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2011 WYOMING MATHEMATICS CONTENT AND PERFORMANCE STANDARDS
RATIONALE
Mathematics is the language that defines the blueprint of the universe. Mathematics is woven into all parts of our lives, and is more than a list of skills to be mastered. The essence of mathematics is the ability to employ critical thinking and reasoning to solve problems. To be successful in mathematics, one must see mathematics as sensible, useful, and worthwhile. The development and maturation of students’ conceptual understanding and application of the processes and procedures of mathematics is the driving force behind the 2011 Wyoming Mathematics Content and Performance Standards.
Why Do We Have Standards for Mathematics? Standards ensure that all students in Wyoming receive a uniform and consistent mathematical education and are prepared for success in and out of the classroom. Therefore, the 2011 Wyoming Mathematics Content and Performance Standards should: Provide students, parents, and educators focus and coherence through application and
understanding of mathematical concepts and processes. Be aligned vertically from K12 with clearly defined goals and outcomes for learning. Emphasize conceptual understanding. Encourage multiple models, representations and strategies. Incorporate technological applications that optimize mathematical understanding and
application. Develop students’ mathematical thinking. Develop reasoning, solving, representing, proving, communicating, and connecting
across contexts and applications. Promote habits of mind recognizing or identifying mathematics in the world around us. Engage students in making sense, building conceptual understanding, developing
procedural fluency, and employing adaptive reasoning. Build constructive attitudes to see mathematics as sensible, useful and worthwhile, and
increase beliefs in one’s own ability to do mathematics. Develop students’ abilities to use mathematical language with care and precision in
communicating concepts, skills, symbols, and vocabulary. Drive professional development to improve student learning.

Why Include the Common Core State Standards into the 2011 Wyoming Mathematics Content and Performance Standards?
The Common Core State Standards for Mathematics is a stateled effort to establish a single set of clear educational standards that states can share and voluntarily adopt. Including the Common Core State Standards into the 2011 Wyoming Mathematics Content and Performance Standards prepares Wyoming students to be competitive on the national and world stage. These standards are a set of specific, rigorous expectations that build students’ conceptual understanding, mathematical language, and application of processes and procedures coherently from one grade to the next so all students will be prepared for postsecondary experiences. The use of technology is expected throughout all levels of the standards. The focus areas for each grade level and conceptual category establish a depth of knowledge as opposed to a breadth of knowledge across multiple standards in each grade level or content area. The Standards for Mathematical Practices describe the essential ways of thinking and habits of mind that are the hallmark of a mathematically literate and informed citizen. The Common Core State Standards for mathematics stress both conceptual understanding and procedural skills to ensure students learn and can apply the critical information needed to succeed at each level. This creates a vertical articulation where the mathematics learned in elementary school provides the foundation for the study of statistics, probability, ratio and proportion, geometry, and algebra in middle school. This is, in turn, the bedrock upon which the knowledge needed for success in colleges and careers can be developed in the high school.

ORGANIZATION OF STANDARDS
The 2011 Wyoming Mathematics Content and Performance Standards, which include the Common Core State Standards for mathematics, define what students should understand and be able to do in their study of mathematics. They do not dictate curriculum or teaching methods. Teachers ensure students achieve standards by using a variety of instructional strategies based on their students’ needs. The Content and Performance Standards are divided into two areas of equal importance: The Standards for Mathematical Practice are embedded at every grade level to establish habits of mind which will empower students to become mathematically literate,
and The Standards for Mathematical Content are gradelevel specific kindergarten through grade eight and conceptual category specific in high school. They provide a scaffold that allows students to become increasingly more proficient in understanding and using mathematics with a steady progression leading to college and career readiness by the time students graduate from high school.
PERFORMANCE LEVEL DESCRIPTORS These statements help teachers judge where students are performing in relation to the standards. They describe student performance at various levels of proficiency. To consider a standard as “met”, students are required to perform at the “proficient” level. A general definition of each level is provided below.
Advanced: Superior academic performance indicating an indepth understanding and exemplary display of the knowledge and skills included in the Wyoming Content and Performance Standards. Proficient: Satisfactory academic performance indicating a solid understanding and display of the knowledge and skills included in the Wyoming Content and Performance Standards. Basic: Marginal academic performance, work approaching, but not yet reaching, satisfactory performance, indicating partial understanding and limited display of the knowledge and skills included in the Wyoming Content and Performance Standards.

THE STANDARDS FOR MATHEMATICAL PRACTICE
What are the Standards for Mathematical Practice?
The Standards for Mathematical Practice bring the complexities of the world into focus and give schema for grappling with authentic and meaningful problems. The Practices define experiences that build understanding of mathematics and ways of thinking through which students develop, apply, and assess their knowledge: Develop Mathematical Practices Students make sense of quantities and their relationships in situations by sifting through
available information to construct an approach for solving the problem. Students persevere in the development of a variety of approaches.
Apply Mathematical Practices Students take complex scenarios and distill important quantities and their relationships by
looking for patterns and making use of structure. They then apply appropriate models and use appropriate tools to derive a solution.
Students maintain oversight of the process, while attending to the details as they work to solve a problem.
Assess Mathematical Practices Students are critical consumers of the practices and processes they adapt from other
sources and are able to consider the efficiency and effectiveness of a variety of methods. Students apply precision in communicating processes and solutions. They explain why
and how various methods succeed or fail. The Standards for Mathematical Practice establish habits of mind and empower students to become mathematically literate and informed citizens.
Why are the Standards of Mathematical Practice important? Algorithmic knowledge is no longer sufficient when preparing our students to become globally competitive. The knowledge of good practitioners goes beyond algorithmic learning and allows them to picture the problem and the many roads that may lead to absolution. They realize that mathematics is applicable outside of the classroom and are confident in their ability to apply mathematical concepts to all aspects of life. The symbiotic nature of the Standards of Mathematical Practice allows students to deepen their understandings of mathematical concepts and cultivates their autonomy as mathematically literate and informed citizens. Employing mathematics as a means of synthesizing complex concepts and making informed decisions is paramount to success in all postsecondary endeavors.

MATHEMATICS PERFORMANCE LEVEL DESCRIPTORS
Because of the importance of the Standards for Mathematical Practice, they were chosen to be the basis for the content level performance descriptors across all grades. The depth of understanding and the level and consistency with which students apply them will vary according to grade level appropriateness. When used in conjunction with standard level performance descriptors, they will provide an appropriate framework to determine a student’s level of mastery of the standards. Advanced: Students at the advanced level consistently apply the Standards for Mathematical Practice to persevere in solving complex problems. They use abstract and quantitative reasoning to model mathematics. Students strategically select appropriate tools and technology. Students express answers with precision appropriate for the context of routine and nonroutine problems. They are able to recognize structure and repeated patterns to make generalizations and transfer them to new applications. They are able to organize and communicate their ideas to others, as well as analyze and evaluate mathematical thinking and strategies of others. Proficient: Students at the proficient level apply the Standards for Mathematical Practice to persevere in solving problems. They use abstract or quantitative reasoning to model mathematics. Students use, and understand the limitations of, appropriate tools and technology. Students express answers with precision appropriate for the context of the problem. They are able to recognize structure and repeated patterns to make generalizations. They are able to organize and communicate their ideas to others, as well as recognize mathematical thinking and strategies of others. Basic: Students at the basic level attempt to apply the Standards for Mathematical Practice in solving problems. They use limited reasoning to model mathematics. Students demonstrate some appropriate use of tools and technology. Students express answers without precision. They are able to recognize structure and repeated patterns. They attempt to organize and communicate their ideas to others.
THE STANDARDS FOR MATHEMATICAL CONTENT
Each grade level in the K – 8 standards is prefaced with an explanation of instructional focus areas for that grade level. Each conceptual category in the high school standards is prefaced with an explanation of the implication of that category to a student’s mastery of mathematics. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by a plus symbol (+). (Not required for all students.)

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 nontraditional 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,researchstudiesofmathematicseducationinhighperforming
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). (emphasisadded)
TheseStandardsendeavortofollowsuchadesign,notonlybystressingconceptual
understandingofkeyideas,butalsobycontinuallyreturningtoorganizing
principlessuchasplacevalueorthepropertiesofoperationstostructurethose
ideas.
Inaddition,the“sequenceoftopicsandperformances”thatisoutlinedinabodyof
mathematicsstandardsmustalsorespectwhatisknownabouthowstudentslearn.
AsConfrey(2007)pointsout,developing“sequencedobstaclesandchallenges
forstudents…absenttheinsightsaboutmeaningthatderivefromcarefulstudyof
learning,wouldbeunfortunateandunwise.”Inrecognitionofthis,thedevelopment
oftheseStandardsbeganwithresearchbasedlearningprogressionsdetailing
whatisknowntodayabouthowstudents’mathematicalknowledge,skill,and
understandingdevelopovertime.
Understanding mathematics
TheseStandardsdefinewhatstudentsshouldunderstandandbeabletodoin
theirstudyofmathematics.Askingastudenttounderstandsomethingmeans
askingateachertoassesswhetherthestudenthasunderstoodit.Butwhatdoes
mathematicalunderstandinglooklike?Onehallmarkofmathematicalunderstanding
istheabilitytojustify,inawayappropriatetothestudent’smathematicalmaturity,
whyaparticularmathematicalstatementistrueorwhereamathematicalrulecomesfrom.Thereisaworldofdifferencebetweenastudentwhocansummona
mnemonicdevicetoexpandaproductsuchas(a+ b)(x+y)andastudentwhocanexplainwherethemnemoniccomesfrom.Thestudentwhocanexplaintherule
understandsthemathematics,andmayhaveabetterchancetosucceedataless
familiartasksuchasexpanding(a+ b+c)(x+y).Mathematicalunderstandingandproceduralskillareequallyimportant,andbothareassessableusingmathematical
tasksofsufficientrichness.
TheStandardssetgradespecificstandardsbutdonotdefinetheintervention
methodsormaterialsnecessarytosupportstudentswhoarewellbeloworwell
abovegradelevelexpectations.ItisalsobeyondthescopeoftheStandardsto
definethefullrangeofsupportsappropriateforEnglishlanguagelearnersand
forstudentswithspecialneeds.Atthesametime,allstudentsmusthavethe
opportunitytolearnandmeetthesamehighstandardsiftheyaretoaccessthe
knowledgeandskillsnecessaryintheirpostschoollives.TheStandardsshould
bereadasallowingforthewidestpossiblerangeofstudentstoparticipatefully
fromtheoutset,alongwithappropriateaccommodationstoensuremaximum
participatonofstudentswithspecialeducationneeds.Forexample,forstudents
withdisabilitiesreadingshouldallowforuseofBraille,screenreadertechnology,or
otherassistivedevices,whilewritingshouldincludetheuseofascribe,computer,
orspeechtotexttechnology.Inasimilarvein,speakingandlisteningshouldbe
interpretedbroadlytoincludesignlanguage.Nosetofgradespecificstandards
canfullyreflectthegreatvarietyinabilities,needs,learningrates,andachievement
levelsofstudentsinanygivenclassroom.However,theStandardsdoprovideclear
signpostsalongthewaytothegoalofcollegeandcareerreadinessforallstudents.
TheStandardsbeginonpage6witheightStandardsforMathematicalPractice.

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How to read the grade level standards
Standards definewhatstudentsshouldunderstandandbeabletodo.
Clusters aregroupsofrelatedstandards.Notethatstandardsfromdifferentclustersmaysometimesbecloselyrelated,becausemathematics
isaconnectedsubject.
domainsarelargergroupsofrelatedstandards.Standardsfromdifferentdomainsmaysometimesbecloselyrelated.
number and operations in Base ten 3.nBtUse place value understanding and properties of operations to perform multidigit arithmetic.
1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.
2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
3. Multiplyonedigitwholenumbersbymultiplesof10intherange1090(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,strategiccompetence,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—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresenting
symbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingto
theirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.
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,twowaytables,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
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7 Look for and make use of structure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.
Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesame
amountassevenandthreemore,ortheymaysortacollectionofshapesaccording
tohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthewellremembered7×5+7×3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2×7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometric
figureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.
Theyalsocanstepbackforanoverviewandshiftperspective.Theycansee
complicatedthings,suchassomealgebraicexpressions,assingleobjectsoras
beingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannot
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)mightleadthemtothegeneralformulaforthesumofageometricseries.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,
anddescribebasictwodimensionalshapes,suchassquares,triangles,
circles,rectangles,andhexagons,presentedinavarietyofways(e.g.,with
differentsizesandorientations),aswellasthreedimensionalshapessuch
ascubes,cones,cylinders,andspheres.Theyusebasicshapesandspatial
reasoningtomodelobjectsintheirenvironmentandtoconstructmore
complexshapes.

Common Core State StandardS for matHematICSK
Ind
er
Ga
rt
en
 10
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

Common Core State StandardS for matHematICSK
Ind
er
Ga
rt
en
 11
Counting and Cardinality K.CC
Know number names and the count sequence.
1. Countto100byonesandbytens.
2. Countforwardbeginningfromagivennumberwithintheknownsequence(insteadofhavingtobeginat1).
3. Writenumbersfrom0to20.Representanumberofobjectswithawrittennumeral020(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 understand 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.)

Common Core State StandardS for matHematICSK
Ind
er
Ga
rt
en
 12
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. Identifyshapesastwodimensional(lyinginaplane,“flat”)orthreedimensional(“solid”).
Analyze, compare, create, and compose shapes.
4. Analyzeandcomparetwoandthreedimensionalshapes,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.

Common Core State StandardS for matHematICSG
ra
de
1  13
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,
includingdiscreteobjectsandlengthbasedmodels(e.g.,cubesconnected
toformlengths),tomodeladdto,takefrom,puttogether,takeapart,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
activityofbuildingupthelengthofanobjectwithequalsizedunits)and
thetransitivityprincipleforindirectmeasurement.1
(4)Studentscomposeanddecomposeplaneorsolidfigures(e.g.,put
twotrianglestogethertomakeaquadrilateral)andbuildunderstanding
ofpartwholerelationshipsaswellasthepropertiesoftheoriginaland
compositeshapes.Astheycombineshapes,theyrecognizethemfrom
differentperspectivesandorientations,describetheirgeometricattributes,
anddeterminehowtheyarealikeanddifferent,todevelopthebackground
formeasurementandforinitialunderstandingsofpropertiessuchas
congruenceandsymmetry.
1Studentsshouldapplytheprincipleoftransitivityofmeasurementtomakeindirectcomparisons,buttheyneednotusethistechnicalterm.

Common Core State StandardS for matHematICSG
ra
de
1  14
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
ra
de
1  15
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. Understandsubtractionasanunknownaddendproblem.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. Understandthatthetwodigitsofatwodigitnumberrepresentamountsoftensandones.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
ra
de
1  16
3. Comparetwotwodigitnumbersbasedonmeaningsofthetensandonesdigits,recordingtheresultsofcomparisonswiththesymbols>,=,and

Common Core State StandardS for matHematICSG
ra
de
2  17
mathematics  Grade 2InGrade2,instructionaltimeshouldfocusonfourcriticalareas:(1)
extendingunderstandingofbasetennotation;(2)buildingfluencywith
additionandsubtraction;(3)usingstandardunitsofmeasure;and(4)
describingandanalyzingshapes.
(1)Studentsextendtheirunderstandingofthebasetensystem.This
includesideasofcountinginfives,tens,andmultiplesofhundreds,tens,
andones,aswellasnumberrelationshipsinvolvingtheseunits,including
comparing.Studentsunderstandmultidigitnumbers(upto1000)written
inbasetennotation,recognizingthatthedigitsineachplacerepresent
amountsofthousands,hundreds,tens,orones(e.g.,853is8hundreds+5
tens+3ones).
(2)Studentsusetheirunderstandingofadditiontodevelopfluencywith
additionandsubtractionwithin100.Theysolveproblemswithin1000
byapplyingtheirunderstandingofmodelsforadditionandsubtraction,
andtheydevelop,discuss,anduseefficient,accurate,andgeneralizable
methodstocomputesumsanddifferencesofwholenumbersinbaseten
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,
andanalyzingtwoandthreedimensionalshapes,studentsdevelopa
foundationforunderstandingarea,volume,congruence,similarity,and
symmetryinlatergrades.

Common Core State StandardS for matHematICSG
ra
de
2  18
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
ra
de
2  19
operations and algebraic thinking 2.oa
Represent and solve problems involving addition and subtraction.
1. Useadditionandsubtractionwithin100tosolveoneandtwostepwordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions, e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1
Add and subtract within 20.
2. Fluentlyaddandsubtractwithin20usingmentalstrategies.2ByendofGrade2,knowfrommemoryallsumsoftwoonedigitnumbers.
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. Understandthatthethreedigitsofathreedigitnumberrepresentamountsofhundreds,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;skipcountby5s,10s,and100s.
3. Readandwritenumbersto1000usingbasetennumerals,numbernames,andexpandedform.
4. Comparetwothreedigitnumbersbasedonmeaningsofthehundreds,tens,andonesdigits,using>,=,and<symbolstorecordtheresultsofcomparisons.
Use place value understanding and properties of operations to add and subtract.
5. Fluentlyaddandsubtractwithin100usingstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
6. Adduptofourtwodigitnumbersusingstrategiesbasedonplacevalueandpropertiesofoperations.
7. Addandsubtractwithin1000,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethod.Understandthatinaddingorsubtractingthreedigitnumbers,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
ra
de
2  20
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,...,andrepresentwholenumbersumsanddifferenceswithin100onanumberlinediagram.
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,wherethehorizontalscaleismarkedoffinwholenumberunits.
10. Drawapicturegraphandabargraph(withsingleunitscale)torepresentadatasetwithuptofourcategories.Solvesimpleputtogether,takeapart,andcompareproblems4usinginformationpresentedinabargraph.
Geometry 2.G
Reason with shapes and their attributes.
1. Recognizeanddrawshapeshavingspecifiedattributes,suchasagivennumberofanglesoragivennumberofequalfaces.5Identifytriangles,quadrilaterals,pentagons,hexagons,andcubes.
2. Partitionarectangleintorowsandcolumnsofsamesizesquaresandcounttofindthetotalnumberofthem.
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
ra
de
3  2
1
MathematicsGrade3
InGrade3,instructionaltimeshouldfocusonfourcriticalareas:(1)
developingunderstandingofmultiplicationanddivisionandstrategies
formultiplicationanddivisionwithin100;(2)developingunderstanding
offractions,especiallyunitfractions(fractionswithnumerator1);(3)
developingunderstandingofthestructureofrectangulararraysandof
area;and(4)describingandanalyzingtwodimensionalshapes.
(1)Studentsdevelopanunderstandingofthemeaningsofmultiplication
anddivisionofwholenumbersthroughactivitiesandproblemsinvolving
equalsizedgroups,arrays,andareamodels;multiplicationisfinding
anunknownproduct,anddivisionisfindinganunknownfactorinthese
situations.Forequalsizedgroupsituations,divisioncanrequirefinding
theunknownnumberofgroupsortheunknowngroupsize.Studentsuse
propertiesofoperationstocalculateproductsofwholenumbers,using
increasinglysophisticatedstrategiesbasedonthesepropertiestosolve
multiplicationanddivisionproblemsinvolvingsingledigitfactors.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)Studentsrecognizeareaasanattributeoftwodimensionalregions.
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
ra
de
3  2
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 multidigit 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
ra
de
3  2
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. Interpretwholenumberquotientsofwholenumbers,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. Understanddivisionasanunknownfactorproblem.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,knowfrommemoryallproductsoftwoonedigitnumbers.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
8. Solvetwostepwordproblemsusingthefouroperations.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.3Thisstandardislimitedtoproblemsposedwithwholenumbersandhavingwholenumberanswers;studentsshouldknowhowtoperformoperationsintheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).

Common Core State StandardS for matHematICSG
ra
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3  2
4
number and operations in Base ten 3.nBt
Use place value understanding and properties of operations to perform multidigit arithmetic.4
1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.
2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
3. Multiplyonedigitwholenumbersbymultiplesof10intherange10–90(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

Common Core State StandardS for matHematICSG
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3  2
5
2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardunitsofgrams(g),kilograms(kg),andliters(l).6Add,subtract,multiply,ordividetosolveonestepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.7
Represent and interpret data.
3. Drawascaledpicturegraphandascaledbargraphtorepresentadatasetwithseveralcategories.Solveoneandtwostep“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. Findtheareaofarectanglewithwholenumbersidelengthsbytilingit,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.
b. Multiplysidelengthstofindareasofrectangleswithwholenumbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwholenumberproductsasrectangularareasinmathematicalreasoning.
c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwholenumbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.
d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposingthemintononoverlappingrectanglesandaddingtheareasofthenonoverlappingparts,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
ra
de
3  2
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
ra
de
4  2
7
mathematics  Grade 4InGrade4,instructionaltimeshouldfocusonthreecriticalareas:(1)
developingunderstandingandfluencywithmultidigitmultiplication,
anddevelopingunderstandingofdividingtofindquotientsinvolving
multidigitdividends;(2)developinganunderstandingoffraction
equivalence,additionandsubtractionoffractionswithlikedenominators,
andmultiplicationoffractionsbywholenumbers;(3)understanding
thatgeometricfigurescanbeanalyzedandclassifiedbasedontheir
properties,suchashavingparallelsides,perpendicularsides,particular
anglemeasures,andsymmetry.
(1)Studentsgeneralizetheirunderstandingofplacevalueto1,000,000,
understandingtherelativesizesofnumbersineachplace.Theyapplytheir
understandingofmodelsformultiplication(equalsizedgroups,arrays,
areamodels),placevalue,andpropertiesofoperations,inparticularthe
distributiveproperty,astheydevelop,discuss,anduseefficient,accurate,
andgeneralizablemethodstocomputeproductsofmultidigitwhole
numbers.Dependingonthenumbersandthecontext,theyselectand
accuratelyapplyappropriatemethodstoestimateormentallycalculate
products.Theydevelopfluencywithefficientproceduresformultiplying
wholenumbers;understandandexplainwhytheproceduresworkbasedon
placevalueandpropertiesofoperations;andusethemtosolveproblems.
Studentsapplytheirunderstandingofmodelsfordivision,placevalue,
propertiesofoperations,andtherelationshipofdivisiontomultiplication
astheydevelop,discuss,anduseefficient,accurate,andgeneralizable
procedurestofindquotientsinvolvingmultidigitdividends.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,andclassifytwodimensional
shapes.Throughbuilding,drawing,andanalyzingtwodimensionalshapes,
studentsdeepentheirunderstandingofpropertiesoftwodimensional
objectsandtheuseofthemtosolveproblemsinvolvingsymmetry.

Common Core State StandardS for matHematICSG
ra
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4  2
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 multidigit whole numbers.
• Use place value understanding and properties of operations to perform multidigit 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
ra
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4  2
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. Solvemultistepwordproblemsposedwithwholenumbersandhavingwholenumberanswersusingthefouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.
Gain familiarity with factors and multiples.
4. Findallfactorpairsforawholenumberintherange1–100.Recognizethatawholenumberisamultipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1–100isamultipleofagivenonedigitnumber.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 multidigit whole numbers.
1. Recognizethatinamultidigitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
2. Readandwritemultidigitwholenumbersusingbasetennumerals,numbernames,andexpandedform.Comparetwomultidigitnumbersbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.
3. Useplacevalueunderstandingtoroundmultidigitwholenumberstoanyplace.
Use place value understanding and properties of operations to perform multidigit arithmetic.
4. Fluentlyaddandsubtractmultidigitwholenumbersusingthestandardalgorithm.
5. Multiplyawholenumberofuptofourdigitsbyaonedigitwholenumber,andmultiplytwotwodigitnumbers,usingstrategiesbasedonplacevalueandthepropertiesofoperations.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.
1SeeGlossary,Table2.2Grade4expectationsinthisdomainarelimitedtowholenumberslessthanorequalto1,000,000.

Common Core State StandardS for matHematICSG
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4  3
0
6. Findwholenumberquotientsandremainderswithuptofourdigitdividendsandonedigitdivisors,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>,=,or1asasumoffractions1/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.

Common Core State StandardS for matHematICSG
ra
de
4  3
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

Common Core State StandardS for matHematICSG
ra
de
4  3
2
6. Measureanglesinwholenumberdegreesusingaprotractor.Sketchanglesofspecifiedmeasure.
7. Recognizeanglemeasureasadditive.Whenanangleisdecomposedintononoverlappingparts,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.Identifytheseintwodimensionalfigures.
2. Classifytwodimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.
3. Recognizealineofsymmetryforatwodimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifylinesymmetricfiguresanddrawlinesofsymmetry.

Common Core State StandardS for matHematICSG
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5  33
mathematics  Grade 5InGrade5,instructionaltimeshouldfocusonthreecriticalareas:(1)
developingfluencywithadditionandsubtractionoffractions,and
developingunderstandingofthemultiplicationoffractionsandofdivision
offractionsinlimitedcases(unitfractionsdividedbywholenumbersand
wholenumbersdividedbyunitfractions);(2)extendingdivisionto2digit
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
basedonthemeaningofbasetennumeralsandpropertiesofoperations.
Theyfinalizefluencywithmultidigitaddition,subtraction,multiplication,
anddivision.Theyapplytheirunderstandingsofmodelsfordecimals,
decimalnotation,andpropertiesofoperationstoaddandsubtract
decimalstohundredths.Theydevelopfluencyinthesecomputations,and
makereasonableestimatesoftheirresults.Studentsusetherelationship
betweendecimalsandfractions,aswellastherelationshipbetween
finitedecimalsandwholenumbers(i.e.,afinitedecimalmultipliedbyan
appropriatepowerof10isawholenumber),tounderstandandexplain
whytheproceduresformultiplyinganddividingfinitedecimalsmake
sense.Theycomputeproductsandquotientsofdecimalstohundredths
efficientlyandaccurately.
(3)Studentsrecognizevolumeasanattributeofthreedimensional
space.Theyunderstandthatvolumecanbemeasuredbyfindingthetotal
numberofsamesizeunitsofvolumerequiredtofillthespacewithout
gapsoroverlaps.Theyunderstandthata1unitby1unitby1unitcube
isthestandardunitformeasuringvolume.Theyselectappropriateunits,
strategies,andtoolsforsolvingproblemsthatinvolveestimatingand
measuringvolume.Theydecomposethreedimensionalshapesandfind
volumesofrightrectangularprismsbyviewingthemasdecomposedinto
layersofarraysofcubes.Theymeasurenecessaryattributesofshapesin
ordertodeterminevolumestosolverealworldandmathematicalproblems.

Common Core State StandardS for matHematICSG
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5  34
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 multidigit 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 realworld and mathematical problems.
• Classify twodimensional 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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5  35
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. Recognizethatinamultidigitnumber,adigitinoneplacerepresents10timesasmuchasitrepresentsintheplacetoitsrightand1/10ofwhatitrepresentsintheplacetoitsleft.
2. Explainpatternsinthenumberofzerosoftheproductwhenmultiplyinganumberbypowersof10,andexplainpatternsintheplacementofthedecimalpointwhenadecimalismultipliedordividedbyapowerof10.Usewholenumberexponentstodenotepowersof10.
3. Read,write,andcomparedecimalstothousandths.
a. Readandwritedecimalstothousandthsusingbasetennumerals,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 multidigit whole numbers and with decimals to hundredths.
5. Fluentlymultiplymultidigitwholenumbersusingthestandardalgorithm.
6. Findwholenumberquotientsofwholenumberswithuptofourdigitdividendsandtwodigitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rec