Common Core Vertical Alignment

CommonCoreStateStandards

VerticalAlignmentMathematicsK‐8

Summer2011

CreatedbyStarkCountyTeachersandTheStarkCountyEducationalServiceCenter

DevelopedbyStarkCountyTeachersandTheStarkCountyESC,Summer2011

TABLE OF CONTENTS

Addition and Subtraction Facts ..............................................................................................................................................................................................1

Coordinate Systems .................................................................................................................................................................................................................2

Plane and Solid Figures ...........................................................................................................................................................................................................3

Equivalent Names for Fractions, Decimals and Percents........................................................................................................................................................4

Meanings and Uses of Fractions .............................................................................................................................................................................................5

Procedures for Addition and Subtraction of Fractions ...........................................................................................................................................................6

Procedures for Multiplication and Division of Fractions .......................................................................................................................................................7

Length and Weight .................................................................................................................................................................................................................9

Equations and Inequalities ......................................................................................................................................................................................................10

Patterns and Functions ............................................................................................................................................................................................................11

Angles .....................................................................................................................................................................................................................................12

Algebraic Notation and Solving Equations ............................................................................................................................................................................13

Time ........................................................................................................................................................................................................................................14

Money .....................................................................................................................................................................................................................................15

Comparing and Ordering Numbers .........................................................................................................................................................................................16

Rote Counting .........................................................................................................................................................................................................................17

Rational Counting ...................................................................................................................................................................................................................18

Place Value and Notation .......................................................................................................................................................................................................19

Radicals and Exponents ..........................................................................................................................................................................................................20

Number Theory ......................................................................................................................................................................................................................21

Ratio and Proportions .............................................................................................................................................................................................................22

Functions ................................................................................................................................................................................................................................23

Pythagorean ............................................................................................................................................................................................................................24

Area, Perimeter, Volume, Surface Area, and Capacity ..........................................................................................................................................................25

Multiplication and Division Facts ..........................................................................................................................................................................................26

Transformations, Congruence, Similarity, and Symmetry .....................................................................................................................................................27

Probability ..............................................................................................................................................................................................................................28

Order of Operations.................................................................................................................................................................................................................29

Units and Systems of Measurement .......................................................................................................................................................................................30

Operations with Decimals ......................................................................................................................................................................................................31

Data Collection........................................................................................................................................................................................................................33

Data Analysis ..........................................................................................................................................................................................................................34

Procedures for Addition and Subtraction ................................................................................................................................................................................35

Procedures for Multiplication and Division ............................................................................................................................................................................36

CommonCore/VerticalAlignment

AdditionandSubtractionFactsK 1 2 3 4 5 6 7 8

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1. Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings2,sounds(e.g.,claps),actingoutsituations,verbalexplanations,expressions,orequations.

2. Solveadditionandsubtractionwordproblems,andaddandsubtractwithin10,e.g.,byusingobjectsordrawingstorepresenttheproblem.

3. Decomposenumberslessthanorequalto10intopairsinmorethanoneway,e.g.,byusingobjectordrawings,andrecordeachdecompositionbyadrawingorequation(e.g,5=2+3and5=4+1).

4. Foranynumberfrom1to9,findthenumberthatmakes10whenaddedtothegivennumber,e.g.,byusingobjectsordrawings,andrecordtheanswerwithadrawingorequation.

5. Fluentlyaddandsubtractwithin5.

3. Applypropertiesofoperationsasstrategiestoaddandsubtract3.Examples:If8+3=11isknown,then3+8=11isalsoknown.(Commutativepropertyofaddition.)Toadd2+6+4,thesecondtwonumberscanbeaddedtomakeaten,so2+6=4=2+1012.(Associativepropertyofaddition.)

4. Understandsubtractionasanunknown‐addendproblem.Forexample,subtract108byfindingthenumberthatmakes10whenaddedto8.

AddandSubtractwithin20.5. Relatecountingtoadditionand

subtraction(e.g.,bycountingon2toadd2).

6. Addandsubtractwithin10,demonstratingfluencyforadditionandsubtractionwithin20.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).

Workwithadditionandsubtractionequations.7. Understandthemeaningofthe

equalsign,anddetermineifequationsinvolvingadditionandsubtractionaretrueorfalse.Forexample,whichofthefollowingequationsaretrueandwhicharefalse?6=6,7=81,5+2=2+5,4+15+2.

8. Determinetheunknownwholenumberinanadditionorsubtractionequationrelatingthreewholenumbers.Forexample,determinetheunknownnumberthatmakestheequationtrueineachoftheequations8+?=11,5=3,6+6[].

2. Fluentlyaddandsubtractwithin20usingmentalstrategies2.ByendofGrade2,knowfrommemoryallsumsoftwoone‐digitnumbers.

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CoordinateSystemsK 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Graphpointsonthecoordinateplanetosolverealworldandmathematicalproblems.1. Useapairof

perpendicularnumberlines,calledaxes,todefineacoordinatesystem,withtheintersectionofthelines(theorigin)arrangedtocoincidewiththe0oneachlineandagivenpointintheplanelocatedbyusinganorderedpairofnumbers,calleditscoordinates.Understandthatthefirstnumberindicateshowfartotravelfromtheorigininthedirectionofoneaxis,andthesecondnumberindicateshowfartotravelinthedirectionofthesecondaxis,withtheconventionthatthenamesofthetwoaxesandthecoordinatescorrespond(e.g.,x‐axisandx‐coordinate,y‐axisandy‐coordinate).

2. Representrealworldandmathematicalproblemsbygraphingpointsinthefirstquadrantofthecoordinateplane,andinterpretcoordinatevaluesofpointsinthecontextofthesituation.

6. Understandarationalnumberasapointonthenumberline.Extendnumberlinediagramsandcoordinateaxesfamiliarfrompreviousgradestorepresentpointsonthelineandintheplanewithnegativenumbercoordinates.

a. Recognizeoppositesignsofnumbersasindicatinglocationsonoppositesidesof0onthenumberline;recognizethattheoppositeoftheoppositeoftheoppositeofanumberisthenumberitself,e.g.,‐(‐3)=3,andthat0isitsownopposite.

b. Understandsignsofnumbersinorderedpairsasindicatinglocationsinquadrantsofthecoordinateplane;recognizethatwhentwoorderedpairsdifferonlybysigns,thelocationsofthepointsarerelatedbyreflectionsacrossoneorbothaxes.

c. Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram;findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane.

8. Solvereal‐worldandmathematicalproblemsbygraphingpointsinallfourquadrantsofthecoordinateplane.Includeuseofcoordinatesandabsolutevaluetofinddistancesbetweenpointswiththesamefirstcoordinateorthesamesecondcoordinate.

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PlaneandSolidFigures

K 1 2 3 4 5 6 7 8PlaneandSolid

Identifyanddescribeshapes(squares,circles,triangles,rectangles,hexagons,cubes,cones,cylinders,andspheres).1. Describeobjectsinthe

environmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.

2. Correctlynameshapesregardlessoftheirorientationsoroverallsize.

3. Identifyshapesastwo‐dimensional(lyinginaplane,“flat”)orthree‐dimensional(“solid”).

Analyze,compare,create,andcomposeshapes.4. Analyzeandcompare

two‐andthree‐dimensionalshapes,indifferentsizesandorientations,usinginformallanguagetodescribetheirsimilarities,differences,parts(e.g.,numberofsidesandvertices/”corners”)andotherattributes(e.g.,havingsidesofequallength).

5. Modelshapesintheworldbybuildingshapesfromcomponents(e.g.,sticksandclayballs)anddrawingshapes.

6. Composesimpleshapestoformlargershapes.Forexample,“Canyoujointhesetwotriangleswithfullsidestouchingtomakearectangle?”

PlanandSolidReasonwithshapesandtheirattributes.1. Distinguishbetween

definingattributes(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

PlaneandOnlyCubesReasonwithshapesandtheirattributes.1. Recognizeanddraw

shapeshavingspecifiedattributes,suchasagivennumberofanglesoragivennumberofequalfaces.5Identifytriangles,quadrilaterals,pentagons,hexagons,andcubes.

PlaneOnlyReasonwithshapesandtheirattributes.1. Understandthat

shapesindifferentcategories(e.g.,rhombuses,rectangles,andothers)mayshareattributes(e.g.,havingfoursides),andthatthesharedattributescandefinealargercategory(e.g.,quadrilaterals).Recognizerhombuses,rectangles,andsquaresasexamplesofquadrilaterals,anddrawexamplesofquadrilateralsthatdonotbelongtoanyofthesesubcategories.

PlaneOnlyDrawandidentifylinesandangles,andclassifyshapesbypropertiesoftheirlinesandangles.1. Drawpoints,lines,

linesegments,rays,angles(right,acute,obtuse),andperpendicularandparallellines.Identifytheseintwo‐dimensionalfigures.

2. Classifytwo‐dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.

3. Recognizealineofsymmetryforatwo‐dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline‐symmetricfiguresanddrawlinesofsymmetry.

PlaneOnlyClassifytwodimensionalfiguresintocategoriesbasedontheirproperties.3. Understandthat

attributesbelongingtoacategoryoftwo‐dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.Forexample,allrectangleshavefourrightanglesandsquaresarerectangles,soallsquareshavefourrightangles.

4. Classifytwo‐dimensionalfiguresinahierarchybasedonproperties.

PlaneandSolid3. Drawpolygonsinthe

coordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal‐worldandmathematicalproblems.

4. Representthree‐dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal‐worldandmathematicalproblems.

PlaneandSolidfigures

Draw,construct,anddescribegeometricalfiguresanddescribetherelationshipsbetweenthem.1. Solveproblems

involvingscaledrawingsofgeometricfigures,includingcomputingactuallengthsandareasfromascaledrawingandreproducingascaledrawingatadifferentscale.

2. Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticingwhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.

3. Describethetwo‐dimensionalfiguresthatresultfromslicingthree‐dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.

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EquivalentNamesforFractions,DecimalsandPercentsK 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards 3. Explainequivalenceoffractionsinspecialcases,andcomparefractionsbyreasoningabouttheirsize.

a. Understandtwofractionsasequivalent(equal)iftheyarethesamesize,orthesamepointonanumberline.

b. Recognizeandgeneratesimpleequivalentfractions,e.g.,½=2/4,4/6=2/3).Explainwhythefractionsareequivalent,e.g.,byusingavisualfractionmodel.

c. Expresswholenumbersasfractions,andrecognizefractionsthatareequivalenttowholenumbers.Examples:Express3intheform3=3/1;recognizethat6/1=6;locate4/4and1atthesamepointonanumberlinediagram.

d. Comparetwofractionswiththesamenumeratororthesamedenominatorbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols.>=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.

1. Explainwhyafractiona/bisequivalenttoafraction(nxa)/(nxb)byusingvisualfractionmodels,withattentiontohowthenumberandsizeofthepartsdiffereventhoughthetwofractionsthemselvesarethesamesize.Usethisprincipletorecognizeandgenerateequivalentfractions.

Understanddecimalnotationforfractions,andcomparedecimalfractions.5. Expressafraction

withdenominator10asanequivalentfractionwithdenominator100,andusethistechniquetoaddtwofractionswithrespectivedenominators10and100.4Forexample,express3/10as30/100,andadd3/10+4/10034/100.

6. Usedecimalnotationforfractionswithdenominators10or100.Forexample,rewrite0.62as62/100;describealengthas0.62meters;locate0.62onanumberlinediagram.

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Meanings and Uses of Fractions

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3. Partitioncirclesandrectanglesintotwoandfourequalshares,describethesharesusingthewordshalves,fourths,andquarters,andusethephraseshalfof,fourthof,andquarterof.Describethewholeastwoof,orfouroftheshares.Understandfortheseexamplesthatdecomposingintomoreequalsharescreatessmallershares.

3. Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.

2. Partitionshapesintopartswithequalareas.Expresstheareaofeachpartasaunitfractionofthewhole.Forexample,partitionashapeinto4partswithequalarea,anddescribetheareaofeachpartas¼oftheareaoftheshape.

Developunderstandingoffractionsasnumbers.1. Understandafraction1/b

asthequantityformedby1partwhenawholeispartitionedintobequalparts;understandafractiona/basthequantityformedbyapartsofsize1/b.

2. Understandafractionasanumberonthenumberline;representfractionsonanumberlinediagram.

a. Representafraction1/bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1/bandthattheendpointofthepartbasedat0locatesthenumber1/bonthenumberline.

b. Representafractiona/bonanumberlinediagrambymarkingoffalengths1/bfrom0.Recognizethattheresultingintervalhassizea/bandthatitsendpointlocatesthenumbera/bonthenumberline.

2. Comparetwofractionswithdifferentnumeratorsanddifferentdenominators,e.g.,bycreatingcommondenominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas½.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswithsymbols>,=,<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.

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;21/8=1+1+1/8=8/8+8/8+1/8.

c. Addandsubtractmixednumberswiththelikedenominators,e.g.,byreplacingeachmixednumberwithanequivalentfraction,and/orbyusingpropertiesofoperationsandtherelationshipbetweenadditionandsubtraction.

d. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.


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Procedures for Addition and Subtraction of Fractions K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 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;21/8=1+1+1/8=8/8+8/8+1/8.

c. Addandsubtractmixednumberswithlikedenominators,e.g.,byreplacingeachmixednumberwithanequivalentfraction,and/orbyusingpropertiesofoperationsandtherelationshipbetweenadditionandsubtraction.

d. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.

Useequivalentfractionsasastrategytoaddandsubtractfractions.1. Addandsubtract

fractionswithunlikedenominators(includingmixednumbers)byreplacinggivenfractionswithequivalentfractionsinsuchawayastoproduceanequivalentsumordifferenceoffractionswithlikedenominators.Forexample,2/3+5/48/12+15/12=23/12.(Ingeneral,a/b+c/d=(ad+bc)/bd.)

2. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewhole,includingcasesofunlikedenominators,e.g.,byusingvisualfractionsmodelsorequationstorepresenttheproblem.Usebenchmarkfractionsandnumbersenseoffractionstoestimatementallyandassessthereasonablenessofanswers.Forexample,recognizeanincorrectresult2/5+1/2=3/7,byobservingthat3/7<1/2.

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Procedures for Multiplication and Division of Fractions K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 4. Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionbyawholenumber.

a. Understandafractiona/basamultipleof1/b.Forexample,useavisualfractionmodeltorepresent5/4astheproductof5x(1/4),recordingtheconclusionbytheequation5/4=5x(1/4).

b. Understandamultipleofa/basamultipleof1/b,andusethisunderstandingtomultiplyafractionbyawholenumber.Forexample,useavisualfractionmodeltoexpress3x2(2/5)as6x(1/4),recognizingthisproductas6/5.(Ingeneral,nx(a/b=(nxa/b.)

c. Solvewordproblemsinvolvingmultiplicationofafractionbyawholenumber,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.Forexample,ifeachpersonatapartywilleat3/8ofapoundofroastbeef,andtherewillbe5peopleattheparty,howmanypoundsofroastbeefwillbeneeded?Betweenwhattwowholenumbersdoesyouranswerlie?

3. Interpretafractionasdivisionofthenumeratorbythedenominator(a/b=a÷b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret ¾ as the result of dividing 3 by 4, noting that ¾ multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size ¾. 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. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q÷b. For example, use a visual fraction model to show (2/3) x 4=8/3, and create a story context for this equation. Do the same with (2/3) x (4/5)=8/15. (In general, (a/b) x (c/d)=ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find area of rectangles, and represent fraction products as rectangular areas.

Applyandextendpreviousunderstandingsofmultiplicationanddivisiontodividefractionsbyfractions.1. Interpretandcompute

quotientsoffractions,andsolveworkproblemsinvolvingdivisionoffractionsbyfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.Forexample,createastorycontextfor(2/3) ÷(3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3)÷(3/4)=8/9 because ¾ of 8/9 is 2/3. (In general, (a/b) ÷(c/d)=ad/bc.) How much chocolate will each person get if 3 people share ½ lb of chocolate equally? How many ¾-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length ¾ mi and area ½ square mi?


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Procedures for Multiplication and Division of Fractions

1 2 3 4 5 6 7 8Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 5. Interpretmultiplicationasscaling

(resizing),by:a. Comparingthesizeofaproducttothesize

ofonefactoronthebasisofthesizeoftheotherfactor,withoutperformingtheindicatedmultiplication.

b. Explainingwhymultiplyingagivennumberbyafractiongreaterthan1resultsinaproductgreaterthanthegivennumber(recognizingmultiplicationbywholenumbersgreaterthan1asafamiliarcase);explainingwhymultiplyingagivennumberbyafractionlessthan1resultsinaproductsmallerthanthegivennumber;andrelatingtheprincipaloffractionequivalencea/b=(nxa)/(nxb)totheeffectofmultiplyinga/bby1.

6. Solverealworldproblemsinvolvingmultiplicationoffractionsandmixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.

7. Applyandextendpreviousunderstandingsofdivisiontodivideunitfractionsbywholenumbersandwholenumbersbyunitfractions.1

a. Interpretdivisionofaunitfractionbyanon‐zerowholenumber,andcomputesuchquotients.Forexample,createastorycontextfor(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 ) x 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4÷(1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4÷(1/5)=20 because 20 x (1/5)=4.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share ½ lb of chocolate equally? How many 1/3 cup servings are in 2 cups of raisins?

Applyandextendpreviousunderstandingsofmultiplicationanddivisiontodividefractionsbyfractions.1. Interpretandcomputequotientsof

fractions,andsolvewordproblemsinvolvingdivisionoffractionsbyfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.Forexample,createastorycontextfor2/3÷(3/4)anduseavisualfractionmodeltoshowthequotient;usetherelationshipbetweenmultiplicationanddivisiontoexplainthat(2/3)+(3/4)=8/9because¾of8/9is2/3.(Ingeneral,(a/b)÷(c/d)–ad/bc.)Howmuchchocolatewilleachpersongetif3peopleshare½lbofchocolateequally?Howmay¾cupservingsarein2/3ofacupofyogurt?Howwideisarectangularstripoflandwithlength¾miandarea½squaremi?


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Length and Weight K 1 2 3 4 5 6 7 8

Describeandcomparemeasurableattributes.1. Describemeasureable

attributesofobjects,suchaslengthorweight.Describeseveralmeasurableattributesofasingleobject.

2. Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“moreof”/”lessof”theattribute,anddescribethedifference.Forexample,directlycomparetheheightsoftwochildrenanddescribeonechildastaller/shorter.

Measurelengthsindirectlyandbyiteratinglengthunits.1. Orderthreeobjectsby

length;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.

2. Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame‐sizelengthunitsthatspanitwithnogapsoroverlaps.Limittocontextswheretheobjectbeingmeasuredisspannedbyawholenumberoflengthunitswithnogapsoroverlaps..

Measureandestimatelengthsinstandardunits.1. Measurethelengthof

anobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.

2. Measurethelengthofanobjecttwice,usinglengthunitsofdifferentlengthsforthetwomeasurements;describehowthetwomeasurementsrelatetothesizeoftheunitchosen.

3. Estimatelengthsusingunitsofinches,feet,centimeters,andmeters.

4. Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthelengthdifferenceintermsofsstandardlengthunit.

5. Useadditionandsubtractionwithin100tosolvewordproblemsinvolvinglengthsthataregiveninthesameunits,e.g.,byusingdrawings(suchasdrawingsofrulers)andequationswithasymbolfortheunknownnumbertorepresenttheproblem.

4. Generatemeasurementdatabymeasuringlengthsusingrulersmarkedwithhalvesandfourthsofaninch.Showthedatabymakingalineplot,wherethehorizontalscaleismarkedoffinappropriateunits‐wholenumbers,halves,orquarters.

Representandinterpretdata.4. Makealinepotto

displayadatasetofmeasurementsinfractionsofaunit(1/2,¼,1/8).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.Forexample,fromalineplotfindandinterpretthedifferenceinlengthbetweenthelongestandshortestspecimensinaninsectcollection.

Representandinterpretdata.2. Makealineplotto

displayadatasetofmeasurementsinfractionsofaunit(1/2,¼,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.Forexample,givendifferentmeasurementsofliquidinidenticalbeakers,findtheamountofliquideachbeakerwouldcontainifthetotalamountinallthebeakerswereredistributedequally.


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Equations and Inequalities K 1 2 3 4 5 6 7 8

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Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 5. Understandsolvinganequationorinequalityasaprocessofansweringaquestion:whichvaluesfromaspecifiedset,ifany,maketheequationorinequalitytrue?Usesubstitutiontodeterminewhetheragivennumberinaspecifiedsetmakesanequationorinequalitytrue.

6. Usevariablestorepresentnumbersandwriteexpressionswhensolvingareal‐worldormathematicalproblem;understandthatavariablecanrepresentanunknownnumber,or,dependingonthepurposeathand,anynumberisaspecifiedset.

7. Solvereal‐worldandmathematicalproblemsbywritingandsolvingequationsoftheformx+p=qandpx=qforcasesinwhichp,qandxareallnonnegativerationalnumbers.

8. Writeaninequalityoftheformx>corx>ctorepresentaconstraintorconditioninareal‐worldormathematicalproblem,Recognizethatinequalitiesoftheformx>corx>chaveinfinitelymanysolutions;representsolutionsofsuchinequalitiesonnumberlinediagrams.

1. Applypropertiesofoperationsasstrategiestoadd,subtract,factorandexpandlinearexpressionswithrationalcoefficients.

b. Solvewordproblemsleadingtoinequalitiesoftheformpx+q>rorpx+q<r,wherep,q,andrarespecificrationalnumbers.Graphthesolutionsetoftheinequalityandinterpretitinthecontextoftheproblem.Forexample:Asasalesperson,youarepaid$50perweekplus$3persale.Thisweekyouwantyourpaytobeatleast$100.Writeaninequalityforthenumberofsalesyouneedtomake,anddescribethesolutions.

7. Solvelinearequationsinonevariable.a. Giveexamplesoflinearequationsin

onevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).

b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.

8. Analyzeandsolvepairsofsimultaneouslinearequations.

a. Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.

b. Solvesystemsoftwolinearequationsintwovariablesalgebraically,andestimatesolutionsbygraphingtheequations.Solvesimplecasesbyinspection,Forexample,3x+2y=5and3x+2y=6havenosolutionbecause3x+2ycannotsimultaneouslybe5and6.

c. Solvereal‐worldandmathematicalproblemsleadingtotwolinearequationsintwovariables.Forexample,givencoordinatesfortwopairsofpoints,determinewhetherthelinethroughthefirstpairofpointsintersectsthelinethroughthesecondpair.

10


Patterns and Functions K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards 9. Identifyarithmeticpatterns(includingpatternsintheadditiontableormultiplicationtable),andexplainthemusingpropertiesofoperations.Forexample,observethat4timesanumberisalwayseven,andexplainwhy4timesanumbercanbedecomposedintotwoequaladdends.

5. Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.Forexample,giventherule“Add3”andthestartingnumber1,generatetermsintheresultingsequenceandobservethatthetermsappeartoalternatebetweenoddandevennumbers.Explaininformallywhythenumberswillcontinuetoalternateinthisway.

3. Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationshipsbetweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwopatterns,andgraphtheorderedpairsonacoordinateplane.Forexample,giventherule“Add3”andthestartingnumber0,andgiventherule“Add6”andthestartingnumber0,generatetermsintheresultingsequences,andobservethatthetermsinonesequencearetwicethecorrespondingtermsintheothersequence.Explaininformallywhythisisso.

Norelatedstandards 2. Understandthatrewritinganexpressionindifferentformsinaproblemcontextcanshedlightontheproblemandhowthequantitiesinitarerelated.Forexample,a+0.05a=1.05ameansthat“increaseby5%isthesameas“multiplyby1.05.”

Solvereallifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations.3. Solvemulti‐stepreal‐liveand

mathematicalproblemsposedwithpositiveandnegativerationalnumbersinanyform(wholenumbers,fractions,anddecimals),usingtoolsstrategically.Applypropertiesofoperationstocalculatewithnumbersinanyform;convertbetweenformsasappropriate;andassessthereasonablenessofanswersusingmentalcomputationandestimationstrategies.Forexample:Ifawomanmaking$25anhourgetsa10%raise,shewillmakeanadditional1/10ofhersalaryanhour,or$2.50,foranewsalaryof$27.50.Ifyouwanttoplaceatowelbar9¾incheslonginthecenterofadoorthatis27½incheswide,youwillneedtoplacethebarabout9inchesfromeachedge;thisestimatecanbeusedasacheckontheexactcomputation.

4. Usevariablestorepresentquantitiesinareal‐worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.

a. Solvewordproblemsleadingtoequationsoftheformpx+q=rand–(x+q)=r,wherep,q,andrarespecificrationalnumbers.Solveequationsoftheseformsfluently.Compareanalgebraicsolutiontoanarithmeticsolution,identifyingthesequenceoftheoperationsusedineachapproach.Forexample,theperimeterofarectangleis54cm.itslengthis6cm.whatisitswidth?

Norelatedstandards

11


Angles K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Geometricmeasurement:understandconceptsofangleandmeasureangles.5. Recognizeanglesas

geometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:

a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one‐degreeangle,”andcanbeusedtomeasureangles.

b. Ananglethatturnsthroughnone‐degreeanglesissaidtohaveananglemeasureofndegrees.

6. Measureanglesinwhole‐numberdegreesusingaprotractor.Sketchanglesofspecifiedmeasure.

7. Recognizeanglemeasureasadditive.Whenanangleisdecomposedintonon‐overlappingparts,theanglemeasureofthewholeisthesumoftheanglemeasuresofthepats.Solveadditionandsubtractionproblemstofindunknownanglesonadiagraminrealworldandmathematicalproblems,e.g.,byusinganequationwithasymbolfortheunknownanglemeasure.

Norelatedstandards Norelatedstandards 5. Usefactsaboutsupplementary,complementary,vertical,andadjacentanglesinamulti‐stepproblemtowriteandsolvesimpleequationsforanunknownangleinafigure.

5. Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle‐anglecriterionforsimilarityoftriangles.Forexample,arrangethreecopiesofthesametrianglesothatthesumofthethreeangelsappearstoformaline,andgiveanargumentintermsoftransversalswhythisisso.

12


Algebraic Notation and Solving Equations K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards 8. Solvetwo‐stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.3

2. Multiplyordividetosolvewordproblemsinvolvingmultiplicativecomparison,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem,distinguishingmultiplicativecomparisonfromadditivecomparison.1

3. Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole‐numberanswersusingthefouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.

2. Writesimpleexpressionsthatrecordcalculationswithnumbers,andinterpretnumericalexpressionswithoutevaluatingthem.Forexample,expressthecalculation“add8and7,thenmultiplyby2”as2X(8+7).Recognizethat3x(18932+921)isthreetimesaslargeas18932+921,withouthavingtocalculatetheindicatedsumorproduct.

2. Write,read,andevaluateexpressionsinwhichlettersstandfornumbers.

3. Writeexpressionsthatrecordoperationswithnumbersandwithlettersstandingfornumbers.Forexample,expressthecalculation“subtractyfrom5”as5y.

4. Applythepropertiesofoperationstogenerateequivalentexpressions.Forexample,applythedistributivepropertytotheexpression3(2+x)toproducetheequivalentexpression6+3x;applythedistributivepropertytotheexpression24x+18ytoproducetheequivalentexpression6(4x+3y);applypropertiesofoperationstoy+y+ytoproducetheequivalentexpression3y.

5. Identifywhentwoexpressionsareequivalent(i.e.,whenthetwoexpressionsnamethesamenumberregardlessofwhichvalueissubstitutedintothem).Forexample,theexpressionsy+y+yand3yareequivalentbecausetheynamethesamenumberregardlessofwhichnumberystandsfor.


13


Time K 1 2 3 4 5 6 7 8

Norelatedstandards

Tellandwritetime.3. Tellandwritetimein

hoursandhalf‐hoursusinganaloganddigitalclocks.

7. Tellandwritetimefromanaloganddigitalclockstothenearestfiveminutes,usinga.m.andp.m.

1. Tellandwritetimetothenearestminuteandmeasuretimeintervalsinminutes.Solvewordproblemsinvolvingadditionandsubtractionoftimeintervalsinminutes,e.g.,byrepresentingtheproblemonanumberlinediagram.

2. Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.Representmeasurementquantitiesusingdiagramssuchasnumberlinediagramsthatfeatureameasurementscale.


14


Money K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards 8. Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbolsappropriately.Example:Ifyouhave2dimesand3pennies,howmanycentsdoyouhave?

Norelatedstandards 2. Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.Representmeasurementquantitiesusingdiagramssuchasnumberlinediagramsthatfeatureameasurementscale.


15


Comparing and Ordering Numbers K 1 2 3 4 5 6 7 8

6. Identifywhetherthenumberofobjectsinonegroupisgraterthan,lessthan,orequaltothenumberofobjectsinanothergroup,e.g.,byusingmatchingandcountingstrategies.1

7. Comparetwonumbersbetween1and10presentedaswrittennumerals.

3. Comparetwotwo‐digitnumbersbasedonmeaningsofthetensandonesdigit,recordingtheresultsofcomparisonswiththesymbols>,+,and<.

4. Comparetwothree‐digitnumbersbasedonmeaningsofthehundreds,tensandonesdigits,using>,+,and<symbolstorecordtheresultsofcomparisons

3.NF3dComparetwofractionswiththesamenumeratororthesamedenominatorbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.

4.NBTReadandwritemulti‐digitwholenumbersusingbase‐tennumerals,numbernames,andexpandedform.Comparetwomulti‐digitnumbersbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.

4.NF2.Comparetwofractionswithdifferentnumeratorsanddifferentdenominators,e.g.,bycreatingcommondenominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas½.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswithsymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.

7. Comparetwodecimalstohundredthsbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwodecimalsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualmodel.

3. Read,write,andcomparedecimalstothethousandths.

a. Readandwritedecimalstothousandthsusingbase‐tennumerals,numbernames,andexpandedform,e.g.,347.392=3x100+4x10+7x1+3x(1/10)+9x(1/100)+2x(1/1000).

b. Comparetwodecimalstothousandthsbasedonmeaningsofthedigitsineachplace,using>,=,<symbolstorecordtheresultsofcomparisons

7. Understandorderingandabsolutevalueofrationalnumbers.

a. Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram.Forexample,interpret‐3>‐7asastatementthat3islocatedtotherightof7onanumberlineorientedfromlefttoright.

b. Write,interpret,andexplainstatementsoforderforrationalnumbersinreal‐worldcontexts.Forexample,write30Ciswarmerthat70C.

c. Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal‐worldsituation.Forexample,foranaccountbalanceof30dollars,write|30|=30todescribethesizeofthedebtindollars.

d. Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder.Forexample,recognizethatanaccountbalancelessthan30dollarsrepresentsadebtgreaterthan30dollars.

Norelatedstandards 2. Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2)Forexample,bytruncatingthedecimalexpansionof2,showthat2isbetween1and2,thenbetween1.4and1.5,andexplainhowtocontinueontogetbetterapproximations.

16


Rote Counting K 1 2 3 4 5 6 7 8

Knownumbernamesandthecountsequence.1. Countto100byones

andbytens.2. Countforward

beginningfromagivennumberwithintheknownsequence(insteadofhavingtobeginat1).

3. Writenumbersfrom0to20.Representanumberofobjectswithawrittennumeral0‐20(with0representingacountofnoobjects).

Extendthecountingsequence.1. Countto120,starting

atanynumberlessthan120.Inthisrange,readandwritenumeralsandrepresentanumberofobjectswithawrittennumeral.

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

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards

17


Rational Counting K 1 2 3 4 5 6 7 8

Counttotellthenumberofobjects.4. Understandthe

relationshipbetweennumbersandquantities;connectcountingtocardinality.

a. Whencountingobjects,saythenumbernamesinthestandardorder,pairingeachobjectwithoneandonlyonenumbernameandeachnumbernamewithoneandonlyoneobject.

b. Understandthatthelastnumbernamesaidtellsthenumberofobjectscounted.Thenumberofobjectsisthesameregardlessoftheirarrangementortheorderinwhichtheywerecounted.

c. Understandthateachsuccessivenumbernamereferstoaquantitythatisonelarger.

5 Counttoanswer“howmany?”questionsaboutasmanyas20thingsarrangedinaline,arectangulararray,oracircle,orasmanyas10thingsinascatteredconfiguration;givenanumberfrom1‐20,countoutthatmanyobjects.

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards

18


Place Value and Notation K 1 2 3 4 5 6 7 8

Workwithnumbers1119togainfoundationsforplacevalue.1. Composeand

decomposenumbersfrom11to19intotenonesandsomefurtherones,e.g.,byusingobjectsordrawings,andrecordeachcompositionordecompositionbyadrawingorequation(e.g.,18=10+8);understandthatthesenumbersarecomposedoftenonesandone,two,three,four,five,six,seven,eight,ornineones.

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).

1. Understandthatthetreedigitsofathree‐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).

3. Readandwritenumberst1000usingbase‐tennumerals,numbernames,andexpandedform.

1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.

1. Recognizethatinamulti‐digitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.Forexample,recognizethat700÷70=10byapplyingconceptsofplacevaleanddivision.

2. Readandwritemulti‐digitwholenumbersusingbase‐tennumerals,numbernames,andexpandedform.Comparetwomulti‐digitnumbersbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.

3. Useplacevalueunderstandingtoroundmulti‐digitwholenumberstoanyplace.

1. Recognizethatinmulti‐digitnumber,adigitinoneplacerepresents10timesassuchasitrepresentsintheplacetoitsrightand1/10ofwhatitrepresentsinthelacetoitsleft.

2. Explainpatternsinthenumberofzerosoftheproductwhenmultiplyinganumberbypowersof10,andexplainpatternsintheplacementofthedecimalpointwhenadecimalismultipliedordividedbyapowerof10.Usewhole‐numberexponentstodenotepowersof10.

3. Read,write,andcomparedecimalstothousandths.

a. Readandwritedecimalstothousandthsusingbase‐tennumerals,numbernames,andexpandedform,e.g.,347,392=3x100+4x10+7x1+3x(1/10)+9x(1/100)+2x(1/1000).

b. Comparetwodecimalstothousandthsbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.

4. Useplacevalueunderstandingtorounddecimalstoanyplace.


19


Radicals and Exponents K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 6.EE1Writeandevaluatenumericalexpressionsinvolvingwhole‐numberexponents.

Norelatedstandards Workwithradicalsandintegerexponents.8.EE1Knowandapplythepropertiesof

integerexponentstogenerateequivalentnumericalexpressions.Forexample,32x35=33=1/33=1/27.

8.EE2Usesquarerootandcuberootsymbolstorepresentsolutionstoequationsoftheformx2=pandx3=p,wherepisapositiverationalnumber.Evaluatesquarerootsofsmallperfectsquaresandcuberootsofsmallperfectcubes.Knowthat✓2is irrational.

8.EE3Usenumbersexpressintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.Forexample,estimatethepopulationoftheUnitedStatesas3x109andthepopulationoftheworldas7x108,anddeterminethattheworldpopulationismorethan20timeslarger.

8.EE4Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothdecimalandscientificnotationareused.Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities(e.g.,usemillimetersperyearforseafloorspreading).Interpretscientificnotationthathasbeengeneratedbytechnology.

8.NS1Knowthatnumbersthatarenotrationalarecalledirrational.Understandinformallythateverynumberhasadecimalexpansion;forrationalnumbersshowthatthedecimalexpansionrepeatseventually,andconvertadecimalexpansionwhichrepeatseventuallyintoarationalnumber.

8.NS2Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2).Forexample,bytruncatingthedecimalexpansionof✓2,showthat✓2isbetween1and2,thenbetween1.4and1.5,andexplainhowtocontinueontogetbetterapproximations.

20


Number Theory K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 4. Findallfactorpairsforwholenumberintherange1‐100.Recognizethatawholenumberisamultipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1‐100ismultipleofagivenone‐digitnumber.Determinewhetheragivenwholenumberintherange1‐100isprimeorcomposite.

Norelatedstandards 5. Understandthatpositiveandnegativenumbersareusedtogethertodescribequantitieshavingoppositedirectionsorvalues(e.g.,temperatureabove/belowzero,elevationabove/belowsealevel,credits/debits,positive/negativeelectriccharge);usepositiveandnegativenumberstorepresentquantitiesinreal‐worldcontexts,explainingthemeaningof0ineachsituation.

7. Understandingorderingandabsolutevalueofrationalnumbers.

a. Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram.Forexample,interpret3>7asastatementthat3islocatedtotherightof7onanumberlineorientedfromlefttoright.

b. Write,interpret,andexplainstatementsoforderforrationalnumbersinreal‐worldcontexts.Forexample,write30C>70Ctoexpressthefactthat30Ciswarmerthan70C.

c. Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal‐worldsituation.Forexample,foranaccountbalanceof30dollars,write|30|=30todescribethesizeofthedebtindollars.

d. Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder.Forexample,recognizethatanaccountbalancelessthan30dollarsrepresentsadebtgreaterthan30dollars.

1. Applyandextendpreviousunderstandingsofadditionandsubtractiontoaddandsubtractrationalnumbers;representadditionandsubtractiononahorizontalorverticalnumberlinediagram.

a. Describesituationsinwhichoppositequantitiescombinetomake0.Forexample,ahydrogenatomhas0chargebecauseitstowconstituentsareoppositelycharged.

b. Understandp+qasthenumberlocatedadistance|q|fromp,inthepositiveornegativedirectiondependingonwhetherqispositiveornegative.Showthatanumberanditsoppositehaveasumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal‐worldcontexts.

c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse,pq=p+(q).Showthatthedistancebetweentworationalnumbersonthenumberlineistheabsolutevalueoftheirdifference,andapplythisprincipalinrealworldcontexts.

d. Applypropertiesofoperationsasstrategiestoaddandsubtractrationalnumbers.

Norelatedstandards

21


Ratio and Proportions K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards

1. Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities.Forexample,“Theratioofwingstobeaksinthebirdhouseatthezoowas2:1,becauseforevery2wingstherewas1beak.”“ForeveryvotecandidateAreceived,candidateCreceivednearlythreevotes.”

2. Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb≠0,anduseratelanguageinthecontextofaratiorelationship.Forexample,“Thisrecipehasaratioof3cupsofflourto4cupsofsugar,sothereis¾cupofflourforeachcupofsugar.”Wepaid$75for15hamburgers,whichisarateof$5perhamburger.”

3. Useratioandratereasoningtosolvereal‐worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.

a. Maketablesofequivalentratiosrelatingquantitieswithwhole‐numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.

b. Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.Forexample,ifittook7hourstomow4lawns,thenatthatrate,howmanylawnscouldbemowedin35hours?Atwhatratewerelawnsbeingmowed?

c. Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.

d. Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.

1. Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.Forexample,ifapersonwalks½mileineach¼hour,computetheunitrateasthecomplexfraction½/¼milesperhour,equivalently2milesperhour.

2. Recognizeandrepresentproportionalrelationshipsbetweenquantities.

a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.

b. Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.

c. Representproportionalrelationshipsbyequations.Forexample,iftotalcostisproportionaltothenumbernofitemspurchasedataconstantpricep,therelationshipbetweenthetotalcostandthenumberitemscanbeexpressedast=pn.

d. Explainwhatapoint(x,y)onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentiontothepoints(0,0)and(1,r)whereristheunitrate.

3. Useproportionalrelationshipstosolvemultistepratioandpercentproblems.Examples:simpleinterest,tax,markupsandmarkdowns,gratuitiesandcommissions,fees,percentincreaseanddecrease,percenterror.

5. Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.Forexample,compareadistancetimegraphtoadistancetimeequationtodeterminewhichoftwomovingobjectshasgreaterspeed.

6. Usesimilartrianglestoexplainwhytheslopemisthesamebetweenanytwodistinctpointsonanon‐verticallineinthecoordinateplane;derivetheequationy=mxforalinethroughtheoriginandtheequationy=mx+bforalineinterceptingtheverticalaxisatb.

22


Functions K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 9. Usevariablestorepresenttwoquantitiesinareal‐worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable,intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariablesusinggraphsandtables,andrelatethesetotheequation.Forexample,inaprobleminvolvingmotionatconstantspeed,listandgraphorderedpairsofdistancesandtimes,andwritetheequationd=65ttorepresenttherelationshipbetweendistanceandtime.

Norelatedstandards Define,evaluate,andcomparefunctions.1. Understandthatafunctionisarulethatassigns

toeachinputexactlyoneoutput.Thegraphofafunctionisthesetoforderedpairsconsistingofaninputandthecorrespondingoutput.1

2. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).Forexample,givenalinearfunctionrepresentedbyatableofvaluesandalinearfunctionrepresentedbyanalgebraicexpression,determinewhichfunctionhasthegreaterrateofchange.

3. Interprettheequationy=mx+basdefiningalinearfunction,whosegraphisastraightline;giveexamplesoffunctionsthatarenotlinear.Forexample,thefunctionA=S2givingtheareaofasquareasafunctionofitssidelengthisnotlinearbecauseitsgraphcontainsthepoints(1,1),(2,4)and(3,9),whicharenotonastraightline.

Usefunctionstomodelrelationshipsbetweenquantities.4. Constructafunctiontomodelalinear

relationshipbetweentwoquantities.Determinetherateofchangeandinitialvalueofthefunctionfromadescriptionofarelationshiporfromtwo(x,y)values,includingreadingthesefromatableorfromagraph,Interprettherateofchangeandinitialvalueofalinearfunctionintermsofthesituationitmodels,andintermsofitsgraphoratableofvalues.

5. Describequalitativelythefunctionalrelationshipbetweentwoquantitiesbyanalyzingagraph(e.g.,wherethefunctionisincreasingordecreasing,linearornonlinear).Sketchagraphthatexhibitsthequalitativefeaturesofafunctionthathasbeendescribedverbally.

23


Pythagorean K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards UnderstandandapplythePythagoreanTheorem.6. Explainaproofofthe

PythagoreanTheoremanditsconverse.

7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal‐worldandmathematicalproblemsintwoandthreedimensions.

8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.

24


Area, Perimeter, Volume, Surface Area, and Capacity K 1 2 3 4 5 6 7 8

Norelatedstandards

Norelatedstandards

2. Partitionarectangleintorowsandcolumnsofsame‐sizesquaresandcounttofindthetotalnumberofthem.

2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardsunitsofgrams(g),kilograms(kg),andliters(1),6Add,subtract,multiply,ordividetosolveone‐stepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.7

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+cisthesumofaxbandaxc.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.

d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposingthemintonon‐overlappingrectanglesandaddingtheareasofthenon‐overlappingparts,applyingthistechniquetosolverealworldproblems.

8 Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.

3. Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.Forexample,findthewidthofarectangularroomgiventheareaoftheflooringandthelength,byviewingtheareaformulaasamultiplicationequationwithanunknownfactor.

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‐numberproductsavolumes,e.g.,torepresenttheassociativepropertyofmultiplication.

b. ApplytheformulasV=lxwxhandv=bxhforrectangularprismstofindvolumesofrightrectangularprismswithwhole‐numberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.

c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwonon‐overlappingrightrectangularprismsbyaddingthevolumesofthenon‐overlappingparts,applyingthistechniquetosolverealworldproblems.

1. Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal‐worldandmathematicalproblems.

2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV=lwhandV=bhtofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal‐worldandmathematicalproblems.

4. Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems;giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle.

6. Solvereal‐worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo‐andthree‐dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.

9. Knowtheformulasforthevolumesofconescylinders,andspheresandusethemtosolvereal‐worldandmathematicalproblems.

25


Multiplication and Division Facts K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards 1. Interpretproductsofwholenumbers,e.g.,interpret5x7asthetotalnumberofobjectsin5groupsof7objectseach.Forexample,describeacontextinwhichatotalnumberofobjectscanbeexpressedas5x7.

2. Interpretwhole‐numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.Forexample,describeacontextinwhichanumberofsharesoranumberofgroupscanbeexpressedas56÷8.

3. Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswithsymbolfortheunknownnumbertorepresenttheproblem.1

4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.Forexample,determinetheunknownnumberthatmakestheequationtrueineachoftheequations8x?=48,5=[]÷3,6x6=?.

5. Applypropertiesofoperationsasstrategiestomultiplyanddivide,2Examples:If6x4=24isknown,then4x6=24isalsoknown.(Commutativepropertyofmultiplication.)3x5x2canbefoundby3x5=15,then15x2=30,orby5x2=10,then3x10=30.(Associativepropertyofmultiplication,)Knowingthat8x5=40and8x2=16,onecanfind8x7as8x(5+2)=(8x5)+(8x2)=40+16=56.(Distributiveproperty.)

6. Understanddivisionasanunknown‐factorproblem.Forexample,find32÷8byfindingthenumberthatmakes32whenmultipliedby8.

7. Fluentlymultiplyanddividewithin100,usingstrategiessuchastherelationshipbetweenmultiplicationanddivision(e.g.,knowingthat8x5=40,oneknows40÷5=8)orpropertiesofoperations.BytheendofGrade3,knowfrommemoryallproductsoftwoone‐digitnumbers.

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards

26


Transformations, Congruence, Similarity, and Symmetry K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 3. Recognizealineofsymmetryforatwo‐dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline‐symmetryfiguresandrawlinesofsymmetry.

Norelatedstandards Norelatedstandards Norelatedstandards 1. Verifyexperimentallythepropertiesofrotations,reflections,andtranslations:

a. Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.

b. Anglesaretakentoanglesofthesamemeasure.

c. Parallellinesaretakentoparallellines.

2. Understandthatatwo‐dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.

3. Describetheeffortofdilations,translations,rotations,andreflectionsontwo‐dimensionalfiguresusingcoordinates.

4. Understandthatatwo‐dimensionalfigureissimilartoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo‐dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.

27


Probability K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 5. Understandthattheprobabilityofachangeeventisanumberbetween0and1thatexpressesthelikelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0indicatesanunlikelyevent,aprobabilityaround½indicatesaneventthatisneitherunlikelynorlikely,andaprobabilitynear1indicatesalikelyevent.

6. Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesitandobservingitslong‐runrelativefrequency,andpredicttheapproximaterelativefrequencygiventheprobability.Forexample,whenrollinganumbercube600times,predictthata3or6wouldberolledroughly200times,butprobablynotexactly200times.

7. Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodeltoobservedfrequencies;iftheagreementsnotgood,explainpossiblesourcesofthediscrepancy.

a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilitiesofevents.Forexample,ifastudentisselectedatrandomfromaclass,findtheprobabilitythatJanewillbeselectedandtheprobabilitythatagirlwillbeselected.

b. Developaprobabilitymodel(whichmaynotbeuniform)byobservingfrequenciesindatageneratedfromachanceprocess.Forexample,findtheapproximateprobabilitythatsspinningpennywilllandheadsuporthatatossedpapercupwilllandopenenddown.Dotheoutcomesforthespinningpennyappeartobeequallylikelybasedontheobservedfrequencies?

8. Findprobabilitiesofcompoundeventsusingorganizelists,tables,treediagrams,andsimulation

a. Understandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs.

b. Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespacewhichcomposetheevent.

c. Designanduseasimulationtogeneratefrequenciesforcompoundevents.Forexample,userandomdigitsasasimulationtooltoapproximatetheanswertothequestion:If40%ofdonorshavetypeablood,whatistheprobabilitythatitwilltakeatleast4donorstofindonewithtypeAblood?

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Order of Operations K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 1. Useparentheses,brackets,orbracesinnumericalexpressions,andevaluateexpressionswiththesesymbols.

6.EE2BIdentifypartsofanexpressionusingmathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofanexpressionasasingleentity.Forexample,describetheexpression2(8+7)asaproductoftwofactors;view(8+8)asbothasingleentityandasumoftwoterms.

6.EE2CEvaluateexpressionsatspecificvaluesoftheirvariables.Includeexpressionsthatarisefromformulasusedinreal‐worldproblems.Performarithmeticoperations,includingthoseinvolvingwhole‐numberexponents,intheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).Forexample,usetheformulasV=S3andA=6S2tofindthevolumeandsurfaceareaofacubewithsidesoflengthS=½.


29


Units and Systems of Measurement K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 4.MD1Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.Recordmeasurementequivalentsinatwo‐columntable.Forexample,knowthat1ftis12timesaslongas1in.Expressthelengthofa4ftsnakeas48in.Generateaconversiontableforfeetandincheslistingthenumberpars(1,12),(2,24),(3,36),…

5.MD1Convertamongdifferent‐sizedstandardsmeasurementunitswithinagivenmeasurementsystem(e.g.,convert5cmto0.05),andusetheseconversionsinsolvingmulti‐step,realworldproblems.


30


Operations with Decimals K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 7. Add,subtract,multiply,anddividedecimalstohundredths,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.

3. Fluentlyadd,subtract,multiply,anddividemulti‐digitdecimalsusingthestandardalgorithmforeachoperation.


31


Operations with Decimals K 1 2 3 4 5 6 7 8

Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards Norelatedstandards 7. Add,subtract,multiply,anddividedecimalstohundredths,usingconcretemodelsordrawingsandstrategiesbasedonplacedvalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.



32


Data Collection K 1 2 3 4 5 6 7 8

Classifyobjectsandcountthenumberofobjectsineachcategory.3. Classifyobjectsinto

givencategories;countthenumbersofobjectsineachcategoryandsortthecategoriesbycount.3

Representandinterpretdata.4. Organize,represent,

andinterpretdatawithuptothreecategories;askandanswerquestionsaboutthetotalnumberofdatapoints,howmanyineachcategory,andhowmanymoreorlessareinonecategorythaninanother.

Representandinterpretdata.9. Generate

measurementdatabymeasuringlengthsofseveralobjectstothenearestwholeunit,orbymakingrepeatedmeasurementsofthesameobject.Showthemeasurementsbymakingalineplot,wherethehorizontalscaleismarkedoffinwhole‐numberunits.

10. Drawapicturegraphandabargraph(withsingle‐unitscale)torepresentadatasetwithuptofourcategories.Solvesimpleput‐together,take‐apart,andcompareproblems4usinginformationpresentedinabargraph.

Representandinterpretdata.3. Drawascaledpicture

graphandascaledbargraphtorepresentadatasetwithseveralcategories.Solveone‐andtwo‐step“howmanymore”and“howmanyless”problemsusinginformationpresentedinscaledbargraphs.Forexample,drawabargraphinwhicheachsquareinthebargraphmightrepresent5pets.

4. Generatemeasurementdatabymeasuringlengthsusingrulersmarkedwithhalvesandfourthsofaninch.Showthedatabymakingalineplot,wherethehorizontalscaleismarkedoffinappropriateunits‐wholenumbers,halves,orquarters.


displayadatasetofmeasurementsinfractionsofaunit(1/2,¼,1/8).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.Forexample,fromalineplotfindandinterpretthedifferenceinlengthbetweenthelongestandshortestspecimensinaninsectcollection.


displayadatasetofmeasurementsinfractionsofaunit(1/2,¼,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedlineplots.Forexample,givendifferentmeasurementsofliquidinidenticalbeakers,findtheamountofliquideachbeakerwouldcontainifthetotalamountinallthebeakerswereredistributedequally.

Summarizeanddescribedistributions.4. Displaynumerical

datainplotsonanumberline,includingdotplots,histograms,andboxplots.


33


Data Analysis K 1 2 3 4 5 6 7 8

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Developunderstandingofstatisticalvariability.1. Recognizeastatisticalquestionasone

thatanticipatesvariabilityinthedatarelatedtothequestionandaccountsforitintheanswers.Forexample,“HowoldamI?”isnotastatisticalquestion,but“Howoldarethestudentsinmyschool?”isastatisticalquestionbecauseoneanticipatesvariabilityinstudents’ages.

2. Understandthatasetofdatacollectedtoanswerastatisticalquestionhasadistributionwhichcanbedescribedbyitscenter,spread,andoverallshape.

3. Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber,whileameasureofvariationdescribeshowitsvaluesvarywithasinglenumber.

5. Summarizenumericaldatasetsinrelationtotheircontext,suchasby:

a. Reportingthenumberofobservations.b. Describingthenatureoftheattribute

underinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.

c. Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.

d. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.

Userandomsamplingtodrawinferencesaboutapopulation.1. Understandthatstatisticscanbeusedtogain

informationaboutapopulationbyexaminingasampleofthepopulation;generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation.Understandthatrandomsamplingtendstoproducerepresentativesamplesandsupportvalidinferences.

2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.Forexample,estimatethemeanwordlengthinabookbyrandomlysamplingwordsfromthebook;predictthewinnerofaschoolelectionbasedonrandomlysampledsurveydata.Gaugehowfarofftheestimateorpredictionmightbe.

Drawinformalcomparativeinferencesabouttwopopulations.3. Informallyassessthedegreeofvisualoverlap

oftwonumericaldatadistributionswithsimilarvariabilities,measuringthedifferencebetweenthecentersbyexpressingitasamultipleofameasureofvariability.Forexample,themeanheightofplayersonthebasketballteamis10cmgreaterthanthemeanheightofplayersonthesoccerteam,abouttwicethevariability(meanabsolutedeviation)oneitherteam;onadotplot,theseparationbetweenthetwodistributionsofheightsisnoticeable.

4. Usemeasuresofcenterandmeasuresofvariabilityfornumericaldatafromrandomsamplestodrawinformalcomparativeinferencesabouttwopopulations.Forexample,decidewhetherthewordsinachapterofaseventhgradesciencebookaregenerallylongerthanthewordsinachapterofafourthgradesciencebook.

Investigatepatternsofassociationinbivariatedata.1. Constructandinterpretscatter

plotsforbivariatemeasurementdatatoinvestigatepatternsofassociationbetweentwoquantities.Describepatternssuchasclustering,outliers,positiveornegativeassociation,linearassociation,andnonlinearassociation.

2. Knowthatstraightlinesarewidelyusedtomodelrelationshipsbetweentwoquantitativevariables.Forscatterplotsthatsuggestalinearassociation,informallyfitastraightline,andinformallyassessthemodelfitbyjudgingtheclosenessofthedatapointstotheline.

3. Usetheequationofalinearmodeltosolveproblemsinthecontextofbivariatemeasurementdata,interpretingtheslopeandintercept.Forexample,inalinearmodelforabiologyexperiment,interpretaslopeof1.5cm/hrasmeaningthatanadditionalhourofsunlighteachdayisassociatedwithanadditional1.5cminmatureplantheight.

4. Understandthatpatternsofassociationcanalsobeseeninbivariatecategoricaldatabydisplayingfrequenciesandrelativefrequenciesinatwo‐waytable.Constructandinterpretatwo‐waytablesummarizingdataontwocategoricalvariablescollectedfromthesamesubjects.Userelativefrequenciescalculatedforrowsorcolumnstodescribepossibleassociationbetweenthetwovariables.Forexample,collectdatafromstudentsinyourclassonwhetherornottheyhaveacurfewoschoolnightsandwhetherornottheyhaveassignedchoresathome.Isthereevidencethatthosewhohaveacurfewalsotendtohavechores.

34


Procedures for addition and subtraction

K 1 2 3 4 5 6 7 8Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartandtakingfrom.1. Representadditionand

subtractionwithobjects,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.

Representandsolveproblemsinvolvingadditionandsubtraction.1. Useadditionandsubtractionwithin20to

solvewordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheprobem.2

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

Useplacevalueunderstandingandpropertiesofoperationstoaddandsubtract.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;relatethestrategytoawrittenmethodandexplainthereasoningused.

Representandsolveproblemsinvolvingadditionandsubtraction.1. Useadditionandsubtractionwithin100to

solveone‐andtwo‐stepwordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1

Useplacevalueunderstandingandpropertiesofoperationstoaddandsubtract.5. Fluentlyaddandsubtractwithin100using

strategiesbasedonplacevalue,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

Relateadditionandsubtractiontolength.5. Useadditionandsubtractionwithin100to

solvewordproblemsinvolvinglengthsthataregiveninthesameunits,e.g.,byusingdrawings(suchasdrawingsofrulers)andequationswithasymbolfortheunknownnumbertorepresenttheproblem.

6. Representwholenumbersaslengthsfrom0onanumberlinediagramwithequallyspacedpointscorrespondingtothenumber0,1,2…,andrepresentwhole‐numbersumsanddifferenceswithin100onanumberlinediagram.

3.OA8Solvetwo‐stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.3

3.NBT2Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.

3. Solvemultistepwordproblemsposeswithwholenumbersandhavingwhole‐numberanswersusingthefuroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.

4. Fluentlyaddandsubtractmulti‐digitwholenumbersusingthestandardalgorithm.

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35


Procedures for Multiplication and Division K 1 2 3 4 5 6 7 8

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Workwithequalgroupsofobjectstogainfoundationsformultiplication2.OA3Determine

whetheragroupofobjects(upto20)hasanoddorevennumberofmembers,e.g.,bypairingobjectsorcountingthemby2s;writeanequationtoexpressanevennumberasasumoftwoequaladdends.

2.OA4Useadditiontofindthetotalnumberofobjectsarrangedinrectangulararrayswithupto5rowsandupto5columns;writeanequationtoexpressthetotalasasumofequaladdends.

2.G2Partitionarectangleintorowsandcolumnsofsame‐sizesquaresandcounttofindthetotalnumberofthem.

Representandsolveproblemsinvolvingmultiplicationanddivision.1. Interpretproductsofwhole

numbers,e.g.,interpret5x7asthetotalnumberofobjectsin5groupsof7objectseach.Forexample,describeacontextinwhichatotalnumberofobjectscanbeexpressedas5x7.

2. Interpretwhole‐numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberfobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.Forexample,describeacontextinwhichanumberofsharesoranumberofgroupscanbeexpressedas56÷8.

3. Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1

4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.Forexample,determinetheunknownnumberthatmakestheequationtrueineachoftheequations8x?=48,5=[]÷3,6x6=?/

3.OA8Solvetwo‐stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.3

3.NBT3Multiplyone‐digitwholenumbersbymultiplesof10intherange10‐90(e.g.,9x80,5x60)usingstrategiesbasedonplacevalueandpropertiesofoperations

1. Interpretamultiplicationequationasacomparison,e.g.,interpret35=5x7asastatementthat35is5timesasmanas7and7timesasmanyas5.Representverbalstatementsofmultiplicativecomparisonsasmultiplicationequations.

2. Multiplyordividetosolvewordproblemsinvolvingmultiplicativecomparison,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem,distinguishingmultiplicativecomparisonfromadditivecomparison.1

3. Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole‐numberanswersusingthefouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.

5. Multiplyawholenumberofuptofourdigitsbyaone‐digitwholenumber,andmultiplytwotwo‐digitnumbers,usingstrategiesbasedonplacevalueandthepropertiesofoperations.Illustrateandexplainthecalculationbyusingequations,rectangulararraysand/orareamodels.

6. Findwhole‐numberquotientsandremainderswithuptofour‐digitdividendsandone‐digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.

5. Interpretmultiplicationasscaling(resizing),by:

a. Comparingthesizeofaproducttothesizeofonefactoronthebasisofthesizeoftheotherfactor,withoutperformingtheindicatedmultiplication.

b. Explainingwhymultiplyingagivennumberbyafractiongraterthan1resultsinaproductgreaterthanthegivennumber(recognizingmultiplicationbywholenumbersgreaterthan1asafamiliarcase);explainingwhymultiplyingagivennumberbyafractionlessthan1resultsinaproductsmallerthanthegivennumber;andrelatingtheprincipleoffractionequivalencea/b=(nxa)/(nxb)totheeffectofmultiplyinga/bby1.

6. Fluentlymultiplymulti‐digitwholenumbersusingthestandardalgorithm.

7. Findwhole‐numberquotientsofwholenumberswithuptofour‐digitdividendsandtwo‐digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.

2. Fluentlydividemulti‐digitnumbersusingthestandardalgorithm.


4. Findthegreatestcommonfactoroftwowholenumberslessthanorequalto100andtheleastcommonmultipleoftwowholenumberslessthanorequalto12.Usethedistributivepropertytoexpressasumoftwowholenumbers1‐100withacommonfactorasamultipleofasumoftwowholenumberswithnocommonfactor.Forexample,express36+8as4(9=2).

2. Applyandextendpreviousunderstandingsofmultiplicationanddivisionoffractionstomultiplyanddividerationalnumbers.

a. Understandthatmultiplicationisextendedfromfractionstorationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(‐1)(‐1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbersbydescribingreal‐worldcontexts.

b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon‐zerodivisor)isarationalnumber.Ifpandqareintegers,then–(p/q)=(p)/q=p/(q).Interpretquotientsofrationalnumbersbydescribingrealworldcontexts.

c. Applypropertiesofoperationsasstrategiestomultiplyanddividerationalnumbers.

d. Convertarationalnumbertoadecimalusinglongdivision;knowthatthedecimalformofarationalnumberterminatesin0soreventuallyrepeats.

3. Solvereal‐worldandmathematicalproblemsinvolvingthefouroperationswithrationalnumbers.1

Norelatedstandards

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Common Core Vertical Alignment

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