Michigan K-12 Standards
Mathematics
R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E R E L A T I O N H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P • R I G O R • • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E
Michigan State Board of Education
Kathleen N. Straus, President
Bloomfield Township
John C. Austin, Vice President Ann Arbor
Carolyn L. Curtin, Secretary
Evart
Marianne Yared McGuire, Treasurer Detroit
Nancy Danhof, NASBE Delegate
East Lansing
Elizabeth W. Bauer Birmingham
Daniel Varner
Detroit
Casandra E. Ulbrich Rochester Hills
Governor Jennifer M. Granholm
Ex Officio
Michael P. Flanagan, Chairman Superintendent of Public Instruction
Ex Officio
MDE Staff
Sally Vaughn, Ph.D.
Deputy Superintendent and Chief Academic Officer
Linda Forward, Director Office of Education Improvement and Innovation
Welcome
Welcome to the Michigan K-12 Standards for Mathematics, adopted by the State Board of Education in 2010. With the reauthorization of the 2001 Elementary and Secondary Education Act (ESEA), commonly known as No Child Left Behind (NCLB), Michigan embarked on a standards revision process, starting with the K-8 mathematics and ELA standards that resulted in the Grade Level Content Expectations (GLCE). These were intended to lay the framework for the grade level testing in these subject areas required under NCLB. These were followed by GLCE for science and social studies, and by High School Content Expectations (HSCE) for all subject areas. Seven years later the revision cycle continued with Michigan working with other states to build on and refine current state standards that would allow states to work collaboratively to develop a repository of quality resources based on a common set of standards. These standards are the result of that collaboration.
Michigan’s K–12 academic standards serve to outline learning expectations for Michigan’s students and are intended to guide local curriculum development. Because these Mathematics standards are shared with other states, local districts have access to a broad set of resources they can call upon as they develop their local curricula and assessments. State standards also serve as a platform for state-level assessments, which are used to measure how well schools are providing opportunities for all students to learn the content required to be career– and college–ready.
Linda Forward, Director, Office of Education Improvement and Innovation
Vanessa Keesler, Deputy Superintendent, Division of Education Services
Mike Flanagan, Superintendent of Public Instruction
table of ContentsIntroduction 3
Standards for mathematical Practice 6
Standards for mathematical Content
Kindergarten 9Grade1 13Grade2 17Grade3 21Grade4 27Grade5 33Grade6 39Grade7 46Grade8 52HighSchool—Introduction
HighSchool—NumberandQuantity 58HighSchool—Algebra 62HighSchool—Functions 67HighSchool—Modeling 72HighSchool—Geometry 74HighSchool—StatisticsandProbability 79
Glossary 85Sample of Works Consulted 91
Int
ro
dU
Ct
Ion
| 3
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
Int
ro
dU
Ct
Ion
| 4
that to be coherent, a set of content standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures inherent in the discipline. These deeper structures then serve as a means for connecting the particulars (such as an understanding of the rational number system and its properties). (emphasis
added)
TheseStandardsendeavortofollowsuchadesign,notonlybystressingconceptual
understandingofkeyideas,butalsobycontinuallyreturningtoorganizing
principlessuchasplacevalueorthepropertiesofoperationstostructurethose
ideas.
Inaddition,the“sequenceoftopicsandperformances”thatisoutlinedinabodyof
mathematicsstandardsmustalsorespectwhatisknownabouthowstudentslearn.
AsConfrey(2007)pointsout,developing“sequencedobstaclesandchallenges
forstudents…absenttheinsightsaboutmeaningthatderivefromcarefulstudyof
learning,wouldbeunfortunateandunwise.”Inrecognitionofthis,thedevelopment
oftheseStandardsbeganwithresearch-basedlearningprogressionsdetailing
whatisknowntodayabouthowstudents’mathematicalknowledge,skill,and
understandingdevelopovertime.
Understanding mathematics
TheseStandardsdefinewhatstudentsshouldunderstandandbeabletodoin
theirstudyofmathematics.Askingastudenttounderstandsomethingmeans
askingateachertoassesswhetherthestudenthasunderstoodit.Butwhatdoes
mathematicalunderstandinglooklike?Onehallmarkofmathematicalunderstanding
istheabilitytojustify,inawayappropriatetothestudent’smathematicalmaturity,
whyaparticularmathematicalstatementistrueorwhereamathematicalrule
comesfrom.Thereisaworldofdifferencebetweenastudentwhocansummona
mnemonicdevicetoexpandaproductsuchas(a+ b)(x+y)andastudentwho
canexplainwherethemnemoniccomesfrom.Thestudentwhocanexplaintherule
understandsthemathematics,andmayhaveabetterchancetosucceedataless
familiartasksuchasexpanding(a+ b+c)(x+y).Mathematicalunderstandingand
proceduralskillareequallyimportant,andbothareassessableusingmathematical
tasksofsufficientrichness.
TheStandardssetgrade-specificstandardsbutdonotdefinetheintervention
methodsormaterialsnecessarytosupportstudentswhoarewellbeloworwell
abovegrade-levelexpectations.ItisalsobeyondthescopeoftheStandardsto
definethefullrangeofsupportsappropriateforEnglishlanguagelearnersand
forstudentswithspecialneeds.Atthesametime,allstudentsmusthavethe
opportunitytolearnandmeetthesamehighstandardsiftheyaretoaccessthe
knowledgeandskillsnecessaryintheirpost-schoollives.TheStandardsshould
bereadasallowingforthewidestpossiblerangeofstudentstoparticipatefully
fromtheoutset,alongwithappropriateaccommodationstoensuremaximum
participatonofstudentswithspecialeducationneeds.Forexample,forstudents
withdisabilitiesreadingshouldallowforuseofBraille,screenreadertechnology,or
otherassistivedevices,whilewritingshouldincludetheuseofascribe,computer,
orspeech-to-texttechnology.Inasimilarvein,speakingandlisteningshouldbe
interpretedbroadlytoincludesignlanguage.Nosetofgrade-specificstandards
canfullyreflectthegreatvarietyinabilities,needs,learningrates,andachievement
levelsofstudentsinanygivenclassroom.However,theStandardsdoprovideclear
signpostsalongthewaytothegoalofcollegeandcareerreadinessforallstudents.
TheStandardsbeginonpage6witheightStandardsforMathematicalPractice.
Int
ro
dU
Ct
Ion
| 5
How to read the grade level standards
Standards definewhatstudentsshouldunderstandandbeabletodo.
Clusters aregroupsofrelatedstandards.Notethatstandardsfromdifferentclusters
maysometimesbecloselyrelated,becausemathematics
isaconnectedsubject.
domainsarelargergroupsofrelatedstandards.Standardsfromdifferentdomains
maysometimesbecloselyrelated.
number and operations in Base ten 3.nBtUse place value understanding and properties of operations to perform multi-digit arithmetic.
1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.
2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
3. Multiplyone-digitwholenumbersbymultiplesof10intherange10-90(e.g.,9×80,5×60)usingstrategiesbasedonplacevalueandpropertiesofoperations.
TheseStandardsdonotdictatecurriculumorteachingmethods.Forexample,just
becausetopicAappearsbeforetopicBinthestandardsforagivengrade,itdoes
notnecessarilymeanthattopicAmustbetaughtbeforetopicB.Ateachermight
prefertoteachtopicBbeforetopicA,ormightchoosetohighlightconnectionsby
teachingtopicAandtopicBatthesametime.Or,ateachermightprefertoteacha
topicofhisorherownchoosingthatleads,asabyproduct,tostudentsreachingthe
standardsfortopicsAandB.
Whatstudentscanlearnatanyparticulargradeleveldependsuponwhatthey
havelearnedbefore.Ideallythen,eachstandardinthisdocumentmighthavebeen
phrasedintheform,“Studentswhoalreadyknow...shouldnextcometolearn....”
Butatpresentthisapproachisunrealistic—notleastbecauseexistingeducation
researchcannotspecifyallsuchlearningpathways.Ofnecessitytherefore,
gradeplacementsforspecifictopicshavebeenmadeonthebasisofstateand
internationalcomparisonsandthecollectiveexperienceandcollectiveprofessional
judgmentofeducators,researchersandmathematicians.Onepromiseofcommon
statestandardsisthatovertimetheywillallowresearchonlearningprogressions
toinformandimprovethedesignofstandardstoamuchgreaterextentthanis
possibletoday.Learningopportunitieswillcontinuetovaryacrossschoolsand
schoolsystems,andeducatorsshouldmakeeveryefforttomeettheneedsof
individualstudentsbasedontheircurrentunderstanding.
TheseStandardsarenotintendedtobenewnamesforoldwaysofdoingbusiness.
Theyareacalltotakethenextstep.Itistimeforstatestoworktogethertobuild
onlessonslearnedfromtwodecadesofstandardsbasedreforms.Itistimeto
recognizethatstandardsarenotjustpromisestoourchildren,butpromiseswe
intendtokeep.
domain
ClusterStandard
Sta
nd
ar
dS
fo
r m
at
He
ma
tIC
al
pr
aC
tIC
e | 6
mathematics | Standards for mathematical PracticeTheStandardsforMathematicalPracticedescribevarietiesofexpertisethat
mathematicseducatorsatalllevelsshouldseektodevelopintheirstudents.
Thesepracticesrestonimportant“processesandproficiencies”withlongstanding
importanceinmathematicseducation.ThefirstofthesearetheNCTMprocess
standardsofproblemsolving,reasoningandproof,communication,representation,
andconnections.Thesecondarethestrandsofmathematicalproficiencyspecified
intheNationalResearchCouncil’sreportAdding It Up:adaptivereasoning,strategic
competence,conceptualunderstanding(comprehensionofmathematicalconcepts,
operationsandrelations),proceduralfluency(skillincarryingoutprocedures
flexibly,accurately,efficientlyandappropriately),andproductivedisposition
(habitualinclinationtoseemathematicsassensible,useful,andworthwhile,coupled
withabeliefindiligenceandone’sownefficacy).
1 Make sense of problems and persevere in solving them.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaning
ofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,
constraints,relationships,andgoals.Theymakeconjecturesabouttheformand
meaningofthesolutionandplanasolutionpathwayratherthansimplyjumpinginto
asolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesand
simplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They
monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudents
might,dependingonthecontextoftheproblem,transformalgebraicexpressionsor
changetheviewingwindowontheirgraphingcalculatortogettheinformationthey
need.Mathematicallyproficientstudentscanexplaincorrespondencesbetween
equations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportant
featuresandrelationships,graphdata,andsearchforregularityortrends.Younger
studentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualize
andsolveaproblem.Mathematicallyproficientstudentschecktheiranswersto
problemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthis
makesense?”Theycanunderstandtheapproachesofotherstosolvingcomplex
problemsandidentifycorrespondencesbetweendifferentapproaches.
2 Reason abstractly and quantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationships
inproblemsituations.Theybringtwocomplementaryabilitiestobearonproblems
involvingquantitativerelationships:theabilitytodecontextualize—toabstract
agivensituationandrepresentitsymbolicallyandmanipulatetherepresenting
symbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingto
theirreferents—andtheabilitytocontextualize,topauseasneededduringthe
manipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.
Quantitativereasoningentailshabitsofcreatingacoherentrepresentationof
theproblemathand;consideringtheunitsinvolved;attendingtothemeaningof
quantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferent
propertiesofoperationsandobjects.
3 Construct viable arguments and critique the reasoning of others.Mathematicallyproficientstudentsunderstandandusestatedassumptions,
definitions,andpreviouslyestablishedresultsinconstructingarguments.They
makeconjecturesandbuildalogicalprogressionofstatementstoexplorethe
truthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingtheminto
cases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,
Sta
nd
ar
dS
fo
r m
at
He
ma
tIC
al
pr
aC
tIC
e | 7
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.
Sta
nd
ar
dS
fo
r m
at
He
ma
tIC
al
pr
aC
tIC
e | 8
7 Look for and make use of structure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.
Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesame
amountassevenandthreemore,ortheymaysortacollectionofshapesaccording
tohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthe
wellremembered7×5+7×3,inpreparationforlearningaboutthedistributive
property.Intheexpressionx2+9x+14,olderstudentscanseethe14as2×7and
the9as2+7.Theyrecognizethesignificanceofanexistinglineinageometric
figureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.
Theyalsocanstepbackforanoverviewandshiftperspective.Theycansee
complicatedthings,suchassomealgebraicexpressions,assingleobjectsoras
beingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5
minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannot
bemorethan5foranyrealnumbersxandy.
8 Look for and express regularity in repeated reasoning.Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlook
bothforgeneralmethodsandforshortcuts.Upperelementarystudentsmight
noticewhendividing25by11thattheyarerepeatingthesamecalculationsover
andoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattention
tothecalculationofslopeastheyrepeatedlycheckwhetherpointsareontheline
through(1,2)withslope3,middleschoolstudentsmightabstracttheequation
(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding
(x–1)(x+1),(x–1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthemtothe
generalformulaforthesumofageometricseries.Astheyworktosolveaproblem,
mathematicallyproficientstudentsmaintainoversightoftheprocess,while
attendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheir
intermediateresults.
Connecting the Standards for Mathematical Practice to the Standards for Mathematical ContentTheStandardsforMathematicalPracticedescribewaysinwhichdevelopingstudent
practitionersofthedisciplineofmathematicsincreasinglyoughttoengagewith
thesubjectmatterastheygrowinmathematicalmaturityandexpertisethroughout
theelementary,middleandhighschoolyears.Designersofcurricula,assessments,
andprofessionaldevelopmentshouldallattendtotheneedtoconnectthe
mathematicalpracticestomathematicalcontentinmathematicsinstruction.
TheStandardsforMathematicalContentareabalancedcombinationofprocedure
andunderstanding.Expectationsthatbeginwiththeword“understand”areoften
especiallygoodopportunitiestoconnectthepracticestothecontent.Students
wholackunderstandingofatopicmayrelyonprocedurestooheavily.Without
aflexiblebasefromwhichtowork,theymaybelesslikelytoconsideranalogous
problems,representproblemscoherently,justifyconclusions,applythemathematics
topracticalsituations,usetechnologymindfullytoworkwiththemathematics,
explainthemathematicsaccuratelytootherstudents,stepbackforanoverview,or
deviatefromaknownproceduretofindashortcut.Inshort,alackofunderstanding
effectivelypreventsastudentfromengaginginthemathematicalpractices.
Inthisrespect,thosecontentstandardswhichsetanexpectationofunderstanding
arepotential“pointsofintersection”betweentheStandardsforMathematical
ContentandtheStandardsforMathematicalPractice.Thesepointsofintersection
areintendedtobeweightedtowardcentralandgenerativeconceptsinthe
schoolmathematicscurriculumthatmostmeritthetime,resources,innovative
energies,andfocusnecessarytoqualitativelyimprovethecurriculum,instruction,
assessment,professionaldevelopment,andstudentachievementinmathematics.
KIn
de
rG
ar
te
n | 9
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.
KIn
de
rG
ar
te
n | 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 andadding to, and understand subtraction astaking apart and taking from.
number and operations in Base ten
• Work with numbers 11–19 to gain foundationsfor place value.
measurement and data
• describe and compare measurable attributes.
• Classify objects and count the number ofobjects in categories.
Geometry
• Identify and describe shapes.
• analyze, compare, create, and composeshapes.
mathematical Practices
1. Makesenseofproblemsandperseverein
solvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
Grade K overview
KIn
de
rG
ar
te
n | 11
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.)
KIn
de
rG
ar
te
n | 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 twochildren and describe one child as taller/shorter.
Classify objects and count the number of objects in each category.
3. Classifyobjectsintogivencategories;countthenumbersofobjectsineachcategoryandsortthecategoriesbycount.3
Geometry K.G
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
1. Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,in front of,behind,andnext to.
2. Correctlynameshapesregardlessoftheirorientationsoroverallsize.
3. Identifyshapesastwo-dimensional(lyinginaplane,“flat”)orthree-dimensional(“solid”).
Analyze, compare, create, and compose shapes.
4. Analyzeandcomparetwo-andthree-dimensionalshapes,indifferentsizesandorientations,usinginformallanguagetodescribetheirsimilarities,differences,parts(e.g.,numberofsidesandvertices/“corners”)andotherattributes(e.g.,havingsidesofequallength).
5. Modelshapesintheworldbybuildingshapesfromcomponents(e.g.,sticksandclayballs)anddrawingshapes.
6. Composesimpleshapestoformlargershapes.For example, “Can youjoin these two triangles with full sides touching to make a rectangle?”
3Limitcategorycountstobelessthanorequalto10.
Gr
ad
e 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,
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.
Gr
ad
e 1 | 14
Grade 1 overviewoperations and algebraic thinking
• represent and solve problems involvingaddition and subtraction.
• Understand and apply properties of operationsand the relationship between addition andsubtraction.
• add and subtract within 20.
• Work with addition and subtraction equations.
number and operations in Base ten
• extend the counting sequence.
• Understand place value.
• Use place value understanding and propertiesof operations to add and subtract.
measurement and data
• measure lengths indirectly and by iteratinglength 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.
Gr
ad
e 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 ofaddition.) To add 2 + 6 + 4, the second two numbers can be added to makea ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
4. Understandsubtractionasanunknown-addendproblem.For example,subtract 10 – 8 by finding the number that makes 10 when added to 8.
Add and subtract within 20.
5. Relatecountingtoadditionandsubtraction(e.g.,bycountingon2to
add2).
6. Addandsubtractwithin20,demonstratingfluencyforadditionandsubtractionwithin10.Usestrategiessuchascountingon;makingten(e.g.,8+6=8+2+4=10+4=14);decomposinganumberleadingtoaten(e.g.,13–4=13–3–1=10–1=9);usingtherelationshipbetweenadditionandsubtraction(e.g.,knowingthat8+4=12,oneknows12–8=4);andcreatingequivalentbuteasierorknownsums(e.g.,adding6+7bycreatingtheknownequivalent6+6+1=12+1=13).
Work with addition and subtraction equations.
7. Understandthemeaningoftheequalsign,anddetermineifequationsinvolvingadditionandsubtractionaretrueorfalse.For example, whichof the following equations are true and which are false? 6 = 6, 7 = 8 – 1,5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
8. Determinetheunknownwholenumberinanadditionorsubtractionequationrelatingthreewholenumbers.For example, determine theunknown number that makes the equation true in each of the equations 8 +? = 11, 5 = � – 3, 6 + 6 = �.
number and operations in Base ten 1.nBt
Extend the counting sequence.
1. Countto120,startingatanynumberlessthan120.Inthisrange,readandwritenumeralsandrepresentanumberofobjectswithawrittennumeral.
Understand place value.
2. Understandthatthetwodigitsofatwo-digitnumberrepresentamountsoftensandones.Understandthefollowingasspecialcases:
a. 10canbethoughtofasabundleoftenones—calleda“ten.”
b. Thenumbersfrom11to19arecomposedofatenandone,two,three,four,five,six,seven,eight,ornineones.
c. Thenumbers10,20,30,40,50,60,70,80,90refertoone,two,three,four,five,six,seven,eight,orninetens(and0ones).
2SeeGlossary,Table1.3Studentsneednotuseformaltermsfortheseproperties.
Gr
ad
e 1 | 16
3. Comparetwotwo-digitnumbersbasedonmeaningsofthetensandonesdigits,recordingtheresultsofcomparisonswiththesymbols>,=,and<.
Use place value understanding and properties of operations to add and subtract.
4. Addwithin100,includingaddingatwo-digitnumberandaone-digitnumber,andaddingatwo-digitnumberandamultipleof10,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.Understandthatinaddingtwo-digitnumbers,oneaddstensandtens,onesandones;andsometimesitisnecessarytocomposeaten.
5. Givenatwo-digitnumber,mentallyfind10moreor10lessthanthenumber,withouthavingtocount;explainthereasoningused.
6. Subtractmultiplesof10intherange10-90frommultiplesof10intherange10-90(positiveorzerodifferences),usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethe
strategytoawrittenmethodandexplainthereasoningused.
measurement and data 1.md
Measure lengths indirectly and by iterating length units.
1. Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.
2. Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limit tocontexts where the object being measured is spanned by a whole number oflength units with no gaps or overlaps.
Tell and write time.
3. Tellandwritetimeinhoursandhalf-hoursusinganaloganddigitalclocks.
Represent and interpret data.
4. Organize,represent,andinterpretdatawithuptothreecategories;askandanswerquestionsaboutthetotalnumberofdatapoints,howmanyineachcategory,andhowmanymoreorlessareinonecategorythaninanother.
Geometry 1.G
Reason with shapes and their attributes.
1. Distinguishbetweendefiningattributes(e.g.,trianglesareclosedandthree-sided)versusnon-definingattributes(e.g.,color,orientation,overallsize);buildanddrawshapestopossessdefiningattributes.
2. Composetwo-dimensionalshapes(rectangles,squares,trapezoids,triangles,half-circles,andquarter-circles)orthree-dimensionalshapes(cubes,rightrectangularprisms,rightcircularcones,andrightcircularcylinders)tocreateacompositeshape,andcomposenewshapesfromthecompositeshape.4
3. Partitioncirclesandrectanglesintotwoandfourequalshares,describethesharesusingthewordshalves,fourths,andquarters,andusethephraseshalf of,fourth of,andquarter of.Describethewholeastwoof,orfouroftheshares.Understandfortheseexamplesthatdecomposingintomoreequalsharescreatessmallershares.
4Studentsdonotneedtolearnformalnamessuchas“rightrectangularprism.”
Gr
ad
e 2 | 17
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.
Gr
ad
e 2 | 18
operations and algebraic thinking
• represent and solve problems involvingaddition and subtraction.
• add and subtract within 20.
• Work with equal groups of objects to gainfoundations for multiplication.
number and operations in Base ten
• Understand place value.
• Use place value understanding andproperties of operations to add and subtract.
measurement and data
• measure and estimate lengths in standardunits.
• 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
Gr
ad
e 2 | 19
operations and algebraic thinking 2.oa
Represent and solve problems involving addition and subtraction.
1. Useadditionandsubtractionwithin100tosolveone-andtwo-stepwordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1
Add and subtract within 20.
2. Fluentlyaddandsubtractwithin20usingmentalstrategies.2ByendofGrade2,knowfrommemoryallsumsoftwoone-digitnumbers.
Work with equal groups of objects to gain foundations for multiplication.
3. Determinewhetheragroupofobjects(upto20)hasanoddorevennumberofmembers,e.g.,bypairingobjectsorcountingthemby2s;writeanequationtoexpressanevennumberasasumoftwoequaladdends.
4. Useadditiontofindthetotalnumberofobjectsarrangedinrectangulararrayswithupto5rowsandupto5columns;writeanequationtoexpressthetotalasasumofequaladdends.
number and operations in Base ten 2.nBt
Understand place value.
1. Understandthatthethreedigitsofathree-digitnumberrepresentamountsofhundreds,tens,andones;e.g.,706equals7hundreds,0tens,and6ones.Understandthefollowingasspecialcases:
a. 100canbethoughtofasabundleoftentens—calleda“hundred.”
b. Thenumbers100,200,300,400,500,600,700,800,900refertoone,two,three,four,five,six,seven,eight,orninehundreds(and0tensand0ones).
2. Countwithin1000;skip-countby5s,10s,and100s.
3. Readandwritenumbersto1000usingbase-tennumerals,numbernames,andexpandedform.
4. Comparetwothree-digitnumbersbasedonmeaningsofthehundreds,tens,andonesdigits,using>,=,and<symbolstorecordtheresultsofcomparisons.
Use place value understanding and properties of operations to add and subtract.
5. Fluentlyaddandsubtractwithin100usingstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
6. Adduptofourtwo-digitnumbersusingstrategiesbasedonplacevalueandpropertiesofoperations.
7. Addandsubtractwithin1000,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethod.Understandthatinaddingorsubtractingthree-digitnumbers,oneaddsorsubtractshundredsandhundreds,tensandtens,onesandones;andsometimesitisnecessarytocomposeordecomposetensorhundreds.
8. Mentallyadd10or100toagivennumber100–900,andmentallysubtract10or100fromagivennumber100–900.
9. Explainwhyadditionandsubtractionstrategieswork,usingplacevalueandthepropertiesofoperations.3
1SeeGlossary,Table1.2Seestandard1.OA.6foralistofmentalstrategies.3Explanationsmaybesupportedbydrawingsorobjects.
Gr
ad
e 2 | 2
0
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 2dimes 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.
Gr
ad
e 3
| 21
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.
Gr
ad
e 3
| 22
operations and algebraic thinking
• represent and solve problems involvingmultiplication and division.
• Understand properties of multiplication andthe relationship between multiplication anddivision.
• multiply and divide within 100.
• Solve problems involving the four operations,and identify and explain patterns in arithmetic.
number and operations in Base ten
• Use place value understanding and propertiesof operations to perform multi-digit arithmetic.
number and operations—fractions
• develop understanding of fractions as numbers.
measurement and data
• Solve problems involving measurement andestimation of intervals of time, liquid volumes,and masses of objects.
• represent and interpret data.
• Geometric measurement: understand conceptsof area and relate area to multiplication and toaddition.
• Geometric measurement: recognize perimeteras an attribute of plane figures and distinguishbetween linear and area measures.
Geometry
• 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.
Gr
ad
e 3
| 23
operations and algebraic thinking 3.oa
Represent and solve problems involving multiplication and division.
1. Interpretproductsofwholenumbers,e.g.,interpret5×7asthetotalnumberofobjectsin5groupsof7objectseach.For example, describea context in which a total number of objects can be expressed as 5 × 7.
2. Interpretwhole-numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.Forexample, describe a context in which a number of shares or a number ofgroups can be expressed as 56 ÷ 8.
3. Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1
4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.For example, determine theunknown number that makes the equation true in each of the equations 8× ? = 48, 5 = � ÷ 3, 6 × 6 = ?.
Understand properties of multiplication and the relationship between multiplication and division.
5. Applypropertiesofoperationsasstrategiestomultiplyanddivide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.(Commutative property of multiplication.) 3 × 5 × 2 can be found by 3× 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associativeproperty of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, onecan find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributiveproperty.)
6. Understanddivisionasanunknown-factorproblem.For example, find32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Multiply and divide within 100.
7. Fluentlymultiplyanddividewithin100,usingstrategiessuchastherelationshipbetweenmultiplicationanddivision(e.g.,knowingthat8×5=40,oneknows40÷5=8)orpropertiesofoperations.BytheendofGrade3,knowfrommemoryallproductsoftwoone-digitnumbers.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
8. Solvetwo-stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.3
9. Identifyarithmeticpatterns(includingpatternsintheadditiontableormultiplicationtable),andexplainthemusingpropertiesofoperations.For example, observe that 4 times a number is always even, and explainwhy 4 times a number can be decomposed into two equal addends.
1SeeGlossary,Table2.2Studentsneednotuseformaltermsfortheseproperties.3Thisstandardislimitedtoproblemsposedwithwholenumbersandhavingwhole-numberanswers;studentsshouldknowhowtoperformoperationsintheconven-tionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).
gr
ad
e 3
| 24
Number and Operations in Base Ten 3.NBT
Use place value understanding and properties of operations to perform multi-digit arithmetic.4
1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.
2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
3. Multiplyone-digitwholenumbersbymultiplesof10intherange10–90(e.g.,9×80,5×60)usingstrategiesbasedonplacevalueandpropertiesofoperations.
Number and Operations—Fractions5 3.NF
Develop understanding of fractions as numbers.1. Understandafraction1/basthequantityformedby1partwhena
wholeispartitionedintob equalparts;understandafractiona/basthequantityformedbyapartsofsize1/b.
2. Understandafractionasanumberonthenumberline;representfractionsonanumberlinediagram.
a. Representafraction1/bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1/bandthattheendpointofthepartbasedat0locatesthenumber1/bonthenumberline.
b. Representafractiona/bonanumberlinediagrambymarkingoffalengths1/bfrom0.Recognizethattheresultingintervalhassizea/bandthatitsendpointlocatesthenumbera/bonthenumberline.
3. Explainequivalenceoffractionsinspecialcases,andcomparefractionsbyreasoningabouttheirsize.
a. Understandtwofractionsasequivalent(equal)iftheyarethesamesize,orthesamepointonanumberline.
b. Recognizeandgeneratesimpleequivalentfractions,e.g.,1/2=2/4,4/6=2/3.Explainwhythefractionsareequivalent,e.g.,byusingavisualfractionmodel.
c. Expresswholenumbersasfractions,andrecognizefractionsthatareequivalenttowholenumbers.Examples: Express 3 in the form3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same pointof a number line diagram.
d. Comparetwofractionswiththesamenumeratororthesamedenominatorbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.
Measurement and Data 3.MD
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
1. Tellandwritetimetothenearestminuteandmeasuretimeintervalsinminutes.Solvewordproblemsinvolvingadditionandsubtractionoftimeintervalsinminutes,e.g.,byrepresentingtheproblemonanumberlinediagram.
4Arangeofalgorithmsmaybeused.5Grade3expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,6,and8.
Gr
ad
e 3
| 25
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 inthe bar graph might represent 5 pets.
4. Generatemeasurementdatabymeasuringlengthsusingrulersmarkedwithhalvesandfourthsofaninch.Showthedatabymakingalineplot,wherethehorizontalscaleismarkedoffinappropriateunits—wholenumbers,halves,orquarters.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
5. Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement.
a. Asquarewithsidelength1unit,called“aunitsquare,”issaidtohave“onesquareunit”ofarea,andcanbeusedtomeasurearea.
b. Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.
6. Measureareasbycountingunitsquares(squarecm,squarem,squarein,squareft,andimprovisedunits).
7. Relateareatotheoperationsofmultiplicationandaddition.
a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.
b. Multiplysidelengthstofindareasofrectangleswithwhole-numbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.
c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.
d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposingthemintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
8. Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.
6Excludescompoundunitssuchascm3andfindingthegeometricvolumeofacontainer.7Excludesmultiplicativecomparisonproblems(problemsinvolvingnotionsof“timesasmuch”;seeGlossary,Table2).
Gr
ad
e 3
| 26
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 4parts with equal area, and describe the area of each part as 1/4 of the areaof the shape.
Gr
ad
e 4
| 27
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.
Gr
ad
e 4
| 28
Grade 4 overviewoperations and algebraic thinking
• Use the four operations with whole numbers tosolve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.
number and operations in Base ten
• Generalize place value understanding for multi-digit whole numbers.
• Use place value understanding and properties ofoperations to perform multi-digit arithmetic.
number and operations—fractions
• extend understanding of fraction equivalenceand ordering.
• Build fractions from unit fractions by applyingand extending previous understandings ofoperations on whole numbers.
• Understand decimal notation for fractions, andcompare decimal fractions.
measurement and data
• Solve problems involving measurement andconversion of measurements from a larger unit toa smaller unit.
• represent and interpret data.
• Geometric measurement: understand concepts ofangle and measure angles.
Geometry
• draw and identify lines and angles, and classifyshapes 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.
Gr
ad
e 4
| 29
operations and algebraic thinking 4.oa
Use the four operations with whole numbers to solve problems.
1. Interpretamultiplicationequationasacomparison,e.g.,interpret35=5×7asastatementthat35is5timesasmanyas7and7timesasmanyas5.Representverbalstatementsofmultiplicativecomparisonsasmultiplicationequations.
2. Multiplyordividetosolvewordproblemsinvolvingmultiplicativecomparison,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem,distinguishingmultiplicativecomparisonfromadditivecomparison.1
3. Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole-numberanswersusingthefouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.
Gain familiarity with factors and multiples.
4. Findallfactorpairsforawholenumberintherange1–100.Recognizethatawholenumberisamultipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1–100isamultipleofagivenone-digitnumber.Determinewhetheragivenwholenumberintherange1–100isprimeorcomposite.
Generate and analyze patterns.
5. Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.For example, given the rule “Add 3” and the starting number 1, generateterms in the resulting sequence and observe that the terms appear toalternate between odd and even numbers. Explain informally why thenumbers will continue to alternate in this way.
number and operations in Base ten2 4.nBt
Generalize place value understanding for multi-digit whole numbers.
1. Recognizethatinamulti-digitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.Forexample, recognize that 700 ÷ 70 = 10 by applying concepts of place valueand division.
2. Readandwritemulti-digitwholenumbersusingbase-tennumerals,numbernames,andexpandedform.Comparetwomulti-digitnumbersbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.
3. Useplacevalueunderstandingtoroundmulti-digitwholenumberstoanyplace.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
4. Fluentlyaddandsubtractmulti-digitwholenumbersusingthestandardalgorithm.
5. Multiplyawholenumberofuptofourdigitsbyaone-digitwholenumber,andmultiplytwotwo-digitnumbers,usingstrategiesbasedonplacevalueandthepropertiesofoperations.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.
1SeeGlossary,Table2.2Grade4expectationsinthisdomainarelimitedtowholenumberslessthanorequalto1,000,000.
Gr
ad
e 4
| 30
6. Findwhole-numberquotientsandremainderswithuptofour-digitdividendsandone-digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.
number and operations—fractions3 4.nf
Extend understanding of fraction equivalence and ordering.
1. Explainwhyafractiona/bisequivalenttoafraction(n×a)/(n×b)byusingvisualfractionmodels,withattentiontohowthenumberandsizeofthepartsdiffereventhoughthetwofractionsthemselvesarethesamesize.Usethisprincipletorecognizeandgenerateequivalentfractions.
2. Comparetwofractionswithdifferentnumeratorsanddifferentdenominators,e.g.,bycreatingcommondenominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas1/2.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswithsymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understandafractiona/bwitha>1asasumoffractions1/b.
a. Understandadditionandsubtractionoffractionsasjoiningandseparatingpartsreferringtothesamewhole.
b. Decomposeafractionintoasumoffractionswiththesamedenominatorinmorethanoneway,recordingeachdecompositionbyanequation.Justifydecompositions,e.g.,byusingavisualfractionmodel.Examples: 3/8 = 1/8 + 1/8 + 1/8 ;3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Addandsubtractmixednumberswithlikedenominators,e.g.,byreplacingeachmixednumberwithanequivalentfraction,and/orbyusingpropertiesofoperationsandtherelationshipbetweenadditionandsubtraction.
d. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.
4. Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionbyawholenumber.
a. Understandafractiona/basamultipleof1/b.For example, usea visual fraction model to represent 5/4 as the product 5 × (1/4),recording the conclusion by the equation 5/4 = 5 × (1/4).
b. Understandamultipleofa/basamultipleof1/b,andusethisunderstandingtomultiplyafractionbyawholenumber.Forexample, use a visual fraction model to express 3 × (2/5) as 6 × (1/5),recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solvewordproblemsinvolvingmultiplicationofafractionbyawholenumber,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, if each person at a party willeat 3/8 of a pound of roast beef, and there will be 5 people at theparty, how many pounds of roast beef will be needed? Between whattwo whole numbers does your answer lie?
3Grade4expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,5,6,8,10,12,and100.
Gr
ad
e 4
| 31
Understand decimal notation for fractions, and compare decimal fractions.
5. Expressafractionwithdenominator10asanequivalentfractionwithdenominator100,andusethistechniquetoaddtwofractionswithrespectivedenominators10and100.4For example, express 3/10 as30/100, and add 3/10 + 4/100 = 34/100.
6. Usedecimalnotationforfractionswithdenominators10or100.Forexample, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate0.62 on a number line diagram.
7. Comparetwodecimalstohundredthsbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwodecimalsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualmodel.
measurement and data 4.md
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
1. Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz.;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.Recordmeasurementequivalentsinatwo-columntable.For example, know that 1 ft is 12 times as long as 1 in.Express the length of a 4 ft snake as 48 in. Generate a conversion table forfeet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
2. Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.Representmeasurementquantitiesusingdiagramssuchasnumberlinediagramsthatfeatureameasurementscale.
3. Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.For example, find the width of a rectangularroom given the area of the flooring and the length, by viewing the areaformula as a multiplication equation with an unknown factor.
Represent and interpret data.
4. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.For example,from a line plot find and interpret the difference in length between thelongest and shortest specimens in an insect collection.
Geometric measurement: understand concepts of angle and measure angles.
5. Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:
a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.
b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.
4Studentswhocangenerateequivalentfractionscandevelopstrategiesforaddingfractionswithunlikedenominatorsingeneral.Butadditionandsubtractionwithun-likedenominatorsingeneralisnotarequirementatthisgrade.
Gr
ad
e 4
| 32
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.
Gr
ad
e 5 | 3
3
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.
Gr
ad
e 5 | 3
4
operations and algebraic thinking
• Write and interpret numerical expressions.
• analyze patterns and relationships.
number and operations in Base ten
• Understand the place value system.
• Perform operations with multi-digit wholenumbers and with decimals to hundredths.
number and operations—fractions
• Use equivalent fractions as a strategy to addand subtract fractions.
• apply and extend previous understandingsof multiplication and division to multiply anddivide fractions.
measurement and data
• Convert like measurement units within a givenmeasurement system.
• represent and interpret data.
• Geometric measurement: understand conceptsof volume and relate volume to multiplicationand to addition.
Geometry
• Graph points on the coordinate plane to solvereal-world and mathematical problems.
• Classify two-dimensional figures into categoriesbased on their properties.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 5 overview
Gr
ad
e 5 | 3
5
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 therule “Add 3” and the starting number 0, and given the rule “Add 6” and thestarting number 0, generate terms in the resulting sequences, and observethat the terms in one sequence are twice the corresponding terms in theother sequence. Explain informally why this is so.
number and operations in Base ten 5.nBt
Understand the place value system.
1. Recognizethatinamulti-digitnumber,adigitinoneplacerepresents10timesasmuchasitrepresentsintheplacetoitsrightand1/10ofwhatitrepresentsintheplacetoitsleft.
2. Explainpatternsinthenumberofzerosoftheproductwhenmultiplyinganumberbypowersof10,andexplainpatternsintheplacementofthedecimalpointwhenadecimalismultipliedordividedbyapowerof10.Usewhole-numberexponentstodenotepowersof10.
3. Read,write,andcomparedecimalstothousandths.
a. Readandwritedecimalstothousandthsusingbase-tennumerals,numbernames,andexpandedform,e.g.,347.392=3×100+4×10+7×1+3×(1/10)+9×(1/100)+2×(1/1000).
b. Comparetwodecimalstothousandthsbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.
4. Useplacevalueunderstandingtorounddecimalstoanyplace.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluentlymultiplymulti-digitwholenumbersusingthestandardalgorithm.
6. Findwhole-numberquotientsofwholenumberswithuptofour-digitdividendsandtwo-digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.
7. Add,subtract,multiply,anddividedecimalstohundredths,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.
Gr
ad
e 5 | 3
6
number and operations—fractions 5.nf
Use equivalent fractions as a strategy to add and subtract fractions.
1. Addandsubtractfractionswithunlikedenominators(includingmixednumbers)byreplacinggivenfractionswithequivalentfractionsinsuchawayastoproduceanequivalentsumordifferenceoffractionswithlikedenominators.For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (Ingeneral, a/b + c/d = (ad + bc)/bd.)
2. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewhole,includingcasesofunlikedenominators,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.Usebenchmarkfractionsandnumbersenseoffractionstoestimatementallyandassessthereasonablenessofanswers.Forexample, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that3/7 < 1/2.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3. Interpretafractionasdivisionofthenumeratorbythedenominator(a/b=a÷b).Solvewordproblemsinvolvingdivisionofwholenumbersleadingtoanswersintheformoffractionsormixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
4. Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionorwholenumberbyafraction.
a. Interprettheproduct(a/b)×qasapartsofapartitionofqintobequalparts;equivalently,astheresultofasequenceofoperationsa×q÷b.For example, use a visual fraction model toshow (2/3) × 4 = 8/3, and create a story context for this equation. Dothe same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Findtheareaofarectanglewithfractionalsidelengthsbytilingitwithunitsquaresoftheappropriateunitfractionsidelengths,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.Multiplyfractionalsidelengthstofindareasofrectangles,andrepresentfractionproductsasrectangularareas.
5. Interpretmultiplicationasscaling(resizing),by:
a. Comparingthesizeofaproducttothesizeofonefactoronthebasisofthesizeoftheotherfactor,withoutperformingtheindicatedmultiplication.
b. Explainingwhymultiplyingagivennumberbyafractiongreaterthan1resultsinaproductgreaterthanthegivennumber(recognizingmultiplicationbywholenumbersgreaterthan1asafamiliarcase);explainingwhymultiplyingagivennumberbyafractionlessthan1resultsinaproductsmallerthanthegivennumber;andrelatingtheprincipleoffractionequivalencea/b=(n×a)/(n×b)totheeffectofmultiplyinga/bby1.
6. Solverealworldproblemsinvolvingmultiplicationoffractionsandmixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.
7. Applyandextendpreviousunderstandingsofdivisiontodivideunitfractionsbywholenumbersandwholenumbersbyunitfractions.1
a. Interpretdivisionofaunitfractionbyanon-zerowholenumber,
1Studentsabletomultiplyfractionsingeneralcandevelopstrategiestodividefrac-tionsingeneral,byreasoningabouttherelationshipbetweenmultiplicationanddivision.Butdivisionofafractionbyafractionisnotarequirementatthisgrade.
Gr
ad
e 5 | 3
7
andcomputesuchquotients.For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient.Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpretdivisionofawholenumberbyaunitfraction,andcomputesuchquotients.For example, create a story context for4 ÷ (1/5), and use a visual fraction model to show the quotient. Usethe relationship between multiplication and division to explain that4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solverealworldproblemsinvolvingdivisionofunitfractionsbynon-zerowholenumbersanddivisionofwholenumbersbyunitfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, how much chocolate will eachperson get if 3 people share 1/2 lb of chocolate equally? How many1/3-cup servings are in 2 cups of raisins?
measurement and data 5.md
Convert like measurement units within a given measurement system.
1. Convertamongdifferent-sizedstandardmeasurementunitswithinagivenmeasurementsystem(e.g.,convert5cmto0.05m),andusetheseconversionsinsolvingmulti-step,realworldproblems.
Represent and interpret data.
2. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.For example,given different measurements of liquid in identical beakers, find theamount of liquid each beaker would contain if the total amount in all thebeakers were redistributed equally.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
3. Recognizevolumeasanattributeofsolidfiguresandunderstandconceptsofvolumemeasurement.
a. Acubewithsidelength1unit,calleda“unitcube,”issaidtohave“onecubicunit”ofvolume,andcanbeusedtomeasurevolume.
b. Asolidfigurewhichcanbepackedwithoutgapsoroverlapsusingnunitcubesissaidtohaveavolumeofncubicunits.
4. Measurevolumesbycountingunitcubes,usingcubiccm,cubicin,cubicft,andimprovisedunits.
5. Relatevolumetotheoperationsofmultiplicationandadditionandsolverealworldandmathematicalproblemsinvolvingvolume.
a. Findthevolumeofarightrectangularprismwithwhole-numbersidelengthsbypackingitwithunitcubes,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengths,equivalentlybymultiplyingtheheightbytheareaofthebase.Representthreefoldwhole-numberproductsasvolumes,e.g.,torepresenttheassociativepropertyofmultiplication.
b. ApplytheformulasV=l×w×handV=b×hforrectangularprismstofindvolumesofrightrectangularprismswithwhole-numberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.
c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwonon-overlappingrightrectangularprismsbyaddingthevolumesofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
Gr
ad
e 5 | 3
8
Geometry 5.G
Graph points on the coordinate plane to solve real-world and mathematical problems.
1. Useapairofperpendicularnumberlines,calledaxes,todefineacoordinatesystem,withtheintersectionofthelines(theorigin)arrangedtocoincidewiththe0oneachlineandagivenpointintheplanelocatedbyusinganorderedpairofnumbers,calleditscoordinates.Understandthatthefirstnumberindicateshowfartotravelfromtheorigininthedirectionofoneaxis,andthesecondnumberindicateshowfartotravelinthedirectionofthesecondaxis,withtheconventionthatthenamesofthetwoaxesandthecoordinatescorrespond(e.g.,x-axisandx-coordinate,y-axisandy-coordinate).
2. Representrealworldandmathematicalproblemsbygraphingpointsinthefirstquadrantofthecoordinateplane,andinterpretcoordinatevaluesofpointsinthecontextofthesituation.
Classify two-dimensional figures into categories based on their properties.
3. Understandthatattributesbelongingtoacategoryoftwo-dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.For example, all rectangles have four right angles and squares arerectangles, so all squares have four right angles.
4. Classifytwo-dimensionalfiguresinahierarchybasedonproperties.
Gr
ad
e 6
| 39
mathematics | Grade 6InGrade6,instructionaltimeshouldfocusonfourcriticalareas:(1)
connectingratioandratetowholenumbermultiplicationanddivision
andusingconceptsofratioandratetosolveproblems;(2)completing
understandingofdivisionoffractionsandextendingthenotionofnumber
tothesystemofrationalnumbers,whichincludesnegativenumbers;
(3)writing,interpreting,andusingexpressionsandequations;and(4)
developingunderstandingofstatisticalthinking.
(1)Studentsusereasoningaboutmultiplicationanddivisiontosolve
ratioandrateproblemsaboutquantities.Byviewingequivalentratios
andratesasderivingfrom,andextending,pairsofrows(orcolumns)in
themultiplicationtable,andbyanalyzingsimpledrawingsthatindicate
therelativesizeofquantities,studentsconnecttheirunderstandingof
multiplicationanddivisionwithratiosandrates.Thusstudentsexpandthe
scopeofproblemsforwhichtheycanusemultiplicationanddivisionto
solveproblems,andtheyconnectratiosandfractions.Studentssolvea
widevarietyofproblemsinvolvingratiosandrates.
(2)Studentsusethemeaningoffractions,themeaningsofmultiplication
anddivision,andtherelationshipbetweenmultiplicationanddivisionto
understandandexplainwhytheproceduresfordividingfractionsmake
sense.Studentsusetheseoperationstosolveproblems.Studentsextend
theirpreviousunderstandingsofnumberandtheorderingofnumbers
tothefullsystemofrationalnumbers,whichincludesnegativerational
numbers,andinparticularnegativeintegers.Theyreasonabouttheorder
andabsolutevalueofrationalnumbersandaboutthelocationofpointsin
allfourquadrantsofthecoordinateplane.
(3)Studentsunderstandtheuseofvariablesinmathematicalexpressions.
Theywriteexpressionsandequationsthatcorrespondtogivensituations,
evaluateexpressions,anduseexpressionsandformulastosolveproblems.
Studentsunderstandthatexpressionsindifferentformscanbeequivalent,
andtheyusethepropertiesofoperationstorewriteexpressionsin
equivalentforms.Studentsknowthatthesolutionsofanequationarethe
valuesofthevariablesthatmaketheequationtrue.Studentsuseproperties
ofoperationsandtheideaofmaintainingtheequalityofbothsidesof
anequationtosolvesimpleone-stepequations.Studentsconstructand
analyzetables,suchastablesofquantitiesthatareinequivalentratios,
andtheyuseequations(suchas3x=y)todescriberelationshipsbetween
quantities.
(4)Buildingonandreinforcingtheirunderstandingofnumber,students
begintodeveloptheirabilitytothinkstatistically.Studentsrecognizethata
datadistributionmaynothaveadefinitecenterandthatdifferentwaysto
measurecenteryielddifferentvalues.Themedianmeasurescenterinthe
sensethatitisroughlythemiddlevalue.Themeanmeasurescenterinthe
sensethatitisthevaluethateachdatapointwouldtakeonifthetotalof
thedatavalueswereredistributedequally,andalsointhesensethatitisa
balancepoint.Studentsrecognizethatameasureofvariability(interquartile
rangeormeanabsolutedeviation)canalsobeusefulforsummarizing
databecausetwoverydifferentsetsofdatacanhavethesamemeanand
Gr
ad
e 6
| 40
medianyetbedistinguishedbytheirvariability.Studentslearntodescribe
andsummarizenumericaldatasets,identifyingclusters,peaks,gaps,and
symmetry,consideringthecontextinwhichthedatawerecollected.
StudentsinGrade6alsobuildontheirworkwithareainelementary
schoolbyreasoningaboutrelationshipsamongshapestodeterminearea,
surfacearea,andvolume.Theyfindareasofrighttriangles,othertriangles,
andspecialquadrilateralsbydecomposingtheseshapes,rearranging
orremovingpieces,andrelatingtheshapestorectangles.Usingthese
methods,studentsdiscuss,develop,andjustifyformulasforareasof
trianglesandparallelograms.Studentsfindareasofpolygonsandsurface
areasofprismsandpyramidsbydecomposingthemintopieceswhose
areatheycandetermine.Theyreasonaboutrightrectangularprisms
withfractionalsidelengthstoextendformulasforthevolumeofaright
rectangularprismtofractionalsidelengths.Theyprepareforworkon
scaledrawingsandconstructionsinGrade7bydrawingpolygonsinthe
coordinateplane.
ratios and Proportional relationships
• Understand ratio concepts and use ratioreasoning to solve problems.
the number System
• apply and extend previous understandings ofmultiplication and division to divide fractionsby fractions.
• Compute fluently with multi-digit numbers andfind common factors and multiples.
• apply and extend previous understandings ofnumbers to the system of rational numbers.
expressions and equations
• apply and extend previous understandings ofarithmetic to algebraic expressions.
• reason about and solve one-variable equationsand inequalities.
• represent and analyze quantitativerelationships between dependent andindependent variables.
Geometry
• Solve real-world and mathematical problemsinvolving area, surface area, and volume.
Statistics and Probability
• develop understanding of statistical variability.
• Summarize and describe distributions.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 6 overview
Gr
ad
e 6
| 42
ratios and Proportional relationships 6.rP
Understand ratio concepts and use ratio reasoning to solve problems.
1. Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities.For example, “The ratioof wings to beaks in the bird house at the zoo was 2:1, because forevery 2 wings there was 1 beak.” “For every vote candidate A received,candidate C received nearly three votes.”
2. Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb≠0,anduseratelanguageinthecontextofaratiorelationship.For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar,so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15hamburgers, which is a rate of $5 per hamburger.”1
3. Useratioandratereasoningtosolvereal-worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.
a. Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.
b. Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.For example, if it took 7 hours to mow 4 lawns, thenat that rate, how many lawns could be mowed in 35 hours? At whatrate were lawns being mowed?
c. Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.
d. Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.
the number System 6.nS
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
1. Interpretandcomputequotientsoffractions,andsolvewordproblemsinvolvingdivisionoffractionsbyfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.Forexample, create a story context for (2/3) ÷ (3/4) and use a visual fractionmodel to show the quotient; use the relationship between multiplicationand division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3.(In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each personget if 3 people share 1/2 lb of chocolate equally? How many 3/4-cupservings are in 2/3 of a cup of yogurt? How wide is a rectangular strip ofland with length 3/4 mi and area 1/2 square mi?
Compute fluently with multi-digit numbers and find common factors and multiples.
2. Fluentlydividemulti-digitnumbersusingthestandardalgorithm.
3. Fluentlyadd,subtract,multiply,anddividemulti-digitdecimalsusingthestandardalgorithmforeachoperation.
4. Findthegreatestcommonfactoroftwowholenumberslessthanorequalto100andtheleastcommonmultipleoftwowholenumberslessthanorequalto12.Usethedistributivepropertytoexpressasumoftwowholenumbers1–100withacommonfactorasamultipleofasumoftwowholenumberswithnocommonfactor.For example,express 36 + 8 as 4 (9 + 2).
1Expectationsforunitratesinthisgradearelimitedtonon-complexfractions.
Gr
ad
e 6
| 43
Apply and extend previous understandings of numbers to the system of rational numbers.
5. Understandthatpositiveandnegativenumbersareusedtogethertodescribequantitieshavingoppositedirectionsorvalues(e.g.,temperatureabove/belowzero,elevationabove/belowsealevel,credits/debits,positive/negativeelectriccharge);usepositiveandnegativenumberstorepresentquantitiesinreal-worldcontexts,explainingthemeaningof0ineachsituation.
6. Understandarationalnumberasapointonthenumberline.Extendnumberlinediagramsandcoordinateaxesfamiliarfrompreviousgradestorepresentpointsonthelineandintheplanewithnegativenumbercoordinates.
a. Recognizeoppositesignsofnumbersasindicatinglocationsonoppositesidesof0onthenumberline;recognizethattheoppositeoftheoppositeofanumberisthenumberitself,e.g.,–(–3)=3,andthat0isitsownopposite.
b. Understandsignsofnumbersinorderedpairsasindicatinglocationsinquadrantsofthecoordinateplane;recognizethatwhentwoorderedpairsdifferonlybysigns,thelocationsofthepointsarerelatedbyreflectionsacrossoneorbothaxes.
c. Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram;findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane.
7. Understandorderingandabsolutevalueofrationalnumbers.
a. Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram.For example,interpret –3 > –7 as a statement that –3 is located to the right of –7 ona number line oriented from left to right.
b. Write,interpret,andexplainstatementsoforderforrationalnumbersinreal-worldcontexts.For example, write –3 oC > –7 oC toexpress the fact that –3 oC is warmer than –7 oC.
c. Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal-worldsituation.Forexample, for an account balance of –30 dollars, write |–30| = 30 todescribe the size of the debt in dollars.
d. Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder.For example, recognize that an account balance less than –30dollars represents a debt greater than 30 dollars.
8. Solvereal-worldandmathematicalproblemsbygraphingpointsinallfourquadrantsofthecoordinateplane.Includeuseofcoordinatesandabsolutevaluetofinddistancesbetweenpointswiththesamefirstcoordinateorthesamesecondcoordinate.
expressions and equations 6.ee
Apply and extend previous understandings of arithmetic to algebraic expressions.
1. Writeandevaluatenumericalexpressionsinvolvingwhole-numberexponents.
2. Write,read,andevaluateexpressionsinwhichlettersstandfornumbers.
a. Writeexpressionsthatrecordoperationswithnumbersandwithlettersstandingfornumbers.For example, express the calculation“Subtract y from 5” as 5 – y.
Gr
ad
e 6
| 44
b. Identifypartsofanexpressionusingmathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofanexpressionasasingleentity.For example, describe theexpression 2 (8 + 7) as a product of two factors; view (8 + 7) as botha single entity and a sum of two terms.
c. Evaluateexpressionsatspecificvaluesoftheirvariables.Includeexpressionsthatarisefromformulasusedinreal-worldproblems.Performarithmeticoperations,includingthoseinvolvingwhole-numberexponents,intheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).For example, use the formulas V = s3 and A = 6 s2 to find the volumeand surface area of a cube with sides of length s = 1/2.
3. Applythepropertiesofoperationstogenerateequivalentexpressions.For example, apply the distributive property to the expression 3 (2 + x) toproduce the equivalent expression 6 + 3x; apply the distributive propertyto the expression 24x + 18y to produce the equivalent expression6 (4x + 3y); apply properties of operations to y + y + y to produce theequivalent expression 3y.
4. Identifywhentwoexpressionsareequivalent(i.e.,whenthetwoexpressionsnamethesamenumberregardlessofwhichvalueissubstitutedintothem).For example, the expressions y + y + y and 3yare equivalent because they name the same number regardless of whichnumber y stands for.
Reason about and solve one-variable equations and inequalities.
5. Understandsolvinganequationorinequalityasaprocessofansweringaquestion:whichvaluesfromaspecifiedset,ifany,maketheequationorinequalitytrue?Usesubstitutiontodeterminewhetheragivennumberinaspecifiedsetmakesanequationorinequalitytrue.
6. Usevariablestorepresentnumbersandwriteexpressionswhensolvingareal-worldormathematicalproblem;understandthatavariablecanrepresentanunknownnumber,or,dependingonthepurposeathand,anynumberinaspecifiedset.
7. Solvereal-worldandmathematicalproblemsbywritingandsolvingequationsoftheformx+p=qandpx=qforcasesinwhichp,qandxareallnonnegativerationalnumbers.
8. Writeaninequalityoftheformx>corx <c torepresentaconstraintorconditioninareal-worldormathematicalproblem.Recognizethatinequalitiesoftheformx>corx<chaveinfinitelymanysolutions;representsolutionsofsuchinequalitiesonnumberlinediagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
9. Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable,intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariablesusinggraphsandtables,andrelatethesetotheequation.For example, in aproblem involving motion at constant speed, list and graph ordered pairsof distances and times, and write the equation d = 65t to represent therelationship between distance and time.
Geometry 6.G
Solve real-world and mathematical problems involving area, surface area, and volume.
1. Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
Gr
ad
e 6
| 45
2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV = l w h and V = b htofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems.
3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
4. Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
Statistics and Probability 6.SP
Develop understanding of statistical variability.
1. Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedatarelatedtothequestionandaccountsforitintheanswers.Forexample, “How old am I?” is not a statistical question, but “How old are thestudents in my school?” is a statistical question because one anticipatesvariability in students’ ages.
2. Understandthatasetofdatacollectedtoanswerastatisticalquestionhasadistributionwhichcanbedescribedbyitscenter,spread,andoverallshape.
3. Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber,whileameasureofvariationdescribeshowitsvaluesvarywithasinglenumber.
Summarize and describe distributions.
4. Displaynumericaldatainplotsonanumberline,includingdotplots,histograms,andboxplots.
5. Summarizenumericaldatasetsinrelationtotheircontext,suchasby:
a. Reportingthenumberofobservations.
b. Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.
c. Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.
d. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.
Gr
ad
e 7 | 4
6
mathematics | Grade 7InGrade7,instructionaltimeshouldfocusonfourcriticalareas:(1)
developingunderstandingofandapplyingproportionalrelationships;
(2)developingunderstandingofoperationswithrationalnumbersand
workingwithexpressionsandlinearequations;(3)solvingproblems
involvingscaledrawingsandinformalgeometricconstructions,and
workingwithtwo-andthree-dimensionalshapestosolveproblems
involvingarea,surfacearea,andvolume;and(4)drawinginferencesabout
populationsbasedonsamples.
(1)Studentsextendtheirunderstandingofratiosanddevelop
understandingofproportionalitytosolvesingle-andmulti-stepproblems.
Studentsusetheirunderstandingofratiosandproportionalitytosolve
awidevarietyofpercentproblems,includingthoseinvolvingdiscounts,
interest,taxes,tips,andpercentincreaseordecrease.Studentssolve
problemsaboutscaledrawingsbyrelatingcorrespondinglengthsbetween
theobjectsorbyusingthefactthatrelationshipsoflengthswithinan
objectarepreservedinsimilarobjects.Studentsgraphproportional
relationshipsandunderstandtheunitrateinformallyasameasureofthe
steepnessoftherelatedline,calledtheslope.Theydistinguishproportional
relationshipsfromotherrelationships.
(2)Studentsdevelopaunifiedunderstandingofnumber,recognizing
fractions,decimals(thathaveafiniteorarepeatingdecimal
representation),andpercentsasdifferentrepresentationsofrational
numbers.Studentsextendaddition,subtraction,multiplication,anddivision
toallrationalnumbers,maintainingthepropertiesofoperationsandthe
relationshipsbetweenadditionandsubtraction,andmultiplicationand
division.Byapplyingtheseproperties,andbyviewingnegativenumbers
intermsofeverydaycontexts(e.g.,amountsowedortemperaturesbelow
zero),studentsexplainandinterprettherulesforadding,subtracting,
multiplying,anddividingwithnegativenumbers.Theyusethearithmetic
ofrationalnumbersastheyformulateexpressionsandequationsinone
variableandusetheseequationstosolveproblems.
(3)StudentscontinuetheirworkwithareafromGrade6,solvingproblems
involvingtheareaandcircumferenceofacircleandsurfaceareaofthree-
dimensionalobjects.Inpreparationforworkoncongruenceandsimilarity
inGrade8theyreasonaboutrelationshipsamongtwo-dimensionalfigures
usingscaledrawingsandinformalgeometricconstructions,andtheygain
familiaritywiththerelationshipsbetweenanglesformedbyintersecting
lines.Studentsworkwiththree-dimensionalfigures,relatingthemtotwo-
dimensionalfiguresbyexaminingcross-sections.Theysolvereal-world
andmathematicalproblemsinvolvingarea,surfacearea,andvolumeof
two-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,
polygons,cubesandrightprisms.
(4)Studentsbuildontheirpreviousworkwithsingledatadistributionsto
comparetwodatadistributionsandaddressquestionsaboutdifferences
betweenpopulations.Theybegininformalworkwithrandomsampling
togeneratedatasetsandlearnabouttheimportanceofrepresentative
samplesfordrawinginferences.
Gr
ad
e 7 | 4
7
ratios and Proportional relationships
• analyze proportional relationships and usethem to solve real-world and mathematicalproblems.
the number System
• apply and extend previous understandingsof operations with fractions to add, subtract,multiply, and divide rational numbers.
expressions and equations
• Use properties of operations to generateequivalent expressions.
• Solve real-life and mathematical problemsusing numerical and algebraic expressions andequations.
Geometry
• draw, construct and describe geometricalfigures and describe the relationships betweenthem.
• Solve real-life and mathematical problemsinvolving angle measure, area, surface area,and volume.
Statistics and Probability
• Use random sampling to draw inferences abouta population.
• draw informal comparative inferences abouttwo populations.
• Investigate chance processes and develop, use,and evaluate probability models.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 7 overview
Gr
ad
e 7 | 4
8
ratios and Proportional relationships 7.rP
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.For example, if a person walks 1/2 mile in each 1/4 hour, computethe unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2miles per hour.
2. Recognizeandrepresentproportionalrelationshipsbetweenquantities.
a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.
b. Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.
c. Representproportionalrelationshipsbyequations.For example, iftotal cost t is proportional to the number n of items purchased ata constant price p, the relationship between the total cost and thenumber of items can be expressed as t = pn.
d. Explainwhatapoint (x, y) onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentiontothepoints(0,0)and(1, r) where r istheunitrate.
3. Useproportionalrelationshipstosolvemultistepratioandpercentproblems.Examples: simple interest, tax, markups and markdowns,gratuities and commissions, fees, percent increase and decrease, percenterror.
the number System 7.nS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
1. Applyandextendpreviousunderstandingsofadditionandsubtractiontoaddandsubtractrationalnumbers;representadditionandsubtractiononahorizontalorverticalnumberlinediagram.
a. Describesituationsinwhichoppositequantitiescombinetomake0.For example, a hydrogen atom has 0 charge because its twoconstituents are oppositely charged.
b. Understandp+qasthenumberlocatedadistance|q|fromp,inthepositiveornegativedirectiondependingonwhetherqispositiveornegative.Showthatanumberanditsoppositehaveasumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.
c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse,p–q=p+(–q).Showthatthedistancebetweentworationalnumbersonthenumberlineistheabsolutevalueoftheirdifference,andapplythisprincipleinreal-worldcontexts.
d. Applypropertiesofoperationsasstrategiestoaddandsubtractrationalnumbers.
2. Applyandextendpreviousunderstandingsofmultiplicationanddivisionandoffractionstomultiplyanddividerationalnumbers.
a. Understandthatmultiplicationisextendedfromfractionstorationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(–1)(–1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbersbydescribingreal-worldcontexts.
Gr
ad
e 7 | 4
9
b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zerodivisor)isarationalnumber.Ifpandqareintegers,then–(p/q)=(–p)/q=p/(–q).Interpretquotientsofrationalnumbersbydescribingreal-worldcontexts.
c. Applypropertiesofoperationsasstrategiestomultiplyanddividerationalnumbers.
d. Convertarationalnumbertoadecimalusinglongdivision;knowthatthedecimalformofarationalnumberterminatesin0soreventuallyrepeats.
3. Solvereal-worldandmathematicalproblemsinvolvingthefouroperationswithrationalnumbers.1
expressions and equations 7.ee
Use properties of operations to generate equivalent expressions.
1. Applypropertiesofoperationsasstrategiestoadd,subtract,factor,andexpandlinearexpressionswithrationalcoefficients.
2. Understandthatrewritinganexpressionindifferentformsinaproblemcontextcanshedlightontheproblemandhowthequantitiesinitarerelated.For example, a + 0.05a = 1.05a means that “increase by5%” is the same as “multiply by 1.05.”
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solvemulti-stepreal-lifeandmathematicalproblemsposedwithpositiveandnegativerationalnumbersinanyform(wholenumbers,fractions,anddecimals),usingtoolsstrategically.Applypropertiesofoperationstocalculatewithnumbersinanyform;convertbetweenformsasappropriate;andassessthereasonablenessofanswersusingmentalcomputationandestimationstrategies.For example: If a womanmaking $25 an hour gets a 10% raise, she will make an additional 1/10 ofher salary an hour, or $2.50, for a new salary of $27.50. If you want to placea towel bar 9 3/4 inches long in the center of a door that is 27 1/2 incheswide, you will need to place the bar about 9 inches from each edge; thisestimate can be used as a check on the exact computation.
4. Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.
a. Solvewordproblemsleadingtoequationsoftheformpx+q=randp(x+q)=r,wherep,q,andrarespecificrationalnumbers.Solveequationsoftheseformsfluently.Compareanalgebraicsolutiontoanarithmeticsolution,identifyingthesequenceoftheoperationsusedineachapproach.For example, the perimeter of arectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solvewordproblemsleadingtoinequalitiesoftheformpx+q>rorpx+q<r,wherep,q,andrarespecificrationalnumbers.Graphthesolutionsetoftheinequalityandinterpretitinthecontextoftheproblem.For example: As a salesperson, you are paid $50 perweek plus $3 per sale. This week you want your pay to be at least$100. Write an inequality for the number of sales you need to make,and describe the solutions.
Geometry 7.G
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solveproblemsinvolvingscaledrawingsofgeometricfigures,includingcomputingactuallengthsandareasfromascaledrawingandreproducingascaledrawingatadifferentscale.
1Computationswithrationalnumbersextendtherulesformanipulatingfractionstocomplexfractions.
Gr
ad
e 7 | 5
0
2. Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticingwhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.
3. Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems;giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle.
5. Usefactsaboutsupplementary,complementary,vertical,andadjacentanglesinamulti-stepproblemtowriteandsolvesimpleequationsforanunknownangleinafigure.
6. Solvereal-worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.
Statistics and Probability 7.SP
Use random sampling to draw inferences about a population.
1. Understandthatstatisticscanbeusedtogaininformationaboutapopulationbyexaminingasampleofthepopulation;generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation.Understandthatrandomsamplingtendstoproducerepresentativesamplesandsupportvalidinferences.
2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.For example, estimate the mean word length ina book by randomly sampling words from the book; predict the winner ofa school election based on randomly sampled survey data. Gauge how faroff the estimate or prediction might be.
Draw informal comparative inferences about two populations.
3. Informallyassessthedegreeofvisualoverlapoftwonumericaldatadistributionswithsimilarvariabilities,measuringthedifferencebetweenthecentersbyexpressingitasamultipleofameasureofvariability.For example, the mean height of players on the basketballteam is 10 cm greater than the mean height of players on the soccer team,about twice the variability (mean absolute deviation) on either team; ona dot plot, the separation between the two distributions of heights isnoticeable.
4. Usemeasuresofcenterandmeasuresofvariabilityfornumericaldatafromrandomsamplestodrawinformalcomparativeinferencesabouttwopopulations.For example, decide whether the words in a chapterof a seventh-grade science book are generally longer than the words in achapter of a fourth-grade science book.
Investigate chance processes and develop, use, and evaluate probability models.
5. Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthelikelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynorlikely,andaprobabilitynear1indicatesalikelyevent.
Gr
ad
e 7 | 5
1
6. Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesitandobservingitslong-runrelativefrequency,andpredicttheapproximaterelativefrequencygiventheprobability.For example, when rolling a number cube 600 times, predictthat a 3 or 6 would be rolled roughly 200 times, but probably not exactly200 times.
7. Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy.
a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilitiesofevents.For example, if a student is selected atrandom from a class, find the probability that Jane will be selectedand the probability that a girl will be selected.
b. Developaprobabilitymodel(whichmaynotbeuniform)byobservingfrequenciesindatageneratedfromachanceprocess.For example, find the approximate probability that a spinning pennywill land heads up or that a tossed paper cup will land open-enddown. Do the outcomes for the spinning penny appear to be equallylikely based on the observed frequencies?
8. Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.
a. Understandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs.
b. Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespacewhichcomposetheevent.
c. Designanduseasimulationtogeneratefrequenciesforcompoundevents.For example, use random digits as a simulationtool to approximate the answer to the question: If 40% of donorshave type A blood, what is the probability that it will take at least 4donors to find one with type A blood?
Gr
ad
e 8
| 52
mathematics | Grade 8InGrade8,instructionaltimeshouldfocusonthreecriticalareas:(1)formulating
andreasoningaboutexpressionsandequations,includingmodelinganassociation
inbivariatedatawithalinearequation,andsolvinglinearequationsandsystems
oflinearequations;(2)graspingtheconceptofafunctionandusingfunctions
todescribequantitativerelationships;(3)analyzingtwo-andthree-dimensional
spaceandfiguresusingdistance,angle,similarity,andcongruence,and
understandingandapplyingthePythagoreanTheorem.
(1)Studentsuselinearequationsandsystemsoflinearequationstorepresent,
analyze,andsolveavarietyofproblems.Studentsrecognizeequationsfor
proportions(y/x=mory=mx)asspeciallinearequations(y=mx+b),
understandingthattheconstantofproportionality(m)istheslope,andthegraphs
arelinesthroughtheorigin.Theyunderstandthattheslope(m)ofalineisa
constantrateofchange,sothatiftheinputorx-coordinatechangesbyanamount
A,theoutputory-coordinatechangesbytheamountm·A.Studentsalsousealinear
equationtodescribetheassociationbetweentwoquantitiesinbivariatedata(such
asarmspanvs.heightforstudentsinaclassroom).Atthisgrade,fittingthemodel,
andassessingitsfittothedataaredoneinformally.Interpretingthemodelinthe
contextofthedatarequiresstudentstoexpressarelationshipbetweenthetwo
quantitiesinquestionandtointerpretcomponentsoftherelationship(suchasslope
andy-intercept)intermsofthesituation.
Studentsstrategicallychooseandefficientlyimplementprocedurestosolvelinear
equationsinonevariable,understandingthatwhentheyusethepropertiesof
equalityandtheconceptoflogicalequivalence,theymaintainthesolutionsofthe
originalequation.Studentssolvesystemsoftwolinearequationsintwovariables
andrelatethesystemstopairsoflinesintheplane;theseintersect,areparallel,or
arethesameline.Studentsuselinearequations,systemsoflinearequations,linear
functions,andtheirunderstandingofslopeofalinetoanalyzesituationsandsolve
problems.
(2)Studentsgrasptheconceptofafunctionasarulethatassignstoeachinput
exactlyoneoutput.Theyunderstandthatfunctionsdescribesituationswhereone
quantitydeterminesanother.Theycantranslateamongrepresentationsandpartial
representationsoffunctions(notingthattabularandgraphicalrepresentations
maybepartialrepresentations),andtheydescribehowaspectsofthefunctionare
reflectedinthedifferentrepresentations.
(3)Studentsuseideasaboutdistanceandangles,howtheybehaveunder
translations,rotations,reflections,anddilations,andideasaboutcongruenceand
similaritytodescribeandanalyzetwo-dimensionalfiguresandtosolveproblems.
Studentsshowthatthesumoftheanglesinatriangleistheangleformedbya
straightline,andthatvariousconfigurationsoflinesgiverisetosimilartriangles
becauseoftheanglescreatedwhenatransversalcutsparallellines.Students
understandthestatementofthePythagoreanTheoremanditsconverse,andcan
explainwhythePythagoreanTheoremholds,forexample,bydecomposinga
squareintwodifferentways.TheyapplythePythagoreanTheoremtofinddistances
betweenpointsonthecoordinateplane,tofindlengths,andtoanalyzepolygons.
Studentscompletetheirworkonvolumebysolvingproblemsinvolvingcones,
cylinders,andspheres.
Gr
ad
e 8
| 53
the number System
• Know that there are numbers that are notrational, and approximate them by rationalnumbers.
expressions and equations
• Work with radicals and integer exponents.
• Understand the connections betweenproportional relationships, lines, and linearequations.
• analyze and solve linear equations and pairs ofsimultaneous linear equations.
functions
• define, evaluate, and compare functions.
• Use functions to model relationships betweenquantities.
Geometry
• Understand congruence and similarity usingphysical models, transparencies, or geometrysoftware.
• Understand and apply the Pythagoreantheorem.
• Solve real-world and mathematical problemsinvolving volume of cylinders, cones andspheres.
Statistics and Probability
• Investigate patterns of association in bivariatedata.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 8 overview
Gr
ad
e 8
| 54
the number System 8.nS
Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Knowthatnumbersthatarenotrationalarecalledirrational.Understandinformallythateverynumberhasadecimalexpansion;forrationalnumbersshowthatthedecimalexpansionrepeatseventually,andconvertadecimalexpansionwhichrepeatseventuallyintoarationalnumber.
2. Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2).For example,by truncating the decimal expansion of √2, show that √2 is between 1 and2, then between 1.4 and 1.5, and explain how to continue on to get betterapproximations.
expressions and equations 8.ee
Work with radicals and integer exponents.
1. Knowandapplythepropertiesofintegerexponentstogenerateequivalentnumericalexpressions.For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
2. Usesquarerootandcuberootsymbolstorepresentsolutionstoequationsoftheformx2=pandx3=p,wherepisapositiverationalnumber.Evaluatesquarerootsofsmallperfectsquaresandcuberootsofsmallperfectcubes.Knowthat√2isirrational.
3. Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.For example,estimate the population of the United States as 3 × 108 and the populationof the world as 7 × 109, and determine that the world population is morethan 20 times larger.
4. Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothdecimalandscientificnotationareused.Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities(e.g.,usemillimetersperyearforseafloorspreading).Interpretscientificnotationthathasbeengeneratedbytechnology.
Understand the connections between proportional relationships, lines, and linear equations.
5. Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.For example, compare a distance-timegraph to a distance-time equation to determine which of two movingobjects has greater speed.
6. Usesimilartrianglestoexplainwhytheslopemisthesamebetweenanytwodistinctpointsonanon-verticallineinthecoordinateplane;derivetheequationy=mxforalinethroughtheoriginandtheequationy=mx+bforalineinterceptingtheverticalaxisatb.
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solvelinearequationsinonevariable.
a. Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).
b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.
Gr
ad
e 8
| 55
8. Analyzeandsolvepairsofsimultaneouslinearequations.
a. Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.
b. Solvesystemsoftwolinearequationsintwovariablesalgebraically,andestimatesolutionsbygraphingtheequations.Solvesimplecasesbyinspection.For example, 3x + 2y = 5 and 3x +2y = 6 have no solution because 3x + 2y cannot simultaneously be 5and 6.
c. Solvereal-worldandmathematicalproblemsleadingtotwolinearequationsintwovariables.For example, given coordinates for twopairs of points, determine whether the line through the first pair ofpoints intersects the line through the second pair.
functions 8.f
Define, evaluate, and compare functions.
1. Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.Thegraphofafunctionisthesetoforderedpairsconsistingofaninputandthecorrespondingoutput.1
2. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a linear function represented by a tableof values and a linear function represented by an algebraic expression,determine which function has the greater rate of change.
3. Interprettheequationy=mx+basdefiningalinearfunction,whosegraphisastraightline;giveexamplesoffunctionsthatarenotlinear.For example, the function A = s2 giving the area of a square as a functionof its side length is not linear because its graph contains the points (1,1),(2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.
4. Constructafunctiontomodelalinearrelationshipbetweentwoquantities.Determinetherateofchangeandinitialvalueofthefunctionfromadescriptionofarelationshiporfromtwo(x,y)values,includingreadingthesefromatableorfromagraph.Interprettherateofchangeandinitialvalueofalinearfunctionintermsofthesituationitmodels,andintermsofitsgraphoratableofvalues.
5. Describequalitativelythefunctionalrelationshipbetweentwoquantitiesbyanalyzingagraph(e.g.,wherethefunctionisincreasingordecreasing,linearornonlinear).Sketchagraphthatexhibitsthequalitativefeaturesofafunctionthathasbeendescribedverbally.
Geometry 8.G
Understand congruence and similarity using physical models, trans-parencies, or geometry software.
1. Verifyexperimentallythepropertiesofrotations,reflections,andtranslations:
a. Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.
b. Anglesaretakentoanglesofthesamemeasure.
c. Parallellinesaretakentoparallellines.
2. Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.
1FunctionnotationisnotrequiredinGrade8.
Gr
ad
e 8
| 56
3. Describetheeffectofdilations,translations,rotations,andreflectionsontwo-dimensionalfiguresusingcoordinates.
4. Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.
5. Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.For example, arrange three copies of the same triangle so thatthe sum of the three angles appears to form a line, and give an argumentin terms of transversals why this is so.
Understand and apply the Pythagorean Theorem.
6. ExplainaproofofthePythagoreanTheoremanditsconverse.
7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.
8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.
Statistics and Probability 8.SP
Investigate patterns of association in bivariate data.
1. Constructandinterpretscatterplotsforbivariatemeasurementdatatoinvestigatepatternsofassociationbetweentwoquantities.Describepatternssuchasclustering,outliers,positiveornegativeassociation,linearassociation,andnonlinearassociation.
2. Knowthatstraightlinesarewidelyusedtomodelrelationshipsbetweentwoquantitativevariables.Forscatterplotsthatsuggestalinearassociation,informallyfitastraightline,andinformallyassessthemodelfitbyjudgingtheclosenessofthedatapointstotheline.
3. Usetheequationofalinearmodeltosolveproblemsinthecontextofbivariatemeasurementdata,interpretingtheslopeandintercept.For example, in a linear model for a biology experiment, interpret a slopeof 1.5 cm/hr as meaning that an additional hour of sunlight each day isassociated with an additional 1.5 cm in mature plant height.
4. Understandthatpatternsofassociationcanalsobeseeninbivariatecategoricaldatabydisplayingfrequenciesandrelativefrequenciesinatwo-waytable.Constructandinterpretatwo-waytablesummarizingdataontwocategoricalvariablescollectedfromthesamesubjects.Userelativefrequenciescalculatedforrowsorcolumnstodescribepossibleassociationbetweenthetwovariables.For example, collectdata from students in your class on whether or not they have a curfew onschool nights and whether or not they have assigned chores at home. Isthere evidence that those who have a curfew also tend to have chores?
HIG
H S
CH
oo
l | 5
7
mathematics Standards for High SchoolThehighschoolstandardsspecifythemathematicsthatallstudentsshould
studyinordertobecollegeandcareerready.Additionalmathematicsthat
studentsshouldlearninordertotakeadvancedcoursessuchascalculus,
advancedstatistics,ordiscretemathematicsisindicatedby(+),asinthis
example:
(+)Representcomplexnumbersonthecomplexplaneinrectangular
andpolarform(includingrealandimaginarynumbers).
Allstandardswithouta(+)symbolshouldbeinthecommonmathematics
curriculumforallcollegeandcareerreadystudents.Standardswitha(+)
symbolmayalsoappearincoursesintendedforallstudents.
Thehighschoolstandardsarelistedinconceptualcategories:
• NumberandQuantity
• Algebra
• Functions
• Modeling
• Geometry
• StatisticsandProbability
Conceptualcategoriesportrayacoherentviewofhighschool
mathematics;astudent’sworkwithfunctions,forexample,crossesa
numberoftraditionalcourseboundaries,potentiallyupthroughand
includingcalculus.
Modelingisbestinterpretednotasacollectionofisolatedtopicsbutin
relationtootherstandards.MakingmathematicalmodelsisaStandardfor
MathematicalPractice,andspecificmodelingstandardsappearthroughout
thehighschoolstandardsindicatedbyastarsymbol(★).Thestarsymbol
sometimesappearsontheheadingforagroupofstandards;inthatcase,it
shouldbeunderstoodtoapplytoallstandardsinthatgroup.
HIG
H S
CH
oo
l —
nU
mb
er
an
d q
Ua
nt
Ity
| 58
mathematics | High School—number and QuantityNumbers and Number Systems.Duringtheyearsfromkindergartentoeighth
grade,studentsmustrepeatedlyextendtheirconceptionofnumber.Atfirst,
“number”means“countingnumber”:1,2,3...Soonafterthat,0isusedtorepresent
“none”andthewholenumbersareformedbythecountingnumberstogether
withzero.Thenextextensionisfractions.Atfirst,fractionsarebarelynumbers
andtiedstronglytopictorialrepresentations.Yetbythetimestudentsunderstand
divisionoffractions,theyhaveastrongconceptoffractionsasnumbersandhave
connectedthem,viatheirdecimalrepresentations,withthebase-tensystemused
torepresentthewholenumbers.Duringmiddleschool,fractionsareaugmentedby
negativefractionstoformtherationalnumbers.InGrade8,studentsextendthis
systemoncemore,augmentingtherationalnumberswiththeirrationalnumbers
toformtherealnumbers.Inhighschool,studentswillbeexposedtoyetanother
extensionofnumber,whentherealnumbersareaugmentedbytheimaginary
numberstoformthecomplexnumbers.
Witheachextensionofnumber,themeaningsofaddition,subtraction,
multiplication,anddivisionareextended.Ineachnewnumbersystem—integers,
rationalnumbers,realnumbers,andcomplexnumbers—thefouroperationsstay
thesameintwoimportantways:Theyhavethecommutative,associative,and
distributivepropertiesandtheirnewmeaningsareconsistentwiththeirprevious
meanings.
Extendingthepropertiesofwhole-numberexponentsleadstonewandproductive
notation.Forexample,propertiesofwhole-numberexponentssuggestthat(51/3)3
shouldbe5(1/3)3=51=5andthat51/3shouldbethecuberootof5.
Calculators,spreadsheets,andcomputeralgebrasystemscanprovidewaysfor
studentstobecomebetteracquaintedwiththesenewnumbersystemsandtheir
notation.Theycanbeusedtogeneratedatafornumericalexperiments,tohelp
understandtheworkingsofmatrix,vector,andcomplexnumberalgebra,andto
experimentwithnon-integerexponents.
Quantities.Inrealworldproblems,theanswersareusuallynotnumbersbut
quantities:numberswithunits,whichinvolvesmeasurement.Intheirworkin
measurementupthroughGrade8,studentsprimarilymeasurecommonlyused
attributessuchaslength,area,andvolume.Inhighschool,studentsencountera
widervarietyofunitsinmodeling,e.g.,acceleration,currencyconversions,derived
quantitiessuchasperson-hoursandheatingdegreedays,socialscienceratessuch
asper-capitaincome,andratesineverydaylifesuchaspointsscoredpergameor
battingaverages.Theyalsoencounternovelsituationsinwhichtheythemselves
mustconceivetheattributesofinterest.Forexample,tofindagoodmeasureof
overallhighwaysafety,theymightproposemeasuressuchasfatalitiesperyear,
fatalitiesperyearperdriver,orfatalitiespervehicle-miletraveled.Suchaconceptual
processissometimescalledquantification.Quantificationisimportantforscience,
aswhensurfaceareasuddenly“standsout”asanimportantvariableinevaporation.
Quantificationisalsoimportantforcompanies,whichmustconceptualizerelevant
attributesandcreateorchoosesuitablemeasuresforthem.
HIG
H S
CH
oo
l —
nU
mb
er
an
d q
Ua
nt
Ity
| 59
The Real Number System
• extend the properties of exponents to rationalexponents
• Use properties of rational and irrationalnumbers.
Quantities
• reason quantitatively and use units to solveproblems
The Complex Number System
• Perform arithmetic operations with complexnumbers
• represent complex numbers and theiroperations on the complex plane
• Use complex numbers in polynomial identitiesand equations
Vector and Matrix Quantities
• represent and model with vector quantities.
• Perform operations on vectors.
• Perform operations on matrices and usematrices in applications.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
number and Quantity overview
HIG
H S
CH
oo
l —
nU
mb
er
an
d q
Ua
nt
Ity
| 60
the real number System n-rn
Extend the properties of exponents to rational exponents.
1. Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesofintegerexponentstothosevalues,allowingforanotationforradicalsintermsofrationalexponents.For example, we define 51/3 to be the cube root of 5because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
2. Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthepropertiesofexponents.
Use properties of rational and irrational numbers.
3. Explainwhythesumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.
Quantities★ n-Q
Reason quantitatively and use units to solve problems.
1. Useunitsasawaytounderstandproblemsandtoguidethesolutionofmulti-stepproblems;chooseandinterpretunitsconsistentlyinformulas;chooseandinterpretthescaleandtheoriginingraphsanddatadisplays.
2. Defineappropriatequantitiesforthepurposeofdescriptivemodeling.
3. Choosealevelofaccuracyappropriatetolimitationsonmeasurementwhenreportingquantities.
the Complex number System n-Cn
Perform arithmetic operations with complex numbers.
1. Knowthereisacomplexnumberisuchthati2=–1,andeverycomplexnumberhastheforma +bi withaandbreal.
2. Usetherelationi2=–1andthecommutative,associative,anddistributivepropertiestoadd,subtract,andmultiplycomplexnumbers.
3. (+)Findtheconjugateofacomplexnumber;useconjugatestofindmoduliandquotientsofcomplexnumbers.
Represent complex numbers and their operations on the complex plane.
4. (+)Representcomplexnumbersonthecomplexplaneinrectangularandpolarform(includingrealandimaginarynumbers),andexplainwhytherectangularandpolarformsofagivencomplexnumberrepresentthesamenumber.
5. (+)Representaddition,subtraction,multiplication,andconjugationofcomplexnumbersgeometricallyonthecomplexplane;usepropertiesofthisrepresentationforcomputation.For example, (–1+√3i)3=8because(–1+√3i)has modulus2and argument120°.
6. (+)Calculatethedistancebetweennumbersinthecomplexplaneasthemodulusofthedifference,andthemidpointofasegmentastheaverageofthenumbersatitsendpoints.
Use complex numbers in polynomial identities and equations.
7. Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions.
8. (+)Extendpolynomialidentitiestothecomplexnumbers.For example,rewrite x2+4as(x+2i)(x–2i).
9. (+)KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.
HIG
H S
CH
oo
l —
nU
mb
er
an
d q
Ua
nt
Ity
| 61
Vector and matrix Quantities n-Vm
Represent and model with vector quantities.
1. (+)Recognizevectorquantitiesashavingbothmagnitudeanddirection.Representvectorquantitiesbydirectedlinesegments,anduseappropriatesymbolsforvectorsandtheirmagnitudes(e.g.,v,|v|,||v||,v).
2. (+)Findthecomponentsofavectorbysubtractingthecoordinatesofaninitialpointfromthecoordinatesofaterminalpoint.
3. (+)Solveproblemsinvolvingvelocityandotherquantitiesthatcanberepresentedbyvectors.
Perform operations on vectors.
4. (+)Addandsubtractvectors.
a. Addvectorsend-to-end,component-wise,andbytheparallelogramrule.Understandthatthemagnitudeofasumoftwovectorsistypicallynotthesumofthemagnitudes.
b. Giventwovectorsinmagnitudeanddirectionform,determinethemagnitudeanddirectionoftheirsum.
c. Understandvectorsubtractionv–wasv+(–w),where–wistheadditiveinverseofw,withthesamemagnitudeaswandpointingintheoppositedirection.Representvectorsubtractiongraphicallybyconnectingthetipsintheappropriateorder,andperformvectorsubtractioncomponent-wise.
5. (+)Multiplyavectorbyascalar.
a. Representscalarmultiplicationgraphicallybyscalingvectorsandpossiblyreversingtheirdirection;performscalarmultiplicationcomponent-wise,e.g.,asc(v
x,v
y)=(cv
x,cv
y).
b. Computethemagnitudeofascalarmultiplecvusing||cv||=|c|v.Computethedirectionofcvknowingthatwhen|c|v≠0,thedirectionofcviseitheralongv(forc>0)oragainstv(forc<0).
Perform operations on matrices and use matrices in applications.
6. (+)Usematricestorepresentandmanipulatedata,e.g.,torepresentpayoffsorincidencerelationshipsinanetwork.
7. (+)Multiplymatricesbyscalarstoproducenewmatrices,e.g.,aswhenallofthepayoffsinagamearedoubled.
8. (+)Add,subtract,andmultiplymatricesofappropriatedimensions.
9. (+)Understandthat,unlikemultiplicationofnumbers,matrixmultiplicationforsquarematricesisnotacommutativeoperation,butstillsatisfiestheassociativeanddistributiveproperties.
10. (+)Understandthatthezeroandidentitymatricesplayaroleinmatrixadditionandmultiplicationsimilartotheroleof0and1intherealnumbers.Thedeterminantofasquarematrixisnonzeroifandonlyifthematrixhasamultiplicativeinverse.
11. (+)Multiplyavector(regardedasamatrixwithonecolumn)byamatrixofsuitabledimensionstoproduceanothervector.Workwithmatricesastransformationsofvectors.
12. (+)Workwith2×2matricesastransformationsoftheplane,andinterprettheabsolutevalueofthedeterminantintermsofarea.
HIG
H S
CH
oo
l —
alG
eb
ra
| 62
mathematics | High School—algebraExpressions.Anexpressionisarecordofacomputationwithnumbers,symbolsthatrepresentnumbers,arithmeticoperations,exponentiation,and,atmoreadvancedlevels,theoperationofevaluatingafunction.Conventionsabouttheuseofparenthesesandtheorderofoperationsassurethateachexpressionisunambiguous.Creatinganexpressionthatdescribesacomputationinvolvingageneralquantityrequirestheabilitytoexpressthecomputationingeneralterms,abstractingfromspecificinstances.
Readinganexpressionwithcomprehensioninvolvesanalysisofitsunderlyingstructure.Thismaysuggestadifferentbutequivalentwayofwritingtheexpressionthatexhibitssomedifferentaspectofitsmeaning.Forexample,p+0.05pcanbeinterpretedastheadditionofa5%taxtoapricep.Rewritingp+0.05pas1.05pshowsthataddingataxisthesameasmultiplyingthepricebyaconstantfactor.
Algebraicmanipulationsaregovernedbythepropertiesofoperationsandexponents,andtheconventionsofalgebraicnotation.Attimes,anexpressionistheresultofapplyingoperationstosimplerexpressions.Forexample,p+0.05pisthesumofthesimplerexpressionspand0.05p.Viewinganexpressionastheresultofoperationonsimplerexpressionscansometimesclarifyitsunderlyingstructure.
Aspreadsheetoracomputeralgebrasystem(CAS)canbeusedtoexperimentwithalgebraicexpressions,performcomplicatedalgebraicmanipulations,andunderstandhowalgebraicmanipulationsbehave.
Equations and inequalities.Anequationisastatementofequalitybetweentwoexpressions,oftenviewedasaquestionaskingforwhichvaluesofthevariablestheexpressionsoneithersideareinfactequal.Thesevaluesarethesolutionstotheequation.Anidentity,incontrast,istrueforallvaluesofthevariables;identitiesareoftendevelopedbyrewritinganexpressioninanequivalentform.
Thesolutionsofanequationinonevariableformasetofnumbers;thesolutionsofanequationintwovariablesformasetoforderedpairsofnumbers,whichcanbeplottedinthecoordinateplane.Twoormoreequationsand/orinequalitiesformasystem.Asolutionforsuchasystemmustsatisfyeveryequationandinequalityinthesystem.
Anequationcanoftenbesolvedbysuccessivelydeducingfromitoneormoresimplerequations.Forexample,onecanaddthesameconstanttobothsideswithoutchangingthesolutions,butsquaringbothsidesmightleadtoextraneoussolutions.Strategiccompetenceinsolvingincludeslookingaheadforproductivemanipulationsandanticipatingthenatureandnumberofsolutions.
Someequationshavenosolutionsinagivennumbersystem,buthaveasolutioninalargersystem.Forexample,thesolutionofx+1=0isaninteger,notawholenumber;thesolutionof2x+1=0isarationalnumber,notaninteger;thesolutionsofx2–2=0arerealnumbers,notrationalnumbers;andthesolutionsofx2+2=0arecomplexnumbers,notrealnumbers.
Thesamesolutiontechniquesusedtosolveequationscanbeusedtorearrangeformulas.Forexample,theformulafortheareaofatrapezoid,A=((b
1+b
2)/2)h,can
besolvedforhusingthesamedeductiveprocess.
Inequalitiescanbesolvedbyreasoningaboutthepropertiesofinequality.Many,butnotall,ofthepropertiesofequalitycontinuetoholdforinequalitiesandcanbeusefulinsolvingthem.
Connections to Functions and Modeling. Expressionscandefinefunctions,andequivalentexpressionsdefinethesamefunction.Askingwhentwofunctionshavethesamevalueforthesameinputleadstoanequation;graphingthetwofunctionsallowsforfindingapproximatesolutionsoftheequation.Convertingaverbaldescriptiontoanequation,inequality,orsystemoftheseisanessentialskillinmodeling.
HIG
H S
CH
oo
l —
alG
eb
ra
| 63
Seeing Structure in Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solveproblems
Arithmetic with Polynomials and Rational Expressions
• Perform arithmetic operations on polynomials
• Understand the relationship between zeros andfactors of polynomials
• Use polynomial identities to solve problems
• rewrite rational expressions
Creating Equations
• Create equations that describe numbers orrelationships
Reasoning with Equations and Inequalities
• Understand solving equations as a process ofreasoning and explain the reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• represent and solve equations and inequalitiesgraphically
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
algebra overview
HIG
H S
CH
oo
l —
alG
eb
ra
| 64
Seeing Structure in expressions a-SSe
Interpret the structure of expressions
1. Interpretexpressionsthatrepresentaquantityintermsofitscontext.★
a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.
b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.For example, interpretP(1+r)nas the productof P and a factor not depending on P.
2. Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample, see x4–y4as(x2)2–(y2)2,thus recognizing it as a difference ofsquares that can be factored as(x2–y2)(x2+y2).
Write expressions in equivalent forms to solve problems
3. Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesofthequantityrepresentedbytheexpression.★
a. Factoraquadraticexpressiontorevealthezerosofthefunctionitdefines.
b. Completethesquareinaquadraticexpressiontorevealthemaximumorminimumvalueofthefunctionitdefines.
c. Usethepropertiesofexponentstotransformexpressionsforexponentialfunctions. For example the expression1.15tcan berewritten as(1.151/12)12t≈1.01212tto reveal the approximate equivalentmonthly interest rate if the annual rate is 15%.
4. Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusetheformulatosolveproblems.Forexample, calculate mortgage payments.★
arithmetic with Polynomials and rational expressions a-aPr
Perform arithmetic operations on polynomials
1. Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyareclosedundertheoperationsofaddition,subtraction,andmultiplication;add,subtract,andmultiplypolynomials.
Understand the relationship between zeros and factors of polynomials
2. KnowandapplytheRemainderTheorem:Forapolynomialp(x)andanumbera,theremainderondivisionbyx–aisp(a),sop(a)=0ifandonlyif(x–a)isafactorofp(x).
3. Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezerostoconstructaroughgraphofthefunctiondefinedbythepolynomial.
Use polynomial identities to solve problems
4. Provepolynomialidentitiesandusethemtodescribenumericalrelationships.For example, the polynomial identity(x2+y2)2=(x2–y2)2+(2xy)2can be used to generate Pythagorean triples.
5. (+)KnowandapplytheBinomialTheoremfortheexpansionof(x+y)ninpowersofxandyforapositiveintegern,wherexandyareanynumbers,withcoefficientsdeterminedforexamplebyPascal’sTriangle.1
1TheBinomialTheoremcanbeprovedbymathematicalinductionorbyacom-binatorialargument.
HIG
H S
CH
oo
l —
alG
eb
ra
| 65
Rewrite rational expressions
6. Rewritesimplerationalexpressionsindifferentforms;writea(x)/b(x)intheformq(x)+r(x)/b(x),wherea(x),b(x),q(x),andr(x)arepolynomialswiththedegreeofr(x)lessthanthedegreeofb(x),usinginspection,longdivision,or,forthemorecomplicatedexamples,acomputeralgebrasystem.
7. (+)Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,closedunderaddition,subtraction,multiplication,anddivisionbyanonzerorationalexpression;add,subtract,multiply,anddividerationalexpressions.
Creating equations★ a-Ced
Create equations that describe numbers or relationships
1. Createequationsandinequalitiesinonevariableandusethemtosolveproblems.Include equations arising from linear and quadraticfunctions, and simple rational and exponential functions.
2. Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswithlabelsandscales.
3. Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornon-viableoptionsinamodelingcontext.For example, represent inequalitiesdescribing nutritional and cost constraints on combinations of differentfoods.
4. Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.For example, rearrange Ohm’s law V =IR to highlight resistance R.
reasoning with equations and Inequalities a-reI
Understand solving equations as a process of reasoning and explain the reasoning
1. Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbersassertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution.Constructaviableargumenttojustifyasolutionmethod.
2. Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneoussolutionsmayarise.
Solve equations and inequalities in one variable
3. Solvelinearequationsandinequalitiesinonevariable,includingequationswithcoefficientsrepresentedbyletters.
4. Solvequadraticequationsinonevariable.
a. Usethemethodofcompletingthesquaretotransformanyquadraticequationinxintoanequationoftheform(x–p)2=qthathasthesamesolutions.Derivethequadraticformulafromthisform.
b. Solvequadraticequationsbyinspection(e.g.,forx2=49),takingsquareroots,completingthesquare,thequadraticformulaandfactoring,asappropriatetotheinitialformoftheequation.Recognizewhenthequadraticformulagivescomplexsolutionsandwritethemasa±biforrealnumbersaandb.
Solve systems of equations
5. Provethat,givenasystemoftwoequationsintwovariables,replacingoneequationbythesumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.
HIG
H S
CH
oo
l —
alG
eb
ra
| 66
6. Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.
7. Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariablesalgebraicallyandgraphically.For example,find the points of intersection between the line y=–3xandthecirclex2+y2=3.
8. (+)Representasystemoflinearequationsasasinglematrixequationinavectorvariable.
9. (+)Findtheinverseofamatrixifitexistsanduseittosolvesystemsoflinearequations(usingtechnologyformatricesofdimension3×3orgreater).
Represent and solve equations and inequalities graphically
10. Understandthatthegraphofanequationintwovariablesisthesetofallitssolutionsplottedinthecoordinateplane,oftenformingacurve(whichcouldbealine).
11. Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,usingtechnologytographthefunctions,maketablesofvalues,orfindsuccessiveapproximations.Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolutevalue,exponential,andlogarithmicfunctions.★
12. Graphthesolutionstoalinearinequalityintwovariablesasahalf-plane(excludingtheboundaryinthecaseofastrictinequality),andgraphthesolutionsettoasystemoflinearinequalitiesintwovariablesastheintersectionofthecorrespondinghalf-planes.
HIG
H S
CH
oo
l —
fU
nC
tIo
nS
| 67
mathematics | High School—functionsFunctionsdescribesituationswhereonequantitydeterminesanother.Forexample,thereturnon$10,000investedatanannualizedpercentagerateof4.25%isafunctionofthelengthoftimethemoneyisinvested.Becausewecontinuallymaketheoriesaboutdependenciesbetweenquantitiesinnatureandsociety,functionsareimportanttoolsintheconstructionofmathematicalmodels.
Inschoolmathematics,functionsusuallyhavenumericalinputsandoutputsandareoftendefinedbyanalgebraicexpression.Forexample,thetimeinhoursittakesforacartodrive100milesisafunctionofthecar’sspeedinmilesperhour,v;theruleT(v)=100/vexpressesthisrelationshipalgebraicallyanddefinesafunctionwhosenameisT.
Thesetofinputstoafunctioniscalleditsdomain.Weofteninferthedomaintobeallinputsforwhichtheexpressiondefiningafunctionhasavalue,orforwhichthefunctionmakessenseinagivencontext.
Afunctioncanbedescribedinvariousways,suchasbyagraph(e.g.,thetraceofaseismograph);byaverbalrule,asin,“I’llgiveyouastate,yougivemethecapitalcity;”byanalgebraicexpressionlikef(x)=a+bx;orbyarecursiverule.Thegraphofafunctionisoftenausefulwayofvisualizingtherelationshipofthefunctionmodels,andmanipulatingamathematicalexpressionforafunctioncanthrowlightonthefunction’sproperties.
Functionspresentedasexpressionscanmodelmanyimportantphenomena.Twoimportantfamiliesoffunctionscharacterizedbylawsofgrowtharelinearfunctions,whichgrowataconstantrate,andexponentialfunctions,whichgrowataconstantpercentrate.Linearfunctionswithaconstanttermofzerodescribeproportionalrelationships.
Agraphingutilityoracomputeralgebrasystemcanbeusedtoexperimentwithpropertiesofthesefunctionsandtheirgraphsandtobuildcomputationalmodelsoffunctions,includingrecursivelydefinedfunctions.
Connections to Expressions, Equations, Modeling, and Coordinates.
Determininganoutputvalueforaparticularinputinvolvesevaluatinganexpression;findinginputsthatyieldagivenoutputinvolvessolvinganequation.Questionsaboutwhentwofunctionshavethesamevalueforthesameinputleadtoequations,whosesolutionscanbevisualizedfromtheintersectionoftheirgraphs.Becausefunctionsdescriberelationshipsbetweenquantities,theyarefrequentlyusedinmodeling.Sometimesfunctionsaredefinedbyarecursiveprocess,whichcanbedisplayedeffectivelyusingaspreadsheetorothertechnology.
HIG
H S
CH
oo
l —
fU
nC
tIo
nS
| 68
Interpreting Functions
• Understand the concept of a function and usefunction notation
• Interpret functions that arise in applications interms of the context
• analyze functions using differentrepresentations
Building Functions
• Build a function that models a relationshipbetween two quantities
• Build new functions from existing functions
Linear, Quadratic, and Exponential Models
• Construct and compare linear, quadratic, andexponential models and solve problems
• Interpret expressions for functions in terms ofthe situation they model
Trigonometric Functions
• extend the domain of trigonometric functionsusing the unit circle
• model periodic phenomena with trigonometricfunctions
• Prove and apply trigonometric identities
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
functions overview
HIG
H S
CH
oo
l —
fU
nC
tIo
nS
| 69
Interpreting functions f-If
Understand the concept of a function and use function notation
1. Understandthatafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange.Iffisafunctionandxisanelementofitsdomain,thenf(x)denotestheoutputoffcorrespondingtotheinputx.Thegraphoffisthegraphoftheequationy=f(x).
2. Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpretstatementsthatusefunctionnotationintermsofacontext.
3. Recognizethatsequencesarefunctions,sometimesdefinedrecursively,whosedomainisasubsetoftheintegers.For example, theFibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) +f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context
4. Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Key features include: intercepts; intervals where thefunction is increasing, decreasing, positive, or negative; relative maximumsand minimums; symmetries; end behavior; and periodicity.★
5. Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipitdescribes.For example, if the functionh(n) gives the number of person-hours it takes to assemble n engines in afactory, then the positive integers would be an appropriate domain for thefunction.★
6. Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval.Estimatetherateofchangefromagraph.★
Analyze functions using different representations
7. Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyformorecomplicatedcases.★
a. Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.
b. Graphsquareroot,cuberoot,andpiecewise-definedfunctions,includingstepfunctionsandabsolutevaluefunctions.
c. Graphpolynomialfunctions,identifyingzeroswhensuitablefactorizationsareavailable,andshowingendbehavior.
d. (+)Graphrationalfunctions,identifyingzerosandasymptoteswhensuitablefactorizationsareavailable,andshowingendbehavior.
e. Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,andamplitude.
8. Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealandexplaindifferentpropertiesofthefunction.
a. Usetheprocessoffactoringandcompletingthesquareinaquadraticfunctiontoshowzeros,extremevalues,andsymmetryofthegraph,andinterprettheseintermsofacontext.
b. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions.For example, identify percent rate of changein functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, andclassify them as representing exponential growth or decay.
HIG
H S
CH
oo
l —
fU
nC
tIo
nS
| 70
9. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a graph of one quadratic function andan algebraic expression for another, say which has the larger maximum.
Building functions f-Bf
Build a function that models a relationship between two quantities
1. Writeafunctionthatdescribesarelationshipbetweentwoquantities.★
a. Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.
b. Combinestandardfunctiontypesusingarithmeticoperations.Forexample, build a function that models the temperature of a coolingbody by adding a constant function to a decaying exponential, andrelate these functions to the model.
c. (+)Composefunctions. For example, if T(y) is the temperature inthe atmosphere as a function of height, and h(t) is the height of aweather balloon as a function of time, then T(h(t)) is the temperatureat the location of the weather balloon as a function of time.
2. Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicitformula,usethemtomodelsituations,andtranslatebetweenthetwoforms.★
Build new functions from existing functions
3. Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Include recognizing even and odd functions from their graphs andalgebraic expressions for them.
4. Findinversefunctions.
a. Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.Forexample, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.
b. (+)Verifybycompositionthatonefunctionistheinverseofanother.
c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasaninverse.
d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.
5. (+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethisrelationshiptosolveproblemsinvolvinglogarithmsandexponents.
Linear, Quadratic, and exponential models★ f-Le
Construct and compare linear, quadratic, and exponential models and solve problems
1. Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwithexponentialfunctions.
a. Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervals,andthatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
b. Recognizesituationsinwhichonequantitychangesataconstantrateperunitintervalrelativetoanother.
c. Recognizesituationsinwhichaquantitygrowsordecaysbyaconstantpercentrateperunitintervalrelativetoanother.
HIG
H S
CH
oo
l —
fU
nC
tIo
nS
| 71
2. Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefromatable).
3. Observeusinggraphsandtablesthataquantityincreasingexponentiallyeventuallyexceedsaquantityincreasinglinearly,quadratically,or(moregenerally)asapolynomialfunction.
4. Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddarenumbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.
Interpret expressions for functions in terms of the situation they model
5. Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
trigonometric functions f-tf
Extend the domain of trigonometric functions using the unit circle
1. Understandradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle.
2. Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoallrealnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwisearoundtheunitcircle.
3. (+)Usespecialtrianglestodeterminegeometricallythevaluesofsine,cosine,tangentforπ/3,π/4andπ/6,andusetheunitcircletoexpressthevaluesofsine,cosine,andtangentforπ–x,π+x,and2π–xintermsoftheirvaluesforx,wherexisanyrealnumber.
4. (+)Usetheunitcircletoexplainsymmetry(oddandeven)andperiodicityoftrigonometricfunctions.
Model periodic phenomena with trigonometric functions
5. Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,frequency,andmidline.★
6. (+)Understandthatrestrictingatrigonometricfunctiontoadomainonwhichitisalwaysincreasingoralwaysdecreasingallowsitsinversetobeconstructed.
7. (+)Useinversefunctionstosolvetrigonometricequationsthatariseinmodelingcontexts;evaluatethesolutionsusingtechnology,andinterpretthemintermsofthecontext.★
Prove and apply trigonometric identities
8. ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseittofindsin(θ),cos(θ),ortan(θ)givensin(θ),cos(θ),ortan(θ)andthequadrantoftheangle.
9. (+)Provetheadditionandsubtractionformulasforsine,cosine,andtangentandusethemtosolveproblems.
HIG
H S
CH
oo
l —
mo
de
lIn
G | 7
2
mathematics | High School—modelingModelinglinksclassroommathematicsandstatisticstoeverydaylife,work,anddecision-making.Modelingistheprocessofchoosingandusingappropriatemathematicsandstatisticstoanalyzeempiricalsituations,tounderstandthembetter,andtoimprovedecisions.Quantitiesandtheirrelationshipsinphysical,economic,publicpolicy,social,andeverydaysituationscanbemodeledusingmathematicalandstatisticalmethods.Whenmakingmathematicalmodels,technologyisvaluableforvaryingassumptions,exploringconsequences,andcomparingpredictionswithdata.
Amodelcanbeverysimple,suchaswritingtotalcostasaproductofunitpriceandnumberbought,orusingageometricshapetodescribeaphysicalobjectlikeacoin.Evensuchsimplemodelsinvolvemakingchoices.Itisuptouswhethertomodelacoinasathree-dimensionalcylinder,orwhetheratwo-dimensionaldiskworkswellenoughforourpurposes.Othersituations—modelingadeliveryroute,aproductionschedule,oracomparisonofloanamortizations—needmoreelaboratemodelsthatuseothertoolsfromthemathematicalsciences.Real-worldsituationsarenotorganizedandlabeledforanalysis;formulatingtractablemodels,representingsuchmodels,andanalyzingthemisappropriatelyacreativeprocess.Likeeverysuchprocess,thisdependsonacquiredexpertiseaswellascreativity.
Someexamplesofsuchsituationsmightinclude:
• Estimatinghowmuchwaterandfoodisneededforemergencyreliefinadevastatedcityof3millionpeople,andhowitmightbedistributed.
• Planningatabletennistournamentfor7playersataclubwith4tables,whereeachplayerplaysagainsteachotherplayer.
• Designingthelayoutofthestallsinaschoolfairsoastoraiseasmuchmoneyaspossible.
• Analyzingstoppingdistanceforacar.
• Modelingsavingsaccountbalance,bacterialcolonygrowth,orinvestmentgrowth.
• Engagingincriticalpathanalysis,e.g.,appliedtoturnaroundofanaircraftatanairport.
• Analyzingriskinsituationssuchasextremesports,pandemics,andterrorism.
• Relatingpopulationstatisticstoindividualpredictions.
Insituationslikethese,themodelsdeviseddependonanumberoffactors:Howpreciseananswerdowewantorneed?Whataspectsofthesituationdowemostneedtounderstand,control,oroptimize?Whatresourcesoftimeandtoolsdowehave?Therangeofmodelsthatwecancreateandanalyzeisalsoconstrainedbythelimitationsofourmathematical,statistical,andtechnicalskills,andourabilitytorecognizesignificantvariablesandrelationshipsamongthem.Diagramsofvariouskinds,spreadsheetsandothertechnology,andalgebraarepowerfultoolsforunderstandingandsolvingproblemsdrawnfromdifferenttypesofreal-worldsituations.
Oneoftheinsightsprovidedbymathematicalmodelingisthatessentiallythesamemathematicalorstatisticalstructurecansometimesmodelseeminglydifferentsituations.Modelscanalsoshedlightonthemathematicalstructuresthemselves,forexample,aswhenamodelofbacterialgrowthmakesmorevividtheexplosivegrowthoftheexponentialfunction.
Thebasicmodelingcycleissummarizedinthediagram.Itinvolves(1)identifyingvariablesinthesituationandselectingthosethatrepresentessentialfeatures,(2)formulatingamodelbycreatingandselectinggeometric,graphical,tabular,algebraic,orstatisticalrepresentationsthatdescriberelationshipsbetweenthevariables,(3)analyzingandperformingoperationsontheserelationshipstodrawconclusions,(4)interpretingtheresultsofthemathematicsintermsoftheoriginalsituation,(5)validatingtheconclusionsbycomparingthemwiththesituation,andtheneitherimprovingthemodelor,ifit
HIG
H S
CH
oo
l —
mo
de
lIn
G | 7
3
isacceptable,(6)reportingontheconclusionsandthereasoningbehindthem.Choices,assumptions,andapproximationsarepresentthroughoutthiscycle.
Indescriptivemodeling,amodelsimplydescribesthephenomenaorsummarizestheminacompactform.Graphsofobservationsareafamiliardescriptivemodel—forexample,graphsofglobaltemperatureandatmosphericCO
2overtime.
Analyticmodelingseekstoexplaindataonthebasisofdeepertheoreticalideas,albeitwithparametersthatareempiricallybased;forexample,exponentialgrowthofbacterialcolonies(untilcut-offmechanismssuchaspollutionorstarvationintervene)followsfromaconstantreproductionrate.Functionsareanimportanttoolforanalyzingsuchproblems.
Graphingutilities,spreadsheets,computeralgebrasystems,anddynamicgeometrysoftwarearepowerfultoolsthatcanbeusedtomodelpurelymathematicalphenomena(e.g.,thebehaviorofpolynomials)aswellasphysicalphenomena.
modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).
HIG
H S
CH
oo
l —
Ge
om
et
ry
| 74
mathematics | High School—GeometryAnunderstandingoftheattributesandrelationshipsofgeometricobjectscanbeappliedindiversecontexts—interpretingaschematicdrawing,estimatingtheamountofwoodneededtoframeaslopingroof,renderingcomputergraphics,ordesigningasewingpatternforthemostefficientuseofmaterial.
Althoughtherearemanytypesofgeometry,schoolmathematicsisdevotedprimarilytoplaneEuclideangeometry,studiedbothsynthetically(withoutcoordinates)andanalytically(withcoordinates).EuclideangeometryischaracterizedmostimportantlybytheParallelPostulate,thatthroughapointnotonagivenlinethereisexactlyoneparallelline.(Sphericalgeometry,incontrast,hasnoparallellines.)
Duringhighschool,studentsbegintoformalizetheirgeometryexperiencesfromelementaryandmiddleschool,usingmoreprecisedefinitionsanddevelopingcarefulproofs.LaterincollegesomestudentsdevelopEuclideanandothergeometriescarefullyfromasmallsetofaxioms.
Theconceptsofcongruence,similarity,andsymmetrycanbeunderstoodfromtheperspectiveofgeometrictransformation.Fundamentalaretherigidmotions:translations,rotations,reflections,andcombinationsofthese,allofwhicharehereassumedtopreservedistanceandangles(andthereforeshapesgenerally).Reflectionsandrotationseachexplainaparticulartypeofsymmetry,andthesymmetriesofanobjectofferinsightintoitsattributes—aswhenthereflectivesymmetryofanisoscelestriangleassuresthatitsbaseanglesarecongruent.
Intheapproachtakenhere,twogeometricfiguresaredefinedtobecongruentifthereisasequenceofrigidmotionsthatcarriesoneontotheother.Thisistheprincipleofsuperposition.Fortriangles,congruencemeanstheequalityofallcorrespondingpairsofsidesandallcorrespondingpairsofangles.Duringthemiddlegrades,throughexperiencesdrawingtrianglesfromgivenconditions,studentsnoticewaystospecifyenoughmeasuresinatriangletoensurethatalltrianglesdrawnwiththosemeasuresarecongruent.Oncethesetrianglecongruencecriteria(ASA,SAS,andSSS)areestablishedusingrigidmotions,theycanbeusedtoprovetheoremsabouttriangles,quadrilaterals,andothergeometricfigures.
Similaritytransformations(rigidmotionsfollowedbydilations)definesimilarityinthesamewaythatrigidmotionsdefinecongruence,therebyformalizingthesimilarityideasof"sameshape"and"scalefactor"developedinthemiddlegrades.Thesetransformationsleadtothecriterionfortrianglesimilaritythattwopairsofcorrespondinganglesarecongruent.
Thedefinitionsofsine,cosine,andtangentforacuteanglesarefoundedonrighttrianglesandsimilarity,and,withthePythagoreanTheorem,arefundamentalinmanyreal-worldandtheoreticalsituations.ThePythagoreanTheoremisgeneralizedtonon-righttrianglesbytheLawofCosines.Together,theLawsofSinesandCosinesembodythetrianglecongruencecriteriaforthecaseswherethreepiecesofinformationsufficetocompletelysolveatriangle.Furthermore,theselawsyieldtwopossiblesolutionsintheambiguouscase,illustratingthatSide-Side-Angleisnotacongruencecriterion.
Analyticgeometryconnectsalgebraandgeometry,resultinginpowerfulmethodsofanalysisandproblemsolving.Justasthenumberlineassociatesnumberswithlocationsinonedimension,apairofperpendicularaxesassociatespairsofnumberswithlocationsintwodimensions.Thiscorrespondencebetweennumericalcoordinatesandgeometricpointsallowsmethodsfromalgebratobeappliedtogeometryandviceversa.Thesolutionsetofanequationbecomesageometriccurve,makingvisualizationatoolfordoingandunderstandingalgebra.Geometricshapescanbedescribedbyequations,makingalgebraicmanipulationintoatoolforgeometricunderstanding,modeling,andproof.Geometrictransformationsofthegraphsofequationscorrespondtoalgebraicchangesintheirequations.
Dynamicgeometryenvironmentsprovidestudentswithexperimentalandmodelingtoolsthatallowthemtoinvestigategeometricphenomenainmuchthesamewayascomputeralgebrasystemsallowthemtoexperimentwithalgebraicphenomena.
Connections to Equations. Thecorrespondencebetweennumericalcoordinatesandgeometricpointsallowsmethodsfromalgebratobeappliedtogeometryandviceversa.Thesolutionsetofanequationbecomesageometriccurve,makingvisualizationatoolfordoingandunderstandingalgebra.Geometricshapescanbedescribedbyequations,makingalgebraicmanipulationintoatoolforgeometricunderstanding,modeling,andproof.
HIG
H S
CH
oo
l —
Ge
om
et
ry
| 75
Congruence
• experiment with transformations in the plane
• Understand congruence in terms of rigidmotions
• Prove geometric theorems
• make geometric constructions
Similarity, Right Triangles, and Trigonometry
• Understand similarity in terms of similaritytransformations
• Prove theorems involving similarity
• define trigonometric ratios and solve problemsinvolving right triangles
• apply trigonometry to general triangles
Circles
• Understand and apply theorems about circles
• find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations
• translate between the geometric descriptionand the equation for a conic section
• Use coordinates to prove simple geometrictheorems algebraically
Geometric Measurement and Dimension
• explain volume formulas and use them to solveproblems
• Visualize relationships between two-dimensional and three-dimensional objects
Modeling with Geometry
• apply geometric concepts in modelingsituations
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Geometry overview
HIG
H S
CH
oo
l —
Ge
om
et
ry
| 76
Congruence G-Co
Experiment with transformations in the plane
1. Knowprecisedefinitionsofangle,circle,perpendicularline,parallelline,andlinesegment,basedontheundefinednotionsofpoint,line,distancealongaline,anddistancearoundacirculararc.
2. Representtransformationsintheplaneusing,e.g.,transparenciesandgeometrysoftware;describetransformationsasfunctionsthattakepointsintheplaneasinputsandgiveotherpointsasoutputs.Comparetransformationsthatpreservedistanceandangletothosethatdonot(e.g.,translationversushorizontalstretch).
3. Givenarectangle,parallelogram,trapezoid,orregularpolygon,describetherotationsandreflectionsthatcarryitontoitself.
4. Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicularlines,parallellines,andlinesegments.
5. Givenageometricfigureandarotation,reflection,ortranslation,drawthetransformedfigureusing,e.g.,graphpaper,tracingpaper,orgeometrysoftware.Specifyasequenceoftransformationsthatwillcarryagivenfigureontoanother.
Understand congruence in terms of rigid motions
6. Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectofagivenrigidmotiononagivenfigure;giventwofigures,usethedefinitionofcongruenceintermsofrigidmotionstodecideiftheyarecongruent.
7. Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.
8. Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthedefinitionofcongruenceintermsofrigidmotions.
Prove geometric theorems
9. Provetheoremsaboutlinesandangles. Theorems include: verticalangles are congruent; when a transversal crosses parallel lines, alternateinterior angles are congruent and corresponding angles are congruent;points on a perpendicular bisector of a line segment are exactly thoseequidistant from the segment’s endpoints.
10. Provetheoremsabouttriangles.Theorems include: measures of interiorangles of a triangle sum to 180°; base angles of isosceles triangles arecongruent; the segment joining midpoints of two sides of a triangle isparallel to the third side and half the length; the medians of a trianglemeet at a point.
11. Provetheoremsaboutparallelograms.Theorems include: oppositesides are congruent, opposite angles are congruent, the diagonalsof a parallelogram bisect each other, and conversely, rectangles areparallelograms with congruent diagonals.
Make geometric constructions
12. Makeformalgeometricconstructionswithavarietyoftoolsandmethods(compassandstraightedge,string,reflectivedevices,paperfolding,dynamicgeometricsoftware,etc.).Copying a segment;copying an angle; bisecting a segment; bisecting an angle; constructingperpendicular lines, including the perpendicular bisector of a line segment;and constructing a line parallel to a given line through a point not on theline.
13. Constructanequilateraltriangle,asquare,andaregularhexagoninscribedinacircle.
HIG
H S
CH
oo
l —
Ge
om
et
ry
| 77
Similarity, right triangles, and trigonometry G-Srt
Understand similarity in terms of similarity transformations
1. Verifyexperimentallythepropertiesofdilationsgivenbyacenterandascalefactor:
a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassingthroughthecenterunchanged.
b. Thedilationofalinesegmentislongerorshorterintheratiogivenbythescalefactor.
2. Giventwofigures,usethedefinitionofsimilarityintermsofsimilaritytransformations todecideiftheyaresimilar;explainusingsimilaritytransformationsthemeaningofsimilarityfortrianglesastheequalityofallcorrespondingpairsofanglesandtheproportionalityofallcorrespondingpairsofsides.
3. UsethepropertiesofsimilaritytransformationstoestablishtheAAcriterionfortwotrianglestobesimilar.
Prove theorems involving similarity
4. Provetheoremsabouttriangles.Theorems include: a line parallel to oneside of a triangle divides the other two proportionally, and conversely; thePythagorean Theorem proved using triangle similarity.
5. Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandtoproverelationshipsingeometricfigures.
Define trigonometric ratios and solve problems involving right triangles
6. Understandthatbysimilarity,sideratiosinrighttrianglesarepropertiesoftheanglesinthetriangle,leadingtodefinitionsoftrigonometricratiosforacuteangles.
7. Explainandusetherelationshipbetweenthesineandcosineofcomplementaryangles.
8. UsetrigonometricratiosandthePythagoreanTheoremtosolverighttrianglesinappliedproblems.★
Apply trigonometry to general triangles
9. (+)DerivetheformulaA=1/2absin(C)fortheareaofatrianglebydrawinganauxiliarylinefromavertexperpendiculartotheoppositeside.
10. (+)ProvetheLawsofSinesandCosinesandusethemtosolveproblems.
11. (+)UnderstandandapplytheLawofSinesandtheLawofCosinestofindunknownmeasurementsinrightandnon-righttriangles(e.g.,surveyingproblems,resultantforces).
Circles G-C
Understand and apply theorems about circles
1. Provethatallcirclesaresimilar.
2. Identifyanddescriberelationshipsamonginscribedangles,radii,andchords.Include the relationship between central, inscribed, andcircumscribed angles; inscribed angles on a diameter are right angles;the radius of a circle is perpendicular to the tangent where the radiusintersects the circle.
3. Constructtheinscribedandcircumscribedcirclesofatriangle,andprovepropertiesofanglesforaquadrilateralinscribedinacircle.
4. (+)Constructatangentlinefromapointoutsideagivencircletothecircle.
HIG
H S
CH
oo
l —
Ge
om
et
ry
| 78
Find arc lengths and areas of sectors of circles
5. Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleisproportionaltotheradius,anddefinetheradianmeasureoftheangleastheconstantofproportionality;derivetheformulafortheareaofasector.
expressing Geometric Properties with equations G-GPe
Translate between the geometric description and the equation for a conic section
1. DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;completethesquaretofindthecenterandradiusofacirclegivenbyanequation.
2. Derivetheequationofaparabolagivenafocusanddirectrix.
3. (+)Derivetheequationsofellipsesandhyperbolasgiventhefoci,usingthefactthatthesumordifferenceofdistancesfromthefociisconstant.
Use coordinates to prove simple geometric theorems algebraically
4. Usecoordinatestoprovesimplegeometrictheoremsalgebraically.Forexample, prove or disprove that a figure defined by four given points in thecoordinate plane is a rectangle; prove or disprove that the point (1, √3) lieson the circle centered at the origin and containing the point (0, 2).
5. Provetheslopecriteriaforparallelandperpendicularlinesandusethemtosolvegeometricproblems(e.g.,findtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint).
6. Findthepointonadirectedlinesegmentbetweentwogivenpointsthatpartitionsthesegmentinagivenratio.
7. Usecoordinatestocomputeperimetersofpolygonsandareasoftrianglesandrectangles,e.g.,usingthedistanceformula.★
Geometric measurement and dimension G-Gmd
Explain volume formulas and use them to solve problems
1. Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofacircle,volumeofacylinder,pyramid,andcone.Usedissection arguments, Cavalieri’s principle, and informal limit arguments.
2. (+)GiveaninformalargumentusingCavalieri’sprinciplefortheformulasforthevolumeofasphereandothersolidfigures.
3. Usevolumeformulasforcylinders,pyramids,cones,andspherestosolveproblems.★
Visualize relationships between two-dimensional and three-dimensional objects
4. Identifytheshapesoftwo-dimensionalcross-sectionsofthree-dimensionalobjects,andidentifythree-dimensionalobjectsgeneratedbyrotationsoftwo-dimensionalobjects.
modeling with Geometry G-mG
Apply geometric concepts in modeling situations
1. Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunkorahumantorsoasacylinder).★
2. Applyconceptsofdensitybasedonareaandvolumeinmodelingsituations(e.g.,personspersquaremile,BTUspercubicfoot).★
3. Applygeometricmethodstosolvedesignproblems(e.g.,designinganobjectorstructuretosatisfyphysicalconstraintsorminimizecost;workingwithtypographicgridsystemsbasedonratios).★
HIG
H S
CH
oo
l —
Sta
tIS
tIC
S | 7
9
mathematics | High School—Statistics and Probability★
Decisionsorpredictionsareoftenbasedondata—numbersincontext.Thesedecisionsorpredictionswouldbeeasyifthedataalwayssentaclearmessage,butthemessageisoftenobscuredbyvariability.Statisticsprovidestoolsfordescribingvariabilityindataandformakinginformeddecisionsthattakeitintoaccount.
Dataaregathered,displayed,summarized,examined,andinterpretedtodiscoverpatternsanddeviationsfrompatterns.Quantitativedatacanbedescribedintermsofkeycharacteristics:measuresofshape,center,andspread.Theshapeofadatadistributionmightbedescribedassymmetric,skewed,flat,orbellshaped,anditmightbesummarizedbyastatisticmeasuringcenter(suchasmeanormedian)andastatisticmeasuringspread(suchasstandarddeviationorinterquartilerange).Differentdistributionscanbecomparednumericallyusingthesestatisticsorcomparedvisuallyusingplots.Knowledgeofcenterandspreadarenotenoughtodescribeadistribution.Whichstatisticstocompare,whichplotstouse,andwhattheresultsofacomparisonmightmean,dependonthequestiontobeinvestigatedandthereal-lifeactionstobetaken.
Randomizationhastwoimportantusesindrawingstatisticalconclusions.First,collectingdatafromarandomsampleofapopulationmakesitpossibletodrawvalidconclusionsaboutthewholepopulation,takingvariabilityintoaccount.Second,randomlyassigningindividualstodifferenttreatmentsallowsafaircomparisonoftheeffectivenessofthosetreatments.Astatisticallysignificantoutcomeisonethatisunlikelytobeduetochancealone,andthiscanbeevaluatedonlyundertheconditionofrandomness.Theconditionsunderwhichdataarecollectedareimportantindrawingconclusionsfromthedata;incriticallyreviewingusesofstatisticsinpublicmediaandotherreports,itisimportanttoconsiderthestudydesign,howthedataweregathered,andtheanalysesemployedaswellasthedatasummariesandtheconclusionsdrawn.
Randomprocessescanbedescribedmathematicallybyusingaprobabilitymodel:alistordescriptionofthepossibleoutcomes(thesamplespace),eachofwhichisassignedaprobability.Insituationssuchasflippingacoin,rollinganumbercube,ordrawingacard,itmightbereasonabletoassumevariousoutcomesareequallylikely.Inaprobabilitymodel,samplepointsrepresentoutcomesandcombinetomakeupevents;probabilitiesofeventscanbecomputedbyapplyingtheAdditionandMultiplicationRules.Interpretingtheseprobabilitiesreliesonanunderstandingofindependenceandconditionalprobability,whichcanbeapproachedthroughtheanalysisoftwo-waytables.
Technologyplaysanimportantroleinstatisticsandprobabilitybymakingitpossibletogenerateplots,regressionfunctions,andcorrelationcoefficients,andtosimulatemanypossibleoutcomesinashortamountoftime.
Connections to Functions and Modeling.Functionsmaybeusedtodescribedata;ifthedatasuggestalinearrelationship,therelationshipcanbemodeledwitharegressionline,anditsstrengthanddirectioncanbeexpressedthroughacorrelationcoefficient.
HIG
H S
CH
oo
l —
Sta
tIS
tIC
S | 8
0
Interpreting Categorical and Quantitative Data
• Summarize, represent, and interpret data on asingle count or measurement variable
• Summarize, represent, and interpret data ontwo categorical and quantitative variables
• Interpret linear models
Making Inferences and Justifying Conclusions
• Understand and evaluate random processesunderlying statistical experiments
• make inferences and justify conclusions fromsample surveys, experiments and observationalstudies
Conditional Probability and the Rules of Prob-ability
• Understand independence and conditionalprobability and use them to interpret data
• Use the rules of probability to computeprobabilities of compound events in a uniformprobability model
Using Probability to Make Decisions
• Calculate expected values and use them tosolve problems
• Use probability to evaluate outcomes ofdecisions
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Statistics and Probability overview
HIG
H S
CH
oo
l —
Sta
tIS
tIC
S | 8
1
Interpreting Categorical and Quantitative data S-Id
Summarize, represent, and interpret data on a single count or measurement variable
1. Representdatawithplotsontherealnumberline(dotplots,histograms,andboxplots).
2. Usestatisticsappropriatetotheshapeofthedatadistributiontocomparecenter(median,mean)andspread(interquartilerange,standarddeviation)oftwoormoredifferentdatasets.
3. Interpretdifferencesinshape,center,andspreadinthecontextofthedatasets,accountingforpossibleeffectsofextremedatapoints(outliers).
4. Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandtoestimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureisnotappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormalcurve.
Summarize, represent, and interpret data on two categorical and quantitative variables
5. Summarizecategoricaldatafortwocategoriesintwo-wayfrequencytables.Interpretrelativefrequenciesinthecontextofthedata(includingjoint,marginal,andconditionalrelativefrequencies).Recognizepossibleassociationsandtrendsinthedata.
6. Representdataontwoquantitativevariablesonascatterplot,anddescribehowthevariablesarerelated.
a. Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsinthecontextofthedata.Use given functions or choosea function suggested by the context. Emphasize linear, quadratic, andexponential models.
b. Informallyassessthefitofafunctionbyplottingandanalyzingresiduals.
c. Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.
Interpret linear models
7. Interprettheslope(rateofchange)andtheintercept(constantterm)ofalinearmodelinthecontextofthedata.
8. Compute(usingtechnology)andinterpretthecorrelationcoefficientofalinearfit.
9. Distinguishbetweencorrelationandcausation.
making Inferences and Justifying Conclusions S-IC
Understand and evaluate random processes underlying statistical experiments
1. Understandstatisticsasaprocessformakinginferencesaboutpopulationparametersbasedonarandomsamplefromthatpopulation.
2. Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-generatingprocess,e.g.,usingsimulation.For example, a modelsays a spinning coin falls heads up with probability 0.5. Would a result of 5tails in a row cause you to question the model?
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
3. Recognizethepurposesofanddifferencesamongsamplesurveys,experiments,andobservationalstudies;explainhowrandomizationrelatestoeach.
HIG
H S
CH
oo
l —
Sta
tIS
tIC
S | 8
2
4. Usedatafromasamplesurveytoestimateapopulationmeanorproportion;developamarginoferrorthroughtheuseofsimulationmodelsforrandomsampling.
5. Usedatafromarandomizedexperimenttocomparetwotreatments;usesimulationstodecideifdifferencesbetweenparametersaresignificant.
6. Evaluatereportsbasedondata.
Conditional Probability and the rules of Probability S-CP
Understand independence and conditional probability and use them to interpret data
1. Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)oftheoutcomes,orasunions,intersections,orcomplementsofotherevents(“or,”“and,”“not”).
2. UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheristheproductoftheirprobabilities,andusethischaracterizationtodetermineiftheyareindependent.
3. UnderstandtheconditionalprobabilityofAgivenBas P(AandB)/P(B),andinterpretindependenceofAandBassayingthattheconditionalprobabilityof Agiven BisthesameastheprobabilityofA,andtheconditionalprobabilityofBgivenAisthesameastheprobabilityofB.
4. Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheachobjectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentandtoapproximateconditionalprobabilities.For example, collectdata from a random sample of students in your school on their favoritesubject among math, science, and English. Estimate the probability that arandomly selected student from your school will favor science given thatthe student is in tenth grade. Do the same for other subjects and comparethe results.
5. Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageandeverydaysituations.Forexample, compare the chance of having lung cancer if you are a smokerwith the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model
6. FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongtoA,andinterprettheanswerintermsofthemodel.
7. ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerintermsofthemodel.
8. (+)ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=P(A)P(B|A)=P(B)P(A|B),andinterprettheanswerintermsofthemodel.
9. (+)Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems.
Using Probability to make decisions S-md
Calculate expected values and use them to solve problems
1. (+)Definearandomvariableforaquantityofinterestbyassigninganumericalvaluetoeacheventinasamplespace;graphthecorrespondingprobabilitydistributionusingthesamegraphicaldisplaysasfordatadistributions.
2. (+)Calculatetheexpectedvalueofarandomvariable;interpretitasthemeanoftheprobabilitydistribution.
HIG
H S
CH
oo
l —
Sta
tIS
tIC
S | 8
3
3. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichtheoreticalprobabilitiescanbecalculated;findtheexpectedvalue.For example, find the theoretical probabilitydistribution for the number of correct answers obtained by guessing onall five questions of a multiple-choice test where each question has fourchoices, and find the expected grade under various grading schemes.
4. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichprobabilitiesareassignedempirically;findtheexpectedvalue.For example, find a current data distribution on thenumber of TV sets per household in the United States, and calculate theexpected number of sets per household. How many TV sets would youexpect to find in 100 randomly selected households?
Use probability to evaluate outcomes of decisions
5. (+)Weighthepossibleoutcomesofadecisionbyassigningprobabilitiestopayoffvaluesandfindingexpectedvalues.
a. Findtheexpectedpayoffforagameofchance.For example, findthe expected winnings from a state lottery ticket or a game at a fast-food restaurant.
b. Evaluateandcomparestrategiesonthebasisofexpectedvalues.For example, compare a high-deductible versus a low-deductibleautomobile insurance policy using various, but reasonable, chances ofhaving a minor or a major accident.
6. (+)Useprobabilitiestomakefairdecisions(e.g.,drawingbylots,usingarandomnumbergenerator).
7. (+)Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,pullingahockeygoalieattheendofagame).
84
note on courses and transitions
ThehighschoolportionoftheStandardsforMathematicalContentspecifiesthemathematicsallstudentsshouldstudyforcollegeandcareerreadiness.Thesestandardsdonotmandatethesequenceofhighschoolcourses.However,theorganizationofhighschoolcoursesisacriticalcomponenttoimplementationofthestandards.Tothatend,samplehighschoolpathwaysformathematics–inbothatraditionalcoursesequence(AlgebraI,Geometry,andAlgebraII)aswellasanintegratedcoursesequence(Mathematics1,Mathematics2,Mathematics3)– willbemadeavailableshortlyafterthereleaseofthefinalCommonCoreStateStandards.Itisexpectedthatadditionalmodelpathwaysbasedonthesestandardswillbecomeavailableaswell.
Thestandardsthemselvesdonotdictatecurriculum,pedagogy,ordeliveryofcontent.Inparticular,statesmayhandlethetransitiontohighschoolindifferentways.Forexample,manystudentsintheU.S.todaytakeAlgebraIinthe8thgrade,andinsomestatesthisisarequirement.TheK-7standardscontaintheprerequisitestopreparestudentsforAlgebraIby8thgrade,andthestandardsaredesignedtopermitstatestocontinueexistingpoliciesconcerningAlgebraIin8thgrade.
Asecondmajortransitionisthetransitionfromhighschooltopost-secondaryeducationforcollegeandcareers.Theevidenceconcerningcollegeandcareerreadinessshowsclearlythattheknowledge,skills,andpracticesimportantforreadinessincludeagreatdealofmathematicspriortotheboundarydefinedby(+)symbolsinthesestandards.Indeed,someofthehighestprioritycontentforcollegeandcareerreadinesscomesfromGrades6-8.Thisbodyofmaterialincludespowerfullyusefulproficienciessuchasapplyingratioreasoninginreal-worldandmathematicalproblems,computingfluentlywithpositiveandnegativefractionsanddecimals,andsolvingreal-worldandmathematicalproblemsinvolvinganglemeasure,area,surfacearea,andvolume.Becauseimportantstandardsforcollegeandcareerreadinessaredistributedacrossgradesandcourses,systemsforevaluatingcollegeandcareerreadinessshouldreachasfarbackinthestandardsasGrades6-8.Itisimportanttonoteaswellthatcutscoresorotherinformationgeneratedbyassessmentsystemsforcollegeandcareerreadinessshouldbedevelopedincollaborationwithrepresentativesfromhighereducationandworkforcedevelopmentprograms,andshouldbevalidatedbysubsequentperformanceofstudentsincollegeandtheworkforce.
Glo
SS
ar
y | 8
5
Addition and subtraction within 5, 10, 20, 100, or 1000.Additionorsubtractionoftwowholenumberswithwholenumberanswers,andwithsumorminuendintherange0-5,0-10,0-20,or0-100,respectively.Example:8+2=10isanadditionwithin10,14–5=9isasubtractionwithin20,and55–18=37isasubtractionwithin100.
Additive inverses.Twonumberswhosesumis0areadditiveinversesofoneanother.Example:3/4and–3/4areadditiveinversesofoneanotherbecause3/4+(–3/4)=(–3/4)+3/4=0.
Associative property of addition.SeeTable3inthisGlossary.
Associative property of multiplication. SeeTable3inthisGlossary.
Bivariate data. Pairsoflinkednumericalobservations.Example:alistofheightsandweightsforeachplayeronafootballteam.
Box plot.Amethodofvisuallydisplayingadistributionofdatavaluesbyusingthemedian,quartiles,andextremesofthedataset.Aboxshowsthemiddle50%ofthedata.1
Commutative property.SeeTable3inthisGlossary.
Complex fraction.AfractionA/BwhereAand/orBarefractions(Bnonzero).
Computation algorithm.Asetofpredefinedstepsapplicabletoaclassofproblemsthatgivesthecorrectresultineverycasewhenthestepsarecarriedoutcorrectly.See also:computationstrategy.
Computation strategy.Purposefulmanipulationsthatmaybechosenforspecificproblems,maynothaveafixedorder,andmaybeaimedatconvertingoneproblemintoanother.See also:computationalgorithm.
Congruent.Twoplaneorsolidfiguresarecongruentifonecanbeobtainedfromtheotherbyrigidmotion(asequenceofrotations,reflections,andtranslations).
Counting on.Astrategyforfindingthenumberofobjectsinagroupwithouthavingtocounteverymemberofthegroup.Forexample,ifastackofbooksisknowntohave8booksand3morebooksareaddedtothetop,itisnotnecessarytocountthestackalloveragain.Onecanfindthetotalbycounting on—pointingtothetopbookandsaying“eight,”followingthiswith“nine,ten,eleven.Thereareelevenbooksnow.”
Dot plot. See: lineplot.
Dilation.Atransformationthatmoveseachpointalongtheraythroughthepointemanatingfromafixedcenter,andmultipliesdistancesfromthecenterbyacommonscalefactor.
Expanded form.Amulti-digitnumberisexpressedinexpandedformwhenitiswrittenasasumofsingle-digitmultiplesofpowersoften.Forexample,643=600+40+3.
Expected value. Forarandomvariable,theweightedaverageofitspossiblevalues,withweightsgivenbytheirrespectiveprobabilities.
First quartile. ForadatasetwithmedianM,thefirstquartileisthemedianofthedatavalueslessthanM.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},thefirstquartileis6.2See also:median,thirdquartile,interquartilerange.
Fraction.Anumberexpressibleintheforma/bwhereaisawholenumberandbisapositivewholenumber.(Thewordfractioninthesestandardsalwaysreferstoanon-negativenumber.)See also:rationalnumber.
Identity property of 0.SeeTable3inthisGlossary.
Independently combined probability models.Twoprobabilitymodelsaresaidtobecombinedindependentlyiftheprobabilityofeachorderedpairinthecombinedmodelequalstheproductoftheoriginalprobabilitiesofthetwoindividualoutcomesintheorderedpair.
1AdaptedfromWisconsinDepartmentofPublicInstruction,http://dpi.wi.gov/standards/mathglos.html,accessedMarch2,2010.2Manydifferentmethodsforcomputingquartilesareinuse.ThemethoddefinedhereissometimescalledtheMooreandMcCabemethod.SeeLangford,E.,“QuartilesinElementaryStatistics,”Journal of Statistics EducationVolume14,Number3(2006).
Glossary
Glo
SS
ar
y | 8
6
Integer.Anumberexpressibleintheformaor–aforsomewholenumbera.
Interquartile Range. Ameasureofvariationinasetofnumericaldata,theinterquartilerangeisthedistancebetweenthefirstandthirdquartilesofthedataset.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},theinterquartilerangeis15–6=9.See also:firstquartile,thirdquartile.
Line plot.Amethodofvisuallydisplayingadistributionofdatavalueswhereeachdatavalueisshownasadotormarkaboveanumberline.Alsoknownasadotplot.3
Mean.Ameasureofcenterinasetofnumericaldata,computedbyaddingthevaluesinalistandthendividingbythenumberofvaluesinthelist.4Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},themeanis21.
Mean absolute deviation.Ameasureofvariationinasetofnumericaldata,computedbyaddingthedistancesbetweeneachdatavalueandthemean,thendividingbythenumberofdatavalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},themeanabsolutedeviationis20.
Median.Ameasureofcenterinasetofnumericaldata.Themedianofalistofvaluesisthevalueappearingatthecenterofasortedversionofthelist—orthemeanofthetwocentralvalues,ifthelistcontainsanevennumberofvalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,90},themedianis11.
Midline. Inthegraphofatrigonometricfunction,thehorizontallinehalfwaybetweenitsmaximumandminimumvalues.
Multiplication and division within 100.Multiplicationordivisionoftwowholenumberswithwholenumberanswers,andwithproductordividendintherange0-100.Example:72÷8=9.
Multiplicative inverses.Twonumberswhoseproductis1aremultiplicativeinversesofoneanother.Example:3/4and4/3aremultiplicativeinversesofoneanotherbecause3/4×4/3=4/3×3/4=1.
Number line diagram. Adiagramofthenumberlineusedtorepresentnumbersandsupportreasoningaboutthem.Inanumberlinediagramformeasurementquantities,theintervalfrom0to1onthediagramrepresentstheunitofmeasureforthequantity.
Percent rate of change.Arateofchangeexpressedasapercent.Example:ifapopulationgrowsfrom50to55inayear,itgrowsby5/50=10%peryear.
Probability distribution.Thesetofpossiblevaluesofarandomvariablewithaprobabilityassignedtoeach.
Properties of operations.SeeTable3inthisGlossary.
Properties of equality.SeeTable4inthisGlossary.
Properties of inequality.SeeTable5inthisGlossary.
Properties of operations.SeeTable3inthisGlossary.
Probability.Anumberbetween0and1usedtoquantifylikelihoodforprocessesthathaveuncertainoutcomes(suchastossingacoin,selectingapersonatrandomfromagroupofpeople,tossingaballatatarget,ortestingforamedicalcondition).
Probability model. Aprobabilitymodelisusedtoassignprobabilitiestooutcomesofachanceprocessbyexaminingthenatureoftheprocess.Thesetofalloutcomesiscalledthesamplespace,andtheirprobabilitiessumto1.See also: uniformprobabilitymodel.
Random variable. Anassignmentofanumericalvaluetoeachoutcomeinasamplespace.
Rational expression.Aquotientoftwopolynomialswithanon-zerodenominator.
Rational number.Anumberexpressibleintheforma/bor– a/bforsomefractiona/b.Therationalnumbersincludetheintegers.
Rectilinear figure. Apolygonallanglesofwhicharerightangles.
Rigid motion.Atransformationofpointsinspaceconsistingofasequenceof
3AdaptedfromWisconsinDepartmentofPublicInstruction,op. cit.4Tobemoreprecise,thisdefinesthearithmetic mean.
Glo
SS
ar
y | 8
7
oneormoretranslations,reflections,and/orrotations.Rigidmotionsarehereassumedtopreservedistancesandanglemeasures.
Repeating decimal.Thedecimalformofarationalnumber.See also:terminatingdecimal.
Sample space.Inaprobabilitymodelforarandomprocess,alistoftheindividualoutcomesthataretobeconsidered.
Scatter plot. Agraphinthecoordinateplanerepresentingasetofbivariatedata.Forexample,theheightsandweightsofagroupofpeoplecouldbedisplayedonascatterplot.5
Similarity transformation.Arigidmotionfollowedbyadilation.
Tape diagram.Adrawingthatlookslikeasegmentoftape,usedtoillustratenumberrelationships.Alsoknownasastripdiagram,barmodel,fractionstrip,orlengthmodel.
Terminating decimal. Adecimaliscalledterminatingifitsrepeatingdigitis0.
Third quartile.ForadatasetwithmedianM,thethirdquartileisthemedianofthedatavaluesgreaterthanM.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},thethirdquartileis15.See also:median,firstquartile,interquartilerange.
Transitivity principle for indirect measurement. IfthelengthofobjectAisgreaterthanthelengthofobjectB,andthelengthofobjectBisgreaterthanthelengthofobjectC,thenthelengthofobjectAisgreaterthanthelengthofobjectC.Thisprincipleappliestomeasurementofotherquantitiesaswell.
Uniform probability model.Aprobabilitymodelwhichassignsequalprobabilitytoalloutcomes.See also:probabilitymodel.
Vector. Aquantitywithmagnitudeanddirectionintheplaneorinspace,definedbyanorderedpairortripleofrealnumbers.
Visual fraction model. Atapediagram,numberlinediagram,orareamodel.
Whole numbers.Thenumbers0,1,2,3,….
5AdaptedfromWisconsinDepartmentofPublicInstruction,op. cit.
Glo
SS
ar
y | 8
8
Table 1.Commonadditionandsubtractionsituations.6
result Unknown Change Unknown Start Unknown
add to
Twobunniessatonthegrass.Threemorebunnieshoppedthere.Howmanybunniesareonthegrassnow?
2+3=?
Twobunniesweresittingonthegrass.Somemorebunnieshoppedthere.Thentherewerefivebunnies.Howmanybunnieshoppedovertothefirsttwo?
2+?=5
Somebunniesweresittingonthegrass.Threemorebunnieshoppedthere.Thentherewerefivebunnies.Howmanybunnieswereonthegrassbefore?
?+3=5
take from
Fiveappleswereonthetable.Iatetwoapples.Howmanyapplesareonthetablenow?
5–2=?
Fiveappleswereonthetable.Iatesomeapples.Thentherewerethreeapples.HowmanyapplesdidIeat?
5–?=3
Someappleswereonthetable.Iatetwoapples.Thentherewerethreeapples.Howmanyappleswereonthetablebefore?
?–2=3
total Unknown addend Unknown Both addends Unknown1
Put together/ take apart2
Threeredapplesandtwogreenapplesareonthetable.Howmanyapplesareonthetable?
3+2=?
Fiveapplesareonthetable.Threeareredandtherestaregreen.Howmanyapplesaregreen?
3+?=5,5–3=?
Grandmahasfiveflowers.Howmanycansheputinherredvaseandhowmanyinherbluevase?
5=0+5,5=5+0
5=1+4,5=4+1
5=2+3,5=3+2
difference Unknown Bigger Unknown Smaller Unknown
Compare3
(“Howmanymore?”version):
Lucyhastwoapples.Juliehasfiveapples.HowmanymoreapplesdoesJuliehavethanLucy?
(“Howmanyfewer?”version):
Lucyhastwoapples.Juliehasfiveapples.HowmanyfewerapplesdoesLucyhavethanJulie?
2+?=5,5–2=?
(Versionwith“more”):
JuliehasthreemoreapplesthanLucy.Lucyhastwoapples.HowmanyapplesdoesJuliehave?
(Versionwith“fewer”):
Lucyhas3fewerapplesthanJulie.Lucyhastwoapples.HowmanyapplesdoesJuliehave?
2+3=?,3+2=?
(Versionwith“more”):
JuliehasthreemoreapplesthanLucy.Juliehasfiveapples.HowmanyapplesdoesLucyhave?
(Versionwith“fewer”):
Lucyhas3fewerapplesthanJulie.Juliehasfiveapples.HowmanyapplesdoesLucyhave?
5–3=?,?+3=5
6AdaptedfromBox2-4ofMathematicsLearninginEarlyChildhood,NationalResearchCouncil(2009,pp.32,33).
1Thesetakeapartsituationscanbeusedtoshowallthedecompositionsofagivennumber.Theassociatedequations,whichhavethetotalontheleftoftheequalsign,helpchildrenunderstandthatthe=signdoesnotalwaysmeanmakesorresultsinbutalwaysdoesmeanisthesamenumberas.2Eitheraddendcanbeunknown,sotherearethreevariationsoftheseproblemsituations.BothAddendsUnknownisapro-ductiveextensionofthisbasicsituation,especiallyforsmallnumberslessthanorequalto10.3FortheBiggerUnknownorSmallerUnknownsituations,oneversiondirectsthecorrectoperation(theversionusingmoreforthebiggerunknownandusinglessforthesmallerunknown).Theotherversionsaremoredifficult.
Glo
SS
ar
y | 8
9
Table 2. Commonmultiplicationanddivisionsituations.7
Unknown ProductGroup Size Unknown
(“Howmanyineachgroup?”Division)
number of Groups Unknown(“Howmanygroups?”Division)
3 × 6 = ? 3 × ? = 18, and 18 ÷ 3 = ? ? × 6 = 18, and 18 ÷ 6 = ?
equal Groups
Thereare3bagswith6plumsineachbag.Howmanyplumsarethereinall?
Measurement example.Youneed3lengthsofstring,each6incheslong.Howmuchstringwillyouneedaltogether?
If18plumsaresharedequallyinto3bags,thenhowmanyplumswillbeineachbag?
Measurement example.Youhave18inchesofstring,whichyouwillcutinto3equalpieces.Howlongwilleachpieceofstringbe?
If18plumsaretobepacked6toabag,thenhowmanybagsareneeded?
Measurement example.Youhave18inchesofstring,whichyouwillcutintopiecesthatare6incheslong.Howmanypiecesofstringwillyouhave?
arrays,4 area5
Thereare3rowsofappleswith6applesineachrow.Howmanyapplesarethere?
Area example.Whatistheareaofa3cmby6cmrectangle?
If18applesarearrangedinto3equalrows,howmanyappleswillbeineachrow?
Area example.Arectanglehasarea18squarecentimeters.Ifonesideis3cmlong,howlongisasidenexttoit?
If18applesarearrangedintoequalrowsof6apples,howmanyrowswilltherebe?
Area example.Arectanglehasarea18squarecentimeters.Ifonesideis6cmlong,howlongisasidenexttoit?
Compare
Abluehatcosts$6.Aredhatcosts3timesasmuchasthebluehat.Howmuchdoestheredhatcost?
Measurement example.Arubberbandis6cmlong.Howlongwilltherubberbandbewhenitisstretchedtobe3timesaslong?
Aredhatcosts$18andthatis3timesasmuchasabluehatcosts.Howmuchdoesabluehatcost?
Measurement example.Arubberbandisstretchedtobe18cmlongandthatis3timesaslongasitwasatfirst.Howlongwastherubberbandatfirst?
Aredhatcosts$18andabluehatcosts$6.Howmanytimesasmuchdoestheredhatcostasthebluehat?
Measurement example.Arubberbandwas6cmlongatfirst.Nowitisstretchedtobe18cmlong.Howmanytimesaslongistherubberbandnowasitwasatfirst?
General a×b=? a×? =p,andp÷ a=? ?×b=p, and p÷ b =?
7Thefirstexamplesineachcellareexamplesofdiscretethings.Theseareeasierforstudentsandshouldbegivenbeforethemeasurementexamples.
4Thelanguageinthearrayexamplesshowstheeasiestformofarrayproblems.Aharderformistousethetermsrowsandcolumns:Theapplesinthegrocerywindowarein3rowsand6columns.Howmanyapplesareinthere?Bothformsarevaluable.5Areainvolvesarraysofsquaresthathavebeenpushedtogethersothattherearenogapsoroverlaps,soarrayproblemsincludetheseespeciallyimportantmeasurementsituations.
Glo
SS
ar
y | 9
0
Table 3.Thepropertiesofoperations.Herea,bandcstandforarbitrarynumbersinagivennumbersystem.Thepropertiesofoperationsapplytotherationalnumbersystem,therealnumbersystem,andthecomplexnumbersystem.
Associative property of addition
Commutative property of addition
Additive identity property of 0
Existence of additive inverses
Associative property of multiplication
Commutative property of multiplication
Multiplicative identity property of 1
Existence of multiplicative inverses
Distributive property of multiplication over addition
(a + b)+ c = a + (b + c)
a + b= b + a
a + 0 = 0+ a= a
Foreveryathereexists–asothata+(–a)= (–a)+a=0.
(a × b)× c = a × (b × c)
a × b= b × a
a × 1= 1×a= a
Foreverya≠0thereexists1/asothata×1/a=1/a× a=1.
a ×(b + c) = a × b + a × c
Table 4.Thepropertiesofequality.Herea,bandcstandforarbitrarynumbersintherational,real,orcomplexnumbersystems.
Reflexive property of equality
Symmetric property of equality
Transitive property of equality
Addition property of equality
Subtraction property of equality
Multiplication property of equality
Division property of equality
Substitution property of equality
a=a
If a = b,then b = a.
If a = b and b = c,then a = c.
If a = b,then a + c = b + c.
If a = b,then a – c = b – c.
If a = b,then a × c = b × c.
If a = b and c ≠ 0,then a ÷ c = b ÷ c.
Ifa=b,thenbmaybesubstitutedfora
inanyexpressioncontaininga.
Table 5.Thepropertiesofinequality.Herea,bandcstandforarbitrarynumbersintherationalorrealnumbersystems.
Exactlyoneofthefollowingistrue:a<b,a=b,a>b.
Ifa>bandb>cthena>c.
Ifa>b,thenb<a.
Ifa>b,then–a<–b.
If a>b,thena±c>b±c.
Ifa>bandc>0,thena ×c>b ×c.
Ifa>bandc<0,thena ×c<b ×c.
Ifa>bandc>0,thena ÷c>b ÷c.
Ifa>bandc<0,thena ÷c<b ÷c.
wo
rK
S C
on
SU
lte
d | 9
1
Existingstatestandardsdocuments.
ResearchsummariesandbriefsprovidedtotheWorkingGroupbyresearchers.
NationalAssessmentGoverningBoard,Mathematics Framework for the 2009 National Assessment of Educational Progress.U.S.DepartmentofEducation,2008.
NAEPValidityStudiesPanel,ValidityStudyoftheNAEPMathematicsAssessment:Grades4and8.Daroetal.,2007.
Mathematicsdocumentsfrom:Alberta,Canada;Belgium;China;ChineseTaipei;Denmark;England;Finland;HongKong;India;Ireland;Japan;Korea;NewZealand;Singapore;Victoria(BritishColumbia).
AddingitUp:HelpingChildrenLearnMathematics.NationalResearchCouncil,MathematicsLearningStudyCommittee,2001.
BenchmarkingforSuccess:EnsuringU.S.StudentsReceiveaWorld-ClassEducation.NationalGovernorsAssociation,CouncilofChiefStateSchoolOfficers,andAchieve,Inc.,2008.
Crossroads in Mathematics(1995)andBeyond Crossroads (2006).AmericanMathematicalAssociationofTwo-YearColleges(AMATYC).
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence.NationalCouncilofTeachersofMathematics,2006.
Focus in High School Mathematics: Reasoning and Sense Making.NationalCouncilofTeachersofMathematics.Reston,VA:NCTM.
Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S.DepartmentofEducation:Washington,DC,2008.
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A PreK-12 Curriculum Framework.
How People Learn: Brain, Mind, Experience, and School. Bransford,J.D.,Brown,A.L.,andCocking,R.R.,eds.CommitteeonDevelopmentsintheScienceofLearning,CommissiononBehavioralandSocialSciencesandEducation,NationalResearchCouncil,1999.
Mathematics and Democracy, The Case for Quantitative Literacy,Steen,L.A.(ed.).NationalCouncilonEducationandtheDisciplines,2001.
Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Cross,C.T.,Woods,T.A.,andSchweingruber,S.,eds.CommitteeonEarlyChildhoodMathematics,NationalResearchCouncil,2009.
The Opportunity Equation: Transforming Mathematics and Science Education for Citizenship and the Global Economy.TheCarnegieCorporationofNewYorkandtheInstituteforAdvancedStudy,2009.Online:http://www.opportunityequation.org/
Principles and Standards for School Mathematics.NationalCouncilofTeachersofMathematics,2000.
The Proficiency Illusion.Cronin,J.,Dahlin,M.,Adkins,D.,andKingsbury,G.G.;forewordbyC.E.Finn,Jr.,andM.J.Petrilli.ThomasB.FordhamInstitute,2007.
Ready or Not: Creating a High School Diploma That Counts.AmericanDiplomaProject,2004.
A Research Companion to Principles and Standards for School Mathematics.NationalCouncilofTeachersofMathematics,2003.
Sizing Up State Standards 2008.AmericanFederationofTeachers,2008.
A Splintered Vision: An Investigation of U.S. Science and Mathematics Education.Schmidt,W.H.,McKnight,C.C.,Raizen,S.A.,etal.U.S.NationalResearchCenterfortheThirdInternationalMathematicsandScienceStudy,MichiganStateUniversity,1997.
Stars By Which to Navigate? Scanning National and International Education Standards in 2009.Carmichael,S.B.,Wilson.W.S,Finn,Jr.,C.E.,Winkler,A.M.,andPalmieri,S.ThomasB.FordhamInstitute,2009.
Askey,R.,“KnowingandTeachingElementaryMathematics,”American Educator,Fall1999.
Aydogan,C.,Plummer,C.,Kang,S.J.,Bilbrey,C.,Farran,D.C.,&Lipsey,M.W.(2005).Aninvestigationofprekindergartencurricula:Influencesonclassroomcharacteristicsandchildengagement.PaperpresentedattheNAEYC.
Blum,W.,Galbraith,P.L.,Henn,H-W.andNiss,M.(Eds)Applications and Modeling in Mathematics Education,ICMIStudy14.Amsterdam:Springer.
Brosterman,N.(1997).Inventing kindergarten.NewYork:HarryN.Abrams.
Clements,D.H.,&Sarama,J.(2009).Learning and teaching early math: The learning trajectories approach.NewYork:Routledge.
Clements,D.H.,Sarama,J.,&DiBiase,A.-M.(2004).Clements,D.H.,Sarama,J.,&DiBiase,A.-M.(2004).Engaging young children in mathematics: Standards for early childhood mathematics education.Mahwah,NJ:LawrenceErlbaumAssociates.
CobbandMoore,“Mathematics,Statistics,andTeaching,”Amer. Math. Monthly104(9),pp.801-823,1997.
Confrey,J.,“TracingtheEvolutionofMathematicsContentStandardsintheUnitedStates:LookingBackandProjectingForward.”K12MathematicsCurriculumStandardsconferenceproceedings,February5-6,2007.
Conley,D.T.Knowledge and Skills for University Success,2008.
Conley,D.T.Toward a More Comprehensive Conception of College Readiness,2007.
Cuoco,A.,Goldenberg,E.P.,andMark,J.,“HabitsofMind:AnOrganizingPrincipleforaMathematicsCurriculum,”Journal of Mathematical Behavior,15(4),375-402,1996.
Carpenter,T.P.,Fennema,E.,Franke,M.L.,Levi,L.,&Empson,S.B.(1999).Children’s Mathematics: Cognitively Guided Instruction.Portsmouth,NH:Heinemann.
VandeWalle,J.A.,Karp,K.,&Bay-Williams,J.M.(2010).Elementary and Middle School Mathematics: Teaching Developmentally(Seventhed.).Boston:AllynandBacon.
Ginsburg,A.,Leinwand,S.,andDecker,K.,“InformingGrades1-6StandardsDevelopment:WhatCanBeLearnedfromHigh-PerformingHongKong,Korea,andSingapore?”AmericanInstitutesforResearch,2009.
Ginsburgetal.,“WhattheUnitedStatesCanLearnFromSingapore’sWorld-ClassMathematicsSystem(andwhatSingaporecanlearnfromtheUnitedStates),”AmericanInstitutesforResearch,2005.
Ginsburgetal.,“ReassessingU.S.InternationalMathematicsPerformance:NewFindingsfromthe2003TIMMSandPISA,”AmericanInstitutesforResearch,2005.
Ginsburg,H.P.,Lee,J.S.,&Stevenson-Boyd,J.(2008).Mathematicseducationforyoungchildren:Whatitisandhowtopromoteit.Social Policy Report,22(1),1-24.
SampleofWorksConsulted
wo
rK
S C
on
SU
lte
d | 9
2
Harel,G.,“WhatisMathematics?APedagogicalAnswertoaPhilosophicalQuestion,”inR.B.GoldandR.Simons(eds.),Current Issues in the Philosophy of Mathematics from the Perspective of Mathematicians.MathematicalAssociationofAmerica,2008.
Henry,V.J.,&Brown,R.S.(2008).First-gradebasicfacts:Aninvestigationintoteachingandlearningofanaccelerated,high-demandmemorizationstandard.Journal for Research in Mathematics Education,39,153-183.
Howe,R.,“FromArithmetictoAlgebra.”
Howe,R.,“StartingOffRightinArithmetic,”http://math.arizona.edu/~ime/2008-09/MIME/BegArith.pdf.
Jordan,N.C.,Kaplan,D.,Ramineni,C.,andLocuniak,M.N.,“Earlymathmatters:kindergartennumbercompetenceandlatermathematicsoutcomes,”Dev. Psychol.45,850–867,2009.
Kader,G.,“MeansandMADS,”Mathematics Teaching in the Middle School,4(6),1999,pp.398-403.
Kilpatrick,J.,Mesa,V.,andSloane,F.,“U.S.AlgebraPerformanceinanInternationalContext,”inLoveless(ed.),Lessons Learned: What International Assessments Tell Us About Math Achievement.Washington,D.C.:BrookingsInstitutionPress,2007.
Leinwand,S.,andGinsburg,A.,“MeasuringUp:HowtheHighestPerformingState(Massachusetts)ComparestotheHighestPerformingCountry(HongKong)inGrade3Mathematics,”AmericanInstitutesforResearch,2009.
Niss,M.,“QuantitativeLiteracyandMathematicalCompetencies,”inQuantitative Literacy: Why Numeracy Matters for Schools and Colleges,Madison,B.L.,andSteen,L.A.(eds.),NationalCouncilonEducationandtheDisciplines.ProceedingsoftheNationalForumonQuantitativeLiteracyheldattheNationalAcademyofSciencesinWashington,D.C.,December1-2,2001.
Pratt,C.(1948).Ilearnfromchildren.NewYork:SimonandSchuster.
Reys,B.(ed.),The Intended Mathematics Curriculum as Represented in State-Level Curriculum Standards: Consensus or Confusion? IAP-InformationAgePublishing,2006.
Sarama,J.,&Clements,D.H.(2009).Early childhood mathematics education research: Learning trajectories for young children.NewYork:Routledge.
Schmidt,W.,Houang,R.,andCogan,L.,“ACoherentCurriculum:TheCaseofMathematics,”American Educator,Summer2002,p.4.
Schmidt,W.H.,andHouang,R.T.,“LackofFocusintheIntendedMathematicsCurriculum:SymptomorCause?”inLoveless(ed.),Lessons Learned: What International Assessments Tell Us About Math Achievement.Washington,D.C.:BrookingsInstitutionPress,2007.
Steen,L.A.,“FacingFacts:AchievingBalanceinHighSchoolMathematics.”Mathematics Teacher,Vol.100.SpecialIssue.
Wu,H.,“Fractions,decimals,andrationalnumbers,”2007,http://math.berkeley.edu/~wu/(March19,2008).
Wu,H.,“LectureNotesforthe2009Pre-AlgebraInstitute,”September15,2009.
Wu,H.,“Preserviceprofessionaldevelopmentofmathematicsteachers,”http://math.berkeley.edu/~wu/pspd2.pdf.
MassachusettsDepartmentofEducation.ProgressReportoftheMathematicsCurriculumFrameworkRevisionPanel,MassachusettsDepartmentofElementaryandSecondaryEducation,2009.
www.doe.mass.edu/boe/docs/0509/item5_report.pdf.
ACTCollegeReadinessBenchmarks™
ACTCollegeReadinessStandards™
ACTNationalCurriculumSurvey™
Adelman,C.,The Toolbox Revisited: Paths to Degree Completion From High School Through College,2006.
Advanced Placement Calculus, Statistics and Computer Science Course Descriptions.May 2009, May 2010. CollegeBoard,2008.
Aligning Postsecondary Expectations and High School Practice: The Gap Defined(ACT:PolicyImplicationsoftheACTNationalCurriculumSurveyResults2005-2006).
Condition of Education, 2004: Indicator 30, Top 30 Postsecondary Courses, U.S.DepartmentofEducation,2004.
Condition of Education, 2007: High School Course-Taking.U.S.DepartmentofEducation,2007.
Crisis at the Core: Preparing All Students for College and Work,ACT.
Achieve,Inc.,FloridaPostsecondarySurvey,2008.
Golfin,Peggy,etal.CNACorporation.Strengthening Mathematics at the Postsecondary Level: Literature Review and Analysis,2005.
Camara,W.J.,Shaw,E.,andPatterson,B.(June13,2009).FirstYearEnglishandMathCollegeCoursework.CollegeBoard:NewYork,NY(Availablefromauthors).
CLEPPrecalculusCurriculumSurvey:SummaryofResults.TheCollegeBoard,2005.
CollegeBoardStandardsforCollegeSuccess:MathematicsandStatistics.CollegeBoard,2006.
Miller,G.E.,Twing,J.,andMeyers,J.“HigherEducationReadinessComponent(HERC)CorrelationStudy.”Austin,TX:Pearson.
On Course for Success: A Close Look at Selected High School Courses That Prepare All Students for College and Work,ACT.
Out of Many, One: Towards Rigorous Common Core Standards from the Ground Up.Achieve,2008.
Ready for College and Ready for Work: Same or Different?ACT.
RigoratRisk:ReaffirmingQualityintheHighSchoolCoreCurriculum,ACT.
The Forgotten Middle: Ensuring that All Students Are on Target for College and Career Readiness before High School,ACT.
Achieve,Inc.,VirginiaPostsecondarySurvey,2004.
ACTJobSkillComparisonCharts.
Achieve,MathematicsatWork,2008.
The American Diploma Project Workplace Study.NationalAllianceofBusinessStudy,2002.
Carnevale,AnthonyandDesrochers,Donna.Connecting Education Standards and Employment: Course-taking Patterns of Young Workers,2002.
ColoradoBusinessLeaders’TopSkills,2006.
Hawai’i Career Ready Study: access to living wage careers from high school,2007.
States’CareerClusterInitiative.Essential Knowledge and Skill Statements,2008.
ACTWorkKeysOccupationalProfiles™.
ProgramforInternationalStudentAssessment(PISA),2006.
TrendsinInternationalMathematicsandScienceStudy(TIMSS),2007.
wo
rK
S C
on
SU
lte
d | 9
3
InternationalBaccalaureate,MathematicsStandardLevel,2006.
UniversityofCambridgeInternationalExaminations:GeneralCertificateofSecondaryEducationinMathematics,2009.
EdExcel,GeneralCertificateofSecondaryEducation,Mathematics,2009.
Blachowicz,Camille,andFisher,Peter.“VocabularyInstruction.”InHandbook of Reading Research,VolumeIII,editedbyMichaelKamil,PeterMosenthal,P.DavidPearson,andRebeccaBarr,pp.503-523.Mahwah,NJ:LawrenceErlbaumAssociates,2000.
Gándara,Patricia,andContreras,Frances.The Latino Education Crisis: The Consequences of Failed Social Policies.Cambridge,Ma:HarvardUniversityPress,2009.
Moschkovich,JuditN.“SupportingtheParticipationofEnglishLanguageLearnersinMathematicalDiscussions.”For the Learning of Mathematics19(March1999):11-19.
Moschkovich,J.N.(inpress).Language,culture,andequityinsecondarymathematicsclassrooms.ToappearinF.Lester&J.Lobato(ed.),TeachingandLearningMathematics:TranslatingResearchtotheSecondaryClassroom, Reston,VA:NCTM.
Moschkovich,JuditN.“ExaminingMathematicalDiscoursePractices,”For the Learning of Mathematics 27 (March2007):24-30.
Moschkovich,JuditN.“UsingTwoLanguageswhenLearningMathematics:HowCanResearchHelpUsUnderstandMathematicsLearnersWhoUseTwoLanguages?”Research Brief and Clip,NationalCouncilofTeachersofMathematics,2009http://www.nctm.org/uploadedFiles/Research_News_and_Advocacy/Research/Clips_and_Briefs/Research_brief_12_Using_2.pdf.(accessedNovember25,2009).
Moschkovich,J.N.(2007)BilingualMathematicsLearners:Howviewsoflanguage,bilinguallearners,andmathematicalcommunicationimpactinstruction.InNasir,N.andCobb,P.(eds.),Diversity, Equity, and Access to Mathematical Ideas.NewYork:TeachersCollegePress,89-104.
Schleppegrell,M.J.(2007).Thelinguisticchallengesofmathematicsteachingandlearning:Aresearchreview.Reading & Writing Quarterly, 23:139-159.
IndividualswithDisabilitiesEducationAct(IDEA),34CFR§300.34(a).(2004).
IndividualswithDisabilitiesEducationAct(IDEA),34CFR§300.39(b)(3).(2004).
OfficeofSpecialEducationPrograms,U.S.DepartmentofEducation.“IDEARegulations:IdentificationofStudentswithSpecificLearningDisabilities,”2006.
Thompson,S.J.,Morse,A.B.,Sharpe,M.,andHall,S.,“AccommodationsManual:HowtoSelect,AdministerandEvaluateUseofAccommodationsandAssessmentforStudentswithDisabilities,”2ndEdition.CouncilofChiefStateSchoolOfficers,2005.