Michigan K-12 Standards...

Michigan K-12 Standards

Mathematics

R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E R E L A T I O N H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P • R I G O R • • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E

Michigan State Board of Education

Kathleen N. Straus, President

Bloomfield Township

John C. Austin, Vice President Ann Arbor

Carolyn L. Curtin, Secretary

Evart

Marianne Yared McGuire, Treasurer Detroit

Nancy Danhof, NASBE Delegate

East Lansing

Elizabeth W. Bauer Birmingham

Daniel Varner

Detroit

Casandra E. Ulbrich Rochester Hills

Governor Jennifer M. Granholm

Ex Officio

Michael P. Flanagan, Chairman Superintendent of Public Instruction

Ex Officio

MDE Staff

Sally Vaughn, Ph.D.

Deputy Superintendent and Chief Academic Officer

Linda Forward, Director Office of Education Improvement and Innovation

Welcome

Welcome to the Michigan K-12 Standards for Mathematics, adopted by the State Board of Education in 2010. With the reauthorization of the 2001 Elementary and Secondary Education Act (ESEA), commonly known as No Child Left Behind (NCLB), Michigan embarked on a standards revision process, starting with the K-8 mathematics and ELA standards that resulted in the Grade Level Content Expectations (GLCE). These were intended to lay the framework for the grade level testing in these subject areas required under NCLB. These were followed by GLCE for science and social studies, and by High School Content Expectations (HSCE) for all subject areas. Seven years later the revision cycle continued with Michigan working with other states to build on and refine current state standards that would allow states to work collaboratively to develop a repository of quality resources based on a common set of standards. These standards are the result of that collaboration.

Michigan’s K–12 academic standards serve to outline learning expectations for Michigan’s students and are intended to guide local curriculum development. Because these Mathematics standards are shared with other states, local districts have access to a broad set of resources they can call upon as they develop their local curricula and assessments. State standards also serve as a platform for state-level assessments, which are used to measure how well schools are providing opportunities for all students to learn the content required to be career– and college–ready.

Linda Forward, Director, Office of Education Improvement and Innovation

Vanessa Keesler, Deputy Superintendent, Division of Education Services

Mike Flanagan, Superintendent of Public Instruction

table of ContentsIntroduction 3

Standards for mathematical Practice 6

Standards for mathematical Content

Kindergarten 9Grade1 13Grade2 17Grade3 21Grade4 27Grade5 33Grade6 39Grade7 46Grade8 52HighSchool—Introduction

HighSchool—NumberandQuantity 58HighSchool—Algebra 62HighSchool—Functions 67HighSchool—Modeling 72HighSchool—Geometry 74HighSchool—StatisticsandProbability 79

Glossary 85Sample of Works Consulted 91

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IntroductionToward greater focus and coherence

Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to number than to other topics. Mathematical process goals should be integrated in these content areas.

—MathematicsLearninginEarlyChildhood,NationalResearchCouncil,2009

The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can inform an international benchmarking process for the development of K–6 mathematics standards in the U.S. First, the composite standards concentrate the early learning of mathematics on the number, measurement, and geometry strands with less emphasis on data analysis and little exposure to algebra. The Hong Kong standards for grades 1–3 devote approximately half the targeted time to numbers and almost all the time remaining to geometry and measurement.

—Ginsburg,LeinwandandDecker,2009

Because the mathematics concepts in [U.S.] textbooks are often weak, the presentation becomes more mechanical than is ideal. We looked at both traditional and non-traditional textbooks used in the US and found this conceptual weakness in both.

—Ginsburgetal.,2005

There are many ways to organize curricula. The challenge, now rarely met, is to avoid those that distort mathematics and turn off students.

—Steen,2007

Foroveradecade,researchstudiesofmathematicseducationinhigh-performing

countrieshavepointedtotheconclusionthatthemathematicscurriculuminthe

UnitedStatesmustbecomesubstantiallymorefocusedandcoherentinorderto

improvemathematicsachievementinthiscountry.Todeliveronthepromiseof

commonstandards,thestandardsmustaddresstheproblemofacurriculumthat

is“amilewideandaninchdeep.”TheseStandardsareasubstantialanswertothat

challenge.

Itisimportanttorecognizethat“fewerstandards”arenosubstituteforfocused

standards.Achieving“fewerstandards”wouldbeeasytodobyresortingtobroad,

generalstatements.Instead,theseStandardsaimforclarityandspecificity.

Assessingthecoherenceofasetofstandardsismoredifficultthanassessing

theirfocus.WilliamSchmidtandRichardHouang(2002)havesaidthatcontent

standardsandcurriculaarecoherentiftheyare:

articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives. That is, what and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. This implies

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that to be coherent, a set of content standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures inherent in the discipline. These deeper structures then serve as a means for connecting the particulars (such as an understanding of the rational number system and its properties). (emphasis

added)

TheseStandardsendeavortofollowsuchadesign,notonlybystressingconceptual

understandingofkeyideas,butalsobycontinuallyreturningtoorganizing

principlessuchasplacevalueorthepropertiesofoperationstostructurethose

ideas.

Inaddition,the“sequenceoftopicsandperformances”thatisoutlinedinabodyof

mathematicsstandardsmustalsorespectwhatisknownabouthowstudentslearn.

AsConfrey(2007)pointsout,developing“sequencedobstaclesandchallenges

forstudents…absenttheinsightsaboutmeaningthatderivefromcarefulstudyof

learning,wouldbeunfortunateandunwise.”Inrecognitionofthis,thedevelopment

oftheseStandardsbeganwithresearch-basedlearningprogressionsdetailing

whatisknowntodayabouthowstudents’mathematicalknowledge,skill,and

understandingdevelopovertime.

Understanding mathematics

TheseStandardsdefinewhatstudentsshouldunderstandandbeabletodoin

theirstudyofmathematics.Askingastudenttounderstandsomethingmeans

askingateachertoassesswhetherthestudenthasunderstoodit.Butwhatdoes

mathematicalunderstandinglooklike?Onehallmarkofmathematicalunderstanding

istheabilitytojustify,inawayappropriatetothestudent’smathematicalmaturity,

whyaparticularmathematicalstatementistrueorwhereamathematicalrule

comesfrom.Thereisaworldofdifferencebetweenastudentwhocansummona

mnemonicdevicetoexpandaproductsuchas(a+ b)(x+y)andastudentwho

canexplainwherethemnemoniccomesfrom.Thestudentwhocanexplaintherule

understandsthemathematics,andmayhaveabetterchancetosucceedataless

familiartasksuchasexpanding(a+ b+c)(x+y).Mathematicalunderstandingand

proceduralskillareequallyimportant,andbothareassessableusingmathematical

tasksofsufficientrichness.

TheStandardssetgrade-specificstandardsbutdonotdefinetheintervention

methodsormaterialsnecessarytosupportstudentswhoarewellbeloworwell

abovegrade-levelexpectations.ItisalsobeyondthescopeoftheStandardsto

definethefullrangeofsupportsappropriateforEnglishlanguagelearnersand

forstudentswithspecialneeds.Atthesametime,allstudentsmusthavethe

opportunitytolearnandmeetthesamehighstandardsiftheyaretoaccessthe

knowledgeandskillsnecessaryintheirpost-schoollives.TheStandardsshould

bereadasallowingforthewidestpossiblerangeofstudentstoparticipatefully

fromtheoutset,alongwithappropriateaccommodationstoensuremaximum

participatonofstudentswithspecialeducationneeds.Forexample,forstudents

withdisabilitiesreadingshouldallowforuseofBraille,screenreadertechnology,or

otherassistivedevices,whilewritingshouldincludetheuseofascribe,computer,

orspeech-to-texttechnology.Inasimilarvein,speakingandlisteningshouldbe

interpretedbroadlytoincludesignlanguage.Nosetofgrade-specificstandards

canfullyreflectthegreatvarietyinabilities,needs,learningrates,andachievement

levelsofstudentsinanygivenclassroom.However,theStandardsdoprovideclear

signpostsalongthewaytothegoalofcollegeandcareerreadinessforallstudents.

TheStandardsbeginonpage6witheightStandardsforMathematicalPractice.

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How to read the grade level standards

Standards definewhatstudentsshouldunderstandandbeabletodo.

Clusters aregroupsofrelatedstandards.Notethatstandardsfromdifferentclusters

maysometimesbecloselyrelated,becausemathematics

isaconnectedsubject.

domainsarelargergroupsofrelatedstandards.Standardsfromdifferentdomains

maysometimesbecloselyrelated.

number and operations in Base ten 3.nBtUse place value understanding and properties of operations to perform multi-digit arithmetic.

1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.

2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.

3. Multiplyone-digitwholenumbersbymultiplesof10intherange10-90(e.g.,9×80,5×60)usingstrategiesbasedonplacevalueandpropertiesofoperations.

TheseStandardsdonotdictatecurriculumorteachingmethods.Forexample,just

becausetopicAappearsbeforetopicBinthestandardsforagivengrade,itdoes

notnecessarilymeanthattopicAmustbetaughtbeforetopicB.Ateachermight

prefertoteachtopicBbeforetopicA,ormightchoosetohighlightconnectionsby

teachingtopicAandtopicBatthesametime.Or,ateachermightprefertoteacha

topicofhisorherownchoosingthatleads,asabyproduct,tostudentsreachingthe

standardsfortopicsAandB.

Whatstudentscanlearnatanyparticulargradeleveldependsuponwhatthey

havelearnedbefore.Ideallythen,eachstandardinthisdocumentmighthavebeen

phrasedintheform,“Studentswhoalreadyknow...shouldnextcometolearn....”

Butatpresentthisapproachisunrealistic—notleastbecauseexistingeducation

researchcannotspecifyallsuchlearningpathways.Ofnecessitytherefore,

gradeplacementsforspecifictopicshavebeenmadeonthebasisofstateand

internationalcomparisonsandthecollectiveexperienceandcollectiveprofessional

judgmentofeducators,researchersandmathematicians.Onepromiseofcommon

statestandardsisthatovertimetheywillallowresearchonlearningprogressions

toinformandimprovethedesignofstandardstoamuchgreaterextentthanis

possibletoday.Learningopportunitieswillcontinuetovaryacrossschoolsand

schoolsystems,andeducatorsshouldmakeeveryefforttomeettheneedsof

individualstudentsbasedontheircurrentunderstanding.

TheseStandardsarenotintendedtobenewnamesforoldwaysofdoingbusiness.

Theyareacalltotakethenextstep.Itistimeforstatestoworktogethertobuild

onlessonslearnedfromtwodecadesofstandardsbasedreforms.Itistimeto

recognizethatstandardsarenotjustpromisestoourchildren,butpromiseswe

intendtokeep.

domain

ClusterStandard

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mathematics | Standards for mathematical PracticeTheStandardsforMathematicalPracticedescribevarietiesofexpertisethat

mathematicseducatorsatalllevelsshouldseektodevelopintheirstudents.

Thesepracticesrestonimportant“processesandproficiencies”withlongstanding

importanceinmathematicseducation.ThefirstofthesearetheNCTMprocess

standardsofproblemsolving,reasoningandproof,communication,representation,

andconnections.Thesecondarethestrandsofmathematicalproficiencyspecified

intheNationalResearchCouncil’sreportAdding It Up:adaptivereasoning,strategic

competence,conceptualunderstanding(comprehensionofmathematicalconcepts,

operationsandrelations),proceduralfluency(skillincarryingoutprocedures

flexibly,accurately,efficientlyandappropriately),andproductivedisposition

(habitualinclinationtoseemathematicsassensible,useful,andworthwhile,coupled

withabeliefindiligenceandone’sownefficacy).

1 Make sense of problems and persevere in solving them.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaning

ofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,

constraints,relationships,andgoals.Theymakeconjecturesabouttheformand

meaningofthesolutionandplanasolutionpathwayratherthansimplyjumpinginto

asolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesand

simplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They

monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudents

might,dependingonthecontextoftheproblem,transformalgebraicexpressionsor

changetheviewingwindowontheirgraphingcalculatortogettheinformationthey

need.Mathematicallyproficientstudentscanexplaincorrespondencesbetween

equations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportant

featuresandrelationships,graphdata,andsearchforregularityortrends.Younger

studentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualize

andsolveaproblem.Mathematicallyproficientstudentschecktheiranswersto

problemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthis

makesense?”Theycanunderstandtheapproachesofotherstosolvingcomplex

problemsandidentifycorrespondencesbetweendifferentapproaches.

2 Reason abstractly and quantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationships

inproblemsituations.Theybringtwocomplementaryabilitiestobearonproblems

involvingquantitativerelationships:theabilitytodecontextualize—toabstract

agivensituationandrepresentitsymbolicallyandmanipulatetherepresenting

symbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingto

theirreferents—andtheabilitytocontextualize,topauseasneededduringthe

manipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.

Quantitativereasoningentailshabitsofcreatingacoherentrepresentationof

theproblemathand;consideringtheunitsinvolved;attendingtothemeaningof

quantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferent

propertiesofoperationsandobjects.

3 Construct viable arguments and critique the reasoning of others.Mathematicallyproficientstudentsunderstandandusestatedassumptions,

definitions,andpreviouslyestablishedresultsinconstructingarguments.They

makeconjecturesandbuildalogicalprogressionofstatementstoexplorethe

truthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingtheminto

cases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,

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communicatethemtoothers,andrespondtotheargumentsofothers.Theyreason

inductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthe

contextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoable

tocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicor

reasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—explain

whatitis.Elementarystudentscanconstructargumentsusingconcretereferents

suchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesense

andbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillater

grades.Later,studentslearntodeterminedomainstowhichanargumentapplies.

Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhether

theymakesense,andaskusefulquestionstoclarifyorimprovethearguments.

4 Model with mathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolve

problemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismight

beassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,

astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzea

probleminthecommunity.Byhighschool,astudentmightusegeometrytosolvea

designproblemoruseafunctiontodescribehowonequantityofinterestdepends

onanother.Mathematicallyproficientstudentswhocanapplywhattheyknoware

comfortablemakingassumptionsandapproximationstosimplifyacomplicated

situation,realizingthatthesemayneedrevisionlater.Theyareabletoidentify

importantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuch

toolsasdiagrams,two-waytables,graphs,flowchartsandformulas.Theycananalyze

thoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheir

mathematicalresultsinthecontextofthesituationandreflectonwhethertheresults

makesense,possiblyimprovingthemodelifithasnotserveditspurpose.

5 Use appropriate tools strategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvinga

mathematicalproblem.Thesetoolsmightincludepencilandpaper,concrete

models,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,

astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsare

sufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesound

decisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththe

insighttobegainedandtheirlimitations.Forexample,mathematicallyproficient

highschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusinga

graphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimation

andothermathematicalknowledge.Whenmakingmathematicalmodels,theyknow

thattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,

exploreconsequences,andcomparepredictionswithdata.Mathematically

proficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternal

mathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethem

toposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreand

deepentheirunderstandingofconcepts.

6 Attend to precision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.They

trytousecleardefinitionsindiscussionwithothersandintheirownreasoning.

Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsign

consistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,

andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.They

calculateaccuratelyandefficiently,expressnumericalanswerswithadegreeof

precisionappropriatefortheproblemcontext.Intheelementarygrades,students

givecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhigh

schooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.

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7 Look for and make use of structure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.

Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesame

amountassevenandthreemore,ortheymaysortacollectionofshapesaccording

tohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthe

wellremembered7×5+7×3,inpreparationforlearningaboutthedistributive

property.Intheexpressionx2+9x+14,olderstudentscanseethe14as2×7and

the9as2+7.Theyrecognizethesignificanceofanexistinglineinageometric

figureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.

Theyalsocanstepbackforanoverviewandshiftperspective.Theycansee

complicatedthings,suchassomealgebraicexpressions,assingleobjectsoras

beingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5

minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannot

bemorethan5foranyrealnumbersxandy.

8 Look for and express regularity in repeated reasoning.Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlook

bothforgeneralmethodsandforshortcuts.Upperelementarystudentsmight

noticewhendividing25by11thattheyarerepeatingthesamecalculationsover

andoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattention

tothecalculationofslopeastheyrepeatedlycheckwhetherpointsareontheline

through(1,2)withslope3,middleschoolstudentsmightabstracttheequation

(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding

(x–1)(x+1),(x–1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthemtothe

generalformulaforthesumofageometricseries.Astheyworktosolveaproblem,

mathematicallyproficientstudentsmaintainoversightoftheprocess,while

attendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheir

intermediateresults.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical ContentTheStandardsforMathematicalPracticedescribewaysinwhichdevelopingstudent

practitionersofthedisciplineofmathematicsincreasinglyoughttoengagewith

thesubjectmatterastheygrowinmathematicalmaturityandexpertisethroughout

theelementary,middleandhighschoolyears.Designersofcurricula,assessments,

andprofessionaldevelopmentshouldallattendtotheneedtoconnectthe

mathematicalpracticestomathematicalcontentinmathematicsinstruction.

TheStandardsforMathematicalContentareabalancedcombinationofprocedure

andunderstanding.Expectationsthatbeginwiththeword“understand”areoften

especiallygoodopportunitiestoconnectthepracticestothecontent.Students

wholackunderstandingofatopicmayrelyonprocedurestooheavily.Without

aflexiblebasefromwhichtowork,theymaybelesslikelytoconsideranalogous

problems,representproblemscoherently,justifyconclusions,applythemathematics

topracticalsituations,usetechnologymindfullytoworkwiththemathematics,

explainthemathematicsaccuratelytootherstudents,stepbackforanoverview,or

deviatefromaknownproceduretofindashortcut.Inshort,alackofunderstanding

effectivelypreventsastudentfromengaginginthemathematicalpractices.

Inthisrespect,thosecontentstandardswhichsetanexpectationofunderstanding

arepotential“pointsofintersection”betweentheStandardsforMathematical

ContentandtheStandardsforMathematicalPractice.Thesepointsofintersection

areintendedtobeweightedtowardcentralandgenerativeconceptsinthe

schoolmathematicscurriculumthatmostmeritthetime,resources,innovative

energies,andfocusnecessarytoqualitativelyimprovethecurriculum,instruction,

assessment,professionaldevelopment,andstudentachievementinmathematics.

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mathematics | KindergartenInKindergarten,instructionaltimeshouldfocusontwocriticalareas:(1)

representing,relating,andoperatingonwholenumbers,initiallywith

setsofobjects;(2)describingshapesandspace.Morelearningtimein

Kindergartenshouldbedevotedtonumberthantoothertopics.

(1)Studentsusenumbers,includingwrittennumerals,torepresent

quantitiesandtosolvequantitativeproblems,suchascountingobjectsin

aset;countingoutagivennumberofobjects;comparingsetsornumerals;

andmodelingsimplejoiningandseparatingsituationswithsetsofobjects,

oreventuallywithequationssuchas5+2=7and7–2=5.(Kindergarten

studentsshouldseeadditionandsubtractionequations,andstudent

writingofequationsinkindergartenisencouraged,butitisnotrequired.)

Studentschoose,combine,andapplyeffectivestrategiesforanswering

quantitativequestions,includingquicklyrecognizingthecardinalitiesof

smallsetsofobjects,countingandproducingsetsofgivensizes,counting

thenumberofobjectsincombinedsets,orcountingthenumberofobjects

thatremaininasetaftersomearetakenaway.

(2)Studentsdescribetheirphysicalworldusinggeometricideas(e.g.,

shape,orientation,spatialrelations)andvocabulary.Theyidentify,name,

anddescribebasictwo-dimensionalshapes,suchassquares,triangles,

circles,rectangles,andhexagons,presentedinavarietyofways(e.g.,with

differentsizesandorientations),aswellasthree-dimensionalshapessuch

ascubes,cones,cylinders,andspheres.Theyusebasicshapesandspatial

reasoningtomodelobjectsintheirenvironmentandtoconstructmore

complexshapes.

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Counting and Cardinality

• Know number names and the count sequence.

• Count to tell the number of objects.

• Compare numbers.

operations and algebraic thinking

• Understand addition as putting together andadding to, and understand subtraction astaking apart and taking from.

number and operations in Base ten

• Work with numbers 11–19 to gain foundationsfor place value.

measurement and data

• describe and compare measurable attributes.

• Classify objects and count the number ofobjects in categories.

Geometry

• Identify and describe shapes.

• analyze, compare, create, and composeshapes.

mathematical Practices

1. Makesenseofproblemsandperseverein

solvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritique

thereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeated

reasoning.

Grade K overview

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Counting and Cardinality K.CC

Know number names and the count sequence.

1. Countto100byonesandbytens.

2. Countforwardbeginningfromagivennumberwithintheknownsequence(insteadofhavingtobeginat1).

3. Writenumbersfrom0to20.Representanumberofobjectswithawrittennumeral0-20(with0representingacountofnoobjects).

Count to tell the number of objects.

4. Understandtherelationshipbetweennumbersandquantities;connectcountingtocardinality.

a. Whencountingobjects,saythenumbernamesinthestandardorder,pairingeachobjectwithoneandonlyonenumbernameandeachnumbernamewithoneandonlyoneobject.

b. Understandthatthelastnumbernamesaidtellsthenumberofobjectscounted.Thenumberofobjectsisthesameregardlessoftheirarrangementortheorderinwhichtheywerecounted.

c. Understandthateachsuccessivenumbernamereferstoaquantitythatisonelarger.

5. Counttoanswer“howmany?”questionsaboutasmanyas20thingsarrangedinaline,arectangulararray,oracircle,orasmanyas10thingsinascatteredconfiguration;givenanumberfrom1–20,countoutthatmanyobjects.

Compare numbers.

6. Identifywhetherthenumberofobjectsinonegroupisgreaterthan,lessthan,orequaltothenumberofobjectsinanothergroup,e.g.,byusingmatchingandcountingstrategies.1

7. Comparetwonumbersbetween1and10presentedaswrittennumerals.

operations and algebraic thinking K.oa

Understand addition as putting together and adding to, and under-stand subtraction as taking apart and taking from.

1. Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings2,sounds(e.g.,claps),actingoutsituations,verbalexplanations,expressions,orequations.

2. Solveadditionandsubtractionwordproblems,andaddandsubtractwithin10,e.g.,byusingobjectsordrawingstorepresenttheproblem.

3. Decomposenumberslessthanorequalto10intopairsinmorethanoneway,e.g.,byusingobjectsordrawings,andrecordeachdecompositionbyadrawingorequation(e.g.,5=2+3and5=4+1).

4. Foranynumberfrom1to9,findthenumberthatmakes10whenaddedtothegivennumber,e.g.,byusingobjectsordrawings,andrecordtheanswerwithadrawingorequation.

5. Fluentlyaddandsubtractwithin5.

1Includegroupswithuptotenobjects.2Drawingsneednotshowdetails,butshouldshowthemathematicsintheproblem.(ThisapplieswhereverdrawingsarementionedintheStandards.)

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number and operations in Base ten K.nBt

Work with numbers 11–19 to gain foundations for place value.

1. Composeanddecomposenumbersfrom11to19intotenonesandsomefurtherones,e.g.,byusingobjectsordrawings,andrecordeachcompositionordecompositionbyadrawingorequation(e.g.,18=10+8);understandthatthesenumbersarecomposedoftenonesandone,two,three,four,five,six,seven,eight,ornineones.

measurement and data K.md

Describe and compare measurable attributes.

1. Describemeasurableattributesofobjects,suchaslengthorweight.Describeseveralmeasurableattributesofasingleobject.

2. Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“moreof”/“lessof”theattribute,anddescribethedifference.For example, directly compare the heights of twochildren and describe one child as taller/shorter.

Classify objects and count the number of objects in each category.

3. Classifyobjectsintogivencategories;countthenumbersofobjectsineachcategoryandsortthecategoriesbycount.3

Geometry K.G

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

1. Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,in front of,behind,andnext to.

2. Correctlynameshapesregardlessoftheirorientationsoroverallsize.

3. Identifyshapesastwo-dimensional(lyinginaplane,“flat”)orthree-dimensional(“solid”).

Analyze, compare, create, and compose shapes.

4. Analyzeandcomparetwo-andthree-dimensionalshapes,indifferentsizesandorientations,usinginformallanguagetodescribetheirsimilarities,differences,parts(e.g.,numberofsidesandvertices/“corners”)andotherattributes(e.g.,havingsidesofequallength).

5. Modelshapesintheworldbybuildingshapesfromcomponents(e.g.,sticksandclayballs)anddrawingshapes.

6. Composesimpleshapestoformlargershapes.For example, “Can youjoin these two triangles with full sides touching to make a rectangle?”

3Limitcategorycountstobelessthanorequalto10.

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mathematics | Grade 1InGrade1,instructionaltimeshouldfocusonfourcriticalareas:(1)

developingunderstandingofaddition,subtraction,andstrategiesfor

additionandsubtractionwithin20;(2)developingunderstandingofwhole

numberrelationshipsandplacevalue,includinggroupingintensand

ones;(3)developingunderstandingoflinearmeasurementandmeasuring

lengthsasiteratinglengthunits;and(4)reasoningaboutattributesof,and

composinganddecomposinggeometricshapes.

(1)Studentsdevelopstrategiesforaddingandsubtractingwholenumbers

basedontheirpriorworkwithsmallnumbers.Theyuseavarietyofmodels,

includingdiscreteobjectsandlength-basedmodels(e.g.,cubesconnected

toformlengths),tomodeladd-to,take-from,put-together,take-apart,and

comparesituationstodevelopmeaningfortheoperationsofadditionand

subtraction,andtodevelopstrategiestosolvearithmeticproblemswith

theseoperations.Studentsunderstandconnectionsbetweencounting

andadditionandsubtraction(e.g.,addingtwoisthesameascountingon

two).Theyusepropertiesofadditiontoaddwholenumbersandtocreate

anduseincreasinglysophisticatedstrategiesbasedontheseproperties

(e.g.,“makingtens”)tosolveadditionandsubtractionproblemswithin

20. Bycomparingavarietyofsolutionstrategies,childrenbuildtheir

understandingoftherelationshipbetweenadditionandsubtraction.

(2)Studentsdevelop,discuss,anduseefficient,accurate,andgeneralizable

methodstoaddwithin100andsubtractmultiplesof10.Theycompare

wholenumbers(atleastto100)todevelopunderstandingofandsolve

problemsinvolvingtheirrelativesizes.Theythinkofwholenumbers

between10and100intermsoftensandones(especiallyrecognizingthe

numbers11to19ascomposedofatenandsomeones).Throughactivities

thatbuildnumbersense,theyunderstandtheorderofthecounting

numbersandtheirrelativemagnitudes.

(3)Studentsdevelopanunderstandingofthemeaningandprocessesof

measurement,includingunderlyingconceptssuchasiterating(themental

activityofbuildingupthelengthofanobjectwithequal-sizedunits)and

thetransitivityprincipleforindirectmeasurement.1

(4)Studentscomposeanddecomposeplaneorsolidfigures(e.g.,put

twotrianglestogethertomakeaquadrilateral)andbuildunderstanding

ofpart-wholerelationshipsaswellasthepropertiesoftheoriginaland

compositeshapes.Astheycombineshapes,theyrecognizethemfrom

differentperspectivesandorientations,describetheirgeometricattributes,

anddeterminehowtheyarealikeanddifferent,todevelopthebackground

formeasurementandforinitialunderstandingsofpropertiessuchas

congruenceandsymmetry.

1Studentsshouldapplytheprincipleoftransitivityofmeasurementtomakeindirectcomparisons,buttheyneednotusethistechnicalterm.

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Grade 1 overviewoperations and algebraic thinking

• represent and solve problems involvingaddition and subtraction.

• Understand and apply properties of operationsand the relationship between addition andsubtraction.

• add and subtract within 20.

• Work with addition and subtraction equations.


• extend the counting sequence.

• Understand place value.

• Use place value understanding and propertiesof operations to add and subtract.


• measure lengths indirectly and by iteratinglength units.

• tell and write time.

• represent and interpret data.

Geometry

• reason with shapes and their attributes.


1. Makesenseofproblemsandpersevereinsolvingthem.









reasoning.

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operations and algebraic thinking 1.oa

Represent and solve problems involving addition and subtraction.

1. Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.2

2. Solvewordproblemsthatcallforadditionofthreewholenumberswhosesumislessthanorequalto20,e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.

Understand and apply properties of operations and the relationship between addition and subtraction.

3. Applypropertiesofoperationsasstrategiestoaddandsubtract.3Examples:If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property ofaddition.) To add 2 + 6 + 4, the second two numbers can be added to makea ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

4. Understandsubtractionasanunknown-addendproblem.For example,subtract 10 – 8 by finding the number that makes 10 when added to 8.

Add and subtract within 20.

5. Relatecountingtoadditionandsubtraction(e.g.,bycountingon2to

add2).

6. Addandsubtractwithin20,demonstratingfluencyforadditionandsubtractionwithin10.Usestrategiessuchascountingon;makingten(e.g.,8+6=8+2+4=10+4=14);decomposinganumberleadingtoaten(e.g.,13–4=13–3–1=10–1=9);usingtherelationshipbetweenadditionandsubtraction(e.g.,knowingthat8+4=12,oneknows12–8=4);andcreatingequivalentbuteasierorknownsums(e.g.,adding6+7bycreatingtheknownequivalent6+6+1=12+1=13).

Work with addition and subtraction equations.

7. Understandthemeaningoftheequalsign,anddetermineifequationsinvolvingadditionandsubtractionaretrueorfalse.For example, whichof the following equations are true and which are false? 6 = 6, 7 = 8 – 1,5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

8. Determinetheunknownwholenumberinanadditionorsubtractionequationrelatingthreewholenumbers.For example, determine theunknown number that makes the equation true in each of the equations 8 +? = 11, 5 = � – 3, 6 + 6 = �.

number and operations in Base ten 1.nBt

Extend the counting sequence.

1. Countto120,startingatanynumberlessthan120.Inthisrange,readandwritenumeralsandrepresentanumberofobjectswithawrittennumeral.

Understand place value.

2. Understandthatthetwodigitsofatwo-digitnumberrepresentamountsoftensandones.Understandthefollowingasspecialcases:

a. 10canbethoughtofasabundleoftenones—calleda“ten.”

b. Thenumbersfrom11to19arecomposedofatenandone,two,three,four,five,six,seven,eight,ornineones.

c. Thenumbers10,20,30,40,50,60,70,80,90refertoone,two,three,four,five,six,seven,eight,orninetens(and0ones).

2SeeGlossary,Table1.3Studentsneednotuseformaltermsfortheseproperties.

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3. Comparetwotwo-digitnumbersbasedonmeaningsofthetensandonesdigits,recordingtheresultsofcomparisonswiththesymbols>,=,and<.

Use place value understanding and properties of operations to add and subtract.

4. Addwithin100,includingaddingatwo-digitnumberandaone-digitnumber,andaddingatwo-digitnumberandamultipleof10,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.Understandthatinaddingtwo-digitnumbers,oneaddstensandtens,onesandones;andsometimesitisnecessarytocomposeaten.

5. Givenatwo-digitnumber,mentallyfind10moreor10lessthanthenumber,withouthavingtocount;explainthereasoningused.

6. Subtractmultiplesof10intherange10-90frommultiplesof10intherange10-90(positiveorzerodifferences),usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethe

strategytoawrittenmethodandexplainthereasoningused.

measurement and data 1.md

Measure lengths indirectly and by iterating length units.

1. Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.

2. Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limit tocontexts where the object being measured is spanned by a whole number oflength units with no gaps or overlaps.

Tell and write time.

3. Tellandwritetimeinhoursandhalf-hoursusinganaloganddigitalclocks.

Represent and interpret data.

4. Organize,represent,andinterpretdatawithuptothreecategories;askandanswerquestionsaboutthetotalnumberofdatapoints,howmanyineachcategory,andhowmanymoreorlessareinonecategorythaninanother.

Geometry 1.G

Reason with shapes and their attributes.

1. Distinguishbetweendefiningattributes(e.g.,trianglesareclosedandthree-sided)versusnon-definingattributes(e.g.,color,orientation,overallsize);buildanddrawshapestopossessdefiningattributes.

2. Composetwo-dimensionalshapes(rectangles,squares,trapezoids,triangles,half-circles,andquarter-circles)orthree-dimensionalshapes(cubes,rightrectangularprisms,rightcircularcones,andrightcircularcylinders)tocreateacompositeshape,andcomposenewshapesfromthecompositeshape.4

3. Partitioncirclesandrectanglesintotwoandfourequalshares,describethesharesusingthewordshalves,fourths,andquarters,andusethephraseshalf of,fourth of,andquarter of.Describethewholeastwoof,orfouroftheshares.Understandfortheseexamplesthatdecomposingintomoreequalsharescreatessmallershares.

4Studentsdonotneedtolearnformalnamessuchas“rightrectangularprism.”

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extendingunderstandingofbase-tennotation;(2)buildingfluencywith

additionandsubtraction;(3)usingstandardunitsofmeasure;and(4)

describingandanalyzingshapes.

(1)Studentsextendtheirunderstandingofthebase-tensystem.This

includesideasofcountinginfives,tens,andmultiplesofhundreds,tens,

andones,aswellasnumberrelationshipsinvolvingtheseunits,including

comparing.Studentsunderstandmulti-digitnumbers(upto1000)written

inbase-tennotation,recognizingthatthedigitsineachplacerepresent

amountsofthousands,hundreds,tens,orones(e.g.,853is8hundreds+5

tens+3ones).

(2)Studentsusetheirunderstandingofadditiontodevelopfluencywith

additionandsubtractionwithin100.Theysolveproblemswithin1000

byapplyingtheirunderstandingofmodelsforadditionandsubtraction,

andtheydevelop,discuss,anduseefficient,accurate,andgeneralizable

methodstocomputesumsanddifferencesofwholenumbersinbase-ten

notation,usingtheirunderstandingofplacevalueandthepropertiesof

operations.Theyselectandaccuratelyapplymethodsthatareappropriate

forthecontextandthenumbersinvolvedtomentallycalculatesumsand

differencesfornumberswithonlytensoronlyhundreds.

(3)Studentsrecognizetheneedforstandardunitsofmeasure(centimeter

andinch)andtheyuserulersandothermeasurementtoolswiththe

understandingthatlinearmeasureinvolvesaniterationofunits.They

recognizethatthesmallertheunit,themoreiterationstheyneedtocovera

givenlength.

(4)Studentsdescribeandanalyzeshapesbyexaminingtheirsidesand

angles.Studentsinvestigate,describe,andreasonaboutdecomposing

andcombiningshapestomakeothershapes.Throughbuilding,drawing,

andanalyzingtwo-andthree-dimensionalshapes,studentsdevelopa

foundationforunderstandingarea,volume,congruence,similarity,and

symmetryinlatergrades.

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• represent and solve problems involvingaddition and subtraction.

• add and subtract within 20.

• Work with equal groups of objects to gainfoundations for multiplication.


• Understand place value.

• Use place value understanding andproperties of operations to add and subtract.


• measure and estimate lengths in standardunits.

• relate addition and subtraction to length.

• Work with time and money.


Geometry




solvingthem.









reasoning.

Grade 2 overview

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Represent and solve problems involving addition and subtraction.

1. Useadditionandsubtractionwithin100tosolveone-andtwo-stepwordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1

Add and subtract within 20.

2. Fluentlyaddandsubtractwithin20usingmentalstrategies.2ByendofGrade2,knowfrommemoryallsumsoftwoone-digitnumbers.

Work with equal groups of objects to gain foundations for multiplication.

3. Determinewhetheragroupofobjects(upto20)hasanoddorevennumberofmembers,e.g.,bypairingobjectsorcountingthemby2s;writeanequationtoexpressanevennumberasasumoftwoequaladdends.

4. Useadditiontofindthetotalnumberofobjectsarrangedinrectangulararrayswithupto5rowsandupto5columns;writeanequationtoexpressthetotalasasumofequaladdends.


Understand place value.

1. Understandthatthethreedigitsofathree-digitnumberrepresentamountsofhundreds,tens,andones;e.g.,706equals7hundreds,0tens,and6ones.Understandthefollowingasspecialcases:

a. 100canbethoughtofasabundleoftentens—calleda“hundred.”

b. Thenumbers100,200,300,400,500,600,700,800,900refertoone,two,three,four,five,six,seven,eight,orninehundreds(and0tensand0ones).

2. Countwithin1000;skip-countby5s,10s,and100s.

3. Readandwritenumbersto1000usingbase-tennumerals,numbernames,andexpandedform.

4. Comparetwothree-digitnumbersbasedonmeaningsofthehundreds,tens,andonesdigits,using>,=,and<symbolstorecordtheresultsofcomparisons.

Use place value understanding and properties of operations to add and subtract.

5. Fluentlyaddandsubtractwithin100usingstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.

6. Adduptofourtwo-digitnumbersusingstrategiesbasedonplacevalueandpropertiesofoperations.

7. Addandsubtractwithin1000,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethod.Understandthatinaddingorsubtractingthree-digitnumbers,oneaddsorsubtractshundredsandhundreds,tensandtens,onesandones;andsometimesitisnecessarytocomposeordecomposetensorhundreds.

8. Mentallyadd10or100toagivennumber100–900,andmentallysubtract10or100fromagivennumber100–900.

9. Explainwhyadditionandsubtractionstrategieswork,usingplacevalueandthepropertiesofoperations.3

1SeeGlossary,Table1.2Seestandard1.OA.6foralistofmentalstrategies.3Explanationsmaybesupportedbydrawingsorobjects.

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0


Measure and estimate lengths in standard units.

1. Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.

2. Measurethelengthofanobjecttwice,usinglengthunitsofdifferentlengthsforthetwomeasurements;describehowthetwomeasurementsrelatetothesizeoftheunitchosen.

3. Estimatelengthsusingunitsofinches,feet,centimeters,andmeters.

4. Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthelengthdifferenceintermsofastandardlengthunit.

Relate addition and subtraction to length.

5. Useadditionandsubtractionwithin100tosolvewordproblemsinvolvinglengthsthataregiveninthesameunits,e.g.,byusingdrawings(suchasdrawingsofrulers)andequationswithasymbolfortheunknownnumbertorepresenttheproblem.

6. Representwholenumbersaslengthsfrom0onanumberlinediagramwithequallyspacedpointscorrespondingtothenumbers0,1,2,...,andrepresentwhole-numbersumsanddifferenceswithin100onanumberlinediagram.

Work with time and money.

7. Tellandwritetimefromanaloganddigitalclockstothenearestfiveminutes,usinga.m.andp.m.

8. Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbolsappropriately.Example: If you have 2dimes and 3 pennies, how many cents do you have?


9. Generatemeasurementdatabymeasuringlengthsofseveralobjectstothenearestwholeunit,orbymakingrepeatedmeasurementsofthesameobject.Showthemeasurementsbymakingalineplot,wherethehorizontalscaleismarkedoffinwhole-numberunits.

10. Drawapicturegraphandabargraph(withsingle-unitscale)torepresentadatasetwithuptofourcategories.Solvesimpleput-together,take-apart,andcompareproblems4usinginformationpresentedinabargraph.

Geometry 2.G


1. Recognizeanddrawshapeshavingspecifiedattributes,suchasagivennumberofanglesoragivennumberofequalfaces.5Identifytriangles,quadrilaterals,pentagons,hexagons,andcubes.

2. Partitionarectangleintorowsandcolumnsofsame-sizesquaresandcounttofindthetotalnumberofthem.

3. Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,half of,a third of,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.

4SeeGlossary,Table1.5Sizesarecompareddirectlyorvisually,notcomparedbymeasuring.

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Mathematics|Grade3

InGrade3,instructionaltimeshouldfocusonfourcriticalareas:(1)

developingunderstandingofmultiplicationanddivisionandstrategies

formultiplicationanddivisionwithin100;(2)developingunderstanding

offractions,especiallyunitfractions(fractionswithnumerator1);(3)

developingunderstandingofthestructureofrectangulararraysandof

area;and(4)describingandanalyzingtwo-dimensionalshapes.

(1)Studentsdevelopanunderstandingofthemeaningsofmultiplication

anddivisionofwholenumbersthroughactivitiesandproblemsinvolving

equal-sizedgroups,arrays,andareamodels;multiplicationisfinding

anunknownproduct,anddivisionisfindinganunknownfactorinthese

situations.Forequal-sizedgroupsituations,divisioncanrequirefinding

theunknownnumberofgroupsortheunknowngroupsize.Studentsuse

propertiesofoperationstocalculateproductsofwholenumbers,using

increasinglysophisticatedstrategiesbasedonthesepropertiestosolve

multiplicationanddivisionproblemsinvolvingsingle-digitfactors.By

comparingavarietyofsolutionstrategies,studentslearntherelationship

betweenmultiplicationanddivision.

(2)Studentsdevelopanunderstandingoffractions,beginningwith

unitfractions.Studentsviewfractionsingeneralasbeingbuiltoutof

unitfractions,andtheyusefractionsalongwithvisualfractionmodels

torepresentpartsofawhole.Studentsunderstandthatthesizeofa

fractionalpartisrelativetothesizeofthewhole.Forexample,1/2ofthe

paintinasmallbucketcouldbelesspaintthan1/3ofthepaintinalarger

bucket,but1/3ofaribbonislongerthan1/5ofthesameribbonbecause

whentheribbonisdividedinto3equalparts,thepartsarelongerthan

whentheribbonisdividedinto5equalparts.Studentsareabletouse

fractionstorepresentnumbersequalto,lessthan,andgreaterthanone.

Theysolveproblemsthatinvolvecomparingfractionsbyusingvisual

fractionmodelsandstrategiesbasedonnoticingequalnumeratorsor

denominators.

(3)Studentsrecognizeareaasanattributeoftwo-dimensionalregions.

Theymeasuretheareaofashapebyfindingthetotalnumberofsame-

sizeunitsofarearequiredtocovertheshapewithoutgapsoroverlaps,

asquarewithsidesofunitlengthbeingthestandardunitformeasuring

area.Studentsunderstandthatrectangulararrayscanbedecomposedinto

identicalrowsorintoidenticalcolumns.Bydecomposingrectanglesinto

rectangulararraysofsquares,studentsconnectareatomultiplication,and

justifyusingmultiplicationtodeterminetheareaofarectangle.

(4)Studentsdescribe,analyze,andcomparepropertiesoftwo-

dimensionalshapes.Theycompareandclassifyshapesbytheirsidesand

angles,andconnectthesewithdefinitionsofshapes.Studentsalsorelate

theirfractionworktogeometrybyexpressingtheareaofpartofashape

asaunitfractionofthewhole.

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• represent and solve problems involvingmultiplication and division.

• Understand properties of multiplication andthe relationship between multiplication anddivision.

• multiply and divide within 100.

• Solve problems involving the four operations,and identify and explain patterns in arithmetic.


• Use place value understanding and propertiesof operations to perform multi-digit arithmetic.

number and operations—fractions

• develop understanding of fractions as numbers.


• Solve problems involving measurement andestimation of intervals of time, liquid volumes,and masses of objects.


• Geometric measurement: understand conceptsof area and relate area to multiplication and toaddition.

• Geometric measurement: recognize perimeteras an attribute of plane figures and distinguishbetween linear and area measures.

Geometry





3. Constructviableargumentsandcritiquethereasoningofothers.





8. Lookforandexpressregularityinrepeatedreasoning.

Grade 3 overviewmathematical Practices


solvingthem.









reasoning.

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Represent and solve problems involving multiplication and division.

1. Interpretproductsofwholenumbers,e.g.,interpret5×7asthetotalnumberofobjectsin5groupsof7objectseach.For example, describea context in which a total number of objects can be expressed as 5 × 7.

2. Interpretwhole-numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.Forexample, describe a context in which a number of shares or a number ofgroups can be expressed as 56 ÷ 8.

3. Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1

4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.For example, determine theunknown number that makes the equation true in each of the equations 8× ? = 48, 5 = � ÷ 3, 6 × 6 = ?.

Understand properties of multiplication and the relationship between multiplication and division.

5. Applypropertiesofoperationsasstrategiestomultiplyanddivide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.(Commutative property of multiplication.) 3 × 5 × 2 can be found by 3× 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associativeproperty of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, onecan find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributiveproperty.)

6. Understanddivisionasanunknown-factorproblem.For example, find32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Multiply and divide within 100.

7. Fluentlymultiplyanddividewithin100,usingstrategiessuchastherelationshipbetweenmultiplicationanddivision(e.g.,knowingthat8×5=40,oneknows40÷5=8)orpropertiesofoperations.BytheendofGrade3,knowfrommemoryallproductsoftwoone-digitnumbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

8. Solvetwo-stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.3

9. Identifyarithmeticpatterns(includingpatternsintheadditiontableormultiplicationtable),andexplainthemusingpropertiesofoperations.For example, observe that 4 times a number is always even, and explainwhy 4 times a number can be decomposed into two equal addends.

1SeeGlossary,Table2.2Studentsneednotuseformaltermsfortheseproperties.3Thisstandardislimitedtoproblemsposedwithwholenumbersandhavingwhole-numberanswers;studentsshouldknowhowtoperformoperationsintheconven-tionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).

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Number and Operations in Base Ten 3.NBT

Use place value understanding and properties of operations to perform multi-digit arithmetic.4

1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.

2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.

3. Multiplyone-digitwholenumbersbymultiplesof10intherange10–90(e.g.,9×80,5×60)usingstrategiesbasedonplacevalueandpropertiesofoperations.

Number and Operations—Fractions5 3.NF

Develop understanding of fractions as numbers.1. Understandafraction1/basthequantityformedby1partwhena

wholeispartitionedintob equalparts;understandafractiona/basthequantityformedbyapartsofsize1/b.

2. Understandafractionasanumberonthenumberline;representfractionsonanumberlinediagram.

a. Representafraction1/bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1/bandthattheendpointofthepartbasedat0locatesthenumber1/bonthenumberline.

b. Representafractiona/bonanumberlinediagrambymarkingoffalengths1/bfrom0.Recognizethattheresultingintervalhassizea/bandthatitsendpointlocatesthenumbera/bonthenumberline.

3. Explainequivalenceoffractionsinspecialcases,andcomparefractionsbyreasoningabouttheirsize.

a. Understandtwofractionsasequivalent(equal)iftheyarethesamesize,orthesamepointonanumberline.

b. Recognizeandgeneratesimpleequivalentfractions,e.g.,1/2=2/4,4/6=2/3.Explainwhythefractionsareequivalent,e.g.,byusingavisualfractionmodel.

c. Expresswholenumbersasfractions,andrecognizefractionsthatareequivalenttowholenumbers.Examples: Express 3 in the form3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same pointof a number line diagram.

d. Comparetwofractionswiththesamenumeratororthesamedenominatorbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.

Measurement and Data 3.MD

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

1. Tellandwritetimetothenearestminuteandmeasuretimeintervalsinminutes.Solvewordproblemsinvolvingadditionandsubtractionoftimeintervalsinminutes,e.g.,byrepresentingtheproblemonanumberlinediagram.

4Arangeofalgorithmsmaybeused.5Grade3expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,6,and8.

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2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardunitsofgrams(g),kilograms(kg),andliters(l).6Add,subtract,multiply,ordividetosolveone-stepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.7


3. Drawascaledpicturegraphandascaledbargraphtorepresentadatasetwithseveralcategories.Solveone-andtwo-step“howmanymore”and“howmanyless”problemsusinginformationpresentedinscaledbargraphs.For example, draw a bar graph in which each square inthe bar graph might represent 5 pets.

4. Generatemeasurementdatabymeasuringlengthsusingrulersmarkedwithhalvesandfourthsofaninch.Showthedatabymakingalineplot,wherethehorizontalscaleismarkedoffinappropriateunits—wholenumbers,halves,orquarters.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

5. Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement.

a. Asquarewithsidelength1unit,called“aunitsquare,”issaidtohave“onesquareunit”ofarea,andcanbeusedtomeasurearea.

b. Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.

6. Measureareasbycountingunitsquares(squarecm,squarem,squarein,squareft,andimprovisedunits).

7. Relateareatotheoperationsofmultiplicationandaddition.

a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.

b. Multiplysidelengthstofindareasofrectangleswithwhole-numbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.

c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.

d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposingthemintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

8. Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.

6Excludescompoundunitssuchascm3andfindingthegeometricvolumeofacontainer.7Excludesmultiplicativecomparisonproblems(problemsinvolvingnotionsof“timesasmuch”;seeGlossary,Table2).

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Geometry 3.G


1. Understandthatshapesindifferentcategories(e.g.,rhombuses,rectangles,andothers)mayshareattributes(e.g.,havingfoursides),andthatthesharedattributescandefinealargercategory(e.g.,quadrilaterals).Recognizerhombuses,rectangles,andsquaresasexamplesofquadrilaterals,anddrawexamplesofquadrilateralsthatdonotbelongtoanyofthesesubcategories.

2. Partitionshapesintopartswithequalareas.Expresstheareaofeachpartasaunitfractionofthewhole.For example, partition a shape into 4parts with equal area, and describe the area of each part as 1/4 of the areaof the shape.

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mathematics | Grade 4InGrade4,instructionaltimeshouldfocusonthreecriticalareas:(1)

developingunderstandingandfluencywithmulti-digitmultiplication,

anddevelopingunderstandingofdividingtofindquotientsinvolving

multi-digitdividends;(2)developinganunderstandingoffraction

equivalence,additionandsubtractionoffractionswithlikedenominators,

andmultiplicationoffractionsbywholenumbers;(3)understanding

thatgeometricfigurescanbeanalyzedandclassifiedbasedontheir

properties,suchashavingparallelsides,perpendicularsides,particular

anglemeasures,andsymmetry.

(1)Studentsgeneralizetheirunderstandingofplacevalueto1,000,000,

understandingtherelativesizesofnumbersineachplace.Theyapplytheir

understandingofmodelsformultiplication(equal-sizedgroups,arrays,

areamodels),placevalue,andpropertiesofoperations,inparticularthe

distributiveproperty,astheydevelop,discuss,anduseefficient,accurate,

andgeneralizablemethodstocomputeproductsofmulti-digitwhole

numbers.Dependingonthenumbersandthecontext,theyselectand

accuratelyapplyappropriatemethodstoestimateormentallycalculate

products.Theydevelopfluencywithefficientproceduresformultiplying

wholenumbers;understandandexplainwhytheproceduresworkbasedon

placevalueandpropertiesofoperations;andusethemtosolveproblems.

Studentsapplytheirunderstandingofmodelsfordivision,placevalue,

propertiesofoperations,andtherelationshipofdivisiontomultiplication

astheydevelop,discuss,anduseefficient,accurate,andgeneralizable

procedurestofindquotientsinvolvingmulti-digitdividends.Theyselect

andaccuratelyapplyappropriatemethodstoestimateandmentally

calculatequotients,andinterpretremaindersbaseduponthecontext.

(2)Studentsdevelopunderstandingoffractionequivalenceand

operationswithfractions.Theyrecognizethattwodifferentfractionscan

beequal(e.g.,15/9=5/3),andtheydevelopmethodsforgeneratingand

recognizingequivalentfractions.Studentsextendpreviousunderstandings

abouthowfractionsarebuiltfromunitfractions,composingfractions

fromunitfractions,decomposingfractionsintounitfractions,andusing

themeaningoffractionsandthemeaningofmultiplicationtomultiplya

fractionbyawholenumber.

(3)Studentsdescribe,analyze,compare,andclassifytwo-dimensional

shapes.Throughbuilding,drawing,andanalyzingtwo-dimensionalshapes,

studentsdeepentheirunderstandingofpropertiesoftwo-dimensional

objectsandtheuseofthemtosolveproblemsinvolvingsymmetry.

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Grade 4 overviewoperations and algebraic thinking

• Use the four operations with whole numbers tosolve problems.

• Gain familiarity with factors and multiples.

• Generate and analyze patterns.


• Generalize place value understanding for multi-digit whole numbers.

• Use place value understanding and properties ofoperations to perform multi-digit arithmetic.


• extend understanding of fraction equivalenceand ordering.

• Build fractions from unit fractions by applyingand extending previous understandings ofoperations on whole numbers.

• Understand decimal notation for fractions, andcompare decimal fractions.


• Solve problems involving measurement andconversion of measurements from a larger unit toa smaller unit.


• Geometric measurement: understand concepts ofangle and measure angles.

Geometry

• draw and identify lines and angles, and classifyshapes by properties of their lines and angles.










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Use the four operations with whole numbers to solve problems.

1. Interpretamultiplicationequationasacomparison,e.g.,interpret35=5×7asastatementthat35is5timesasmanyas7and7timesasmanyas5.Representverbalstatementsofmultiplicativecomparisonsasmultiplicationequations.

2. Multiplyordividetosolvewordproblemsinvolvingmultiplicativecomparison,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem,distinguishingmultiplicativecomparisonfromadditivecomparison.1

3. Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole-numberanswersusingthefouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.

Gain familiarity with factors and multiples.

4. Findallfactorpairsforawholenumberintherange1–100.Recognizethatawholenumberisamultipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1–100isamultipleofagivenone-digitnumber.Determinewhetheragivenwholenumberintherange1–100isprimeorcomposite.

Generate and analyze patterns.

5. Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.For example, given the rule “Add 3” and the starting number 1, generateterms in the resulting sequence and observe that the terms appear toalternate between odd and even numbers. Explain informally why thenumbers will continue to alternate in this way.

number and operations in Base ten2 4.nBt

Generalize place value understanding for multi-digit whole numbers.

1. Recognizethatinamulti-digitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.Forexample, recognize that 700 ÷ 70 = 10 by applying concepts of place valueand division.

2. Readandwritemulti-digitwholenumbersusingbase-tennumerals,numbernames,andexpandedform.Comparetwomulti-digitnumbersbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.

3. Useplacevalueunderstandingtoroundmulti-digitwholenumberstoanyplace.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

4. Fluentlyaddandsubtractmulti-digitwholenumbersusingthestandardalgorithm.

5. Multiplyawholenumberofuptofourdigitsbyaone-digitwholenumber,andmultiplytwotwo-digitnumbers,usingstrategiesbasedonplacevalueandthepropertiesofoperations.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.

1SeeGlossary,Table2.2Grade4expectationsinthisdomainarelimitedtowholenumberslessthanorequalto1,000,000.

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6. Findwhole-numberquotientsandremainderswithuptofour-digitdividendsandone-digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.

number and operations—fractions3 4.nf

Extend understanding of fraction equivalence and ordering.

1. Explainwhyafractiona/bisequivalenttoafraction(n×a)/(n×b)byusingvisualfractionmodels,withattentiontohowthenumberandsizeofthepartsdiffereventhoughthetwofractionsthemselvesarethesamesize.Usethisprincipletorecognizeandgenerateequivalentfractions.

2. Comparetwofractionswithdifferentnumeratorsanddifferentdenominators,e.g.,bycreatingcommondenominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas1/2.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswithsymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

3. Understandafractiona/bwitha>1asasumoffractions1/b.

a. Understandadditionandsubtractionoffractionsasjoiningandseparatingpartsreferringtothesamewhole.

b. Decomposeafractionintoasumoffractionswiththesamedenominatorinmorethanoneway,recordingeachdecompositionbyanequation.Justifydecompositions,e.g.,byusingavisualfractionmodel.Examples: 3/8 = 1/8 + 1/8 + 1/8 ;3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c. Addandsubtractmixednumberswithlikedenominators,e.g.,byreplacingeachmixednumberwithanequivalentfraction,and/orbyusingpropertiesofoperationsandtherelationshipbetweenadditionandsubtraction.

d. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.

4. Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionbyawholenumber.

a. Understandafractiona/basamultipleof1/b.For example, usea visual fraction model to represent 5/4 as the product 5 × (1/4),recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understandamultipleofa/basamultipleof1/b,andusethisunderstandingtomultiplyafractionbyawholenumber.Forexample, use a visual fraction model to express 3 × (2/5) as 6 × (1/5),recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solvewordproblemsinvolvingmultiplicationofafractionbyawholenumber,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, if each person at a party willeat 3/8 of a pound of roast beef, and there will be 5 people at theparty, how many pounds of roast beef will be needed? Between whattwo whole numbers does your answer lie?

3Grade4expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,5,6,8,10,12,and100.

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Understand decimal notation for fractions, and compare decimal fractions.

5. Expressafractionwithdenominator10asanequivalentfractionwithdenominator100,andusethistechniquetoaddtwofractionswithrespectivedenominators10and100.4For example, express 3/10 as30/100, and add 3/10 + 4/100 = 34/100.

6. Usedecimalnotationforfractionswithdenominators10or100.Forexample, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate0.62 on a number line diagram.

7. Comparetwodecimalstohundredthsbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwodecimalsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualmodel.


Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

1. Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz.;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.Recordmeasurementequivalentsinatwo-columntable.For example, know that 1 ft is 12 times as long as 1 in.Express the length of a 4 ft snake as 48 in. Generate a conversion table forfeet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

2. Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.Representmeasurementquantitiesusingdiagramssuchasnumberlinediagramsthatfeatureameasurementscale.

3. Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.For example, find the width of a rectangularroom given the area of the flooring and the length, by viewing the areaformula as a multiplication equation with an unknown factor.


4. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.For example,from a line plot find and interpret the difference in length between thelongest and shortest specimens in an insect collection.

Geometric measurement: understand concepts of angle and measure angles.

5. Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:

a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.

b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.

4Studentswhocangenerateequivalentfractionscandevelopstrategiesforaddingfractionswithunlikedenominatorsingeneral.Butadditionandsubtractionwithun-likedenominatorsingeneralisnotarequirementatthisgrade.

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6. Measureanglesinwhole-numberdegreesusingaprotractor.Sketchanglesofspecifiedmeasure.

7. Recognizeanglemeasureasadditive.Whenanangleisdecomposedintonon-overlappingparts,theanglemeasureofthewholeisthesumoftheanglemeasuresoftheparts.Solveadditionandsubtractionproblemstofindunknownanglesonadiagraminrealworldandmathematicalproblems,e.g.,byusinganequationwithasymbolfortheunknownanglemeasure.

Geometry 4.G

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

1. Drawpoints,lines,linesegments,rays,angles(right,acute,obtuse),andperpendicularandparallellines.Identifytheseintwo-dimensionalfigures.

2. Classifytwo-dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.

3. Recognizealineofsymmetryforatwo-dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline-symmetricfiguresanddrawlinesofsymmetry.

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mathematics | Grade 5InGrade5,instructionaltimeshouldfocusonthreecriticalareas:(1)

developingfluencywithadditionandsubtractionoffractions,and

developingunderstandingofthemultiplicationoffractionsandofdivision

offractionsinlimitedcases(unitfractionsdividedbywholenumbersand

wholenumbersdividedbyunitfractions);(2)extendingdivisionto2-digit

divisors,integratingdecimalfractionsintotheplacevaluesystemand

developingunderstandingofoperationswithdecimalstohundredths,and

developingfluencywithwholenumberanddecimaloperations;and(3)

developingunderstandingofvolume.

(1)Studentsapplytheirunderstandingoffractionsandfractionmodelsto

representtheadditionandsubtractionoffractionswithunlikedenominators

asequivalentcalculationswithlikedenominators.Theydevelopfluency

incalculatingsumsanddifferencesoffractions,andmakereasonable

estimatesofthem.Studentsalsousethemeaningoffractions,of

multiplicationanddivision,andtherelationshipbetweenmultiplicationand

divisiontounderstandandexplainwhytheproceduresformultiplyingand

dividingfractionsmakesense.(Note:thisislimitedtothecaseofdividing

unitfractionsbywholenumbersandwholenumbersbyunitfractions.)

(2)Studentsdevelopunderstandingofwhydivisionprocedureswork

basedonthemeaningofbase-tennumeralsandpropertiesofoperations.

Theyfinalizefluencywithmulti-digitaddition,subtraction,multiplication,

anddivision.Theyapplytheirunderstandingsofmodelsfordecimals,

decimalnotation,andpropertiesofoperationstoaddandsubtract

decimalstohundredths.Theydevelopfluencyinthesecomputations,and

makereasonableestimatesoftheirresults.Studentsusetherelationship

betweendecimalsandfractions,aswellastherelationshipbetween

finitedecimalsandwholenumbers(i.e.,afinitedecimalmultipliedbyan

appropriatepowerof10isawholenumber),tounderstandandexplain

whytheproceduresformultiplyinganddividingfinitedecimalsmake

sense.Theycomputeproductsandquotientsofdecimalstohundredths

efficientlyandaccurately.

(3)Studentsrecognizevolumeasanattributeofthree-dimensional

space.Theyunderstandthatvolumecanbemeasuredbyfindingthetotal

numberofsame-sizeunitsofvolumerequiredtofillthespacewithout

gapsoroverlaps.Theyunderstandthata1-unitby1-unitby1-unitcube

isthestandardunitformeasuringvolume.Theyselectappropriateunits,

strategies,andtoolsforsolvingproblemsthatinvolveestimatingand

measuringvolume.Theydecomposethree-dimensionalshapesandfind

volumesofrightrectangularprismsbyviewingthemasdecomposedinto

layersofarraysofcubes.Theymeasurenecessaryattributesofshapesin

ordertodeterminevolumestosolverealworldandmathematicalproblems.

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• Write and interpret numerical expressions.

• analyze patterns and relationships.


• Understand the place value system.

• Perform operations with multi-digit wholenumbers and with decimals to hundredths.


• Use equivalent fractions as a strategy to addand subtract fractions.

• apply and extend previous understandingsof multiplication and division to multiply anddivide fractions.


• Convert like measurement units within a givenmeasurement system.


• Geometric measurement: understand conceptsof volume and relate volume to multiplicationand to addition.

Geometry

• Graph points on the coordinate plane to solvereal-world and mathematical problems.

• Classify two-dimensional figures into categoriesbased on their properties.










Grade 5 overview

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Write and interpret numerical expressions.

1. Useparentheses,brackets,orbracesinnumericalexpressions,andevaluateexpressionswiththesesymbols.

2. Writesimpleexpressionsthatrecordcalculationswithnumbers,andinterpretnumericalexpressionswithoutevaluatingthem.For example,express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7).Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921,without having to calculate the indicated sum or product.

Analyze patterns and relationships.

3. Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationshipsbetweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwopatterns,andgraphtheorderedpairsonacoordinateplane.For example, given therule “Add 3” and the starting number 0, and given the rule “Add 6” and thestarting number 0, generate terms in the resulting sequences, and observethat the terms in one sequence are twice the corresponding terms in theother sequence. Explain informally why this is so.


Understand the place value system.

1. Recognizethatinamulti-digitnumber,adigitinoneplacerepresents10timesasmuchasitrepresentsintheplacetoitsrightand1/10ofwhatitrepresentsintheplacetoitsleft.

2. Explainpatternsinthenumberofzerosoftheproductwhenmultiplyinganumberbypowersof10,andexplainpatternsintheplacementofthedecimalpointwhenadecimalismultipliedordividedbyapowerof10.Usewhole-numberexponentstodenotepowersof10.

3. Read,write,andcomparedecimalstothousandths.

a. Readandwritedecimalstothousandthsusingbase-tennumerals,numbernames,andexpandedform,e.g.,347.392=3×100+4×10+7×1+3×(1/10)+9×(1/100)+2×(1/1000).

b. Comparetwodecimalstothousandthsbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.

4. Useplacevalueunderstandingtorounddecimalstoanyplace.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

5. Fluentlymultiplymulti-digitwholenumbersusingthestandardalgorithm.

6. Findwhole-numberquotientsofwholenumberswithuptofour-digitdividendsandtwo-digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.

7. Add,subtract,multiply,anddividedecimalstohundredths,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.

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number and operations—fractions 5.nf

Use equivalent fractions as a strategy to add and subtract fractions.

1. Addandsubtractfractionswithunlikedenominators(includingmixednumbers)byreplacinggivenfractionswithequivalentfractionsinsuchawayastoproduceanequivalentsumordifferenceoffractionswithlikedenominators.For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (Ingeneral, a/b + c/d = (ad + bc)/bd.)

2. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewhole,includingcasesofunlikedenominators,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.Usebenchmarkfractionsandnumbersenseoffractionstoestimatementallyandassessthereasonablenessofanswers.Forexample, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that3/7 < 1/2.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

3. Interpretafractionasdivisionofthenumeratorbythedenominator(a/b=a÷b).Solvewordproblemsinvolvingdivisionofwholenumbersleadingtoanswersintheformoffractionsormixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

4. Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionorwholenumberbyafraction.

a. Interprettheproduct(a/b)×qasapartsofapartitionofqintobequalparts;equivalently,astheresultofasequenceofoperationsa×q÷b.For example, use a visual fraction model toshow (2/3) × 4 = 8/3, and create a story context for this equation. Dothe same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

b. Findtheareaofarectanglewithfractionalsidelengthsbytilingitwithunitsquaresoftheappropriateunitfractionsidelengths,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.Multiplyfractionalsidelengthstofindareasofrectangles,andrepresentfractionproductsasrectangularareas.

5. Interpretmultiplicationasscaling(resizing),by:

a. Comparingthesizeofaproducttothesizeofonefactoronthebasisofthesizeoftheotherfactor,withoutperformingtheindicatedmultiplication.

b. Explainingwhymultiplyingagivennumberbyafractiongreaterthan1resultsinaproductgreaterthanthegivennumber(recognizingmultiplicationbywholenumbersgreaterthan1asafamiliarcase);explainingwhymultiplyingagivennumberbyafractionlessthan1resultsinaproductsmallerthanthegivennumber;andrelatingtheprincipleoffractionequivalencea/b=(n×a)/(n×b)totheeffectofmultiplyinga/bby1.

6. Solverealworldproblemsinvolvingmultiplicationoffractionsandmixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.

7. Applyandextendpreviousunderstandingsofdivisiontodivideunitfractionsbywholenumbersandwholenumbersbyunitfractions.1

a. Interpretdivisionofaunitfractionbyanon-zerowholenumber,

1Studentsabletomultiplyfractionsingeneralcandevelopstrategiestodividefrac-tionsingeneral,byreasoningabouttherelationshipbetweenmultiplicationanddivision.Butdivisionofafractionbyafractionisnotarequirementatthisgrade.

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andcomputesuchquotients.For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient.Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpretdivisionofawholenumberbyaunitfraction,andcomputesuchquotients.For example, create a story context for4 ÷ (1/5), and use a visual fraction model to show the quotient. Usethe relationship between multiplication and division to explain that4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solverealworldproblemsinvolvingdivisionofunitfractionsbynon-zerowholenumbersanddivisionofwholenumbersbyunitfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, how much chocolate will eachperson get if 3 people share 1/2 lb of chocolate equally? How many1/3-cup servings are in 2 cups of raisins?


Convert like measurement units within a given measurement system.

1. Convertamongdifferent-sizedstandardmeasurementunitswithinagivenmeasurementsystem(e.g.,convert5cmto0.05m),andusetheseconversionsinsolvingmulti-step,realworldproblems.


2. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.For example,given different measurements of liquid in identical beakers, find theamount of liquid each beaker would contain if the total amount in all thebeakers were redistributed equally.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

3. Recognizevolumeasanattributeofsolidfiguresandunderstandconceptsofvolumemeasurement.

a. Acubewithsidelength1unit,calleda“unitcube,”issaidtohave“onecubicunit”ofvolume,andcanbeusedtomeasurevolume.

b. Asolidfigurewhichcanbepackedwithoutgapsoroverlapsusingnunitcubesissaidtohaveavolumeofncubicunits.

4. Measurevolumesbycountingunitcubes,usingcubiccm,cubicin,cubicft,andimprovisedunits.

5. Relatevolumetotheoperationsofmultiplicationandadditionandsolverealworldandmathematicalproblemsinvolvingvolume.

a. Findthevolumeofarightrectangularprismwithwhole-numbersidelengthsbypackingitwithunitcubes,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengths,equivalentlybymultiplyingtheheightbytheareaofthebase.Representthreefoldwhole-numberproductsasvolumes,e.g.,torepresenttheassociativepropertyofmultiplication.

b. ApplytheformulasV=l×w×handV=b×hforrectangularprismstofindvolumesofrightrectangularprismswithwhole-numberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.

c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwonon-overlappingrightrectangularprismsbyaddingthevolumesofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.

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Geometry 5.G

Graph points on the coordinate plane to solve real-world and mathematical problems.

1. Useapairofperpendicularnumberlines,calledaxes,todefineacoordinatesystem,withtheintersectionofthelines(theorigin)arrangedtocoincidewiththe0oneachlineandagivenpointintheplanelocatedbyusinganorderedpairofnumbers,calleditscoordinates.Understandthatthefirstnumberindicateshowfartotravelfromtheorigininthedirectionofoneaxis,andthesecondnumberindicateshowfartotravelinthedirectionofthesecondaxis,withtheconventionthatthenamesofthetwoaxesandthecoordinatescorrespond(e.g.,x-axisandx-coordinate,y-axisandy-coordinate).

2. Representrealworldandmathematicalproblemsbygraphingpointsinthefirstquadrantofthecoordinateplane,andinterpretcoordinatevaluesofpointsinthecontextofthesituation.

Classify two-dimensional figures into categories based on their properties.

3. Understandthatattributesbelongingtoacategoryoftwo-dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.For example, all rectangles have four right angles and squares arerectangles, so all squares have four right angles.

4. Classifytwo-dimensionalfiguresinahierarchybasedonproperties.

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connectingratioandratetowholenumbermultiplicationanddivision

andusingconceptsofratioandratetosolveproblems;(2)completing

understandingofdivisionoffractionsandextendingthenotionofnumber

tothesystemofrationalnumbers,whichincludesnegativenumbers;

(3)writing,interpreting,andusingexpressionsandequations;and(4)

developingunderstandingofstatisticalthinking.

(1)Studentsusereasoningaboutmultiplicationanddivisiontosolve

ratioandrateproblemsaboutquantities.Byviewingequivalentratios

andratesasderivingfrom,andextending,pairsofrows(orcolumns)in

themultiplicationtable,andbyanalyzingsimpledrawingsthatindicate

therelativesizeofquantities,studentsconnecttheirunderstandingof

multiplicationanddivisionwithratiosandrates.Thusstudentsexpandthe

scopeofproblemsforwhichtheycanusemultiplicationanddivisionto

solveproblems,andtheyconnectratiosandfractions.Studentssolvea

widevarietyofproblemsinvolvingratiosandrates.

(2)Studentsusethemeaningoffractions,themeaningsofmultiplication

anddivision,andtherelationshipbetweenmultiplicationanddivisionto

understandandexplainwhytheproceduresfordividingfractionsmake

sense.Studentsusetheseoperationstosolveproblems.Studentsextend

theirpreviousunderstandingsofnumberandtheorderingofnumbers

tothefullsystemofrationalnumbers,whichincludesnegativerational

numbers,andinparticularnegativeintegers.Theyreasonabouttheorder

andabsolutevalueofrationalnumbersandaboutthelocationofpointsin

allfourquadrantsofthecoordinateplane.

(3)Studentsunderstandtheuseofvariablesinmathematicalexpressions.

Theywriteexpressionsandequationsthatcorrespondtogivensituations,

evaluateexpressions,anduseexpressionsandformulastosolveproblems.

Studentsunderstandthatexpressionsindifferentformscanbeequivalent,

andtheyusethepropertiesofoperationstorewriteexpressionsin

equivalentforms.Studentsknowthatthesolutionsofanequationarethe

valuesofthevariablesthatmaketheequationtrue.Studentsuseproperties

ofoperationsandtheideaofmaintainingtheequalityofbothsidesof

anequationtosolvesimpleone-stepequations.Studentsconstructand

analyzetables,suchastablesofquantitiesthatareinequivalentratios,

andtheyuseequations(suchas3x=y)todescriberelationshipsbetween

quantities.

(4)Buildingonandreinforcingtheirunderstandingofnumber,students

begintodeveloptheirabilitytothinkstatistically.Studentsrecognizethata

datadistributionmaynothaveadefinitecenterandthatdifferentwaysto

measurecenteryielddifferentvalues.Themedianmeasurescenterinthe

sensethatitisroughlythemiddlevalue.Themeanmeasurescenterinthe

sensethatitisthevaluethateachdatapointwouldtakeonifthetotalof

thedatavalueswereredistributedequally,andalsointhesensethatitisa

balancepoint.Studentsrecognizethatameasureofvariability(interquartile

rangeormeanabsolutedeviation)canalsobeusefulforsummarizing

databecausetwoverydifferentsetsofdatacanhavethesamemeanand

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medianyetbedistinguishedbytheirvariability.Studentslearntodescribe

andsummarizenumericaldatasets,identifyingclusters,peaks,gaps,and

symmetry,consideringthecontextinwhichthedatawerecollected.

StudentsinGrade6alsobuildontheirworkwithareainelementary

schoolbyreasoningaboutrelationshipsamongshapestodeterminearea,

surfacearea,andvolume.Theyfindareasofrighttriangles,othertriangles,

andspecialquadrilateralsbydecomposingtheseshapes,rearranging

orremovingpieces,andrelatingtheshapestorectangles.Usingthese

methods,studentsdiscuss,develop,andjustifyformulasforareasof

trianglesandparallelograms.Studentsfindareasofpolygonsandsurface

areasofprismsandpyramidsbydecomposingthemintopieceswhose

areatheycandetermine.Theyreasonaboutrightrectangularprisms

withfractionalsidelengthstoextendformulasforthevolumeofaright

rectangularprismtofractionalsidelengths.Theyprepareforworkon

scaledrawingsandconstructionsinGrade7bydrawingpolygonsinthe

coordinateplane.

ratios and Proportional relationships

• Understand ratio concepts and use ratioreasoning to solve problems.

the number System

• apply and extend previous understandings ofmultiplication and division to divide fractionsby fractions.

• Compute fluently with multi-digit numbers andfind common factors and multiples.

• apply and extend previous understandings ofnumbers to the system of rational numbers.

expressions and equations

• apply and extend previous understandings ofarithmetic to algebraic expressions.

• reason about and solve one-variable equationsand inequalities.

• represent and analyze quantitativerelationships between dependent andindependent variables.

Geometry

• Solve real-world and mathematical problemsinvolving area, surface area, and volume.

Statistics and Probability

• develop understanding of statistical variability.

• Summarize and describe distributions.










Grade 6 overview

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ratios and Proportional relationships 6.rP

Understand ratio concepts and use ratio reasoning to solve problems.

1. Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities.For example, “The ratioof wings to beaks in the bird house at the zoo was 2:1, because forevery 2 wings there was 1 beak.” “For every vote candidate A received,candidate C received nearly three votes.”

2. Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb≠0,anduseratelanguageinthecontextofaratiorelationship.For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar,so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15hamburgers, which is a rate of $5 per hamburger.”1

3. Useratioandratereasoningtosolvereal-worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.

a. Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.

b. Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.For example, if it took 7 hours to mow 4 lawns, thenat that rate, how many lawns could be mowed in 35 hours? At whatrate were lawns being mowed?

c. Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.

d. Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.

the number System 6.nS

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

1. Interpretandcomputequotientsoffractions,andsolvewordproblemsinvolvingdivisionoffractionsbyfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.Forexample, create a story context for (2/3) ÷ (3/4) and use a visual fractionmodel to show the quotient; use the relationship between multiplicationand division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3.(In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each personget if 3 people share 1/2 lb of chocolate equally? How many 3/4-cupservings are in 2/3 of a cup of yogurt? How wide is a rectangular strip ofland with length 3/4 mi and area 1/2 square mi?

Compute fluently with multi-digit numbers and find common factors and multiples.

2. Fluentlydividemulti-digitnumbersusingthestandardalgorithm.

3. Fluentlyadd,subtract,multiply,anddividemulti-digitdecimalsusingthestandardalgorithmforeachoperation.

4. Findthegreatestcommonfactoroftwowholenumberslessthanorequalto100andtheleastcommonmultipleoftwowholenumberslessthanorequalto12.Usethedistributivepropertytoexpressasumoftwowholenumbers1–100withacommonfactorasamultipleofasumoftwowholenumberswithnocommonfactor.For example,express 36 + 8 as 4 (9 + 2).

1Expectationsforunitratesinthisgradearelimitedtonon-complexfractions.

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Apply and extend previous understandings of numbers to the system of rational numbers.

5. Understandthatpositiveandnegativenumbersareusedtogethertodescribequantitieshavingoppositedirectionsorvalues(e.g.,temperatureabove/belowzero,elevationabove/belowsealevel,credits/debits,positive/negativeelectriccharge);usepositiveandnegativenumberstorepresentquantitiesinreal-worldcontexts,explainingthemeaningof0ineachsituation.

6. Understandarationalnumberasapointonthenumberline.Extendnumberlinediagramsandcoordinateaxesfamiliarfrompreviousgradestorepresentpointsonthelineandintheplanewithnegativenumbercoordinates.

a. Recognizeoppositesignsofnumbersasindicatinglocationsonoppositesidesof0onthenumberline;recognizethattheoppositeoftheoppositeofanumberisthenumberitself,e.g.,–(–3)=3,andthat0isitsownopposite.

b. Understandsignsofnumbersinorderedpairsasindicatinglocationsinquadrantsofthecoordinateplane;recognizethatwhentwoorderedpairsdifferonlybysigns,thelocationsofthepointsarerelatedbyreflectionsacrossoneorbothaxes.

c. Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram;findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane.

7. Understandorderingandabsolutevalueofrationalnumbers.

a. Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram.For example,interpret –3 > –7 as a statement that –3 is located to the right of –7 ona number line oriented from left to right.

b. Write,interpret,andexplainstatementsoforderforrationalnumbersinreal-worldcontexts.For example, write –3 oC > –7 oC toexpress the fact that –3 oC is warmer than –7 oC.

c. Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal-worldsituation.Forexample, for an account balance of –30 dollars, write |–30| = 30 todescribe the size of the debt in dollars.

d. Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder.For example, recognize that an account balance less than –30dollars represents a debt greater than 30 dollars.

8. Solvereal-worldandmathematicalproblemsbygraphingpointsinallfourquadrantsofthecoordinateplane.Includeuseofcoordinatesandabsolutevaluetofinddistancesbetweenpointswiththesamefirstcoordinateorthesamesecondcoordinate.

expressions and equations 6.ee

Apply and extend previous understandings of arithmetic to algebraic expressions.

1. Writeandevaluatenumericalexpressionsinvolvingwhole-numberexponents.

2. Write,read,andevaluateexpressionsinwhichlettersstandfornumbers.

a. Writeexpressionsthatrecordoperationswithnumbersandwithlettersstandingfornumbers.For example, express the calculation“Subtract y from 5” as 5 – y.

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b. Identifypartsofanexpressionusingmathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofanexpressionasasingleentity.For example, describe theexpression 2 (8 + 7) as a product of two factors; view (8 + 7) as botha single entity and a sum of two terms.

c. Evaluateexpressionsatspecificvaluesoftheirvariables.Includeexpressionsthatarisefromformulasusedinreal-worldproblems.Performarithmeticoperations,includingthoseinvolvingwhole-numberexponents,intheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).For example, use the formulas V = s3 and A = 6 s2 to find the volumeand surface area of a cube with sides of length s = 1/2.

3. Applythepropertiesofoperationstogenerateequivalentexpressions.For example, apply the distributive property to the expression 3 (2 + x) toproduce the equivalent expression 6 + 3x; apply the distributive propertyto the expression 24x + 18y to produce the equivalent expression6 (4x + 3y); apply properties of operations to y + y + y to produce theequivalent expression 3y.

4. Identifywhentwoexpressionsareequivalent(i.e.,whenthetwoexpressionsnamethesamenumberregardlessofwhichvalueissubstitutedintothem).For example, the expressions y + y + y and 3yare equivalent because they name the same number regardless of whichnumber y stands for.

Reason about and solve one-variable equations and inequalities.

5. Understandsolvinganequationorinequalityasaprocessofansweringaquestion:whichvaluesfromaspecifiedset,ifany,maketheequationorinequalitytrue?Usesubstitutiontodeterminewhetheragivennumberinaspecifiedsetmakesanequationorinequalitytrue.

6. Usevariablestorepresentnumbersandwriteexpressionswhensolvingareal-worldormathematicalproblem;understandthatavariablecanrepresentanunknownnumber,or,dependingonthepurposeathand,anynumberinaspecifiedset.

7. Solvereal-worldandmathematicalproblemsbywritingandsolvingequationsoftheformx+p=qandpx=qforcasesinwhichp,qandxareallnonnegativerationalnumbers.

8. Writeaninequalityoftheformx>corx <c torepresentaconstraintorconditioninareal-worldormathematicalproblem.Recognizethatinequalitiesoftheformx>corx<chaveinfinitelymanysolutions;representsolutionsofsuchinequalitiesonnumberlinediagrams.

Represent and analyze quantitative relationships between dependent and independent variables.

9. Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable,intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariablesusinggraphsandtables,andrelatethesetotheequation.For example, in aproblem involving motion at constant speed, list and graph ordered pairsof distances and times, and write the equation d = 65t to represent therelationship between distance and time.

Geometry 6.G

Solve real-world and mathematical problems involving area, surface area, and volume.

1. Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

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2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV = l w h and V = b htofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems.

3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

4. Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

Statistics and Probability 6.SP

Develop understanding of statistical variability.

1. Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedatarelatedtothequestionandaccountsforitintheanswers.Forexample, “How old am I?” is not a statistical question, but “How old are thestudents in my school?” is a statistical question because one anticipatesvariability in students’ ages.

2. Understandthatasetofdatacollectedtoanswerastatisticalquestionhasadistributionwhichcanbedescribedbyitscenter,spread,andoverallshape.

3. Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber,whileameasureofvariationdescribeshowitsvaluesvarywithasinglenumber.

Summarize and describe distributions.

4. Displaynumericaldatainplotsonanumberline,includingdotplots,histograms,andboxplots.

5. Summarizenumericaldatasetsinrelationtotheircontext,suchasby:

a. Reportingthenumberofobservations.

b. Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.

c. Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.

d. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.

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developingunderstandingofandapplyingproportionalrelationships;

(2)developingunderstandingofoperationswithrationalnumbersand

workingwithexpressionsandlinearequations;(3)solvingproblems

involvingscaledrawingsandinformalgeometricconstructions,and

workingwithtwo-andthree-dimensionalshapestosolveproblems

involvingarea,surfacearea,andvolume;and(4)drawinginferencesabout

populationsbasedonsamples.

(1)Studentsextendtheirunderstandingofratiosanddevelop

understandingofproportionalitytosolvesingle-andmulti-stepproblems.

Studentsusetheirunderstandingofratiosandproportionalitytosolve

awidevarietyofpercentproblems,includingthoseinvolvingdiscounts,

interest,taxes,tips,andpercentincreaseordecrease.Studentssolve

problemsaboutscaledrawingsbyrelatingcorrespondinglengthsbetween

theobjectsorbyusingthefactthatrelationshipsoflengthswithinan

objectarepreservedinsimilarobjects.Studentsgraphproportional

relationshipsandunderstandtheunitrateinformallyasameasureofthe

steepnessoftherelatedline,calledtheslope.Theydistinguishproportional

relationshipsfromotherrelationships.

(2)Studentsdevelopaunifiedunderstandingofnumber,recognizing

fractions,decimals(thathaveafiniteorarepeatingdecimal

representation),andpercentsasdifferentrepresentationsofrational

numbers.Studentsextendaddition,subtraction,multiplication,anddivision

toallrationalnumbers,maintainingthepropertiesofoperationsandthe

relationshipsbetweenadditionandsubtraction,andmultiplicationand

division.Byapplyingtheseproperties,andbyviewingnegativenumbers

intermsofeverydaycontexts(e.g.,amountsowedortemperaturesbelow

zero),studentsexplainandinterprettherulesforadding,subtracting,

multiplying,anddividingwithnegativenumbers.Theyusethearithmetic

ofrationalnumbersastheyformulateexpressionsandequationsinone

variableandusetheseequationstosolveproblems.

(3)StudentscontinuetheirworkwithareafromGrade6,solvingproblems

involvingtheareaandcircumferenceofacircleandsurfaceareaofthree-

dimensionalobjects.Inpreparationforworkoncongruenceandsimilarity

inGrade8theyreasonaboutrelationshipsamongtwo-dimensionalfigures

usingscaledrawingsandinformalgeometricconstructions,andtheygain

familiaritywiththerelationshipsbetweenanglesformedbyintersecting

lines.Studentsworkwiththree-dimensionalfigures,relatingthemtotwo-

dimensionalfiguresbyexaminingcross-sections.Theysolvereal-world

andmathematicalproblemsinvolvingarea,surfacearea,andvolumeof

two-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,

polygons,cubesandrightprisms.

(4)Studentsbuildontheirpreviousworkwithsingledatadistributionsto

comparetwodatadistributionsandaddressquestionsaboutdifferences

betweenpopulations.Theybegininformalworkwithrandomsampling

togeneratedatasetsandlearnabouttheimportanceofrepresentative

samplesfordrawinginferences.

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ratios and Proportional relationships

• analyze proportional relationships and usethem to solve real-world and mathematicalproblems.

the number System

• apply and extend previous understandingsof operations with fractions to add, subtract,multiply, and divide rational numbers.


• Use properties of operations to generateequivalent expressions.

• Solve real-life and mathematical problemsusing numerical and algebraic expressions andequations.

Geometry

• draw, construct and describe geometricalfigures and describe the relationships betweenthem.

• Solve real-life and mathematical problemsinvolving angle measure, area, surface area,and volume.


• Use random sampling to draw inferences abouta population.

• draw informal comparative inferences abouttwo populations.

• Investigate chance processes and develop, use,and evaluate probability models.










Grade 7 overview

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ratios and Proportional relationships 7.rP

Analyze proportional relationships and use them to solve real-world and mathematical problems.

1. Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.For example, if a person walks 1/2 mile in each 1/4 hour, computethe unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2miles per hour.

2. Recognizeandrepresentproportionalrelationshipsbetweenquantities.

a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.

b. Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.

c. Representproportionalrelationshipsbyequations.For example, iftotal cost t is proportional to the number n of items purchased ata constant price p, the relationship between the total cost and thenumber of items can be expressed as t = pn.

d. Explainwhatapoint (x, y) onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentiontothepoints(0,0)and(1, r) where r istheunitrate.

3. Useproportionalrelationshipstosolvemultistepratioandpercentproblems.Examples: simple interest, tax, markups and markdowns,gratuities and commissions, fees, percent increase and decrease, percenterror.


Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

1. Applyandextendpreviousunderstandingsofadditionandsubtractiontoaddandsubtractrationalnumbers;representadditionandsubtractiononahorizontalorverticalnumberlinediagram.

a. Describesituationsinwhichoppositequantitiescombinetomake0.For example, a hydrogen atom has 0 charge because its twoconstituents are oppositely charged.

b. Understandp+qasthenumberlocatedadistance|q|fromp,inthepositiveornegativedirectiondependingonwhetherqispositiveornegative.Showthatanumberanditsoppositehaveasumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.

c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse,p–q=p+(–q).Showthatthedistancebetweentworationalnumbersonthenumberlineistheabsolutevalueoftheirdifference,andapplythisprincipleinreal-worldcontexts.

d. Applypropertiesofoperationsasstrategiestoaddandsubtractrationalnumbers.

2. Applyandextendpreviousunderstandingsofmultiplicationanddivisionandoffractionstomultiplyanddividerationalnumbers.

a. Understandthatmultiplicationisextendedfromfractionstorationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(–1)(–1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbersbydescribingreal-worldcontexts.

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b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zerodivisor)isarationalnumber.Ifpandqareintegers,then–(p/q)=(–p)/q=p/(–q).Interpretquotientsofrationalnumbersbydescribingreal-worldcontexts.

c. Applypropertiesofoperationsasstrategiestomultiplyanddividerationalnumbers.

d. Convertarationalnumbertoadecimalusinglongdivision;knowthatthedecimalformofarationalnumberterminatesin0soreventuallyrepeats.

3. Solvereal-worldandmathematicalproblemsinvolvingthefouroperationswithrationalnumbers.1


Use properties of operations to generate equivalent expressions.

1. Applypropertiesofoperationsasstrategiestoadd,subtract,factor,andexpandlinearexpressionswithrationalcoefficients.

2. Understandthatrewritinganexpressionindifferentformsinaproblemcontextcanshedlightontheproblemandhowthequantitiesinitarerelated.For example, a + 0.05a = 1.05a means that “increase by5%” is the same as “multiply by 1.05.”

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

3. Solvemulti-stepreal-lifeandmathematicalproblemsposedwithpositiveandnegativerationalnumbersinanyform(wholenumbers,fractions,anddecimals),usingtoolsstrategically.Applypropertiesofoperationstocalculatewithnumbersinanyform;convertbetweenformsasappropriate;andassessthereasonablenessofanswersusingmentalcomputationandestimationstrategies.For example: If a womanmaking $25 an hour gets a 10% raise, she will make an additional 1/10 ofher salary an hour, or $2.50, for a new salary of $27.50. If you want to placea towel bar 9 3/4 inches long in the center of a door that is 27 1/2 incheswide, you will need to place the bar about 9 inches from each edge; thisestimate can be used as a check on the exact computation.

4. Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.

a. Solvewordproblemsleadingtoequationsoftheformpx+q=randp(x+q)=r,wherep,q,andrarespecificrationalnumbers.Solveequationsoftheseformsfluently.Compareanalgebraicsolutiontoanarithmeticsolution,identifyingthesequenceoftheoperationsusedineachapproach.For example, the perimeter of arectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solvewordproblemsleadingtoinequalitiesoftheformpx+q>rorpx+q<r,wherep,q,andrarespecificrationalnumbers.Graphthesolutionsetoftheinequalityandinterpretitinthecontextoftheproblem.For example: As a salesperson, you are paid $50 perweek plus $3 per sale. This week you want your pay to be at least$100. Write an inequality for the number of sales you need to make,and describe the solutions.

Geometry 7.G

Draw, construct, and describe geometrical figures and describe the relationships between them.

1. Solveproblemsinvolvingscaledrawingsofgeometricfigures,includingcomputingactuallengthsandareasfromascaledrawingandreproducingascaledrawingatadifferentscale.

1Computationswithrationalnumbersextendtherulesformanipulatingfractionstocomplexfractions.

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2. Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticingwhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.

3. Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

4. Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems;giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle.

5. Usefactsaboutsupplementary,complementary,vertical,andadjacentanglesinamulti-stepproblemtowriteandsolvesimpleequationsforanunknownangleinafigure.

6. Solvereal-worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.


Use random sampling to draw inferences about a population.

1. Understandthatstatisticscanbeusedtogaininformationaboutapopulationbyexaminingasampleofthepopulation;generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation.Understandthatrandomsamplingtendstoproducerepresentativesamplesandsupportvalidinferences.

2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.For example, estimate the mean word length ina book by randomly sampling words from the book; predict the winner ofa school election based on randomly sampled survey data. Gauge how faroff the estimate or prediction might be.

Draw informal comparative inferences about two populations.

3. Informallyassessthedegreeofvisualoverlapoftwonumericaldatadistributionswithsimilarvariabilities,measuringthedifferencebetweenthecentersbyexpressingitasamultipleofameasureofvariability.For example, the mean height of players on the basketballteam is 10 cm greater than the mean height of players on the soccer team,about twice the variability (mean absolute deviation) on either team; ona dot plot, the separation between the two distributions of heights isnoticeable.

4. Usemeasuresofcenterandmeasuresofvariabilityfornumericaldatafromrandomsamplestodrawinformalcomparativeinferencesabouttwopopulations.For example, decide whether the words in a chapterof a seventh-grade science book are generally longer than the words in achapter of a fourth-grade science book.

Investigate chance processes and develop, use, and evaluate probability models.

5. Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthelikelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynorlikely,andaprobabilitynear1indicatesalikelyevent.

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6. Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesitandobservingitslong-runrelativefrequency,andpredicttheapproximaterelativefrequencygiventheprobability.For example, when rolling a number cube 600 times, predictthat a 3 or 6 would be rolled roughly 200 times, but probably not exactly200 times.

7. Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy.

a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilitiesofevents.For example, if a student is selected atrandom from a class, find the probability that Jane will be selectedand the probability that a girl will be selected.

b. Developaprobabilitymodel(whichmaynotbeuniform)byobservingfrequenciesindatageneratedfromachanceprocess.For example, find the approximate probability that a spinning pennywill land heads up or that a tossed paper cup will land open-enddown. Do the outcomes for the spinning penny appear to be equallylikely based on the observed frequencies?

8. Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.

a. Understandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs.

b. Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespacewhichcomposetheevent.

c. Designanduseasimulationtogeneratefrequenciesforcompoundevents.For example, use random digits as a simulationtool to approximate the answer to the question: If 40% of donorshave type A blood, what is the probability that it will take at least 4donors to find one with type A blood?

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mathematics | Grade 8InGrade8,instructionaltimeshouldfocusonthreecriticalareas:(1)formulating

andreasoningaboutexpressionsandequations,includingmodelinganassociation

inbivariatedatawithalinearequation,andsolvinglinearequationsandsystems

oflinearequations;(2)graspingtheconceptofafunctionandusingfunctions

todescribequantitativerelationships;(3)analyzingtwo-andthree-dimensional

spaceandfiguresusingdistance,angle,similarity,andcongruence,and

understandingandapplyingthePythagoreanTheorem.

(1)Studentsuselinearequationsandsystemsoflinearequationstorepresent,

analyze,andsolveavarietyofproblems.Studentsrecognizeequationsfor

proportions(y/x=mory=mx)asspeciallinearequations(y=mx+b),

understandingthattheconstantofproportionality(m)istheslope,andthegraphs

arelinesthroughtheorigin.Theyunderstandthattheslope(m)ofalineisa

constantrateofchange,sothatiftheinputorx-coordinatechangesbyanamount

A,theoutputory-coordinatechangesbytheamountm·A.Studentsalsousealinear

equationtodescribetheassociationbetweentwoquantitiesinbivariatedata(such

asarmspanvs.heightforstudentsinaclassroom).Atthisgrade,fittingthemodel,

andassessingitsfittothedataaredoneinformally.Interpretingthemodelinthe

contextofthedatarequiresstudentstoexpressarelationshipbetweenthetwo

quantitiesinquestionandtointerpretcomponentsoftherelationship(suchasslope

andy-intercept)intermsofthesituation.

Studentsstrategicallychooseandefficientlyimplementprocedurestosolvelinear

equationsinonevariable,understandingthatwhentheyusethepropertiesof

equalityandtheconceptoflogicalequivalence,theymaintainthesolutionsofthe

originalequation.Studentssolvesystemsoftwolinearequationsintwovariables

andrelatethesystemstopairsoflinesintheplane;theseintersect,areparallel,or

arethesameline.Studentsuselinearequations,systemsoflinearequations,linear

functions,andtheirunderstandingofslopeofalinetoanalyzesituationsandsolve

problems.

(2)Studentsgrasptheconceptofafunctionasarulethatassignstoeachinput

exactlyoneoutput.Theyunderstandthatfunctionsdescribesituationswhereone

quantitydeterminesanother.Theycantranslateamongrepresentationsandpartial

representationsoffunctions(notingthattabularandgraphicalrepresentations

maybepartialrepresentations),andtheydescribehowaspectsofthefunctionare

reflectedinthedifferentrepresentations.

(3)Studentsuseideasaboutdistanceandangles,howtheybehaveunder

translations,rotations,reflections,anddilations,andideasaboutcongruenceand

similaritytodescribeandanalyzetwo-dimensionalfiguresandtosolveproblems.

Studentsshowthatthesumoftheanglesinatriangleistheangleformedbya

straightline,andthatvariousconfigurationsoflinesgiverisetosimilartriangles

becauseoftheanglescreatedwhenatransversalcutsparallellines.Students

understandthestatementofthePythagoreanTheoremanditsconverse,andcan

explainwhythePythagoreanTheoremholds,forexample,bydecomposinga

squareintwodifferentways.TheyapplythePythagoreanTheoremtofinddistances

betweenpointsonthecoordinateplane,tofindlengths,andtoanalyzepolygons.

Studentscompletetheirworkonvolumebysolvingproblemsinvolvingcones,

cylinders,andspheres.

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the number System

• Know that there are numbers that are notrational, and approximate them by rationalnumbers.


• Work with radicals and integer exponents.

• Understand the connections betweenproportional relationships, lines, and linearequations.

• analyze and solve linear equations and pairs ofsimultaneous linear equations.

functions

• define, evaluate, and compare functions.

• Use functions to model relationships betweenquantities.

Geometry

• Understand congruence and similarity usingphysical models, transparencies, or geometrysoftware.

• Understand and apply the Pythagoreantheorem.

• Solve real-world and mathematical problemsinvolving volume of cylinders, cones andspheres.


• Investigate patterns of association in bivariatedata.










Grade 8 overview

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Know that there are numbers that are not rational, and approximate them by rational numbers.

1. Knowthatnumbersthatarenotrationalarecalledirrational.Understandinformallythateverynumberhasadecimalexpansion;forrationalnumbersshowthatthedecimalexpansionrepeatseventually,andconvertadecimalexpansionwhichrepeatseventuallyintoarationalnumber.

2. Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2).For example,by truncating the decimal expansion of √2, show that √2 is between 1 and2, then between 1.4 and 1.5, and explain how to continue on to get betterapproximations.


Work with radicals and integer exponents.

1. Knowandapplythepropertiesofintegerexponentstogenerateequivalentnumericalexpressions.For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

2. Usesquarerootandcuberootsymbolstorepresentsolutionstoequationsoftheformx2=pandx3=p,wherepisapositiverationalnumber.Evaluatesquarerootsofsmallperfectsquaresandcuberootsofsmallperfectcubes.Knowthat√2isirrational.

3. Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.For example,estimate the population of the United States as 3 × 108 and the populationof the world as 7 × 109, and determine that the world population is morethan 20 times larger.

4. Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothdecimalandscientificnotationareused.Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities(e.g.,usemillimetersperyearforseafloorspreading).Interpretscientificnotationthathasbeengeneratedbytechnology.

Understand the connections between proportional relationships, lines, and linear equations.

5. Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.For example, compare a distance-timegraph to a distance-time equation to determine which of two movingobjects has greater speed.

6. Usesimilartrianglestoexplainwhytheslopemisthesamebetweenanytwodistinctpointsonanon-verticallineinthecoordinateplane;derivetheequationy=mxforalinethroughtheoriginandtheequationy=mx+bforalineinterceptingtheverticalaxisatb.

Analyze and solve linear equations and pairs of simultaneous linear equations.

7. Solvelinearequationsinonevariable.

a. Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).

b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.

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8. Analyzeandsolvepairsofsimultaneouslinearequations.

a. Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.

b. Solvesystemsoftwolinearequationsintwovariablesalgebraically,andestimatesolutionsbygraphingtheequations.Solvesimplecasesbyinspection.For example, 3x + 2y = 5 and 3x +2y = 6 have no solution because 3x + 2y cannot simultaneously be 5and 6.

c. Solvereal-worldandmathematicalproblemsleadingtotwolinearequationsintwovariables.For example, given coordinates for twopairs of points, determine whether the line through the first pair ofpoints intersects the line through the second pair.

functions 8.f

Define, evaluate, and compare functions.

1. Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.Thegraphofafunctionisthesetoforderedpairsconsistingofaninputandthecorrespondingoutput.1

2. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a linear function represented by a tableof values and a linear function represented by an algebraic expression,determine which function has the greater rate of change.

3. Interprettheequationy=mx+basdefiningalinearfunction,whosegraphisastraightline;giveexamplesoffunctionsthatarenotlinear.For example, the function A = s2 giving the area of a square as a functionof its side length is not linear because its graph contains the points (1,1),(2,4) and (3,9), which are not on a straight line.

Use functions to model relationships between quantities.

4. Constructafunctiontomodelalinearrelationshipbetweentwoquantities.Determinetherateofchangeandinitialvalueofthefunctionfromadescriptionofarelationshiporfromtwo(x,y)values,includingreadingthesefromatableorfromagraph.Interprettherateofchangeandinitialvalueofalinearfunctionintermsofthesituationitmodels,andintermsofitsgraphoratableofvalues.

5. Describequalitativelythefunctionalrelationshipbetweentwoquantitiesbyanalyzingagraph(e.g.,wherethefunctionisincreasingordecreasing,linearornonlinear).Sketchagraphthatexhibitsthequalitativefeaturesofafunctionthathasbeendescribedverbally.

Geometry 8.G

Understand congruence and similarity using physical models, trans-parencies, or geometry software.

1. Verifyexperimentallythepropertiesofrotations,reflections,andtranslations:

a. Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.

b. Anglesaretakentoanglesofthesamemeasure.

c. Parallellinesaretakentoparallellines.

2. Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.

1FunctionnotationisnotrequiredinGrade8.

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3. Describetheeffectofdilations,translations,rotations,andreflectionsontwo-dimensionalfiguresusingcoordinates.

4. Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.

5. Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.For example, arrange three copies of the same triangle so thatthe sum of the three angles appears to form a line, and give an argumentin terms of transversals why this is so.

Understand and apply the Pythagorean Theorem.

6. ExplainaproofofthePythagoreanTheoremanditsconverse.

7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.

8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

9. Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.


Investigate patterns of association in bivariate data.

1. Constructandinterpretscatterplotsforbivariatemeasurementdatatoinvestigatepatternsofassociationbetweentwoquantities.Describepatternssuchasclustering,outliers,positiveornegativeassociation,linearassociation,andnonlinearassociation.

2. Knowthatstraightlinesarewidelyusedtomodelrelationshipsbetweentwoquantitativevariables.Forscatterplotsthatsuggestalinearassociation,informallyfitastraightline,andinformallyassessthemodelfitbyjudgingtheclosenessofthedatapointstotheline.

3. Usetheequationofalinearmodeltosolveproblemsinthecontextofbivariatemeasurementdata,interpretingtheslopeandintercept.For example, in a linear model for a biology experiment, interpret a slopeof 1.5 cm/hr as meaning that an additional hour of sunlight each day isassociated with an additional 1.5 cm in mature plant height.

4. Understandthatpatternsofassociationcanalsobeseeninbivariatecategoricaldatabydisplayingfrequenciesandrelativefrequenciesinatwo-waytable.Constructandinterpretatwo-waytablesummarizingdataontwocategoricalvariablescollectedfromthesamesubjects.Userelativefrequenciescalculatedforrowsorcolumnstodescribepossibleassociationbetweenthetwovariables.For example, collectdata from students in your class on whether or not they have a curfew onschool nights and whether or not they have assigned chores at home. Isthere evidence that those who have a curfew also tend to have chores?

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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 rationalexponents

• Use properties of rational and irrationalnumbers.

Quantities

• reason quantitatively and use units to solveproblems

The Complex Number System

• Perform arithmetic operations with complexnumbers

• represent complex numbers and theiroperations on the complex plane

• Use complex numbers in polynomial identitiesand equations

Vector and Matrix Quantities

• represent and model with vector quantities.

• Perform operations on vectors.

• Perform operations on matrices and usematrices in applications.










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 5because 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 solveproblems

Arithmetic with Polynomials and Rational Expressions

• Perform arithmetic operations on polynomials

• Understand the relationship between zeros andfactors of polynomials

• Use polynomial identities to solve problems

• rewrite rational expressions

Creating Equations

• Create equations that describe numbers orrelationships

Reasoning with Equations and Inequalities

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

• Solve equations and inequalities in one variable

• Solve systems of equations

• represent and solve equations and inequalitiesgraphically










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 productof P and a factor not depending on P.

2. Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample, see x4–y4as(x2)2–(y2)2,thus recognizing it as a difference ofsquares 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 berewritten as(1.151/12)12t≈1.01212tto reveal the approximate equivalentmonthly interest rate if the annual rate is 15%.

4. Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusetheformulatosolveproblems.Forexample, 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 quadraticfunctions, and simple rational and exponential functions.

2. Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswithlabelsandscales.

3. Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornon-viableoptionsinamodelingcontext.For example, represent inequalitiesdescribing nutritional and cost constraints on combinations of differentfoods.

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 usefunction notation

• Interpret functions that arise in applications interms of the context

• analyze functions using differentrepresentations

Building Functions

• Build a function that models a relationshipbetween two quantities

• Build new functions from existing functions

Linear, Quadratic, and Exponential Models

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

• Interpret expressions for functions in terms ofthe situation they model

Trigonometric Functions

• extend the domain of trigonometric functionsusing the unit circle

• model periodic phenomena with trigonometricfunctions

• 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, theFibonacci 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 thefunction is increasing, decreasing, positive, or negative; relative maximumsand minimums; symmetries; end behavior; and periodicity.★

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

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 changein functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, andclassify 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 andan 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.Forexample, build a function that models the temperature of a coolingbody by adding a constant function to a decaying exponential, andrelate these functions to the model.

c. (+)Composefunctions. For example, if T(y) is the temperature inthe atmosphere as a function of height, and h(t) is the height of aweather balloon as a function of time, then T(h(t)) is the temperatureat 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 andalgebraic expressions for them.

4. Findinversefunctions.

a. Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.Forexample, 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 rigidmotions

• Prove geometric theorems

• make geometric constructions

Similarity, Right Triangles, and Trigonometry

• Understand similarity in terms of similaritytransformations

• Prove theorems involving similarity

• define trigonometric ratios and solve problemsinvolving 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 descriptionand the equation for a conic section

• Use coordinates to prove simple geometrictheorems algebraically

Geometric Measurement and Dimension

• explain volume formulas and use them to solveproblems

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

Modeling with Geometry

• apply geometric concepts in modelingsituations










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: verticalangles are congruent; when a transversal crosses parallel lines, alternateinterior angles are congruent and corresponding angles are congruent;points on a perpendicular bisector of a line segment are exactly thoseequidistant from the segment’s endpoints.

10. Provetheoremsabouttriangles.Theorems include: measures of interiorangles of a triangle sum to 180°; base angles of isosceles triangles arecongruent; the segment joining midpoints of two sides of a triangle isparallel to the third side and half the length; the medians of a trianglemeet at a point.

11. Provetheoremsaboutparallelograms.Theorems include: oppositesides are congruent, opposite angles are congruent, the diagonalsof a parallelogram bisect each other, and conversely, rectangles areparallelograms 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; constructingperpendicular lines, including the perpendicular bisector of a line segment;and constructing a line parallel to a given line through a point not on theline.

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 oneside of a triangle divides the other two proportionally, and conversely; thePythagorean 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, andcircumscribed angles; inscribed angles on a diameter are right angles;the radius of a circle is perpendicular to the tangent where the radiusintersects 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.Forexample, prove or disprove that a figure defined by four given points in thecoordinate plane is a rectangle; prove or disprove that the point (1, √3) lieson 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.Usedissection 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 asingle count or measurement variable

• Summarize, represent, and interpret data ontwo categorical and quantitative variables

• Interpret linear models

Making Inferences and Justifying Conclusions

• Understand and evaluate random processesunderlying statistical experiments

• make inferences and justify conclusions fromsample surveys, experiments and observationalstudies

Conditional Probability and the Rules of Prob-ability

• Understand independence and conditionalprobability and use them to interpret data

• Use the rules of probability to computeprobabilities of compound events in a uniformprobability model

Using Probability to Make Decisions

• Calculate expected values and use them tosolve problems

• Use probability to evaluate outcomes ofdecisions










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 choosea function suggested by the context. Emphasize linear, quadratic, andexponential 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 modelsays a spinning coin falls heads up with probability 0.5. Would a result of 5tails 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, collectdata from a random sample of students in your school on their favoritesubject among math, science, and English. Estimate the probability that arandomly selected student from your school will favor science given thatthe student is in tenth grade. Do the same for other subjects and comparethe results.

5. Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageandeverydaysituations.Forexample, compare the chance of having lung cancer if you are a smokerwith 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 probabilitydistribution for the number of correct answers obtained by guessing onall five questions of a multiple-choice test where each question has fourchoices, and find the expected grade under various grading schemes.

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

Use probability to evaluate outcomes of decisions

5. (+)Weighthepossibleoutcomesofadecisionbyassigningprobabilitiestopayoffvaluesandfindingexpectedvalues.

a. Findtheexpectedpayoffforagameofchance.For example, findthe 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-deductibleautomobile insurance policy using various, but reasonable, chances ofhaving a minor or a major accident.

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

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

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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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Addition and subtraction within 5, 10, 20, 100, or 1000.Additionorsubtractionoftwowholenumberswithwholenumberanswers,andwithsumorminuendintherange0-5,0-10,0-20,or0-100,respectively.Example:8+2=10isanadditionwithin10,14–5=9isasubtractionwithin20,and55–18=37isasubtractionwithin100.

Additive inverses.Twonumberswhosesumis0areadditiveinversesofoneanother.Example:3/4and–3/4areadditiveinversesofoneanotherbecause3/4+(–3/4)=(–3/4)+3/4=0.

Associative property of addition.SeeTable3inthisGlossary.

Associative property of multiplication. SeeTable3inthisGlossary.

Bivariate data. Pairsoflinkednumericalobservations.Example:alistofheightsandweightsforeachplayeronafootballteam.

Box plot.Amethodofvisuallydisplayingadistributionofdatavaluesbyusingthemedian,quartiles,andextremesofthedataset.Aboxshowsthemiddle50%ofthedata.1

Commutative property.SeeTable3inthisGlossary.

Complex fraction.AfractionA/BwhereAand/orBarefractions(Bnonzero).

Computation algorithm.Asetofpredefinedstepsapplicabletoaclassofproblemsthatgivesthecorrectresultineverycasewhenthestepsarecarriedoutcorrectly.See also:computationstrategy.

Computation strategy.Purposefulmanipulationsthatmaybechosenforspecificproblems,maynothaveafixedorder,andmaybeaimedatconvertingoneproblemintoanother.See also:computationalgorithm.

Congruent.Twoplaneorsolidfiguresarecongruentifonecanbeobtainedfromtheotherbyrigidmotion(asequenceofrotations,reflections,andtranslations).

Counting on.Astrategyforfindingthenumberofobjectsinagroupwithouthavingtocounteverymemberofthegroup.Forexample,ifastackofbooksisknowntohave8booksand3morebooksareaddedtothetop,itisnotnecessarytocountthestackalloveragain.Onecanfindthetotalbycounting on—pointingtothetopbookandsaying“eight,”followingthiswith“nine,ten,eleven.Thereareelevenbooksnow.”

Dot plot. See: lineplot.

Dilation.Atransformationthatmoveseachpointalongtheraythroughthepointemanatingfromafixedcenter,andmultipliesdistancesfromthecenterbyacommonscalefactor.

Expanded form.Amulti-digitnumberisexpressedinexpandedformwhenitiswrittenasasumofsingle-digitmultiplesofpowersoften.Forexample,643=600+40+3.

Expected value. Forarandomvariable,theweightedaverageofitspossiblevalues,withweightsgivenbytheirrespectiveprobabilities.

First quartile. ForadatasetwithmedianM,thefirstquartileisthemedianofthedatavalueslessthanM.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},thefirstquartileis6.2See also:median,thirdquartile,interquartilerange.

Fraction.Anumberexpressibleintheforma/bwhereaisawholenumberandbisapositivewholenumber.(Thewordfractioninthesestandardsalwaysreferstoanon-negativenumber.)See also:rationalnumber.

Identity property of 0.SeeTable3inthisGlossary.

Independently combined probability models.Twoprobabilitymodelsaresaidtobecombinedindependentlyiftheprobabilityofeachorderedpairinthecombinedmodelequalstheproductoftheoriginalprobabilitiesofthetwoindividualoutcomesintheorderedpair.

1AdaptedfromWisconsinDepartmentofPublicInstruction,http://dpi.wi.gov/standards/mathglos.html,accessedMarch2,2010.2Manydifferentmethodsforcomputingquartilesareinuse.ThemethoddefinedhereissometimescalledtheMooreandMcCabemethod.SeeLangford,E.,“QuartilesinElementaryStatistics,”Journal of Statistics EducationVolume14,Number3(2006).

Glossary

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Integer.Anumberexpressibleintheformaor–aforsomewholenumbera.

Interquartile Range. Ameasureofvariationinasetofnumericaldata,theinterquartilerangeisthedistancebetweenthefirstandthirdquartilesofthedataset.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},theinterquartilerangeis15–6=9.See also:firstquartile,thirdquartile.

Line plot.Amethodofvisuallydisplayingadistributionofdatavalueswhereeachdatavalueisshownasadotormarkaboveanumberline.Alsoknownasadotplot.3

Mean.Ameasureofcenterinasetofnumericaldata,computedbyaddingthevaluesinalistandthendividingbythenumberofvaluesinthelist.4Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},themeanis21.

Mean absolute deviation.Ameasureofvariationinasetofnumericaldata,computedbyaddingthedistancesbetweeneachdatavalueandthemean,thendividingbythenumberofdatavalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},themeanabsolutedeviationis20.

Median.Ameasureofcenterinasetofnumericaldata.Themedianofalistofvaluesisthevalueappearingatthecenterofasortedversionofthelist—orthemeanofthetwocentralvalues,ifthelistcontainsanevennumberofvalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,90},themedianis11.

Midline. Inthegraphofatrigonometricfunction,thehorizontallinehalfwaybetweenitsmaximumandminimumvalues.

Multiplication and division within 100.Multiplicationordivisionoftwowholenumberswithwholenumberanswers,andwithproductordividendintherange0-100.Example:72÷8=9.

Multiplicative inverses.Twonumberswhoseproductis1aremultiplicativeinversesofoneanother.Example:3/4and4/3aremultiplicativeinversesofoneanotherbecause3/4×4/3=4/3×3/4=1.

Number line diagram. Adiagramofthenumberlineusedtorepresentnumbersandsupportreasoningaboutthem.Inanumberlinediagramformeasurementquantities,theintervalfrom0to1onthediagramrepresentstheunitofmeasureforthequantity.

Percent rate of change.Arateofchangeexpressedasapercent.Example:ifapopulationgrowsfrom50to55inayear,itgrowsby5/50=10%peryear.

Probability distribution.Thesetofpossiblevaluesofarandomvariablewithaprobabilityassignedtoeach.

Properties of operations.SeeTable3inthisGlossary.

Properties of equality.SeeTable4inthisGlossary.

Properties of inequality.SeeTable5inthisGlossary.

Properties of operations.SeeTable3inthisGlossary.

Probability.Anumberbetween0and1usedtoquantifylikelihoodforprocessesthathaveuncertainoutcomes(suchastossingacoin,selectingapersonatrandomfromagroupofpeople,tossingaballatatarget,ortestingforamedicalcondition).

Probability model. Aprobabilitymodelisusedtoassignprobabilitiestooutcomesofachanceprocessbyexaminingthenatureoftheprocess.Thesetofalloutcomesiscalledthesamplespace,andtheirprobabilitiessumto1.See also: uniformprobabilitymodel.

Random variable. Anassignmentofanumericalvaluetoeachoutcomeinasamplespace.

Rational expression.Aquotientoftwopolynomialswithanon-zerodenominator.

Rational number.Anumberexpressibleintheforma/bor– a/bforsomefractiona/b.Therationalnumbersincludetheintegers.

Rectilinear figure. Apolygonallanglesofwhicharerightangles.

Rigid motion.Atransformationofpointsinspaceconsistingofasequenceof

3AdaptedfromWisconsinDepartmentofPublicInstruction,op. cit.4Tobemoreprecise,thisdefinesthearithmetic mean.

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oneormoretranslations,reflections,and/orrotations.Rigidmotionsarehereassumedtopreservedistancesandanglemeasures.

Repeating decimal.Thedecimalformofarationalnumber.See also:terminatingdecimal.

Sample space.Inaprobabilitymodelforarandomprocess,alistoftheindividualoutcomesthataretobeconsidered.

Scatter plot. Agraphinthecoordinateplanerepresentingasetofbivariatedata.Forexample,theheightsandweightsofagroupofpeoplecouldbedisplayedonascatterplot.5

Similarity transformation.Arigidmotionfollowedbyadilation.

Tape diagram.Adrawingthatlookslikeasegmentoftape,usedtoillustratenumberrelationships.Alsoknownasastripdiagram,barmodel,fractionstrip,orlengthmodel.

Terminating decimal. Adecimaliscalledterminatingifitsrepeatingdigitis0.

Third quartile.ForadatasetwithmedianM,thethirdquartileisthemedianofthedatavaluesgreaterthanM.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},thethirdquartileis15.See also:median,firstquartile,interquartilerange.

Transitivity principle for indirect measurement. IfthelengthofobjectAisgreaterthanthelengthofobjectB,andthelengthofobjectBisgreaterthanthelengthofobjectC,thenthelengthofobjectAisgreaterthanthelengthofobjectC.Thisprincipleappliestomeasurementofotherquantitiesaswell.

Uniform probability model.Aprobabilitymodelwhichassignsequalprobabilitytoalloutcomes.See also:probabilitymodel.

Vector. Aquantitywithmagnitudeanddirectionintheplaneorinspace,definedbyanorderedpairortripleofrealnumbers.

Visual fraction model. Atapediagram,numberlinediagram,orareamodel.

Whole numbers.Thenumbers0,1,2,3,….

5AdaptedfromWisconsinDepartmentofPublicInstruction,op. cit.

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Table 1.Commonadditionandsubtractionsituations.6

result Unknown Change Unknown Start Unknown

add to

Twobunniessatonthegrass.Threemorebunnieshoppedthere.Howmanybunniesareonthegrassnow?

2+3=?

Twobunniesweresittingonthegrass.Somemorebunnieshoppedthere.Thentherewerefivebunnies.Howmanybunnieshoppedovertothefirsttwo?

2+?=5

Somebunniesweresittingonthegrass.Threemorebunnieshoppedthere.Thentherewerefivebunnies.Howmanybunnieswereonthegrassbefore?

?+3=5

take from

Fiveappleswereonthetable.Iatetwoapples.Howmanyapplesareonthetablenow?

5–2=?

Fiveappleswereonthetable.Iatesomeapples.Thentherewerethreeapples.HowmanyapplesdidIeat?

5–?=3

Someappleswereonthetable.Iatetwoapples.Thentherewerethreeapples.Howmanyappleswereonthetablebefore?

?–2=3

total Unknown addend Unknown Both addends Unknown1

Put together/ take apart2

Threeredapplesandtwogreenapplesareonthetable.Howmanyapplesareonthetable?

3+2=?

Fiveapplesareonthetable.Threeareredandtherestaregreen.Howmanyapplesaregreen?

3+?=5,5–3=?

Grandmahasfiveflowers.Howmanycansheputinherredvaseandhowmanyinherbluevase?

5=0+5,5=5+0

5=1+4,5=4+1

5=2+3,5=3+2

difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“Howmanymore?”version):

Lucyhastwoapples.Juliehasfiveapples.HowmanymoreapplesdoesJuliehavethanLucy?

(“Howmanyfewer?”version):

Lucyhastwoapples.Juliehasfiveapples.HowmanyfewerapplesdoesLucyhavethanJulie?

2+?=5,5–2=?

(Versionwith“more”):

JuliehasthreemoreapplesthanLucy.Lucyhastwoapples.HowmanyapplesdoesJuliehave?

(Versionwith“fewer”):

Lucyhas3fewerapplesthanJulie.Lucyhastwoapples.HowmanyapplesdoesJuliehave?

2+3=?,3+2=?

(Versionwith“more”):

JuliehasthreemoreapplesthanLucy.Juliehasfiveapples.HowmanyapplesdoesLucyhave?

(Versionwith“fewer”):

Lucyhas3fewerapplesthanJulie.Juliehasfiveapples.HowmanyapplesdoesLucyhave?

5–3=?,?+3=5

6AdaptedfromBox2-4ofMathematicsLearninginEarlyChildhood,NationalResearchCouncil(2009,pp.32,33).

1Thesetakeapartsituationscanbeusedtoshowallthedecompositionsofagivennumber.Theassociatedequations,whichhavethetotalontheleftoftheequalsign,helpchildrenunderstandthatthe=signdoesnotalwaysmeanmakesorresultsinbutalwaysdoesmeanisthesamenumberas.2Eitheraddendcanbeunknown,sotherearethreevariationsoftheseproblemsituations.BothAddendsUnknownisapro-ductiveextensionofthisbasicsituation,especiallyforsmallnumberslessthanorequalto10.3FortheBiggerUnknownorSmallerUnknownsituations,oneversiondirectsthecorrectoperation(theversionusingmoreforthebiggerunknownandusinglessforthesmallerunknown).Theotherversionsaremoredifficult.

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Table 2. Commonmultiplicationanddivisionsituations.7

Unknown ProductGroup Size Unknown

(“Howmanyineachgroup?”Division)

number of Groups Unknown(“Howmanygroups?”Division)

3 × 6 = ? 3 × ? = 18, and 18 ÷ 3 = ? ? × 6 = 18, and 18 ÷ 6 = ?

equal Groups

Thereare3bagswith6plumsineachbag.Howmanyplumsarethereinall?

Measurement example.Youneed3lengthsofstring,each6incheslong.Howmuchstringwillyouneedaltogether?

If18plumsaresharedequallyinto3bags,thenhowmanyplumswillbeineachbag?

Measurement example.Youhave18inchesofstring,whichyouwillcutinto3equalpieces.Howlongwilleachpieceofstringbe?

If18plumsaretobepacked6toabag,thenhowmanybagsareneeded?

Measurement example.Youhave18inchesofstring,whichyouwillcutintopiecesthatare6incheslong.Howmanypiecesofstringwillyouhave?

arrays,4 area5

Thereare3rowsofappleswith6applesineachrow.Howmanyapplesarethere?

Area example.Whatistheareaofa3cmby6cmrectangle?

If18applesarearrangedinto3equalrows,howmanyappleswillbeineachrow?

Area example.Arectanglehasarea18squarecentimeters.Ifonesideis3cmlong,howlongisasidenexttoit?

If18applesarearrangedintoequalrowsof6apples,howmanyrowswilltherebe?

Area example.Arectanglehasarea18squarecentimeters.Ifonesideis6cmlong,howlongisasidenexttoit?

Compare

Abluehatcosts$6.Aredhatcosts3timesasmuchasthebluehat.Howmuchdoestheredhatcost?

Measurement example.Arubberbandis6cmlong.Howlongwilltherubberbandbewhenitisstretchedtobe3timesaslong?

Aredhatcosts$18andthatis3timesasmuchasabluehatcosts.Howmuchdoesabluehatcost?

Measurement example.Arubberbandisstretchedtobe18cmlongandthatis3timesaslongasitwasatfirst.Howlongwastherubberbandatfirst?

Aredhatcosts$18andabluehatcosts$6.Howmanytimesasmuchdoestheredhatcostasthebluehat?

Measurement example.Arubberbandwas6cmlongatfirst.Nowitisstretchedtobe18cmlong.Howmanytimesaslongistherubberbandnowasitwasatfirst?

General a×b=? a×? =p,andp÷ a=? ?×b=p, and p÷ b =?

7Thefirstexamplesineachcellareexamplesofdiscretethings.Theseareeasierforstudentsandshouldbegivenbeforethemeasurementexamples.

4Thelanguageinthearrayexamplesshowstheeasiestformofarrayproblems.Aharderformistousethetermsrowsandcolumns:Theapplesinthegrocerywindowarein3rowsand6columns.Howmanyapplesareinthere?Bothformsarevaluable.5Areainvolvesarraysofsquaresthathavebeenpushedtogethersothattherearenogapsoroverlaps,soarrayproblemsincludetheseespeciallyimportantmeasurementsituations.

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Table 3.Thepropertiesofoperations.Herea,bandcstandforarbitrarynumbersinagivennumbersystem.Thepropertiesofoperationsapplytotherationalnumbersystem,therealnumbersystem,andthecomplexnumbersystem.

Associative property of addition

Commutative property of addition

Additive identity property of 0

Existence of additive inverses

Associative property of multiplication

Commutative property of multiplication

Multiplicative identity property of 1

Existence of multiplicative inverses

Distributive property of multiplication over addition

(a + b)+ c = a + (b + c)

a + b= b + a

a + 0 = 0+ a= a

Foreveryathereexists–asothata+(–a)= (–a)+a=0.

(a × b)× c = a × (b × c)

a × b= b × a

a × 1= 1×a= a

Foreverya≠0thereexists1/asothata×1/a=1/a× a=1.

a ×(b + c) = a × b + a × c

Table 4.Thepropertiesofequality.Herea,bandcstandforarbitrarynumbersintherational,real,orcomplexnumbersystems.

Reflexive property of equality

Symmetric property of equality

Transitive property of equality

Addition property of equality

Subtraction property of equality

Multiplication property of equality

Division property of equality

Substitution property of equality

a=a

If a = b,then b = a.

If a = b and b = c,then a = c.

If a = b,then a + c = b + c.

If a = b,then a – c = b – c.

If a = b,then a × c = b × c.

If a = b and c ≠ 0,then a ÷ c = b ÷ c.

Ifa=b,thenbmaybesubstitutedfora

inanyexpressioncontaininga.

Table 5.Thepropertiesofinequality.Herea,bandcstandforarbitrarynumbersintherationalorrealnumbersystems.

Exactlyoneofthefollowingistrue:a<b,a=b,a>b.

Ifa>bandb>cthena>c.

Ifa>b,thenb<a.

Ifa>b,then–a<–b.

If a>b,thena±c>b±c.

Ifa>bandc>0,thena ×c>b ×c.

Ifa>bandc<0,thena ×c<b ×c.

Ifa>bandc>0,thena ÷c>b ÷c.

Ifa>bandc<0,thena ÷c<b ÷c.

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