common core state standards - cobb county school district...common core state standards for...
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Common Core State StandardS for
mathematics
Common Core State StandardS for matHematICS
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 and adding to, and understand subtraction as taking apart and taking from.
number and operations in Base ten
• Work with numbers 11–19 to gain foundations for place value.
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.
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 two children 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 you join 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.
Common Core State StandardS for matHematICSG
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Grade 1 overviewoperations 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.
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.
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.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
Common Core State StandardS for matHematICSG
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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 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. 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, 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. Determinetheunknownwholenumberinanadditionorsubtractionequationrelatingthreewholenumbers.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 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.
Common Core State StandardS for matHematICSG
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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 to contexts where the object being measured is spanned by a whole number of length 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.”
Common Core State StandardS for matHematICSG
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mathematics | Grade 2InGrade2,instructionaltimeshouldfocusonfourcriticalareas:(1)
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.
Common Core State StandardS for matHematICSG
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operations and algebraic thinking
• represent and solve problems involving addition and subtraction.
• add and subtract within 20.
• Work with equal groups of objects to gain foundations for multiplication.
number and operations in Base ten
• Understand place value.
• Use place value understanding and properties of operations to add and subtract.
measurement and data
• measure and estimate lengths in standard units.
• relate addition and subtraction to length.
• Work with time and money.
• represent and interpret data.
Geometry
• reason with shapes and their attributes.
mathematical Practices
1. Makesenseofproblemsandperseverein
solvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
Grade 2 overview
Common Core State StandardS for matHematICSG
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operations and algebraic thinking 2.oa
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.
number and operations in Base ten 2.nBt
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.
Common Core State StandardS for matHematICSG
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measurement and data 2.md
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 2 dimes and 3 pennies, how many cents do you have?
Represent and interpret data.
9. Generatemeasurementdatabymeasuringlengthsofseveralobjectstothenearestwholeunit,orbymakingrepeatedmeasurementsofthesameobject.Showthemeasurementsbymakingalineplot,wherethehorizontalscaleismarkedoffinwhole-numberunits.
10. Drawapicturegraphandabargraph(withsingle-unitscale)torepresentadatasetwithuptofourcategories.Solvesimpleput-together,take-apart,andcompareproblems4usinginformationpresentedinabargraph.
Geometry 2.G
Reason with shapes and their attributes.
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.
Common Core State StandardS for matHematICSG
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1
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.
Common Core State StandardS for matHematICSG
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2
operations and algebraic thinking
• represent and solve problems involving multiplication and division.
• Understand properties of multiplication and the relationship between multiplication and division.
• multiply and divide within 100.
• Solve problems involving the four operations, and identify and explain patterns in arithmetic.
number and operations in Base ten
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
number and operations—fractions
• develop understanding of fractions as numbers.
measurement and data
• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
• represent and interpret data.
• Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
• Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Geometry
• reason with shapes and their attributes.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 3 overviewmathematical Practices
1. Makesenseofproblemsandperseverein
solvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
Common Core State StandardS for matHematICSG
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3
operations and algebraic thinking 3.oa
Represent and solve problems involving multiplication and division.
1. Interpretproductsofwholenumbers,e.g.,interpret5×7asthetotalnumberofobjectsin5groupsof7objectseach.For example, describe a context in which a total number of objects can be expressed as 5 × 7.
2. Interpretwhole-numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3. Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1
4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.For example, determine the unknown 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. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
6. Understanddivisionasanunknown-factorproblem.For example, find 32 ÷ 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 explain why 4 times a number can be decomposed into two equal addends.
1SeeGlossary,Table2.2Studentsneednotuseformaltermsfortheseproperties.3Thisstandardislimitedtoproblemsposedwithwholenumbersandhavingwhole-numberanswers;studentsshouldknowhowtoperformoperationsintheconven-tionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).
Common Core State StandardS for matHematICSG
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4
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/basthequantityformedby1partwhenawholeispartitionedintob 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 form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of 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.
Common Core State StandardS for matHematICSG
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5
2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardunitsofgrams(g),kilograms(kg),andliters(l).6Add,subtract,multiply,ordividetosolveone-stepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.7
Represent and interpret data.
3. Drawascaledpicturegraphandascaledbargraphtorepresentadatasetwithseveralcategories.Solveone-andtwo-step“howmanymore”and“howmanyless”problemsusinginformationpresentedinscaledbargraphs.For example, draw a bar graph in which each square in the 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).
Common Core State StandardS for matHematICSG
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6
Geometry 3.G
Reason with shapes and their attributes.
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 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
Common Core State StandardS for matHematICSG
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7
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.
Common Core State StandardS for matHematICSG
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8
Grade 4 overviewoperations and algebraic thinking
• Use the four operations with whole numbers to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.
number and operations in Base ten
• Generalize place value understanding for multi-digit whole numbers.
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
number and operations—fractions
• extend understanding of fraction equivalence and ordering.
• Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
• Understand decimal notation for fractions, and compare decimal fractions.
measurement and data
• Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
• represent and interpret data.
• Geometric measurement: understand concepts of angle and measure angles.
Geometry
• draw and identify lines and angles, and classify shapes by properties of their lines and angles.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Common Core State StandardS for matHematICSG
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9
operations and algebraic thinking 4.oa
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, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers 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.For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and 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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0
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, use a 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.For example, 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 will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
3Grade4expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,5,6,8,10,12,and100.
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1
Understand decimal notation for fractions, and compare decimal fractions.
5. Expressafractionwithdenominator10asanequivalentfractionwithdenominator100,andusethistechniquetoaddtwofractionswithrespectivedenominators10and100.4For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
6. Usedecimalnotationforfractionswithdenominators10or100.For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
7. Comparetwodecimalstohundredthsbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwodecimalsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualmodel.
measurement and data 4.md
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 for feet 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 rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
Represent and interpret data.
4. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.For example, from a line plot find and interpret the difference in length between the longest 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.
Common Core State StandardS for matHematICSG
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4 | 3
2
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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operations and algebraic thinking
• Write and interpret numerical expressions.
• analyze patterns and relationships.
number and operations in Base ten
• Understand the place value system.
• Perform operations with multi-digit whole numbers and with decimals to hundredths.
number and operations—fractions
• Use equivalent fractions as a strategy to add and subtract fractions.
• apply and extend previous understandings of multiplication and division to multiply and divide fractions.
measurement and data
• Convert like measurement units within a given measurement system.
• represent and interpret data.
• Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Geometry
• Graph points on the coordinate plane to solve real-world and mathematical problems.
• Classify two-dimensional figures into categories based on their properties.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 5 overview
Common Core State StandardS for matHematICSG
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operations and algebraic thinking 5.oa
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 the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
number and operations in Base ten 5.nBt
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.
Common Core State StandardS for matHematICSG
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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. (In general, a/b + c/d = (ad + bc)/bd.)
2. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewhole,includingcasesofunlikedenominators,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.Usebenchmarkfractionsandnumbersenseoffractionstoestimatementallyandassessthereasonablenessofanswers.For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 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 to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the 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 for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solverealworldproblemsinvolvingdivisionofunitfractionsbynon-zerowholenumbersanddivisionofwholenumbersbyunitfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
measurement and data 5.md
Convert like measurement units within a given measurement system.
1. Convertamongdifferent-sizedstandardmeasurementunitswithinagivenmeasurementsystem(e.g.,convert5cmto0.05m),andusetheseconversionsinsolvingmulti-step,realworldproblems.
Represent and interpret data.
2. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers 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.
Common Core State StandardS for matHematICSG
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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 are rectangles, so all squares have four right angles.
4. Classifytwo-dimensionalfiguresinahierarchybasedonproperties.