2011 wyoming mathematics content and …2011 wyoming mathematics content and performance standards ....

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 K-12 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 state-led 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 post-secondary 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 grade-level 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 in-depth 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 post-secondary 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 non-routine 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

Common Core State StandardS for matHematICSIn

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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). (emphasisadded)

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,

whyaparticularmathematicalstatementistrueorwhereamathematicalrulecomesfrom.Thereisaworldofdifferencebetweenastudentwhocansummona

mnemonicdevicetoexpandaproductsuchas(a+ b)(x+y)andastudentwhocanexplainwherethemnemoniccomesfrom.Thestudentwhocanexplaintherule

understandsthemathematics,andmayhaveabetterchancetosucceedataless

familiartasksuchasexpanding(a+ b+c)(x+y).Mathematicalunderstandingandproceduralskillareequallyimportant,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.Notethatstandardsfromdifferentclustersmaysometimesbecloselyrelated,becausemathematics

isaconnectedsubject.

domainsarelargergroupsofrelatedstandards.Standardsfromdifferentdomainsmaysometimesbecloselyrelated.

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

Common Core State StandardS for matHematICSS

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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,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,studentswillsee7×8equalsthewellremembered7×5+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.

Common Core State StandardS for matHematICSK

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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.

Common Core State StandardS for matHematICSG

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mathematics | Grade 1InGrade1,instructionaltimeshouldfocusonfourcriticalareas:(1)

developingunderstandingofaddition,subtraction,andstrategiesfor

additionandsubtractionwithin20;(2)developingunderstandingofwhole

numberrelationshipsandplacevalue,includinggroupingintensand

ones;(3)developingunderstandingoflinearmeasurementandmeasuring

lengthsasiteratinglengthunits;and(4)reasoningaboutattributesof,and

composinganddecomposinggeometricshapes.

(1)Studentsdevelopstrategiesforaddingandsubtractingwholenumbers

basedontheirpriorworkwithsmallnumbers.Theyuseavarietyofmodels,

includingdiscreteobjectsandlength-basedmodels(e.g.,cubesconnected

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

comparesituationstodevelopmeaningfortheoperationsofadditionand

subtraction,andtodevelopstrategiestosolvearithmeticproblemswith

theseoperations.Studentsunderstandconnectionsbetweencounting

andadditionandsubtraction(e.g.,addingtwoisthesameascountingon

two).Theyusepropertiesofadditiontoaddwholenumbersandtocreate

anduseincreasinglysophisticatedstrategiesbasedontheseproperties

(e.g.,“makingtens”)tosolveadditionandsubtractionproblemswithin

20.Bycomparingavarietyofsolutionstrategies,childrenbuildtheir

understandingoftherelationshipbetweenadditionandsubtraction.

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

methodstoaddwithin100andsubtractmultiplesof10.Theycompare

wholenumbers(atleastto100)todevelopunderstandingofandsolve

problemsinvolvingtheirrelativesizes.Theythinkofwholenumbers

between10and100intermsoftensandones(especiallyrecognizingthe

numbers11to19ascomposedofatenandsomeones).Throughactivities

thatbuildnumbersense,theyunderstandtheorderofthecounting

numbersandtheirrelativemagnitudes.

(3)Studentsdevelopanunderstandingofthemeaningandprocessesof

measurement,includingunderlyingconceptssuchasiterating(themental

activityofbuildingupthelengthofanobjectwithequal-sizedunits)and

thetransitivityprincipleforindirectmeasurement.1

(4)Studentscomposeanddecomposeplaneorsolidfigures(e.g.,put

twotrianglestogethertomakeaquadrilateral)andbuildunderstanding

ofpart-wholerelationshipsaswellasthepropertiesoftheoriginaland

compositeshapes.Astheycombineshapes,theyrecognizethemfrom

differentperspectivesandorientations,describetheirgeometricattributes,

anddeterminehowtheyarealikeanddifferent,todevelopthebackground

formeasurementandforinitialunderstandingsofpropertiessuchas

congruenceandsymmetry.

1Studentsshouldapplytheprincipleoftransitivityofmeasurementtomakeindirectcomparisons,buttheyneednotusethistechnicalterm.


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Grade 1 overviewoperations and algebraic thinking

• represent and solve problems 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.


• extend the counting sequence.

• Understand place value.

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


• measure lengths indirectly and by iterating length units.

• tell and write time.

• represent and interpret data.

Geometry

• reason with shapes and their attributes.


1. Makesenseofproblemsandpersevereinsolvingthem.









reasoning.


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operations and algebraic thinking 1.oa

Represent and solve problems involving addition and subtraction.

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

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

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

3. Applypropertiesofoperationsasstrategiestoaddandsubtract.3Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property 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.


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


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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.


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• represent and solve problems involving addition and subtraction.

• add and subtract within 20.

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


• Understand place value.

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


• measure and estimate lengths in standard units.

• relate addition and subtraction to length.

• Work with time and money.


Geometry




solvingthem.









reasoning.

Grade 2 overview


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Represent and solve problems involving addition and subtraction.

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

Add and subtract within 20.

2. Fluentlyaddandsubtractwithin20usingmentalstrategies.2ByendofGrade2,knowfrommemoryallsumsoftwoone-digitnumbers.

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

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

4. Useadditiontofindthetotalnumberofobjectsarrangedinrectangulararrayswithupto5rowsandupto5columns;writeanequationtoexpressthetotalasasumofequaladdends.


Understand place value.

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

a. 100canbethoughtofasabundleoftentens—calleda“hundred.”

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

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

3. Readandwritenumbersto1000usingbase-tennumerals,numbernames,andexpandedform.

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

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

5. Fluentlyaddandsubtractwithin100usingstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.

6. Adduptofourtwo-digitnumbersusingstrategiesbasedonplacevalueandpropertiesofoperations.

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

8. Mentallyadd10or100toagivennumber100–900,andmentallysubtract10or100fromagivennumber100–900.

9. Explainwhyadditionandsubtractionstrategieswork,usingplacevalueandthepropertiesofoperations.3

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


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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.


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Mathematics|Grade3

InGrade3,instructionaltimeshouldfocusonfourcriticalareas:(1)

developingunderstandingofmultiplicationanddivisionandstrategies

formultiplicationanddivisionwithin100;(2)developingunderstanding

offractions,especiallyunitfractions(fractionswithnumerator1);(3)

developingunderstandingofthestructureofrectangulararraysandof

area;and(4)describingandanalyzingtwo-dimensionalshapes.

(1)Studentsdevelopanunderstandingofthemeaningsofmultiplication

anddivisionofwholenumbersthroughactivitiesandproblemsinvolving

equal-sizedgroups,arrays,andareamodels;multiplicationisfinding

anunknownproduct,anddivisionisfindinganunknownfactorinthese

situations.Forequal-sizedgroupsituations,divisioncanrequirefinding

theunknownnumberofgroupsortheunknowngroupsize.Studentsuse

propertiesofoperationstocalculateproductsofwholenumbers,using

increasinglysophisticatedstrategiesbasedonthesepropertiestosolve

multiplicationanddivisionproblemsinvolvingsingle-digitfactors.By

comparingavarietyofsolutionstrategies,studentslearntherelationship

betweenmultiplicationanddivision.

(2)Studentsdevelopanunderstandingoffractions,beginningwith

unitfractions.Studentsviewfractionsingeneralasbeingbuiltoutof

unitfractions,andtheyusefractionsalongwithvisualfractionmodels

torepresentpartsofawhole.Studentsunderstandthatthesizeofa

fractionalpartisrelativetothesizeofthewhole.Forexample,1/2ofthe

paintinasmallbucketcouldbelesspaintthan1/3ofthepaintinalarger

bucket,but1/3ofaribbonislongerthan1/5ofthesameribbonbecause

whentheribbonisdividedinto3equalparts,thepartsarelongerthan

whentheribbonisdividedinto5equalparts.Studentsareabletouse

fractionstorepresentnumbersequalto,lessthan,andgreaterthanone.

Theysolveproblemsthatinvolvecomparingfractionsbyusingvisual

fractionmodelsandstrategiesbasedonnoticingequalnumeratorsor

denominators.

(3)Studentsrecognizeareaasanattributeoftwo-dimensionalregions.

Theymeasuretheareaofashapebyfindingthetotalnumberofsame-

sizeunitsofarearequiredtocovertheshapewithoutgapsoroverlaps,

asquarewithsidesofunitlengthbeingthestandardunitformeasuring

area.Studentsunderstandthatrectangulararrayscanbedecomposedinto

identicalrowsorintoidenticalcolumns.Bydecomposingrectanglesinto

rectangulararraysofsquares,studentsconnectareatomultiplication,and

justifyusingmultiplicationtodeterminetheareaofarectangle.

(4)Studentsdescribe,analyze,andcomparepropertiesoftwo-

dimensionalshapes.Theycompareandclassifyshapesbytheirsidesand

angles,andconnectthesewithdefinitionsofshapes.Studentsalsorelate

theirfractionworktogeometrybyexpressingtheareaofpartofashape

asaunitfractionofthewhole.


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2


• 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.


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

number and operations—fractions

• develop understanding of fractions as numbers.


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


• 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





3. Constructviableargumentsandcritiquethereasoningofothers.





8. Lookforandexpressregularityinrepeatedreasoning.

Grade 3 overviewmathematical Practices


solvingthem.









reasoning.


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3


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).


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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


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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).


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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.


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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.


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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.


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

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


• 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.


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


• 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.











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9


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>,=,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.


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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


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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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• Write and interpret numerical expressions.

• analyze patterns and relationships.


• Understand the place value system.

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


• 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.


• Convert like measurement units within a given measurement system.


• 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.










Grade 5 overview


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Write and interpret numerical expressions.

1. Useparentheses,brackets,orbracesinnumericalexpressions,andevaluateexpressionswiththesesymbols.

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

Analyze patterns and relationships.

3. Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationshipsbetweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwopatterns,andgraphtheorderedpairsonacoordinateplane.For example, given 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.


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,rec

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