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common core state standards for mathematics

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common core state standards for mathematics table of contents introduction standards for mathematical practice standards for mathematical content kindergarten grade1 grade2 grade3 grade4 grade5 grade6 grade7 grade8 highschool introduction highschool numberandquantity highschool algebra highschool functions highschool modeling highschool geometry highschool statisticsandprobability 3 6 9 13 17 21 27 33 39 46 52 58 62 67 72 74 79 85 91 glossary sample of works consulted

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common core state standards for mathematics introduction toward 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 k­6 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 1­3 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.williamschmidtandrichardhouang2002havesaidthatcontent 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 introduction 3

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common core state standards for mathematics 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 asconfrey2007pointsout,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 mnemonicdevicetoexpandaproductsuchasa b x+yandastudentwho canexplainwherethemnemoniccomesfrom.thestudentwhocanexplaintherule understandsthemathematics,andmayhaveabetterchancetosucceedataless familiartasksuchasexpandinga 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 introduction 4

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common core state standards for mathematics how to read the grade level standards standards definewhatstudentsshouldunderstandandbeabletodo clusters aregroupsofrelatedstandards.notethatstandardsfromdifferentclusters maysometimesbecloselyrelated,becausemathematics isaconnectedsubject domainsarelargergroupsofrelatedstandards.standardsfromdifferentdomains maysometimesbecloselyrelated domain number and operations in base ten 3.nbt use place value understanding and properties of operations to perform multi-digit arithmetic 1 useplacevalueunderstandingtoroundwholenumberstothenearest 10or100 standard 2 fluentlyaddandsubtractwithin1000usingstrategiesandalgorithms basedonplacevalue,propertiesofoperations,and/ortherelationship betweenadditionandsubtraction 3 multiplyone-digitwholenumbersbymultiplesof10intherange 10-90e.g 9×80,5×60usingstrategiesbasedonplacevalueand propertiesofoperations cluster 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 introduction 5

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common core state standards for mathematics mathematics standards for mathematical practice thestandardsformathematicalpracticedescribevarietiesofexpertisethat mathematicseducatorsatalllevelsshouldseektodevelopintheirstudents thesepracticesrestonimportant processesandproficiencies withlongstanding importanceinmathematicseducation.thefirstofthesearethenctmprocess standardsofproblemsolving,reasoningandproof,communication,representation andconnections.thesecondarethestrandsofmathematicalproficiencyspecified inthenationalresearchcouncil sreportadding it up:adaptivereasoning,strategic competence,conceptualunderstandingcomprehensionofmathematicalconcepts operationsandrelations proceduralfluencyskillincarryingoutprocedures 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 standards for mathematical practice 3 construct viable arguments and critique the reasoning of others mathematicallyproficientstudentsunderstandandusestatedassumptions definitions,andpreviouslyestablishedresultsinconstructingarguments.they makeconjecturesandbuildalogicalprogressionofstatementstoexplorethe truthoftheirconjectures.theyareabletoanalyzesituationsbybreakingtheminto cases,andcanrecognizeandusecounterexamples.theyjustifytheirconclusions 6

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common core state standards for mathematics 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 standards for mathematical practice 6 attend to precision mathematicallyproficientstudentstrytocommunicatepreciselytoothers.they trytousecleardefinitionsindiscussionwithothersandintheirownreasoning theystatethemeaningofthesymbolstheychoose,includingusingtheequalsign consistentlyandappropriately.theyarecarefulaboutspecifyingunitsofmeasure andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.they calculateaccuratelyandefficiently,expressnumericalanswerswithadegreeof precisionappropriatefortheproblemcontext.intheelementarygrades,students givecarefullyformulatedexplanationstoeachother.bythetimetheyreachhigh schooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions 7

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common core state standards for mathematics 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,theycansee5­3x­y2as5 minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannot bemorethan5foranyrealnumbersxandy 8 look for and express regularity in repeated reasoning mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlook bothforgeneralmethodsandforshortcuts.upperelementarystudentsmight noticewhendividing25by11thattheyarerepeatingthesamecalculationsover andoveragain,andconcludetheyhavearepeatingdecimal.bypayingattention tothecalculationofslopeastheyrepeatedlycheckwhetherpointsareontheline through1,2withslope3,middleschoolstudentsmightabstracttheequation y­2 x­1 3.noticingtheregularityinthewaytermscancelwhenexpanding x­1 x+1 x­1 x2+x+1 andx­1 x3+x2+x+1mightleadthemtothe generalformulaforthesumofageometricseries.astheyworktosolveaproblem mathematicallyproficientstudentsmaintainoversightoftheprocess,while attendingtothedetails.theycontinuallyevaluatethereasonablenessoftheir intermediateresults connecting the standards for mathematical practice to the standards for mathematical content thestandardsformathematicalpracticedescribewaysinwhichdevelopingstudent 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 standards for mathematical practice 8

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common core state standards for mathematics mathematics kindergarten inkindergarten,instructionaltimeshouldfocusontwocriticalareas 1 representing,relating,andoperatingonwholenumbers,initiallywith setsofobjects 2describingshapesandspace.morelearningtimein kindergartenshouldbedevotedtonumberthantoothertopics 1studentsusenumbers,includingwrittennumerals,torepresent quantitiesandtosolvequantitativeproblems,suchascountingobjectsin aset;countingoutagivennumberofobjects;comparingsetsornumerals andmodelingsimplejoiningandseparatingsituationswithsetsofobjects oreventuallywithequationssuchas5+2=7and7­2=5 kindergarten studentsshouldseeadditionandsubtractionequations,andstudent writingofequationsinkindergartenisencouraged,butitisnotrequired studentschoose,combine,andapplyeffectivestrategiesforanswering quantitativequestions,includingquicklyrecognizingthecardinalitiesof smallsetsofobjects,countingandproducingsetsofgivensizes,counting thenumberofobjectsincombinedsets,orcountingthenumberofobjects thatremaininasetaftersomearetakenaway 2studentsdescribetheirphysicalworldusinggeometricidease.g shape,orientation,spatialrelationsandvocabulary.theyidentify,name anddescribebasictwo-dimensionalshapes,suchassquares,triangles circles,rectangles,andhexagons,presentedinavarietyofwayse.g with differentsizesandorientations aswellasthree-dimensionalshapessuch ascubes,cones,cylinders,andspheres.theyusebasicshapesandspatial reasoningtomodelobjectsintheirenvironmentandtoconstructmore complexshapes kindergarten 9

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common core state standards for mathematics grade k overview counting and cardinality · know number names and the count sequence · count to tell the number of objects · compare numbers mathematical practices 1 makesenseofproblemsandperseverein solvingthem 2 reasonabstractlyandquantitatively 3 constructviableargumentsandcritique thereasoningofothers 4 modelwithmathematics 5 useappropriatetoolsstrategically 6 attendtoprecision 7 lookforandmakeuseofstructure number and operations in base ten · work with numbers 11­19 to gain foundations for place value 8 lookforandexpressregularityinrepeated reasoning operations and algebraic thinking · understand addition as putting together and adding to and understand subtraction as taking apart and taking from measurement and data · describe and compare measurable attributes · classify objects and count the number of objects in categories geometry · identify and describe shapes · analyze compare create and compose shapes kindergarten 10

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common core state standards for mathematics counting and cardinality know number names and the count sequence 1 countto100byonesandbytens 2 countforwardbeginningfromagivennumberwithintheknown sequenceinsteadofhavingtobeginat1 3 writenumbersfrom0to20.representanumberofobjectswitha writtennumeral0-20with0representingacountofnoobjects k.cc count to tell the number of objects 4 understandtherelationshipbetweennumbersandquantities;connect countingtocardinality a b c whencountingobjects,saythenumbernamesinthestandard order,pairingeachobjectwithoneandonlyonenumbername andeachnumbernamewithoneandonlyoneobject understandthatthelastnumbernamesaidtellsthenumberof objectscounted.thenumberofobjectsisthesameregardlessof theirarrangementortheorderinwhichtheywerecounted understandthateachsuccessivenumbernamereferstoaquantity thatisonelarger 5 counttoanswer howmany questionsaboutasmanyas20things arrangedinaline,arectangulararray,oracircle,orasmanyas10 thingsinascatteredconfiguration;givenanumberfrom1­20,count outthatmanyobjects compare numbers 6 identifywhetherthenumberofobjectsinonegroupisgreaterthan lessthan,orequaltothenumberofobjectsinanothergroup,e.g by usingmatchingandcountingstrategies.1 7 comparetwonumbersbetween1and10presentedaswritten numerals operations and algebraic thinking k.oa understand addition as putting together and adding to and understand subtraction as taking apart and taking from 1 representadditionandsubtractionwithobjects,fingers,mental images,drawings2,soundse.g claps actingoutsituations,verbal explanations,expressions,orequations 2 solveadditionandsubtractionwordproblems,andaddandsubtract within10,e.g byusingobjectsordrawingstorepresenttheproblem 3 decomposenumberslessthanorequalto10intopairsinmore thanoneway,e.g byusingobjectsordrawings,andrecordeach decompositionbyadrawingorequatione.g 5=2+3and5=4+1 4 foranynumberfrom1to9,findthenumberthatmakes10when addedtothegivennumber,e.g byusingobjectsordrawings,and recordtheanswerwithadrawingorequation 5 fluentlyaddandsubtractwithin5 1 2 includegroupswithuptotenobjects drawingsneednotshowdetails,butshouldshowthemathematicsintheproblem thisapplieswhereverdrawingsarementionedinthestandards kindergarten 11

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common core state standards for mathematics number and operations in base ten work with numbers 11­19 to gain foundations for place value k.nbt 1 composeanddecomposenumbersfrom11to19intotenonesand somefurtherones,e.g byusingobjectsordrawings,andrecordeach compositionordecompositionbyadrawingorequatione.g 18=10 8 understandthatthesenumbersarecomposedoftenonesandone two,three,four,five,six,seven,eight,ornineones measurement and data describe and compare measurable attributes k.md 1 describemeasurableattributesofobjects,suchaslengthorweight describeseveralmeasurableattributesofasingleobject 2 directlycomparetwoobjectswithameasurableattributeincommon toseewhichobjecthas moreof lessof theattribute,anddescribe thedifference.for example directly compare the heights of two children and describe one child as taller/shorter classify objects and count the number of objects in each category 3 classifyobjectsintogivencategories;countthenumbersofobjectsin eachcategoryandsortthecategoriesbycount.3 geometry k.g identify and describe shapes squares circles triangles rectangles hexagons cubes cones cylinders and spheres 1 describeobjectsintheenvironmentusingnamesofshapes,and describetherelativepositionsoftheseobjectsusingtermssuchas above,below,beside,in front of,behind,andnext to 2 correctlynameshapesregardlessoftheirorientationsoroverallsize 3 identifyshapesastwo-dimensionallyinginaplane flat orthreedimensional solid analyze compare create and compose shapes 4 analyzeandcomparetwo-andthree-dimensionalshapes,in differentsizesandorientations,usinginformallanguagetodescribe theirsimilarities,differences,partse.g numberofsidesand vertices corners andotherattributese.g havingsidesofequal length 5 modelshapesintheworldbybuildingshapesfromcomponentse.g sticksandclayballsanddrawingshapes 6 composesimpleshapestoformlargershapes.for example can you join these two triangles with full sides touching to make a rectangle kindergarten 3 limitcategorycountstobelessthanorequalto10 12

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common core state standards for mathematics mathematics grade 1 ingrade1,instructionaltimeshouldfocusonfourcriticalareas 1 developingunderstandingofaddition,subtraction,andstrategiesfor additionandsubtractionwithin20 2developingunderstandingofwhole numberrelationshipsandplacevalue,includinggroupingintensand ones 3developingunderstandingoflinearmeasurementandmeasuring lengthsasiteratinglengthunits;and4reasoningaboutattributesof,and composinganddecomposinggeometricshapes 1studentsdevelopstrategiesforaddingandsubtractingwholenumbers basedontheirpriorworkwithsmallnumbers.theyuseavarietyofmodels includingdiscreteobjectsandlength-basedmodelse.g cubesconnected toformlengths tomodeladd-to,take-from,put-together,take-apart,and comparesituationstodevelopmeaningfortheoperationsofadditionand subtraction,andtodevelopstrategiestosolvearithmeticproblemswith theseoperations.studentsunderstandconnectionsbetweencounting andadditionandsubtractione.g addingtwoisthesameascountingon two theyusepropertiesofadditiontoaddwholenumbersandtocreate anduseincreasinglysophisticatedstrategiesbasedontheseproperties e.g makingtens tosolveadditionandsubtractionproblemswithin 20.bycomparingavarietyofsolutionstrategies,childrenbuildtheir understandingoftherelationshipbetweenadditionandsubtraction 2studentsdevelop,discuss,anduseefficient,accurate,andgeneralizable methodstoaddwithin100andsubtractmultiplesof10.theycompare wholenumbersatleastto100todevelopunderstandingofandsolve problemsinvolvingtheirrelativesizes.theythinkofwholenumbers between10and100intermsoftensandonesespeciallyrecognizingthe numbers11to19ascomposedofatenandsomeones throughactivities thatbuildnumbersense,theyunderstandtheorderofthecounting numbersandtheirrelativemagnitudes 3studentsdevelopanunderstandingofthemeaningandprocessesof measurement,includingunderlyingconceptssuchasiteratingthemental activityofbuildingupthelengthofanobjectwithequal-sizedunitsand thetransitivityprincipleforindirectmeasurement.1 4studentscomposeanddecomposeplaneorsolidfigurese.g put twotrianglestogethertomakeaquadrilateralandbuildunderstanding ofpart-wholerelationshipsaswellasthepropertiesoftheoriginaland compositeshapes.astheycombineshapes,theyrecognizethemfrom differentperspectivesandorientations,describetheirgeometricattributes anddeterminehowtheyarealikeanddifferent,todevelopthebackground formeasurementandforinitialunderstandingsofpropertiessuchas congruenceandsymmetry studentsshouldapplytheprincipleoftransitivityofmeasurementtomakeindirect comparisons,buttheyneednotusethistechnicalterm 1 grade 1 13

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common core state standards for mathematics grade 1 overview operations and algebraic thinking · represent and solve problems involving addition and subtraction · understand and apply properties of operations and the relationship between addition and subtraction · add and subtract within 20 · work with addition and subtraction equations mathematical practices 1 makesenseofproblemsandperseverein solvingthem 2 reasonabstractlyandquantitatively 3 constructviableargumentsandcritique thereasoningofothers 4 modelwithmathematics 5 useappropriatetoolsstrategically 6 attendtoprecision number and operations in base ten · extend the counting sequence · understand place value · use place value understanding and properties of operations to add and subtract 7 lookforandmakeuseofstructure 8 lookforandexpressregularityinrepeated reasoning measurement and data · measure lengths indirectly and by iterating length units · tell and write time · represent and interpret data geometry · reason with shapes and their attributes grade 1 14

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common core state standards for mathematics operations and algebraic thinking 1.oa represent and solve problems involving addition and subtraction 1 useadditionandsubtractionwithin20tosolvewordproblemsinvolving situationsofaddingto,takingfrom,puttingtogether,takingapart andcomparing,withunknownsinallpositions,e.g byusingobjects drawings,andequationswithasymbolfortheunknownnumberto representtheproblem.2 2 solvewordproblemsthatcallforadditionofthreewholenumbers whosesumislessthanorequalto20,e.g byusingobjects,drawings andequationswithasymbolfortheunknownnumbertorepresentthe problem 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 of addition to add 2 6 4 the second two numbers can be added to make a 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 relatecountingtoadditionandsubtractione.g bycountingon2to add2 6 addandsubtractwithin20,demonstratingfluencyforadditionand subtractionwithin10.usestrategiessuchascountingon;makingten e.g 8+6=8+2+4=10+4=14 decomposinganumberleadingto atene.g 13­4=13­3­1=10­1=9 usingtherelationshipbetween additionandsubtractione.g knowingthat8+4=12,oneknows12­8 =4 andcreatingequivalentbuteasierorknownsumse.g adding6 7bycreatingtheknownequivalent6+6+1=12+1=13 work with addition and subtraction equations 7 understandthemeaningoftheequalsign,anddetermineifequations involvingadditionandsubtractionaretrueorfalse.for example which of the following equations are true and which are false 6 6 7 8 ­ 1 5 2 2 5 4 1 5 2 8 determinetheunknownwholenumberinanadditionorsubtraction equationrelatingthreewholenumbers.for example determine the unknown number that makes the equation true in each of the equations 8 11 5 ­ 3 6 6 number and operations in base ten extend the counting sequence 1.nbt 1 countto120,startingatanynumberlessthan120.inthisrange,read andwritenumeralsandrepresentanumberofobjectswithawritten numeral understand place value 2 understandthatthetwodigitsofatwo-digitnumberrepresentamounts oftensandones.understandthefollowingasspecialcases a b c 2 10canbethoughtofasabundleoftenones calleda ten thenumbersfrom11to19arecomposedofatenandone,two three,four,five,six,seven,eight,ornineones thenumbers10,20,30,40,50,60,70,80,90refertoone,two three,four,five,six,seven,eight,orninetensand0ones grade 1 3 seeglossary,table1 studentsneednotuseformaltermsfortheseproperties 15

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