singapore math scope & sequence [math in focus]

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Deep Math Understanding. Real-World Problem Solvers.

Grades K–5 Scope and Sequence

Concept and skill development through hands-on instruction and practice

3

Key Differences and Distinguishing Characteristics

• basicfacts• placevalue• mentalmath• geometryconcepts

• fractions• decimals

• ratios• modeldrawing

• expressions,equations,andinequalities

Scope and Sequence Grades K–5

Articulated SequenceMath in Focusanswersthecallforacoherentsequenceof topicsgivingstudentstimetomasterfoundationaltopics, sothatlittlerepetitionisrequiredthenextyear.Thus,eachgradelevelcoversfewertopicsbutinmoredepth,andyouwon’tfindalltopicsineverygradelevel.

• “Missingtopics” Whenatopicappearstobe“missing,”youcanbeassuredthatitisfoundineitheranearlierorlatergradelevel.ForexampleyouwillfindcalendarconceptsinGradesKand1,butnotrepeatedinGrade2.

• Moreadvanced Asaresultof notrepeatingtopicsyearafteryear,studentswhouseMath in Focus willadvancefasterthanstudentsinotherprograms.Asaresult, youmayfindtopicsthatseemtobe“tooadvanced.”However,youwillfindyourstudentseasilyabletohandlethechallengeaslongastheyhavehadtheappropriatepreliminaryinstruction.

Developmental ContinuumKindergarten Grades1–2 Grades3–5

Preparation for AlgebraMath in Focusanswersthecalltopreparestudentsforalgebra.AsrecommendedbytheNationalMathPanel,theMath in Focussequenceof topicsemphasizes:

• Numbersense,basicfacts,andcomputation Anearlyunderstandingof compositionanddecompositionof numbersisdevelopedintandemwithmasteryof basic factsandcomputationalgorithmsinGradesK–2.

• Fractionsandproportionalreasoning Significanttimeisallocatedforin-depthworkwithfractionsinGrades3–5.

• Problemsolving Challengingproblemsolvingisbuilt intoeachchapterineverygradelevel.

Foundationalconceptsthroughsongs,rhymes,andhands-onactivities

Emphasisonproblemsolving,skillconsolidation,andadeepunderstandinginpreparationforalgebra

Math in Focus™:The Singapore Approach is the U.S. Edition of Singapore’s most widely used program.

Grade K

Grade 3

Grade 1

Grade 4

Grade 2

Grade 5

4 5

Kindergarten Grade1 Grade2 Grade3 Grade4 Grade5

NumberandOperations NumberandOperations

Setsand Numbers

Use concrete models to createasetwithagivennumberof objects(upto20).

Use concrete and pictorial modelstocreateasetwithagivennumberof objects(upto100).

Use concrete and pictorial modelstocreateasetwithagivennumberof objects(upto1,000).

Setsand Numbers

Groupobjectsandnumbersupto100intensandones.

Groupobjectsandnumbersupto1,000intohundreds,tens,andones.

Groupobjectsintoequalsizedgroups.

Use cardinal and ordinal numbers.

Usecardinalnumbersupto100andordinalnumbersupto10th.

Explorenegativenumbersincontext.

Number Representation

Usenumberstorepresentquantitiesupto20.

Usenumberbondstorepresentnumbercombinations.

Use place-value models to create equivalent representationsof numbers.

Number Representation

Representnumbersto10,000indifferentequivalentforms.

Representnumbersto100,000invariouscontexts.

Understand place value conceptsthroughmillions.

Representnumbersto100onanumberline.

Representnumbersto1,000onanumberline.

Count Countupto20objectsinaset.

Countto100. Countto1,000. Count Countto10,000.

Countonandbackto20. Countby1s,2s,5s,and10sforwardandbackwardto100.

Countbymultiplesof ones,tens,andhundreds.

Countbyhundredsandthousands.

Countbythousandsandtenthousands.

Countbyhundredthousandsandmillions.

Countin2sand5supto20.

CompareandOrder

Compare and order sets and numbersupto20.

Compareandorderwholenumbersto100.

Compareandorderwholenumbersto1,000.

CompareandOrder



Compareandorderwholenumbersto10,000,000.

Compare and order using the terms fewer,more,andless.

Compare and order using the terms same,more,fewer,greater than,less than,equal to,greatest,andleast.

Use<,>,and=tocomparewholenumbers.

PlaceValue Use place value models and place-value charts to representnumbersto100.

Usebase-tenmodelsand place-value charts to representnumbersto1,000.

PlaceValue Use place-value models to read,write,andrepresentnumbersto10,000.

Expressnumbersto100,000instandard,expanded,andwordforms.

Expressnumbersto10,000,000invariousforms.

6 7


NumberandOperations(continued) NumberandOperations(continued)

PlaceValue(continued)

Expressnumbersto100instandardandwordforms.

Expressnumbersto1,000intermsof placevalue.

PlaceValue(continued)

Compose and decompose multi-digitnumbers(includingexpandedform).

Fraction Concepts

Connect geometric concepts withunitfractionshalves,thirds,andfourths.

Fraction Concepts

Understand the meanings andusesof fractionsincludingfractionof aset.

Recognize,write,name,andillustratemixednumbersandimproperfractions.

Convertfractionstodecimals.

Understand the relationship betweenafractionandawhole.

Understandthatthesizeof afractionalpartisrelativetothesizeof thewhole.

Findafractionof aset. Relatefractionsanddivisionexpressions.

Compareandorderhalves,thirds,andfourthsusingbarmodels.

Comparefractionsusingmodels,numberlines.

Generate equivalent fractions.

Identifyequivalentfractionsthroughtheuseof models,multiplication,division,andnumberlines.

Convertamongmixednumbersandimproperfractions.

Addandsubtractlikefractions.

Money Identifyandrelatecoinvalues(penny,nickel,dime,quarter).

Identifyandrelatecoinvalues(penny,nickel,dime,quarter).

Identify$1,$5,$10,and$20bills.

Money Addandsubtractmoney.

Count and make coin combinations.

Count and make coin combinations.

Count and make combinationsof coinsandbills.

Comparemoneyamounts. Solvereal-worldproblemsinvolving addition and subtractionof money.

Decimal Concepts

Use the dollar sign and decimalpoint.

Decimal Concepts

Use the dollar sign and decimalpointinmoneyamounts.

Modeldecimalsusingtenthsandhundredths.

Modeldecimalsusingthousandths.

8 9



Decimal Concepts (continued)

Decimal Concepts (continued)

Understand decimal notation through hundredthsasanextensionof thebase-tensystem.

Understand place value concepts through thousandths.

Readandwritedecimalsthat are greater than or less than1.

Compare and order decimals.

Identifyequivalent decimals.

Connectequivalentfractionsanddecimals.

Convertdecimalstofractions.

Ratio, Proportion,andPercent

Ratio, Proportion,andPercent

Useratiostosolveproblems.

Findequivalentratios.

Solveproblemswithpercent.

Convertfractionstopercents.

Findapercentof anumber.

WholeNumberComputation:AdditionandSubtraction

Modeljoiningandseparatingsets.

Modeladditionandsubtractionsituations.

Modeladditionandsubtractionwithplacevalue.

WholeNumberComputation:AdditionandSubtraction

Modelregroupinginadditionandsubtractionwithplace-value.

Use+,-,and=towritenumbersentencesforadditionandsubtractionstories.

Usemodels,numbers,andsymbolsforadditionandsubtractionfactsto20.

Recall addition and subtractionfacts.

Usetheorder,grouping,andzeropropertiestodevelopadditionandsubtractionfactstrategies.

Usedifferentmethodstodevelopfluencyinaddingandsubtractingmulti-digitnumbers.

Addandsubtractupto2-digitnumberswithandwithoutregrouping

Addandsubtractwholenumbersto1,000.

Addandsubtractwholenumbersto10,000.

10 11



WholeNumberComputation:AdditionandSubtractionReal-WorldProblems

Represent addition and subtractionstories.

Formulateadditionandsubtractionstories.

WholeNumberComputation:AdditionandSubtractionReal-WorldProblems

Solve addition and subtractionproblemsusingbasicfacts.

Solve multi-digit addition andsubtractionproblemsbyusingabarmodel.

Solve addition and subtractionproblemswithgreaternumbersbyusingabarmodel.

WholeNumberComputation:MultiplicationandDivisionConcepts

Countby2sand5supto20.

Countby2s,5s,and10s. Multiplyanddividewith2,3,4,5,and10.

WholeNumberComputation:MultiplicationandDivisionConcepts

Multiplyanddividewith6,7,8,and9.

Addingthesamenumbertomultiply.

Represent multiplication as repeatedaddition.

Represent multiplication in differentways.

Applyunderstandingof modelsformultiplicationanddivision.

Representsharingequallyandmakingequalgroups.

Represent division as repeatedsubtraction.

Represent division in differentways.

Recallmultiplicationfactsandrelateddivisionfacts.

Usethe×,÷,and=symbolsto represent multiplication anddivisionsituations.

WholeNumberComputation:MultiplicationandDivisionAlgorithms

WholeNumberComputation:MultiplicationandDivisionAlgorithms

Multiply1s,10s,and100swithandwithoutregrouping.

Developfluencyinmultiplyingmulti-digitnumbers.

Multiplymulti-digitnumbers.

Use addition and multiplication properties to multiply.

Divide10sand1swithand withoutregrouping,noremainder.

Dividebya1-digitnumber,witharemainder.

Findquotientsinvolvingmulti-digitdividends.

12 13



WholeNumberComputation:MultiplicationandDivisionReal-WorldProblems

Usebarmodelstorepresentmultiplication and division situations.

WholeNumberComputation:MultiplicationandDivisionReal-WorldProblems

Usebarmodelstorepresentmultiplication and division situations.

Solve multiplication and divisionfactproblems.

Solveoneandtwo-stepmultiplication and division problems.

Solve multi-digit multiplication and division problems.

Solve multiplication and divisionproblems

Selectthemostusefulformof thequotientandinterprettheremainder.

Fraction Computation

Addandsubtractlikefractions(halves,thirds,fourths).

Fraction Computation

Addandsubtractlikefractions.

Addandsubtractunlikefractions.

Addandsubtractunlikefractionandmixednumbers.

Multiplyproperfractions,improperfractions,mixednumbers,andwholenumbers.

Dividefractionsbywholenumbers.

Solvewordproblemswithaddition,subtraction,multiplication,anddivisionof fractions.

Decimal Computation

Addandsubtractmoney. Solve addition and subtractionmoneyproblems.

Decimal Computation

Addandsubtractmoneyamounts.

Addandsubtractdecimals. Addandsubtractdecimals.

Multiplyanddividedecimalsbywholenumbers.

Solveproblemswithadditionandsubtractionof decimals.

Solveproblemswithmultiplicationanddivisionof decimals.

EstimationandMentalMath

Use mental math strategies toaddandsubtract.

Use mental math strategies toaddandsubtract.

EstimationandMentalMath

Use mental math strategies toaddsubtract,multiply,anddivide.

Use mental math and estimationstrategiestofindsums,differences,products,andquotients.

Use estimation and mental mathtoestimatesums,differences,products,andquotients.

14 15



EstimationandMentalMath(continued)

EstimationandMentalMath(continued)

Decidewhetheranestimateorexactanswerisneeded.

Estimatequantitybyusingreferents.

Round to the nearest ten to estimate sums and differences.

Usefront-endestimationand rounding to estimate sumsanddifferences.

Use estimation in determiningrelativesizesof amountsordistances.

Estimate sums and differenceswithfractionsanddecimals.

Roundandestimatewithdecimals.

Estimate products and quotientswithdecimals.

Algebra Algebra

Patterns Describeandextendrepeatingshapepatterns.

Identify,describe,andextendtwo-andthree-dimensionalshapepatterns.

Describe,extend,andcreatetwo-dimensionalshapepatterns.

Patterns Createandanalyzemultiplication and division patterns.

Identify,describe,andextendnumericandnonnumericpatterns.

Identify,describe,andextendnumeric patterns involving all operations.

Countby2sand5s. Skipcountby2s,5s,and10s.

Skipcountby2s,3s,4s,5s,and10s.

Skipcountby6s,7s,8s,and9s.

Describearuleforsortingobjects.

Identifyaruleforsortingobjects.

Findmissingtermsinrepeatingpatterns.

Identifyandextendgrowingandrepeatingpatterns.

Identifyrulesfornumberpatterns.

Analyzenumberandcountingpatterns.

Usearuletodescribeasequenceof numbersorobjects.

Findrulestocompletenumberpatterns.

Findmissingtermsingrowingandrepeatingpatterns.

Findmissingtermsintablepatterns.

Properties Identify0astheidentityelementforadditionandsubtraction.

Understand that addition andsubtractionareinverseoperations.

Properties Understand that multiplication and division arerelated.

Represent division as the inverseof multiplication.

Use the Associative and CommutativePropertiesof Addition.

Applypropertiesof addition. Createandanalyzemultiplication and division patterns.

UsetheDistributivePropertyasamultiplicationstrategy.

Model,define,andexplainpropertiesof multiplication.

Number Theory

Identifyoddandevennumbers.

Number Theory

Identifyoddandevennumbers.

Findthegreatestcommonfactorandleastcommonmultiple.

Identifyprimeandcompositenumbers.

16 17


Algebra(continued) Alegbra(continued)

Functional Relationships

Understand the relationships betweenthenumbersinfactfamilies.

Recognizehowbarmodelsshowrelationshipsbetweennumbersandunknownsinnumbersentences.

Functional Relationships

Understand the relationshipsbetweenthenumbersinmultiplication-divisionfactfamilies.

Understand the relationships betweenthenumbersandsymbolsinformulasforareaandperimeter.

Understand the relationships betweenthenumbersandsymbolsinformulasforsurfaceareaandvolume.

Describenumberrelationshipsincontext.



Expressions/Models

Useavarietyof concrete,pictorial,andsymbolicmodelsforadditionandsubtraction.

Useavarietyof concrete,pictorial,andsymbolicmodelsforaddition,subtraction,multiplication,anddivision.

Expressions/Models

Useavarietyof concrete,pictorial,andsymbolicmodelsformulti-digitaddition,subtraction,multiplication,anddivision.

Useavarietyof concrete,pictorial,andsymbolicmodelsformultiplicationand division; and addition andsubtractionwithfractionsanddecimals.

Uselettersasvariables.

Simplifyalgebraicexpressions.

Usetheorderof operationsinnumericexpressionswithtwoormoreoperations.

Number Sentences andEquations

Modeladditionandsubtractionstorieswithadditionandsubtractionnumbersentences.

Modeladditionandsubtractionsituationsbywritingadditionandsubtractionnumbersentences.

Modelmultiplicationanddivisionsituationsbywritingmultiplication and division numbersentences.

Number Sentences andEquations

Write multiplication and divisionnumbersentences.

Writeandsolvenumbersentencesforone-,two-,andthree-stepreal-worldproblems.

Writeandsolvenumbersentencesandequationsforone-andtwo-stepreal-worldproblems.

Usebarmodelsandnumbersentences to represent real-worldproblems.

Writeandsolvenumbersentencesforone-andtwostepreal-worldproblems.

Usebarmodelsandnumbersentencesforone-,two-,andthree-stepreal-worldproblems.

Writeandsolveequations.

Determinethevalueof missingquantitiesinnumbersentences.

Determine the missing parts (quantitiesorsymbols)innumbersentences.


Graphlinearequations.

EqualityandInequality

Understand the meaning of the=signinnumbersentences.

Understandthedifferencebetweenequalityandinequality.

Use and create models that demonstrateequalityorinequality.

EqualityandInequality

Understandequalityandinequality.



Use<,>,and=towritenumbersentences.

Writeandsolveinequalities.

Geometry Geometry

Sizeand Position

Understandbig,middlesized,andsmall.

Describepositionwithleftandright.

Sizeand Position

18 19


Geometry(continued) Geometry(continued)

Sizeand Position (continued)

Describeandcompareobjectsbyposition.

Usepositionalwordstodescribelocation.

Sizeand Position (continued)

Linesand Angles

Identifypartsof linesandcurves.

Linesand Angles

Identifyperpendicularandparallellines.

Drawperpendicularandparallellines.

Workwithanglesonastraightline.

Identifyrightanglesandcompare angles to right angles.

Construct and measure angles.

Workwithanglesatapoint.

Two- DimensionalShapes

Identifysimilaritiesanddifferences.

Two- DimensionalShapes

Nameflatshapesthatmakeupreal-worldobjects.

Identifyreal-worldtwo-dimensionalshapes.

Identify,describe,sort,andclassifytwo-dimensionalshapes.

Identifyanddescribeattributesandpropertiesof two-dimensionalshapes.

Identify,describe,sort,andclassifytwo-dimensionalshapes.

Describe,analyze,compare,andclassifytwo-dimensionalshapesbytheirsidesandangles.

Applythepropertiesof squaresandrectangles.

Applythepropertiesof right,isosceles,andequilateraltriangles.

Sortandclassifytwo-dimensionalshapes.

Identifypartsof linesandcurves.

Classifyandsortpolygonsandquadrilateralsbyattributesandproperties.

Findunknownanglemeasures and side lengths of squaresandrectangles.

Applythesumof theanglemeasuresof atriangle.

Makeflatshapepictures. Compose and decompose two-dimensionalshapes.

Compose and decompose two-dimensionalshapes.

Investigate composing anddecomposingtwo-dimensionalshapes.

Identifyfiguresthatformtessellations.

Applythepropertiesof aparallelogram,rhombus,andtrapezoid.

Useattributesandproperties to solve problems.

Demonstrate that the sum of anytwosidelengthsof a triangle is greater than the lengthof thethirdside.

Compare areas using non-standardunits.

Developfoundationsforunderstandingarea.

Findandcomparetheareaof planefiguresindifferentsquareunits.

Understand the relationships betweenthenumbersandsymbolsinformulasforareaandperimeter.

Findtheareaof atriangle.

Three- DimensionalShapes

Nameandsortsolidshapes. Identifyreal-worldthree-dimensionalshapes.

Three- DimensionalShapes

Identifyandclassifyprismsandpyramids.

Understand that three-dimensional shapes are madeupof two-dimensionalshapes.

Identifytwo-dimensionalshapes in three-dimensional shapes.

Identify,describe,sort,andclassifythree-dimensionalshapes.

Identifythesolidthatcanbemadefromanet.

20 21


Geometry(continued) Geometry(continued)

Three- DimensionalShapes (continued)

Sortandclassifythree-dimensionalshapes.

Identifysurfacesthatslide,stack,androll.

Three- DimensionalShapes (continued)

Identifycylinders,spheres,andcones.

Recognizeshapesfromdifferentperspectives.

Describecylinders,spheres,andconesbythenumberof andtypesof faces,andthenumberof edgesandvertices.

Compose and decompose three-dimensionalshapes.

Buildsolidsusingunitcubes.

CongruenceandSymmetry

Develop initial understandingof congruenceandsymmetry.

CongruenceandSymmetry

Identifysymmetricalfiguresandonelineof symmetry.

Identifylineandrotationalsymmetry.

Solveproblemsinvolvingcongruency.

Relaterotationalsymmetrytoturnsandcongruency.

Transformations Transformations Identifypairsof shapesthatshowaflip,slide,andturn.

Usetransformationstoformtessellations.

Demonstratethatfiguresandtheirflip,slide,andturnimagesarecongruent.

Coordinate Geometry

Coordinate Geometry

Develop coordinate readinesswithtablesandlinegraphs.

Plotpointsonacoordinategrid.

Measurement Measurement

LengthandDistance

Compare lengths and heights using non-standard units.

Comparetwolengthsbycomparingeachwithathirdlength(transitivity).

Demonstrate linear measure asaniterationof units.

LengthandDistance

Select appropriate units and tools to estimate and measurelength.

Compare and order lengths (long,short,longest,shortest).

Using a start line to measure length.

Userulerstomeasurelength. Usemetersticks,12-inchrulers,andyardstickstomeasurelength.

Developabackgroundformeasurement using non-standardunits.

Measurelengths,using non-standardunits.

Measurelengthsinmeters,centimeters,feet,andinches.

Measurelengthtothenearesthalf inchandinch.

Explaintheneedforequal-lengthunitstomeasure.

Usereferentstoestimatedistance.

22 23


Measurement(continued) Measurement(continued)

LengthandDistance (continued)

Explaintheneedforequal-lengthunitstomeasure.

LengthandDistance (continued)

Count length units in groups of 10sand1s.

Compare and measure lengthsusingcustomaryandmetricunits.

Estimate and measure length,distance,andheightinmeterscentimeters,andkilometers.

Compare measurements madeusingdifferentunits.

Demonstrate partitioning andtransitivityinrelationtolength.

Convert among metric units of length.

Understand the inverse relationshipbetweenthesizeof aunitandthenumberof units

Solveproblemsinvolvingestimating,measuring,andcomputinglength.

Solveone-andtwo-stepreal-worldproblemsinmeasurement.

Weight/Mass Orderobjectsbyweight. Compare and measure weightsusingnon-standardunits.

Compare and measure masses.

Weight/Mass Select appropriate units and tools to estimate and measureweight.

Compareweightsusingnon-standardunits.

Comparetwomassesbycomparingeachwithathirdmass(transitivity).

Usereferentstoestimateweight.

Estimateandfindmassesof objects.

Solveweightproblems. Solvemassproblems. Convertamongunitsof mass.

Capacity/ Volume

Compare capacities using non-standardunits.

Measurevolume(capacity)inliters.

Capacity/ Volume

Select appropriate tools and units to estimate and measure volume and capacity.

Solvevolumeproblems. Determine the volume and capacityof acontainer.

Relatetheunitsof customarycapacitytooneanother.

Usereferentstoestimatecapacity.

24 25



Capacity/ Volume (continued)

Capacity/ Volume (continued)

Estimate and measure capacityinlitersandmilliliters.

Convert among metric units of capacity.

Time Nameandorderthedaysof theweekandthemonthsof theyear.

Readacalendartoidentifythedaysof theweek,monthsandseasonsof theyear.

Time

Recognizethecorrectwaytowritethedate.

UseA.M.andP.Mtowritetime.

Readtimeonadigitalclock.

Telltimetothehourandhalf hour.

Telltimetofiveminutes. Convertbetweenhoursandminutes.

Comparedurationsof events.

Findelapsedtime. Determineelapsedtime.

Addandsubtractunitsof time.

Temperature Temperature ReadaFahrenheitthermometer.

Choose the appropriate tool and unit to measure temperature.

Usereferentstoestimatetemperature.

Angles Angles Compare angles to right angles.

Estimate and measure angleswithaprotractor.

Applytheideathatthesumof anglesonastraightlineis180°.

Classifyanglesbyanglemeasure.

Applytheideathatverticalanglesareequalinmeasure.

Relate1/4-,1/2-,3/4-,andfullturnstothenumberof rightangles.

Applytheideathatthesumof anglesatapointis360°.

Perimeter Perimeter Measureperimeterof planefigures.

Findtheperimeterof compositefigures.

26 27



Perimeter(continued)

Perimeter(continued)

Choose the appropriate tool,unit,andstrategytomeasureperimeter.

Solveproblemsinvolvingtheperimeterof squares,rectangles,andcompositefigures.

Estimatetheperimeterof surfacesandobjects.

Area Compare areas using non-standardunits.

Compose and decompose two-dimensionalshapes (foundationforunderstandingarea).

Developfoundationsforunderstandingarea.

Area Findandcomparetheareaof planefiguresindifferentsquareunits.

Explainareaasanattributeof two-dimensionalfigures.

Makedifferentplanefigureswiththesamearea.

Connect area measure totheareamodelformultiplication; use it to justifytheformulafortheareaof arectangle.

Findtheareaof triangles.

Estimateareaof smallandlargesurfaces.

Estimate and measure area insquareunits.

Compare the area and perimeterof twoplanefigures.

Selectappropriateunits,strategies,andtoolstosolveareaproblems.

Findtheareaof rectanglesandcompositefigures.

Explaintherelationshipsamongareaformulasof differentpolygons.

SurfaceAreaandVolume

SurfaceAreaandVolume

Decomposesolidfigurestofindthesurfacearea.

Estimate and measure volumeincubicunits.

DataAnalysis DataAnalysis

ClassifyingandSorting

Understanding similarities anddifferencesinobjectsandshapes.

Sortandclassifygeometricshapes.

Sortandclassifytwo-andthree-dimensionalshapesbyproperties.

ClassifyingandSorting

Classifyandsortpolygonsandquadrilateralsbyattributesandproperties.

Sortingandclassifyingobjectsusingoneortwoattributes.

Sortingandclassifyingdatainordertomakegraphs.

Collectandorganizedatainpicturegraphs.

Collectandorganizedatainbargraphsandlineplots.

Constructlineplots,stem-and-leaf plots,tablesandlinegraphs.

Representdatainadoublebargraph.

28 29


DataAnalysis(continued) DataAnalysis(continued)

Collectand OrganizeData

Organizedataforapicturegraph.

Collectandorganizedataindifferentways.

Collectandorganizedataindifferentways.

Collectand OrganizeData

RepresentData

Represent data in pictographs.

Represent measurements anddatainpicturegraphs,tallycharts,andbargraphs.

Represent data in picture graphs.

RepresentData

Interpret/ AnalyzeData

Interpretdataintallychartsandpictographs.

Interpret data in picture graphs,tallycharts,andbargraphs.

Interpretpicturegraphswithscales.

Interpret/ AnalyzeData

Interpretpictureandbargraphswithscales

Interprettallycharts,bargraphs,picturegraphs,tables,andlinegraphs

Readbargraphswithscales. Analyzedatainadoublebargraph.

Solveproblemsinvolvingdata.

Solvereal-worldproblemsusingpicturegraphs.

Usefrequencytables,bargraphs,picturegraphs,andlineplotstosolvereal-worldproblems.

Findthemean(average),median,mode,andrangeof adataset.

Probability Probability

Outcomes Outcomes Decidewhetheranoutcomeiscertain,morelikely,equallylikely,lesslikely,orimpossible.

Determineexperimentalprobabilityof anoutcome.

ExpressingProbability

ExpressingProbability

Expresstheprobabilityof aneventasafraction.

Comparetheresultsof anexperimentwiththeoreticalprobability.

Findallpossiblecombinationsbylisting,makingatreediagram,andmultiplying.

ProblemSolving ProblemSolving

BuildSkillsThrough Problem Solving

Build skills in addition and subtractionthroughproblemsolving.

Buildskillsinaddition,subtraction,andmeasurement through problemsolving.

Buildskillsinaddition,subtraction,multiplication,division,andmeasurementthroughproblemsolving.

BuildSkillsThrough Problem Solving

Buildskillsinaddition,subtraction,multiplication,division,andmeasurementthroughproblemsolving.

Buildskillsinmultiplication,division,fractionconcepts,dataanalysis,andmeasurement through problemsolving.

Build skills in multiplication; division;fractionconcepts,decimals,ratios,andpercents;dataanalysis;and measurement through problemsolving.

30 31


ProblemSolving(continued) ProblemSolving(continued)

Solve Real-World Problems

Solvereal-worldproblemsinvolving addition and subtraction.

Solvereal-worldproblemsinvolvingaddition,subtraction.

Solvereal-worldproblemsinvolvingaddition,subtraction,multiplication,division,andmeasurement.

Solve Real-WorldProblems


Solvereal-worldproblemsinvolvingmultiplication,division,fractionconcepts,dataanalysis,andmeasurement.

Solvereal-worldproblemsinvolving multiplication; division;conceptswithfractions,decimals,ratios,andpercents;dataanalysis;andmeasurement.

Use AppropriateStrategiesandThinkingSkillstoSolve Problems

Applyproblem-solvingstrategiesinPutonYourThinkingCap!andProblemSolvingactivities.

Applyproblem-solvingstrategiesinPutonYourThinkingCapIandProblemSolvingactivities.

Use AppropriateStrategiesandThinkingSkillstoSolve Problems

Applyproblem-solvingstrategiesinPutonYourThinkingCap!andProblemSolvingactivities.

Use appropriate strategies tosolvereal-worldproblems.

Use appropriate strategies to solvereal-worldproblems.

Applyand Explain ProblemSolving

Solvereal-worldproblems. Applyandexplainproblem-solvingprocessesinPutonYourThinkingCap!andotheractivities.

Applyandexplainproblem-solvingprocessesinPutonYourThinkingCap!andotheractivities.

Applyand Explain Problem Solving

Applyandexplainproblem-solvingprocessesinPutonYourThinkingCap!andotheractivities.

Applyandexplainproblem-solvingprocessinPutonYourThinkingCapandotheractivities.

Applyandexplainproblem-solvingprocessinPutonYourThinkingCapandotheractivities.

Explore Concepts

Usemodelstoexplainreasoning.

ExploreconceptsmoredeeplyandjustifyreasoninginLet’sExploreandHands-Onactivities.


Explore Concepts




ApplyThinkingSkills,PutonYourThinkingCap!,ChallengingPractice,andProblemSolvingactivities.



ApplyThinkingSkills,PutonYourThinkingCap,ChallengingPractice,andProblemSolvingactivities

ApplyThinkingSkills,PutonYourThinkingCap,ChallengingPractice,andProblemSolvingactivities

InvestigateMathematicalIdeas

Investigateideaswithtwo-dimensionalshapes.

Furtherinvestigatemathematicalideasbycompleting critical thinking skillsactivities.


InvestigateMathematicalIdeas




Identify, Demonstrate,andExplainMathematicalProof

Demonstratethatonlyafewbigthingsfitintosmallspacesandmanysmallthingsfitintobigspaces.

Exploretransitivitybycomparing lengths and weightsof threedifferentobjects.

Demonstrate the inverse relationshipbetweenthesizeof aunitandthenumberof units.

Identify, Demonstrate,andExplainMathematicalProof

Demonstratethatfiguresandtheirflip,slide,andturnimagesarecongruent.

Showthatsomefigurescanbeturnedandnotchangeshapeorsize(rotationalsymmetry).

Applytheideathatthesumof anglesonastraightlineis180°.

Applytheideathatthesumof anglesatapointis360°.

Describe,sort,andclassifytwo-andthreedimensionalshapes.

Identifyanddescribeattributesandpropertiesof two-andthreedimensionalshapes.

Identify,describe,sort,andclassifytwo-andthreedimensionalshapes.

Identifypairsof shapesthatshowaflip,side,andturn.

Usepropertiesof squaresand rectangles to solve problems.


32 33


ProblemSolving(continued) ProblemSolving(continued)

Identify, Demonstrate,andExplainMathematicalProof (continued)

Interpretdataintallychartsandpictographs.

Interpretpicturegraphs,tallycharts,andbargraphs.

Interpretpicturegraphswithscales.

Identify, Demonstrate,andExplainMathematicalProof (continued)

Interpretbargraphswithscales.

Analyzeadatasetbyfindingitsmean,median,mode,andrange.

Comparetheresultsof anexperimenttovalidatetheuseof theoreticalprobability.

Identifyandextendrepeatingshapepatterns.

Identifyandextendgrowingnumberpatternsandrepeatingshapepatterns.


Createandanalyzemultiplication and division patterns.

Identify,describe,andextendnumericandnonnumericpatterns.

Identify,describe,andextendnumeric patterns involving all operations.

UseaVarietyofReasoningSkills

Sortandclassifyusingattributes.

Recognizeshapesfromdifferentperspectives.

Identifysurfacesthatslide,stack,androll.

UseaVarietyofReasoningSkills

Model,define,andexplainpropertiesof multiplication.

Usepropertiesof squaresand rectangles to solve problemsaboutareaandperimeter.

Exploretherelationshipamonglists,treediagrams,and multiplication to calculatecombinations.

Identifysimilaritiesanddifferences.

Use the Commutative and Associativeproperties,and10sand1stosolvetwo-digitadditionandsubtractionproblems.

Exploretheinverserelationshipbetweenadditionandsubtraction.

Exploretheinverserelationshipbetweenmultiplicationanddivision.

Usepropertiesof multiplication(including theDistributiveProperty)inestimationandmentalmath.

Use estimation to check reasonableness.

Use estimation to checkreasonableness(whole-numberaddition,subtraction,multiplicationanddivision).

Communication Communication

ConsolidateMathematicalThinking

Consolidate thinking in independentactivities.

PresentmathematicalthinkingthroughMathJournalactivities.


ConsolidateMathematicalThinking




CommunicatewithPeers,Teachers,andOthers

Discuss mathematical ideas in paired and small group activities.

Discuss mathematical ideas inLet’sExploreactivities.


CommunicatewithPeers,Teachers,andOthers



Discuss mathematical ideas in Let’sExploreactivities.

Work together in pairs or groupsinLet’sExplore,Games,andotheractivities.





Share MathematicalThinking

Share mathematical ideas in paired and small group activities.

Share mathematical ideas withothersduringLet’sExploreandHands-Onactivities.


Share MathematicalThinking




34 35


Communication(continued) Communication (continued)

Express MathematicalIdeas

Expressideasinpairedandsmallgroupactivities.

ExpressideasinMathJournalactivities,usinglessonvocabulary.


Express MathematicalIdeas




Use chapter and lesson vocabularycorrectly.





Connections Connections

RecognizeConnectionsinMathematicalIdeas

Understand the connection betweenquantitiesandwrittennumerals.

Understand the relationship betweencountingandadditionandsubtraction.

Examineandapplytheinverserelationshipbetweenadditionandsubtraction.

RecognizeConnectionsinMathematicalIdeas

Applytheinverserelationshipbetweenmultiplicationanddivision.

Demonstrate that decimal notationisanextensionof thebase-tensystem.

Relatefractionsanddivision.

Understand the relationships amongthenumbersinfactfamilies.

Connect geometric concepts withunitfractions.

Understandthatthesizeof afractionalpartisrelativetothesizeof thewhole.

Examinetherelationshipbetweenfractionsanddecimals.

Understand the connection amongfractions,decimals,ratios,andpercentsaswaystorepresentpartsof awhole.

Connect addition and multiplication(repeatedaddition).

Connectsubtractionanddivision(repeatedsubtraction).

Connecttheunitsof customarycapacitytooneanother.

Makeconnectionsamongmultiplication,division,factors,andmultiples.

Examinetherelationshipsbetweenthree-dimensionalfiguresandthetwo-dimensionalfiguresthatformthem.

Recognizeandapplydifferentstrategiesforaddingandsubtracting one-andtwo-digitnumbers

Recognizeandapplydifferentstrategiesformultiplication and division facts.

Understand the relationshipsbetweenthenumbersinmultiplication-divisionfactfamilies.

Convertamongmixednumbersandimproperfractions.

Relatefractionsanddivision.

UnderstandHowConceptsBuildonOneAnother

Explorerelationshipsamongcounting,ordering,andordinalnumbers.

Learnhowplace-valueconceptsapplytoregrouping in addition and subtraction.

Understandhowpatternscanbedescribedusingnumbers,operations,anddatadisplays.

UnderstandHowConceptsBuildonOneAnother

Understand the meanings andusesof fractionsincludingfractionof aset.



Connectequivalentfractionsanddecimals.

Connectequivalentfractions,decimals,andpercents.

Recognizetherelationshipbetweenbarmodels,numbersentences,andnumberpatterns.

Useaddition,subtraction,multiplication,anddivisiontoconstructandanalyzegraphs,frequencytables,andlineplots.

Makeconnectionsamongthegreatestcommonfactor,leastcommonmultiple,andoperationswithfractions.

36 37


Connections(continued) Connections(continued)

Solve Real-WorldProblemsinContexts OutsideofMathematics

Solvereal-worldproblemsinvolvingmoreandless.

Solvereal-worldproblems involvingaddition,subtraction,andmeasurement.

Solvereal-worldproblemsinvolvingaddition,subtraction,multiplication,division,measurement,anddataanalysis.

Solve Real-WorldProblemsinContexts OutsideofMathematics



Solvereal-worldproblemsinvolving multiplication; division;fraction,decimal,ratio,andpercentconcepts;dataanalysis;andmeasurement.

Solvereal-worldproblemsrelatedtomoney.

Compareexperimentalresultsandtheoreticalprobability.

Representation Representation

Use RepresentationstoModel, Organize,andRecord

Use concrete models to createasetwithagivennumberof objects(upto20).

Use concrete and pictorial modelstocreateasetwithagivennumberof objects(upto100).

Use concrete and pictorial modelstocreateasetwithagivennumberof objectsupto1,000).

Use RepresentationstoModel, Organize,andRecord

Use place-value models to read,write,andrepresentnumbersto10,000.

Representnumbersto100,000invariouscontexts.

Explorenegativenumbersincontext.

Usenumbersandnumeralsto represent quantities up to20.

Representnumbersto100onanumberline.

Representnumbersto1,000onanumberline.

Representnumbersindifferentequivalentforms.

Expressnumbersto100,000instandard,expanded,andwordforms.

Expressnumbersto10,000,000invariousforms.

Use picture cards to communicate understanding of comparisons(biggerandsmaller).

Usenumberbondstorepresentnumbers.

Understand the meaning of the=signinnumbersentences.


Usesymbolicnotation(<,>)tocomparenumbers.

Use the dollar sign and decimalpointinmoneyamounts.

Modeldecimalstotenthsandhundredths.

Findequivalentratios.

Modeladditionandsubtractionstorieswithadditionandsubtractionnumbersentences.

Usethe+,−,and=symbolstorepresentreal-worldadditionandsubtractionsituations.

Usebarmodelstorepresentadditionandsubtractionsituations.

Solve addition and subtractionproblemswithgreaternumbersbyusingabarmodel

Write addition and subtractionnumbersentencesforreal-worldproblemswithfractionsanddecimals.

Exploretheuseof lettersasvariablesinexpressionsandinequalities.

Represent addition and subtractionstories.

Represent numerical data usingpicturegraphs,tallycharts,andbargraphs.

Represent numerical data usingpicturegraphswithscales,tallycharts,andbargraphs.

Represent multiplication and divisionindifferentways.

Usemodelstoshowrelationshipsbetweenimproperfractionsandmixednumbers.

Convertfractionsanddecimalstopercents.

Representsharingequallyandmakingequalgroups.

Usethe×,÷,and=symbolsto represent multiplication anddivisionsituations.

Useavarietyof representationsformultiplicationanddivision,suchasarrays,areamodels,numberlines,grouping,andsharing.

Applyunderstandingof modelsformultiplicationanddivision.

Representcombinationswithlists,treediagrams,andmultiplication.

38 39


Representation(continued) Representation(continued)

Use RepresentationstoModel, Organize,andRecord (continued)

Representmultiplicationwithskipcounting,dotpaperarrays,andbarmodels.

Use RepresentationstoModel, Organize,andRecord (continued)


Write addition and subtractionnumbersentencesforreal-worldproblemswithfractionsanddecimals.

Writeandsolveequations.

Represent division as repeatedsubtractionsentences.

Describeandextendshapepatterns.

Identify,describe,andextendtwo-andthree-dimensionalshapepatterns.

Describe,extend,andcreatetwo-dimensionalshapepatterns.

Createandanalyzemultiplication and division patterns.

Describearuleforsortingobjects.

Identifyaruleforsortingobjects.

Identifyandextendgrowingandrepeatingpatterns.


Identifyarulefornumberandcountingpatterns.

Usearuletodescribeasequenceof numbersorobjects.

Findrulestocompletenumberpatterns.

Selectand Apply Representations toModel Problems

Representquantitieswithobjects,numbercubes,andnumerals.

Usenumberbondstorepresentnumbercombinations.

Use place value models to create equivalent representationsof numbers.

Selectand Apply Representations toModel Problems

Useavarietyof modelstorepresentfractionsandequivalentfractions.

Translatebetweenequivalent improper fractionsandmixednumbers.

Translatebetweenfractionsandpercents.

Useavarietyof concrete,pictorial,andsymbolicmodelsforadditionandsubtraction.

Useavarietyof concrete,pictorial,andsymbolicmodelsforaddition,subtraction,multiplication,anddivision.

Useavarietyof concrete,pictorial,andsymbolicmodelsformulti-digitaddition,subtraction,multiplication,anddivision.

Useavarietyof modelsformulti-digit multiplication anddivisionof wholenumbers.

Selectthemostusefulformof thequotient.

Representmultiplicationwithskipcountingandarrays.

Usecustomaryunits(includingfractions)tomeasurelength,weight,andcapacity.

Useavarietyof modelsforadditionandsubtractionof fractionsanddecimals.

Useanettofindthesurfaceareaof aprism.

InterpretPhenomenathroughRepresentations

Showunderstandingof big,middlesized,small,andsamesize.

Measureandcomparelengthsandweightsusingnon-standardunits.

Usemetricandcustomaryunitstomeasurelength,volume(capacity),weight,andmass.

InterpretPhenomenathrough Representations

Usereferentstoestimatelength,capacity,andweight.

Measureperimeterandareaincustomaryandmetricunits.

Measurevolumeof arectangularprism.

Describeandcompareobjectsbyposition.

Usepositionalwordstodescribelocation.

40 41


Representation(continued) Representation(continued)

InterpretPhenomenathrough Representations(continued)

Nameflatshapesthatmakeupreal-worldobjects.

Identifyreal-worldtwo-andthree-dimensionalshapes.

InterpretPhenomenathrough Representations(continued)

Usefrequencytables,bargraphs,picturegraphs,andlineplotstosolveproblems.

Collectdataandorganizeitinatable.

Representdatainadoublebargraph.

Represent measurements and data in picture graphs andbargraphs.

Represent data in picture graphs.

Representdatainbargraphsandpicturegraphs.

Usefrequencytables,bargraphs,picturegraphs,andlineplotstosolveproblems.

Createalinegraphfromdatainatable.

Represent an equation as a graphedline.

Orderanumberof objectsaccordingtolength,height,orweight.

Solveproblemsaboutsharingequallyandmakingequalgroups.

Solvereal-worldproblemsaboutsocialphenomena.

Solvereal-worldproblemsinvolvingsocialsituations.

Usemeasuresof centraltendencytodescribetypicalvaluesof datasets(social).

Use one-to-one correspondence.

Useavarietyof modelsforaddingandsubtracting.

Usebarmodelstorepresentaddition,subtraction,multiplication,anddivisionsituations.

Solvereal-worldproblemsrelatedtomoney.


Solvereal-worldproblemsinvolvingsocialsituations.

Usetechnology(virtualmanipulatives and computers)tomodelanddraw.




Usetechnology(virtualmanipulativesandcomputers)tomodelanddraw.

Useacalculatortomodel,compute,andsolveproblems.

42 43

Aligned with National and International Research Recommendations

NationalMathAdvisoryPanel

“Useshouldbemadeof whatisclearlyknownfromrigorousresearchabouthowchildrenlearn,especiallybyrecognizingthemutually-reinforcingbenefitsof conceptualunderstanding,proceduralfluency,andautomatic(i.e.,quickandeffortless)recallof facts.”

—Foundations for Success,2008

SingaporeMinistryof Education

“Mathematicalproblemsolvingiscentraltomathematicslearning.Itinvolvestheacquisitionandapplicationof mathematicsconceptsandskillsinawiderangeof situations,includingnon-routine, open-ended,andreal-worldproblems.”

—Mathematics Syllabus: Primary,2006

Math in Focusaddresses fewer topics in greater depth

at each level.

• Knowledgeisbuiltcarefullyandthoroughlywithbothmulti-page lessons and multi-daylessons.

• Timeisbuiltintotheprogramtodevelopunderstandingwithhands-on activitieswithmanipulativesaswellasextensive skills practice.

Math in Focusdevelops concepts and skills in tandem.

• Manipulativesandvisualrepresentationsprovideaconceptualbackbone.

• Skills are connected to concepts through visual representations.

• Extensiveproblemsolvingmerges conceptual understanding with computational skills.

BarModel: a visual representationof awordproblem(Grade2)

Math in Focususes a scaffolded approach to solving word

problems, focusing on model drawing to build success and confidence.

• Thevisualrepresentationof wordproblemsleadstosymbolicsolutionsof rich and complex problems.

• Studentsdrawonpriorknowledgeaswellasrecentlyacquiredconceptsandskillsastheycombineproblem-solving strategies with critical thinking skills.

NationalCouncilof Teachersof Mathematics

“Acurriculumismorethanacollectionof activities:it

mustbecoherent,focusedonimportantmathematics,

andwellarticulatedacrossthegrades.”

—Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics,2006

Focus and Depth Interlocking Concepts and Skills Emphasis on Problem Solving

1 Ways to Add . . . . . . . . . . . . . . . . . . . . . 42

Learn Addbycountingon•Countto findhowmanymore•Addusing numberbonds

Hands-On Activities Count on to addusingconnectingcubes•Useten framestoadd(CommutativeProperty)

Game CardFun!

Let’s Practice and Practice and Apply WorkbookA:Practice1 . . . . . . . . . . . . 46–47

Let’s Practice and Practice and Apply WorkbookA:Practice2 . . . . . . . . . . . . 50–51

Grade1,Chapter3,Lesson1

Skill Building

Concept Building

the WHY

the HOW

NationalResearchCouncil

“Opportunitiesshouldinvolveconnectingsymbolicrepresentationsandoperationswithphysicalorpictorialrepresentations,aswellastranslatingbetweenvarioussymbolicrepresentations.”

—Adding It Up: Helping Children Learn Mathematics,2001

Math in Focususes clear and engaging visuals that present

concepts and model solutions.

• Minimal textandsimple,directvisualsallowallstudentsregardlessof languageskillstofocusonthemathlesson.

• Theuseof model drawingsofferavisualrepresentationof wordproblems,leadingtosymbolicsolutionsof richandcomplexproblems.

• Consistentuseof theconcrete–pictorial–abstract pedagogyrepeatedly“models”themodel-drawingproblem-solvingstrategy.

NumberBonds: a visual forcomposinganddecomposingnumbers(Grade1)

Clear Visuals and Use of Models

104

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