares that can be factored as(x2–y2)(x2+y2).

Write expressions in equivalent forms to solve problems

3. Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesofthequantityrepresentedbytheexpression.★

a. Factoraquadraticexpressiontorevealthezerosofthefunctionitdefines.

b. Completethesquareinaquadraticexpressiontorevealthemaximumorminimumvalueofthefunctionitdefines.

c. Usethepropertiesofexponentstotransformexpressionsforexponentialfunctions. For example the expression1.15tcan be rewritten as(1.151/12)12t≈1.01212tto reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

4. Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusetheformulatosolveproblems.For example, calculate mortgage payments.★

arithmetic with Polynomials and rational expressions a-aPr

Perform arithmetic operations on polynomials

1. Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyareclosedundertheoperationsofaddition,subtraction,andmultiplication;add,subtract,andmultiplypolynomials.

Understand the relationship between zeros and factors of polynomials

2. KnowandapplytheRemainderTheorem:Forapolynomialp(x)andanumbera,theremainderondivisionbyx–aisp(a),sop(a)=0ifandonlyif(x–a)isafactorofp(x).

3. Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezerostoconstructaroughgraphofthefunctiondefinedbythepolynomial.

Use polynomial identities to solve problems

4. Provepolynomialidentitiesandusethemtodescribenumericalrelationships.For example, the polynomial identity(x2+y2)2=(x2–y2)2+(2xy)2can be used to generate Pythagorean triples.

5. (+)KnowandapplytheBinomialTheoremfortheexpansionof(x+y)ninpowersofxandyforapositiveintegern,wherexandyareanynumbers,withcoefficientsdeterminedforexamplebyPascal’sTriangle.1

1TheBinomialTheoremcanbeprovedbymathematicalinductionorbyacom-binatorialargument.


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Rewrite rational expressions

6. Rewritesimplerationalexpressionsindifferentforms;writea(x)/b(x)intheformq(x)+r(x)/b(x),wherea(x),b(x),q(x),andr(x)arepolynomialswiththedegreeofr(x)lessthanthedegreeofb(x),usinginspection,longdivision,or,forthemorecomplicatedexamples,acomputeralgebrasystem.

7. (+)Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,closedunderaddition,subtraction,multiplication,anddivisionbyanonzerorationalexpression;add,subtract,multiply,anddividerationalexpressions.

Creating equations★ a-Ced

Create equations that describe numbers or relationships

1. Createequationsandinequalitiesinonevariableandusethemtosolveproblems.Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

2. Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswithlabelsandscales.

3. Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornon-viableoptionsinamodelingcontext.For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

4. Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.For example, rearrange Ohm’s law V = IR to highlight resistance R.

reasoning with equations and Inequalities a-reI

Understand solving equations as a process of reasoning and explain the reasoning

1. Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbersassertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution.Constructaviableargumenttojustifyasolutionmethod.

2. Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneoussolutionsmayarise.

Solve equations and inequalities in one variable

3. Solvelinearequationsandinequalitiesinonevariable,includingequationswithcoefficientsrepresentedbyletters.

4. Solvequadraticequationsinonevariable.

a. Usethemethodofcompletingthesquaretotransformanyquadraticequationinxintoanequationoftheform(x–p)2=qthathasthesamesolutions.Derivethequadraticformulafromthisform.

b. Solvequadraticequationsbyinspection(e.g.,forx2=49),takingsquareroots,completingthesquare,thequadraticformulaandfactoring,asappropriatetotheinitialformoftheequation.Recognizewhenthequadraticformulagivescomplexsolutionsandwritethemasa±biforrealnumbersaandb.

Solve systems of equations

5. Provethat,givenasystemoftwoequationsintwovariables,replacingoneequationbythesumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.


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6. Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.

7. Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariablesalgebraicallyandgraphically.For example, find the points of intersection between the line y=–3xandthecirclex2+y2=3.

8. (+)Representasystemoflinearequationsasasinglematrixequationinavectorvariable.

9. (+)Findtheinverseofamatrixifitexistsanduseittosolvesystemsoflinearequations(usingtechnologyformatricesofdimension3×3orgreater).

Represent and solve equations and inequalities graphically

10. Understandthatthegraphofanequationintwovariablesisthesetofallitssolutionsplottedinthecoordinateplane,oftenformingacurve(whichcouldbealine).

11. Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,usingtechnologytographthefunctions,maketablesofvalues,orfindsuccessiveapproximations.Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolutevalue,exponential,andlogarithmicfunctions.★

12. Graphthesolutionstoalinearinequalityintwovariablesasahalf-plane(excludingtheboundaryinthecaseofastrictinequality),andgraphthesolutionsettoasystemoflinearinequalitiesintwovariablesastheintersectionofthecorrespondinghalf-planes.


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mathematics | High School—functionsFunctionsdescribesituationswhereonequantitydeterminesanother.Forexample,thereturnon$10,000investedatanannualizedpercentagerateof4.25%isafunctionofthelengthoftimethemoneyisinvested.Becausewecontinuallymaketheoriesaboutdependenciesbetweenquantitiesinnatureandsociety,functionsareimportanttoolsintheconstructionofmathematicalmodels.

Inschoolmathematics,functionsusuallyhavenumericalinputsandoutputsandareoftendefinedbyanalgebraicexpression.Forexample,thetimeinhoursittakesforacartodrive100milesisafunctionofthecar’sspeedinmilesperhour,v;theruleT(v)=100/vexpressesthisrelationshipalgebraicallyanddefinesafunctionwhosenameisT.

Thesetofinputstoafunctioniscalleditsdomain.Weofteninferthedomaintobeallinputsforwhichtheexpressiondefiningafunctionhasavalue,orforwhichthefunctionmakessenseinagivencontext.

Afunctioncanbedescribedinvariousways,suchasbyagraph(e.g.,thetraceofaseismograph);byaverbalrule,asin,“I’llgiveyouastate,yougivemethecapitalcity;”byanalgebraicexpressionlikef(x)=a+bx;orbyarecursiverule.Thegraphofafunctionisoftenausefulwayofvisualizingtherelationshipofthefunctionmodels,andmanipulatingamathematicalexpressionforafunctioncanthrowlightonthefunction’sproperties.

Functionspresentedasexpressionscanmodelmanyimportantphenomena.Twoimportantfamiliesoffunctionscharacterizedbylawsofgrowtharelinearfunctions,whichgrowataconstantrate,andexponentialfunctions,whichgrowataconstantpercentrate.Linearfunctionswithaconstanttermofzerodescribeproportionalrelationships.

Agraphingutilityoracomputeralgebrasystemcanbeusedtoexperimentwithpropertiesofthesefunctionsandtheirgraphsandtobuildcomputationalmodelsoffunctions,includingrecursivelydefinedfunctions.

Connections to Expressions, Equations, Modeling, and Coordinates.

Determininganoutputvalueforaparticularinputinvolvesevaluatinganexpression;findinginputsthatyieldagivenoutputinvolvessolvinganequation.Questionsaboutwhentwofunctionshavethesamevalueforthesameinputleadtoequations,whosesolutionscanbevisualizedfromtheintersectionoftheirgraphs.Becausefunctionsdescriberelationshipsbetweenquantities,theyarefrequentlyusedinmodeling.Sometimesfunctionsaredefinedbyarecursiveprocess,whichcanbedisplayedeffectivelyusingaspreadsheetorothertechnology.


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Interpreting Functions

• Understand the concept of a function and use function notation

• Interpret functions that arise in applications in terms of the context

• analyze functions using different representations

Building Functions

• Build a function that models a relationship between two quantities

• Build new functions from existing functions

Linear, Quadratic, and Exponential Models

• Construct and compare linear, quadratic, and exponential models and solve problems

• Interpret expressions for functions in terms of the situation they model

Trigonometric Functions

• extend the domain of trigonometric functions using the unit circle

• model periodic phenomena with trigonometric functions

• Prove and apply trigonometric identities










functions overview


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Interpreting functions f-If

Understand the concept of a function and use function notation

1. Understandthatafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange.Iffisafunctionandxisanelementofitsdomain,thenf(x)denotestheoutputoffcorrespondingtotheinputx.Thegraphoffisthegraphoftheequationy=f(x).

2. Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpretstatementsthatusefunctionnotationintermsofacontext.

3. Recognizethatsequencesarefunctions,sometimesdefinedrecursively,whosedomainisasubsetoftheintegers.For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

Interpret functions that arise in applications in terms of the context

4. Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

5. Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipitdescribes.For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

6. Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval.Estimatetherateofchangefromagraph.★

Analyze functions using different representations

7. Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyformorecomplicatedcases.★

a. Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.

b. Graphsquareroot,cuberoot,andpiecewise-definedfunctions,includingstepfunctionsandabsolutevaluefunctions.

c. Graphpolynomialfunctions,identifyingzeroswhensuitablefactorizationsareavailable,andshowingendbehavior.

d. (+)Graphrationalfunctions,identifyingzerosandasymptoteswhensuitablefactorizationsareavailable,andshowingendbehavior.

e. Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,andamplitude.

8. Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealandexplaindifferentpropertiesofthefunction.

a. Usetheprocessoffactoringandcompletingthesquareinaquadraticfunctiontoshowzeros,extremevalues,andsymmetryofthegraph,andinterprettheseintermsofacontext.

b. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions.For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.


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9. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Building functions f-Bf

Build a function that models a relationship between two quantities

1. Writeafunctionthatdescribesarelationshipbetweentwoquantities.★

a. Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.

b. Combinestandardfunctiontypesusingarithmeticoperations.For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

c. (+)Composefunctions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

2. Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicitformula,usethemtomodelsituations,andtranslatebetweenthetwoforms.★

Build new functions from existing functions

3. Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Include recognizing even and odd functions from their graphs and algebraic expressions for them.

4. Findinversefunctions.

a. Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.

b. (+)Verifybycompositionthatonefunctionistheinverseofanother.

c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasaninverse.

d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.

5. (+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethisrelationshiptosolveproblemsinvolvinglogarithmsandexponents.

Linear, Quadratic, and exponential models★ f-Le

Construct and compare linear, quadratic, and exponential models and solve problems

1. Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwithexponentialfunctions.

a. Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervals,andthatexponentialfunctionsgrowbyequalfactorsoverequalintervals.

b. Recognizesituationsinwhichonequantitychangesataconstantrateperunitintervalrelativetoanother.

c. Recognizesituationsinwhichaquantitygrowsordecaysbyaconstantpercentrateperunitintervalrelativetoanother.


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2. Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefromatable).

3. Observeusinggraphsandtablesthataquantityincreasingexponentiallyeventuallyexceedsaquantityincreasinglinearly,quadratically,or(moregenerally)asapolynomialfunction.

4. Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddarenumbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.

Interpret expressions for functions in terms of the situation they model

5. Interprettheparametersinalinearorexponentialfunctionintermsofacontext.

trigonometric functions f-tf

Extend the domain of trigonometric functions using the unit circle

1. Understandradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle.

2. Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoallrealnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwisearoundtheunitcircle.

3. (+)Usespecialtrianglestodeterminegeometricallythevaluesofsine,cosine,tangentforπ/3,π/4andπ/6,andusetheunitcircletoexpressthevaluesofsine,cosine,andtangentforπ–x,π+x,and2π–xintermsoftheirvaluesforx,wherexisanyrealnumber.

4. (+)Usetheunitcircletoexplainsymmetry(oddandeven)andperiodicityoftrigonometricfunctions.

Model periodic phenomena with trigonometric functions

5. Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,frequency,andmidline.★

6. (+)Understandthatrestrictingatrigonometricfunctiontoadomainonwhichitisalwaysincreasingoralwaysdecreasingallowsitsinversetobeconstructed.

7. (+)Useinversefunctionstosolvetrigonometricequationsthatariseinmodelingcontexts;evaluatethesolutionsusingtechnology,andinterpretthemintermsofthecontext.★

Prove and apply trigonometric identities

8. ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseittofindsin(θ),cos(θ),ortan(θ)givensin(θ),cos(θ),ortan(θ)andthequadrantoftheangle.

9. (+)Provetheadditionandsubtractionformulasforsine,cosine,andtangentandusethemtosolveproblems.


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mathematics | High School—modelingModelinglinksclassroommathematicsandstatisticstoeverydaylife,work,anddecision-making.Modelingistheprocessofchoosingandusingappropriatemathematicsandstatisticstoanalyzeempiricalsituations,tounderstandthembetter,andtoimprovedecisions.Quantitiesandtheirrelationshipsinphysical,economic,publicpolicy,social,andeverydaysituationscanbemodeledusingmathematicalandstatisticalmethods.Whenmakingmathematicalmodels,technologyisvaluableforvaryingassumptions,exploringconsequences,andcomparingpredictionswithdata.

Amodelcanbeverysimple,suchaswritingtotalcostasaproductofunitpriceandnumberbought,orusingageometricshapetodescribeaphysicalobjectlikeacoin.Evensuchsimplemodelsinvolvemakingchoices.Itisuptouswhethertomodelacoinasathree-dimensionalcylinder,orwhetheratwo-dimensionaldiskworkswellenoughforourpurposes.Othersituations—modelingadeliveryroute,aproductionschedule,oracomparisonofloanamortizations—needmoreelaboratemodelsthatuseothertoolsfromthemathematicalsciences.Real-worldsituationsarenotorganizedandlabeledforanalysis;formulatingtractablemodels,representingsuchmodels,andanalyzingthemisappropriatelyacreativeprocess.Likeeverysuchprocess,thisdependsonacquiredexpertiseaswellascreativity.

Someexamplesofsuchsituationsmightinclude:

• Estimatinghowmuchwaterandfoodisneededforemergencyreliefinadevastatedcityof3millionpeople,andhowitmightbedistributed.

• Planningatabletennistournamentfor7playersataclubwith4tables,whereeachplayerplaysagainsteachotherplayer.

• Designingthelayoutofthestallsinaschoolfairsoastoraiseasmuchmoneyaspossible.

• Analyzingstoppingdistanceforacar.

• Modelingsavingsaccountbalance,bacterialcolonygrowth,orinvestmentgrowth.

• Engagingincriticalpathanalysis,e.g.,appliedtoturnaroundofanaircraftatanairport.

• Analyzingriskinsituationssuchasextremesports,pandemics,andterrorism.

• Relatingpopulationstatisticstoindividualpredictions.

Insituationslikethese,themodelsdeviseddependonanumberoffactors:Howpreciseananswerdowewantorneed?Whataspectsofthesituationdowemostneedtounderstand,control,oroptimize?Whatresourcesoftimeandtoolsdowehave?Therangeofmodelsthatwecancreateandanalyzeisalsoconstrainedbythelimitationsofourmathematical,statistical,andtechnicalskills,andourabilitytorecognizesignificantvariablesandrelationshipsamongthem.Diagramsofvariouskinds,spreadsheetsandothertechnology,andalgebraarepowerfultoolsforunderstandingandsolvingproblemsdrawnfromdifferenttypesofreal-worldsituations.

Oneoftheinsightsprovidedbymathematicalmodelingisthatessentiallythesamemathematicalorstatisticalstructurecansometimesmodelseeminglydifferentsituations.Modelscanalsoshedlightonthemathematicalstructuresthemselves,forexample,aswhenamodelofbacterialgrowthmakesmorevividtheexplosivegrowthoftheexponentialfunction.

Thebasicmodelingcycleissummarizedinthediagram.Itinvolves(1)identifyingvariablesinthesituationandselectingthosethatrepresentessentialfeatures,(2)formulatingamodelbycreatingandselectinggeometric,graphical,tabular,algebraic,orstatisticalrepresentationsthatdescriberelationshipsbetweenthevariables,(3)analyzingandperformingoperationsontheserelationshipstodrawconclusions,(4)interpretingtheresultsofthemathematicsintermsoftheoriginalsituation,(5)validatingtheconclusionsbycomparingthemwiththesituation,andtheneitherimprovingthemodelor,ifit


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isacceptable,(6)reportingontheconclusionsandthereasoningbehindthem.Choices,assumptions,andapproximationsarepresentthroughoutthiscycle.

Indescriptivemodeling,amodelsimplydescribesthephenomenaorsummarizestheminacompactform.Graphsofobservationsareafamiliardescriptivemodel—forexample,graphsofglobaltemperatureandatmosphericCO

2overtime.

Analyticmodelingseekstoexplaindataonthebasisofdeepertheoreticalideas,albeitwithparametersthatareempiricallybased;forexample,exponentialgrowthofbacterialcolonies(untilcut-offmechanismssuchaspollutionorstarvationintervene)followsfromaconstantreproductionrate.Functionsareanimportanttoolforanalyzingsuchproblems.

Graphingutilities,spreadsheets,computeralgebrasystems,anddynamicgeometrysoftwarearepowerfultoolsthatcanbeusedtomodelpurelymathematicalphenomena(e.g.,thebehaviorofpolynomials)aswellasphysicalphenomena.

modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).


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mathematics | High School—GeometryAnunderstandingoftheattributesandrelationshipsofgeometricobjectscanbeappliedindiversecontexts—interpretingaschematicdrawing,estimatingtheamountofwoodneededtoframeaslopingroof,renderingcomputergraphics,ordesigningasewingpatternforthemostefficientuseofmaterial.

Althoughtherearemanytypesofgeometry,schoolmathematicsisdevotedprimarilytoplaneEuclideangeometry,studiedbothsynthetically(withoutcoordinates)andanalytically(withcoordinates).EuclideangeometryischaracterizedmostimportantlybytheParallelPostulate,thatthroughapointnotonagivenlinethereisexactlyoneparallelline.(Sphericalgeometry,incontrast,hasnoparallellines.)

Duringhighschool,studentsbegintoformalizetheirgeometryexperiencesfromelementaryandmiddleschool,usingmoreprecisedefinitionsanddevelopingcarefulproofs.LaterincollegesomestudentsdevelopEuclideanandothergeometriescarefullyfromasmallsetofaxioms.

Theconceptsofcongruence,similarity,andsymmetrycanbeunderstoodfromtheperspectiveofgeometrictransformation.Fundamentalaretherigidmotions:translations,rotations,reflections,andcombinationsofthese,allofwhicharehereassumedtopreservedistanceandangles(andthereforeshapesgenerally).Reflectionsandrotationseachexplainaparticulartypeofsymmetry,andthesymmetriesofanobjectofferinsightintoitsattributes—aswhenthereflectivesymmetryofanisoscelestriangleassuresthatitsbaseanglesarecongruent.

Intheapproachtakenhere,twogeometricfiguresaredefinedtobecongruentifthereisasequenceofrigidmotionsthatcarriesoneontotheother.Thisistheprincipleofsuperposition.Fortriangles,congruencemeanstheequalityofallcorrespondingpairsofsidesandallcorrespondingpairsofangles.Duringthemiddlegrades,throughexperiencesdrawingtrianglesfromgivenconditions,studentsnoticewaystospecifyenoughmeasuresinatriangletoensurethatalltrianglesdrawnwiththosemeasuresarecongruent.Oncethesetrianglecongruencecriteria(ASA,SAS,andSSS)areestablishedusingrigidmotions,theycanbeusedtoprovetheoremsabouttriangles,quadrilaterals,andothergeometricfigures.

Similaritytransformations(rigidmotionsfollowedbydilations)definesimilarityinthesamewaythatrigidmotionsdefinecongruence,therebyformalizingthesimilarityideasof"sameshape"and"scalefactor"developedinthemiddlegrades.Thesetransformationsleadtothecriterionfortrianglesimilaritythattwopairsofcorrespondinganglesarecongruent.

Thedefinitionsofsine,cosine,andtangentforacuteanglesarefoundedonrighttrianglesandsimilarity,and,withthePythagoreanTheorem,arefundamentalinmanyreal-worldandtheoreticalsituations.ThePythagoreanTheoremisgeneralizedtonon-righttrianglesbytheLawofCosines.Together,theLawsofSinesandCosinesembodythetrianglecongruencecriteriaforthecaseswherethreepiecesofinformationsufficetocompletelysolveatriangle.Furthermore,theselawsyieldtwopossiblesolutionsintheambiguouscase,illustratingthatSide-Side-Angleisnotacongruencecriterion.

Analyticgeometryconnectsalgebraandgeometry,resultinginpowerfulmethodsofanalysisandproblemsolving.Justasthenumberlineassociatesnumberswithlocationsinonedimension,apairofperpendicularaxesassociatespairsofnumberswithlocationsintwodimensions.Thiscorrespondencebetweennumericalcoordinatesandgeometricpointsallowsmethodsfromalgebratobeappliedtogeometryandviceversa.Thesolutionsetofanequationbecomesageometriccurve,makingvisualizationatoolfordoingandunderstandingalgebra.Geometricshapescanbedescribedbyequations,makingalgebraicmanipulationintoatoolforgeometricunderstanding,modeling,andproof.Geometrictransformationsofthegraphsofequationscorrespondtoalgebraicchangesintheirequations.

Dynamicgeometryenvironmentsprovidestudentswithexperimentalandmodelingtoolsthatallowthemtoinvestigategeometricphenomenainmuchthesamewayascomputeralgebrasystemsallowthemtoexperimentwithalgebraicphenomena.

Connections to Equations. Thecorrespondencebetweennumericalcoordinatesandgeometricpointsallowsmethodsfromalgebratobeappliedtogeometryandviceversa.Thesolutionsetofanequationbecomesageometriccurve,makingvisualizationatoolfordoingandunderstandingalgebra.Geometricshapescanbedescribedbyequations,makingalgebraicmanipulationintoatoolforgeometricunderstanding,modeling,andproof.


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Congruence

• experiment with transformations in the plane

• Understand congruence in terms of rigid motions

• Prove geometric theorems

• make geometric constructions

Similarity, Right Triangles, and Trigonometry

• Understand similarity in terms of similarity transformations

• Prove theorems involving similarity

• define trigonometric ratios and solve problems involving right triangles

• apply trigonometry to general triangles

Circles

• Understand and apply theorems about circles

• find arc lengths and areas of sectors of circles

Expressing Geometric Properties with Equations

• translate between the geometric description and the equation for a conic section

• Use coordinates to prove simple geometric theorems algebraically

Geometric Measurement and Dimension

• explain volume formulas and use them to solve problems

• Visualize relationships between two-dimensional and three-dimensional objects

Modeling with Geometry

• apply geometric concepts in modeling situations










Geometry overview


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Congruence G-Co

Experiment with transformations in the plane

1. Knowprecisedefinitionsofangle,circle,perpendicularline,parallelline,andlinesegment,basedontheundefinednotionsofpoint,line,distancealongaline,anddistancearoundacirculararc.

2. Representtransformationsintheplaneusing,e.g.,transparenciesandgeometrysoftware;describetransformationsasfunctionsthattakepointsintheplaneasinputsandgiveotherpointsasoutputs.Comparetransformationsthatpreservedistanceandangletothosethatdonot(e.g.,translationversushorizontalstretch).

3. Givenarectangle,parallelogram,trapezoid,orregularpolygon,describetherotationsandreflectionsthatcarryitontoitself.

4. Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicularlines,parallellines,andlinesegments.

5. Givenageometricfigureandarotation,reflection,ortranslation,drawthetransformedfigureusing,e.g.,graphpaper,tracingpaper,orgeometrysoftware.Specifyasequenceoftransformationsthatwillcarryagivenfigureontoanother.

Understand congruence in terms of rigid motions

6. Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectofagivenrigidmotiononagivenfigure;giventwofigures,usethedefinitionofcongruenceintermsofrigidmotionstodecideiftheyarecongruent.

7. Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.

8. Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthedefinitionofcongruenceintermsofrigidmotions.

Prove geometric theorems

9. Provetheoremsaboutlinesandangles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

10. Provetheoremsabouttriangles.Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

11. Provetheoremsaboutparallelograms.Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Make geometric constructions

12. Makeformalgeometricconstructionswithavarietyoftoolsandmethods(compassandstraightedge,string,reflectivedevices,paperfolding,dynamicgeometricsoftware,etc.).Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

13. Constructanequilateraltriangle,asquare,andaregularhexagoninscribedinacircle.


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Similarity, right triangles, and trigonometry G-Srt

Understand similarity in terms of similarity transformations

1. Verifyexperimentallythepropertiesofdilationsgivenbyacenterandascalefactor:

a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassingthroughthecenterunchanged.

b. Thedilationofalinesegmentislongerorshorterintheratiogivenbythescalefactor.

2. Giventwofigures,usethedefinitionofsimilarityintermsofsimilaritytransformations todecideiftheyaresimilar;explainusingsimilaritytransformationsthemeaningofsimilarityfortrianglesastheequalityofallcorrespondingpairsofanglesandtheproportionalityofallcorrespondingpairsofsides.

3. UsethepropertiesofsimilaritytransformationstoestablishtheAAcriterionfortwotrianglestobesimilar.

Prove theorems involving similarity

4. Provetheoremsabouttriangles.Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

5. Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandtoproverelationshipsingeometricfigures.

Define trigonometric ratios and solve problems involving right triangles

6. Understandthatbysimilarity,sideratiosinrighttrianglesarepropertiesoftheanglesinthetriangle,leadingtodefinitionsoftrigonometricratiosforacuteangles.

7. Explainandusetherelationshipbetweenthesineandcosineofcomplementaryangles.

8. UsetrigonometricratiosandthePythagoreanTheoremtosolverighttrianglesinappliedproblems.★

Apply trigonometry to general triangles

9. (+)DerivetheformulaA=1/2absin(C)fortheareaofatrianglebydrawinganauxiliarylinefromavertexperpendiculartotheoppositeside.

10. (+)ProvetheLawsofSinesandCosinesandusethemtosolveproblems.

11. (+)UnderstandandapplytheLawofSinesandtheLawofCosinestofindunknownmeasurementsinrightandnon-righttriangles(e.g.,surveyingproblems,resultantforces).

Circles G-C

Understand and apply theorems about circles

1. Provethatallcirclesaresimilar.

2. Identifyanddescriberelationshipsamonginscribedangles,radii,andchords.Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

3. Constructtheinscribedandcircumscribedcirclesofatriangle,andprovepropertiesofanglesforaquadrilateralinscribedinacircle.

4. (+)Constructatangentlinefromapointoutsideagivencircletothecircle.


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Find arc lengths and areas of sectors of circles

5. Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleisproportionaltotheradius,anddefinetheradianmeasureoftheangleastheconstantofproportionality;derivetheformulafortheareaofasector.

expressing Geometric Properties with equations G-GPe

Translate between the geometric description and the equation for a conic section

1. DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;completethesquaretofindthecenterandradiusofacirclegivenbyanequation.

2. Derivetheequationofaparabolagivenafocusanddirectrix.

3. (+)Derivetheequationsofellipsesandhyperbolasgiventhefoci,usingthefactthatthesumordifferenceofdistancesfromthefociisconstant.

Use coordinates to prove simple geometric theorems algebraically

4. Usecoordinatestoprovesimplegeometrictheoremsalgebraically.For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

5. Provetheslopecriteriaforparallelandperpendicularlinesandusethemtosolvegeometricproblems(e.g.,findtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint).

6. Findthepointonadirectedlinesegmentbetweentwogivenpointsthatpartitionsthesegmentinagivenratio.

7. Usecoordinatestocomputeperimetersofpolygonsandareasoftrianglesandrectangles,e.g.,usingthedistanceformula.★

Geometric measurement and dimension G-Gmd

Explain volume formulas and use them to solve problems

1. Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofacircle,volumeofacylinder,pyramid,andcone.Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

2. (+)GiveaninformalargumentusingCavalieri’sprinciplefortheformulasforthevolumeofasphereandothersolidfigures.

3. Usevolumeformulasforcylinders,pyramids,cones,andspherestosolveproblems.★

Visualize relationships between two-dimensional and three-dimensional objects

4. Identifytheshapesoftwo-dimensionalcross-sectionsofthree-dimensionalobjects,andidentifythree-dimensionalobjectsgeneratedbyrotationsoftwo-dimensionalobjects.

modeling with Geometry G-mG

Apply geometric concepts in modeling situations

1. Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunkorahumantorsoasacylinder).★

2. Applyconceptsofdensitybasedonareaandvolumeinmodelingsituations(e.g.,personspersquaremile,BTUspercubicfoot).★

3. Applygeometricmethodstosolvedesignproblems(e.g.,designinganobjectorstructuretosatisfyphysicalconstraintsorminimizecost;workingwithtypographicgridsystemsbasedonratios).★


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mathematics | High School—Statistics and Probability★

Decisionsorpredictionsareoftenbasedondata—numbersincontext.Thesedecisionsorpredictionswouldbeeasyifthedataalwayssentaclearmessage,butthemessageisoftenobscuredbyvariability.Statisticsprovidestoolsfordescribingvariabilityindataandformakinginformeddecisionsthattakeitintoaccount.

Dataaregathered,displayed,summarized,examined,andinterpretedtodiscoverpatternsanddeviationsfrompatterns.Quantitativedatacanbedescribedintermsofkeycharacteristics:measuresofshape,center,andspread.Theshapeofadatadistributionmightbedescribedassymmetric,skewed,flat,orbellshaped,anditmightbesummarizedbyastatisticmeasuringcenter(suchasmeanormedian)andastatisticmeasuringspread(suchasstandarddeviationorinterquartilerange).Differentdistributionscanbecomparednumericallyusingthesestatisticsorcomparedvisuallyusingplots.Knowledgeofcenterandspreadarenotenoughtodescribeadistribution.Whichstatisticstocompare,whichplotstouse,andwhattheresultsofacomparisonmightmean,dependonthequestiontobeinvestigatedandthereal-lifeactionstobetaken.

Randomizationhastwoimportantusesindrawingstatisticalconclusions.First,collectingdatafromarandomsampleofapopulationmakesitpossibletodrawvalidconclusionsaboutthewholepopulation,takingvariabilityintoaccount.Second,randomlyassigningindividualstodifferenttreatmentsallowsafaircomparisonoftheeffectivenessofthosetreatments.Astatisticallysignificantoutcomeisonethatisunlikelytobeduetochancealone,andthiscanbeevaluatedonlyundertheconditionofrandomness.Theconditionsunderwhichdataarecollectedareimportantindrawingconclusionsfromthedata;incriticallyreviewingusesofstatisticsinpublicmediaandotherreports,itisimportanttoconsiderthestudydesign,howthedataweregathered,andtheanalysesemployedaswellasthedatasummariesandtheconclusionsdrawn.

Randomprocessescanbedescribedmathematicallybyusingaprobabilitymodel:alistordescriptionofthepossibleoutcomes(thesamplespace),eachofwhichisassignedaprobability.Insituationssuchasflippingacoin,rollinganumbercube,ordrawingacard,itmightbereasonabletoassumevariousoutcomesareequallylikely.Inaprobabilitymodel,samplepointsrepresentoutcomesandcombinetomakeupevents;probabilitiesofeventscanbecomputedbyapplyingtheAdditionandMultiplicationRules.Interpretingtheseprobabilitiesreliesonanunderstandingofindependenceandconditionalprobability,whichcanbeapproachedthroughtheanalysisoftwo-waytables.

Technologyplaysanimportantroleinstatisticsandprobabilitybymakingitpossibletogenerateplots,regressionfunctions,andcorrelationcoefficients,andtosimulatemanypossibleoutcomesinashortamountoftime.

Connections to Functions and Modeling.Functionsmaybeusedtodescribedata;ifthedatasuggestalinearrelationship,therelationshipcanbemodeledwitharegressionline,anditsstrengthanddirectioncanbeexpressedthroughacorrelationcoefficient.


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Interpreting Categorical and Quantitative Data

• Summarize, represent, and interpret data on a single count or measurement variable

• Summarize, represent, and interpret data on two categorical and quantitative variables

• Interpret linear models

Making Inferences and Justifying Conclusions

• Understand and evaluate random processes underlying statistical experiments

• make inferences and justify conclusions from sample surveys, experiments and observational studies

Conditional Probability and the Rules of Prob-ability

• Understand independence and conditional probability and use them to interpret data

• Use the rules of probability to compute probabilities of compound events in a uniform probability model

Using Probability to Make Decisions

• Calculate expected values and use them to solve problems

• Use probability to evaluate outcomes of decisions










Statistics and Probability overview


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Interpreting Categorical and Quantitative data S-Id

Summarize, represent, and interpret data on a single count or measurement variable

1. Representdatawithplotsontherealnumberline(dotplots,histograms,andboxplots).

2. Usestatisticsappropriatetotheshapeofthedatadistributiontocomparecenter(median,mean)andspread(interquartilerange,standarddeviation)oftwoormoredifferentdatasets.

3. Interpretdifferencesinshape,center,andspreadinthecontextofthedatasets,accountingforpossibleeffectsofextremedatapoints(outliers).

4. Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandtoestimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureisnotappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormalcurve.

Summarize, represent, and interpret data on two categorical and quantitative variables

5. Summarizecategoricaldatafortwocategoriesintwo-wayfrequencytables.Interpretrelativefrequenciesinthecontextofthedata(includingjoint,marginal,andconditionalrelativefrequencies).Recognizepossibleassociationsandtrendsinthedata.

6. Representdataontwoquantitativevariablesonascatterplot,anddescribehowthevariablesarerelated.

a. Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsinthecontextofthedata.Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

b. Informallyassessthefitofafunctionbyplottingandanalyzingresiduals.

c. Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.

Interpret linear models

7. Interprettheslope(rateofchange)andtheintercept(constantterm)ofalinearmodelinthecontextofthedata.

8. Compute(usingtechnology)andinterpretthecorrelationcoefficientofalinearfit.

9. Distinguishbetweencorrelationandcausation.

making Inferences and Justifying Conclusions S-IC

Understand and evaluate random processes underlying statistical experiments

1. Understandstatisticsasaprocessformakinginferencesaboutpopulationparametersbasedonarandomsamplefromthatpopulation.

2. Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-generatingprocess,e.g.,usingsimulation.For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

3. Recognizethepurposesofanddifferencesamongsamplesurveys,experiments,andobservationalstudies;explainhowrandomizationrelatestoeach.


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4. Usedatafromasamplesurveytoestimateapopulationmeanorproportion;developamarginoferrorthroughtheuseofsimulationmodelsforrandomsampling.

5. Usedatafromarandomizedexperimenttocomparetwotreatments;usesimulationstodecideifdifferencesbetweenparametersaresignificant.

6. Evaluatereportsbasedondata.

Conditional Probability and the rules of Probability S-CP

Understand independence and conditional probability and use them to interpret data

1. Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)oftheoutcomes,orasunions,intersections,orcomplementsofotherevents(“or,”“and,”“not”).

2. UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheristheproductoftheirprobabilities,andusethischaracterizationtodetermineiftheyareindependent.

3. UnderstandtheconditionalprobabilityofAgivenBas P(AandB)/P(B),andinterpretindependenceofAandBassayingthattheconditionalprobabilityof Agiven BisthesameastheprobabilityofA,andtheconditionalprobabilityofBgivenAisthesameastheprobabilityofB.

4. Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheachobjectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentandtoapproximateconditionalprobabilities.For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

5. Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageandeverydaysituations.For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Use the rules of probability to compute probabilities of compound events in a uniform probability model

6. FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongtoA,andinterprettheanswerintermsofthemodel.

7. ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerintermsofthemodel.

8. (+)ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=P(A)P(B|A)=P(B)P(A|B),andinterprettheanswerintermsofthemodel.

9. (+)Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems.

Using Probability to make decisions S-md

Calculate expected values and use them to solve problems

1. (+)Definearandomvariableforaquantityofinterestbyassigninganumericalvaluetoeacheventinasamplespace;graphthecorrespondingprobabilitydistributionusingthesamegraphicaldisplaysasfordatadistributions.

2. (+)Calculatetheexpectedvalueofarandomvariable;interpretitasthemeanoftheprobabilitydistribution.


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3. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichtheoreticalprobabilitiescanbecalculated;findtheexpectedvalue.For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

4. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichprobabilitiesareassignedempirically;findtheexpectedvalue.For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Use probability to evaluate outcomes of decisions

5. (+)Weighthepossibleoutcomesofadecisionbyassigningprobabilitiestopayoffvaluesandfindingexpectedvalues.

a. Findtheexpectedpayoffforagameofchance.For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

b. Evaluateandcomparestrategiesonthebasisofexpectedvalues.For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

6. (+)Useprobabilitiestomakefairdecisions(e.g.,drawingbylots,usingarandomnumbergenerator).

7. (+)Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,pullingahockeygoalieattheendofagame).

Common Core State StandardS for matHematICS8

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note on courses and transitions

ThehighschoolportionoftheStandardsforMathematicalContentspecifiesthemathematicsallstudentsshouldstudyforcollegeandcareerreadiness.Thesestandardsdonotmandatethesequenceofhighschoolcourses.However,theorganizationofhighschoolcoursesisacriticalcomponenttoimplementationofthestandards.Tothatend,samplehighschoolpathwaysformathematics–inbothatraditionalcoursesequence(AlgebraI,Geometry,andAlgebraII)aswellasanintegratedcoursesequence(Mathematics1,Mathematics2,Mathematics3)–willbemadeavailableshortlyafterthereleaseofthefinalCommonCoreStateStandards.Itisexpectedthatadditionalmodelpathwaysbasedonthesestandardswillbecomeavailableaswell.

Thestandardsthemselvesdonotdictatecurriculum,pedagogy,ordeliveryofcontent.Inparticular,statesmayhandlethetransitiontohighschoolindifferentways.Forexample,manystudentsintheU.S.todaytakeAlgebraIinthe8thgrade,andinsomestatesthisisarequirement.TheK-7standardscontaintheprerequisitestopreparestudentsforAlgebraIby8thgrade,andthestandardsaredesignedtopermitstatestocontinueexistingpoliciesconcerningAlgebraIin8thgrade.

Asecondmajortransitionisthetransitionfromhighschooltopost-secondaryeducationforcollegeandcareers.Theevidenceconcerningcollegeandcareerreadinessshowsclearlythattheknowledge,skills,andpracticesimportantforreadinessincludeagreatdealofmathematicspriortotheboundarydefinedby(+)symbolsinthesestandards.Indeed,someofthehighestprioritycontentforcollegeandcareerreadinesscomesfromGrades6-8.Thisbodyofmaterialincludespowerfullyusefulproficienciessuchasapplyingratioreasoninginreal-worldandmathematicalproblems,computingfluentlywithpositiveandnegativefractionsanddecimals,andsolvingreal-worldandmathematicalproblemsinvolvinganglemeasure,area,surfacearea,andvolume.Becauseimportantstandardsforcollegeandcareerreadinessaredistributedacrossgradesandcourses,systemsforevaluatingcollegeandcareerreadinessshouldreachasfarbackinthestandardsasGrades6-8.Itisimportanttonoteaswellthatcutscoresorotherinformationgeneratedbyassessmentsystemsforcollegeandcareerreadinessshouldbedevelopedincollaborationwithrepresentativesfromhighereducationandworkforcedevelopmentprograms,andshouldbevalidatedbysubsequentperformanceofstudentsincollegeandtheworkforce.

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mathematics Standards for High School · mathematics Standards for High School ... 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents