mathematics | kindergartendoe.sd.gov/contentstandards/documents/math/k_8_math_standards.pdf ·...
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Common Core State StandardS for matHematICSK
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mathematics | KindergartenInKindergarten,instructionaltimeshouldfocusontwocriticalareas:(1)
representing,relating,andoperatingonwholenumbers,initiallywith
setsofobjects;(2)describingshapesandspace.Morelearningtimein
Kindergartenshouldbedevotedtonumberthantoothertopics.
(1)Studentsusenumbers,includingwrittennumerals,torepresent
quantitiesandtosolvequantitativeproblems,suchascountingobjectsin
aset;countingoutagivennumberofobjects;comparingsetsornumerals;
andmodelingsimplejoiningandseparatingsituationswithsetsofobjects,
oreventuallywithequationssuchas5+2=7and7–2=5.(Kindergarten
studentsshouldseeadditionandsubtractionequations,andstudent
writingofequationsinkindergartenisencouraged,butitisnotrequired.)
Studentschoose,combine,andapplyeffectivestrategiesforanswering
quantitativequestions,includingquicklyrecognizingthecardinalitiesof
smallsetsofobjects,countingandproducingsetsofgivensizes,counting
thenumberofobjectsincombinedsets,orcountingthenumberofobjects
thatremaininasetaftersomearetakenaway.
(2)Studentsdescribetheirphysicalworldusinggeometricideas(e.g.,
shape,orientation,spatialrelations)andvocabulary.Theyidentify,name,
anddescribebasictwo-dimensionalshapes,suchassquares,triangles,
circles,rectangles,andhexagons,presentedinavarietyofways(e.g.,with
differentsizesandorientations),aswellasthree-dimensionalshapessuch
ascubes,cones,cylinders,andspheres.Theyusebasicshapesandspatial
reasoningtomodelobjectsintheirenvironmentandtoconstructmore
complexshapes.
Common Core State StandardS for matHematICSK
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Counting and Cardinality
• Know number names and the count sequence.
• Count to tell the number of objects.
• Compare numbers.
operations and algebraic thinking
• Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
number and operations in Base ten
• Work with numbers 11–19 to gain foundations for place value.
measurement and data
• describe and compare measurable attributes.
• Classify objects and count the number of objects in categories.
Geometry
• Identify and describe shapes.
• analyze, compare, create, and compose shapes.
mathematical Practices
1. Makesenseofproblemsandperseverein
solvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
Grade K overview
Common Core State StandardS for matHematICSK
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Counting and Cardinality K.CC
Know number names and the count sequence.
1. Countto100byonesandbytens.
2. Countforwardbeginningfromagivennumberwithintheknownsequence(insteadofhavingtobeginat1).
3. Writenumbersfrom0to20.Representanumberofobjectswithawrittennumeral0-20(with0representingacountofnoobjects).
Count to tell the number of objects.
4. Understandtherelationshipbetweennumbersandquantities;connectcountingtocardinality.
a. Whencountingobjects,saythenumbernamesinthestandardorder,pairingeachobjectwithoneandonlyonenumbernameandeachnumbernamewithoneandonlyoneobject.
b. Understandthatthelastnumbernamesaidtellsthenumberofobjectscounted.Thenumberofobjectsisthesameregardlessoftheirarrangementortheorderinwhichtheywerecounted.
c. Understandthateachsuccessivenumbernamereferstoaquantitythatisonelarger.
5. Counttoanswer“howmany?”questionsaboutasmanyas20thingsarrangedinaline,arectangulararray,oracircle,orasmanyas10thingsinascatteredconfiguration;givenanumberfrom1–20,countoutthatmanyobjects.
Compare numbers.
6. Identifywhetherthenumberofobjectsinonegroupisgreaterthan,lessthan,orequaltothenumberofobjectsinanothergroup,e.g.,byusingmatchingandcountingstrategies.1
7. Comparetwonumbersbetween1and10presentedaswrittennumerals.
operations and algebraic thinking K.oa
Understand addition as putting together and adding to, and under-stand subtraction as taking apart and taking from.
1. Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings2,sounds(e.g.,claps),actingoutsituations,verbalexplanations,expressions,orequations.
2. Solveadditionandsubtractionwordproblems,andaddandsubtractwithin10,e.g.,byusingobjectsordrawingstorepresenttheproblem.
3. Decomposenumberslessthanorequalto10intopairsinmorethanoneway,e.g.,byusingobjectsordrawings,andrecordeachdecompositionbyadrawingorequation(e.g.,5=2+3and5=4+1).
4. Foranynumberfrom1to9,findthenumberthatmakes10whenaddedtothegivennumber,e.g.,byusingobjectsordrawings,andrecordtheanswerwithadrawingorequation.
5. Fluentlyaddandsubtractwithin5.
1Includegroupswithuptotenobjects.2Drawingsneednotshowdetails,butshouldshowthemathematicsintheproblem.(ThisapplieswhereverdrawingsarementionedintheStandards.)
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number and operations in Base ten K.nBt
Work with numbers 11–19 to gain foundations for place value.
1. Composeanddecomposenumbersfrom11to19intotenonesandsomefurtherones,e.g.,byusingobjectsordrawings,andrecordeachcompositionordecompositionbyadrawingorequation(e.g.,18=10+8);understandthatthesenumbersarecomposedoftenonesandone,two,three,four,five,six,seven,eight,ornineones.
measurement and data K.md
Describe and compare measurable attributes.
1. Describemeasurableattributesofobjects,suchaslengthorweight.Describeseveralmeasurableattributesofasingleobject.
2. Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“moreof”/“lessof”theattribute,anddescribethedifference.For example, directly compare the heights of two children and describe one child as taller/shorter.
Classify objects and count the number of objects in each category.
3. Classifyobjectsintogivencategories;countthenumbersofobjectsineachcategoryandsortthecategoriesbycount.3
Geometry K.G
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
1. Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,in front of,behind,andnext to.
2. Correctlynameshapesregardlessoftheirorientationsoroverallsize.
3. Identifyshapesastwo-dimensional(lyinginaplane,“flat”)orthree-dimensional(“solid”).
Analyze, compare, create, and compose shapes.
4. Analyzeandcomparetwo-andthree-dimensionalshapes,indifferentsizesandorientations,usinginformallanguagetodescribetheirsimilarities,differences,parts(e.g.,numberofsidesandvertices/“corners”)andotherattributes(e.g.,havingsidesofequallength).
5. Modelshapesintheworldbybuildingshapesfromcomponents(e.g.,sticksandclayballs)anddrawingshapes.
6. Composesimpleshapestoformlargershapes.For example, “Can you join these two triangles with full sides touching to make a rectangle?”
3Limitcategorycountstobelessthanorequalto10.
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mathematics | Grade 1InGrade1,instructionaltimeshouldfocusonfourcriticalareas:(1)
developingunderstandingofaddition,subtraction,andstrategiesfor
additionandsubtractionwithin20;(2)developingunderstandingofwhole
numberrelationshipsandplacevalue,includinggroupingintensand
ones;(3)developingunderstandingoflinearmeasurementandmeasuring
lengthsasiteratinglengthunits;and(4)reasoningaboutattributesof,and
composinganddecomposinggeometricshapes.
(1)Studentsdevelopstrategiesforaddingandsubtractingwholenumbers
basedontheirpriorworkwithsmallnumbers.Theyuseavarietyofmodels,
includingdiscreteobjectsandlength-basedmodels(e.g.,cubesconnected
toformlengths),tomodeladd-to,take-from,put-together,take-apart,and
comparesituationstodevelopmeaningfortheoperationsofadditionand
subtraction,andtodevelopstrategiestosolvearithmeticproblemswith
theseoperations.Studentsunderstandconnectionsbetweencounting
andadditionandsubtraction(e.g.,addingtwoisthesameascountingon
two).Theyusepropertiesofadditiontoaddwholenumbersandtocreate
anduseincreasinglysophisticatedstrategiesbasedontheseproperties
(e.g.,“makingtens”)tosolveadditionandsubtractionproblemswithin
20.Bycomparingavarietyofsolutionstrategies,childrenbuildtheir
understandingoftherelationshipbetweenadditionandsubtraction.
(2)Studentsdevelop,discuss,anduseefficient,accurate,andgeneralizable
methodstoaddwithin100andsubtractmultiplesof10.Theycompare
wholenumbers(atleastto100)todevelopunderstandingofandsolve
problemsinvolvingtheirrelativesizes.Theythinkofwholenumbers
between10and100intermsoftensandones(especiallyrecognizingthe
numbers11to19ascomposedofatenandsomeones).Throughactivities
thatbuildnumbersense,theyunderstandtheorderofthecounting
numbersandtheirrelativemagnitudes.
(3)Studentsdevelopanunderstandingofthemeaningandprocessesof
measurement,includingunderlyingconceptssuchasiterating(themental
activityofbuildingupthelengthofanobjectwithequal-sizedunits)and
thetransitivityprincipleforindirectmeasurement.1
(4)Studentscomposeanddecomposeplaneorsolidfigures(e.g.,put
twotrianglestogethertomakeaquadrilateral)andbuildunderstanding
ofpart-wholerelationshipsaswellasthepropertiesoftheoriginaland
compositeshapes.Astheycombineshapes,theyrecognizethemfrom
differentperspectivesandorientations,describetheirgeometricattributes,
anddeterminehowtheyarealikeanddifferent,todevelopthebackground
formeasurementandforinitialunderstandingsofpropertiessuchas
congruenceandsymmetry.
1Studentsshouldapplytheprincipleoftransitivityofmeasurementtomakeindirectcomparisons,buttheyneednotusethistechnicalterm.
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Grade 1 overviewoperations and algebraic thinking
• represent and solve problems involving addition and subtraction.
• Understand and apply properties of operations and the relationship between addition and subtraction.
• add and subtract within 20.
• Work with addition and subtraction equations.
number and operations in Base ten
• extend the counting sequence.
• Understand place value.
• Use place value understanding and properties of operations to add and subtract.
measurement and data
• measure lengths indirectly and by iterating length units.
• tell and write time.
• represent and interpret data.
Geometry
• reason with shapes and their attributes.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
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operations and algebraic thinking 1.oa
Represent and solve problems involving addition and subtraction.
1. Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.2
2. Solvewordproblemsthatcallforadditionofthreewholenumberswhosesumislessthanorequalto20,e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
Understand and apply properties of operations and the relationship between addition and subtraction.
3. Applypropertiesofoperationsasstrategiestoaddandsubtract.3Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
4. Understandsubtractionasanunknown-addendproblem.For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
Add and subtract within 20.
5. Relatecountingtoadditionandsubtraction(e.g.,bycountingon2to
add2).
6. Addandsubtractwithin20,demonstratingfluencyforadditionandsubtractionwithin10.Usestrategiessuchascountingon;makingten(e.g.,8+6=8+2+4=10+4=14);decomposinganumberleadingtoaten(e.g.,13–4=13–3–1=10–1=9);usingtherelationshipbetweenadditionandsubtraction(e.g.,knowingthat8+4=12,oneknows12–8=4);andcreatingequivalentbuteasierorknownsums(e.g.,adding6+7bycreatingtheknownequivalent6+6+1=12+1=13).
Work with addition and subtraction equations.
7. Understandthemeaningoftheequalsign,anddetermineifequationsinvolvingadditionandsubtractionaretrueorfalse.For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
8. Determinetheunknownwholenumberinanadditionorsubtractionequationrelatingthreewholenumbers.For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = � – 3, 6 + 6 = �.
number and operations in Base ten 1.nBt
Extend the counting sequence.
1. Countto120,startingatanynumberlessthan120.Inthisrange,readandwritenumeralsandrepresentanumberofobjectswithawrittennumeral.
Understand place value.
2. Understandthatthetwodigitsofatwo-digitnumberrepresentamountsoftensandones.Understandthefollowingasspecialcases:
a. 10canbethoughtofasabundleoftenones—calleda“ten.”
b. Thenumbersfrom11to19arecomposedofatenandone,two,three,four,five,six,seven,eight,ornineones.
c. Thenumbers10,20,30,40,50,60,70,80,90refertoone,two,three,four,five,six,seven,eight,orninetens(and0ones).
2SeeGlossary,Table1.3Studentsneednotuseformaltermsfortheseproperties.
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3. Comparetwotwo-digitnumbersbasedonmeaningsofthetensandonesdigits,recordingtheresultsofcomparisonswiththesymbols>,=,and<.
Use place value understanding and properties of operations to add and subtract.
4. Addwithin100,includingaddingatwo-digitnumberandaone-digitnumber,andaddingatwo-digitnumberandamultipleof10,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.Understandthatinaddingtwo-digitnumbers,oneaddstensandtens,onesandones;andsometimesitisnecessarytocomposeaten.
5. Givenatwo-digitnumber,mentallyfind10moreor10lessthanthenumber,withouthavingtocount;explainthereasoningused.
6. Subtractmultiplesof10intherange10-90frommultiplesof10intherange10-90(positiveorzerodifferences),usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethe
strategytoawrittenmethodandexplainthereasoningused.
measurement and data 1.md
Measure lengths indirectly and by iterating length units.
1. Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.
2. Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.
Tell and write time.
3. Tellandwritetimeinhoursandhalf-hoursusinganaloganddigitalclocks.
Represent and interpret data.
4. Organize,represent,andinterpretdatawithuptothreecategories;askandanswerquestionsaboutthetotalnumberofdatapoints,howmanyineachcategory,andhowmanymoreorlessareinonecategorythaninanother.
Geometry 1.G
Reason with shapes and their attributes.
1. Distinguishbetweendefiningattributes(e.g.,trianglesareclosedandthree-sided)versusnon-definingattributes(e.g.,color,orientation,overallsize);buildanddrawshapestopossessdefiningattributes.
2. Composetwo-dimensionalshapes(rectangles,squares,trapezoids,triangles,half-circles,andquarter-circles)orthree-dimensionalshapes(cubes,rightrectangularprisms,rightcircularcones,andrightcircularcylinders)tocreateacompositeshape,andcomposenewshapesfromthecompositeshape.4
3. Partitioncirclesandrectanglesintotwoandfourequalshares,describethesharesusingthewordshalves,fourths,andquarters,andusethephraseshalf of,fourth of,andquarter of.Describethewholeastwoof,orfouroftheshares.Understandfortheseexamplesthatdecomposingintomoreequalsharescreatessmallershares.
4Studentsdonotneedtolearnformalnamessuchas“rightrectangularprism.”
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mathematics | Grade 2InGrade2,instructionaltimeshouldfocusonfourcriticalareas:(1)
extendingunderstandingofbase-tennotation;(2)buildingfluencywith
additionandsubtraction;(3)usingstandardunitsofmeasure;and(4)
describingandanalyzingshapes.
(1)Studentsextendtheirunderstandingofthebase-tensystem.This
includesideasofcountinginfives,tens,andmultiplesofhundreds,tens,
andones,aswellasnumberrelationshipsinvolvingtheseunits,including
comparing.Studentsunderstandmulti-digitnumbers(upto1000)written
inbase-tennotation,recognizingthatthedigitsineachplacerepresent
amountsofthousands,hundreds,tens,orones(e.g.,853is8hundreds+5
tens+3ones).
(2)Studentsusetheirunderstandingofadditiontodevelopfluencywith
additionandsubtractionwithin100.Theysolveproblemswithin1000
byapplyingtheirunderstandingofmodelsforadditionandsubtraction,
andtheydevelop,discuss,anduseefficient,accurate,andgeneralizable
methodstocomputesumsanddifferencesofwholenumbersinbase-ten
notation,usingtheirunderstandingofplacevalueandthepropertiesof
operations.Theyselectandaccuratelyapplymethodsthatareappropriate
forthecontextandthenumbersinvolvedtomentallycalculatesumsand
differencesfornumberswithonlytensoronlyhundreds.
(3)Studentsrecognizetheneedforstandardunitsofmeasure(centimeter
andinch)andtheyuserulersandothermeasurementtoolswiththe
understandingthatlinearmeasureinvolvesaniterationofunits.They
recognizethatthesmallertheunit,themoreiterationstheyneedtocovera
givenlength.
(4)Studentsdescribeandanalyzeshapesbyexaminingtheirsidesand
angles.Studentsinvestigate,describe,andreasonaboutdecomposing
andcombiningshapestomakeothershapes.Throughbuilding,drawing,
andanalyzingtwo-andthree-dimensionalshapes,studentsdevelopa
foundationforunderstandingarea,volume,congruence,similarity,and
symmetryinlatergrades.
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operations and algebraic thinking
• represent and solve problems involving addition and subtraction.
• add and subtract within 20.
• Work with equal groups of objects to gain foundations for multiplication.
number and operations in Base ten
• Understand place value.
• Use place value understanding and properties of operations to add and subtract.
measurement and data
• measure and estimate lengths in standard units.
• relate addition and subtraction to length.
• Work with time and money.
• represent and interpret data.
Geometry
• reason with shapes and their attributes.
mathematical Practices
1. Makesenseofproblemsandperseverein
solvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
Grade 2 overview
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operations and algebraic thinking 2.oa
Represent and solve problems involving addition and subtraction.
1. Useadditionandsubtractionwithin100tosolveone-andtwo-stepwordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions, e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1
Add and subtract within 20.
2. Fluentlyaddandsubtractwithin20usingmentalstrategies.2ByendofGrade2,knowfrommemoryallsumsoftwoone-digitnumbers.
Work with equal groups of objects to gain foundations for multiplication.
3. Determinewhetheragroupofobjects(upto20)hasanoddorevennumberofmembers,e.g.,bypairingobjectsorcountingthemby2s;writeanequationtoexpressanevennumberasasumoftwoequaladdends.
4. Useadditiontofindthetotalnumberofobjectsarrangedinrectangulararrayswithupto5rowsandupto5columns;writeanequationtoexpressthetotalasasumofequaladdends.
number and operations in Base ten 2.nBt
Understand place value.
1. Understandthatthethreedigitsofathree-digitnumberrepresentamountsofhundreds,tens,andones;e.g.,706equals7hundreds,0tens,and6ones.Understandthefollowingasspecialcases:
a. 100canbethoughtofasabundleoftentens—calleda“hundred.”
b. Thenumbers100,200,300,400,500,600,700,800,900refertoone,two,three,four,five,six,seven,eight,orninehundreds(and0tensand0ones).
2. Countwithin1000;skip-countby5s,10s,and100s.
3. Readandwritenumbersto1000usingbase-tennumerals,numbernames,andexpandedform.
4. Comparetwothree-digitnumbersbasedonmeaningsofthehundreds,tens,andonesdigits,using>,=,and<symbolstorecordtheresultsofcomparisons.
Use place value understanding and properties of operations to add and subtract.
5. Fluentlyaddandsubtractwithin100usingstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
6. Adduptofourtwo-digitnumbersusingstrategiesbasedonplacevalueandpropertiesofoperations.
7. Addandsubtractwithin1000,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethod.Understandthatinaddingorsubtractingthree-digitnumbers,oneaddsorsubtractshundredsandhundreds,tensandtens,onesandones;andsometimesitisnecessarytocomposeordecomposetensorhundreds.
8. Mentallyadd10or100toagivennumber100–900,andmentallysubtract10or100fromagivennumber100–900.
9. Explainwhyadditionandsubtractionstrategieswork,usingplacevalueandthepropertiesofoperations.3
1SeeGlossary,Table1.2Seestandard1.OA.6foralistofmentalstrategies.3Explanationsmaybesupportedbydrawingsorobjects.
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measurement and data 2.md
Measure and estimate lengths in standard units.
1. Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.
2. Measurethelengthofanobjecttwice,usinglengthunitsofdifferentlengthsforthetwomeasurements;describehowthetwomeasurementsrelatetothesizeoftheunitchosen.
3. Estimatelengthsusingunitsofinches,feet,centimeters,andmeters.
4. Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthelengthdifferenceintermsofastandardlengthunit.
Relate addition and subtraction to length.
5. Useadditionandsubtractionwithin100tosolvewordproblemsinvolvinglengthsthataregiveninthesameunits,e.g.,byusingdrawings(suchasdrawingsofrulers)andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
6. Representwholenumbersaslengthsfrom0onanumberlinediagramwithequallyspacedpointscorrespondingtothenumbers0,1,2,...,andrepresentwhole-numbersumsanddifferenceswithin100onanumberlinediagram.
Work with time and money.
7. Tellandwritetimefromanaloganddigitalclockstothenearestfiveminutes,usinga.m.andp.m.
8. Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbolsappropriately.Example: If you have 2 dimes and 3 pennies, how many cents do you have?
Represent and interpret data.
9. Generatemeasurementdatabymeasuringlengthsofseveralobjectstothenearestwholeunit,orbymakingrepeatedmeasurementsofthesameobject.Showthemeasurementsbymakingalineplot,wherethehorizontalscaleismarkedoffinwhole-numberunits.
10. Drawapicturegraphandabargraph(withsingle-unitscale)torepresentadatasetwithuptofourcategories.Solvesimpleput-together,take-apart,andcompareproblems4usinginformationpresentedinabargraph.
Geometry 2.G
Reason with shapes and their attributes.
1. Recognizeanddrawshapeshavingspecifiedattributes,suchasagivennumberofanglesoragivennumberofequalfaces.5Identifytriangles,quadrilaterals,pentagons,hexagons,andcubes.
2. Partitionarectangleintorowsandcolumnsofsame-sizesquaresandcounttofindthetotalnumberofthem.
3. Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,half of,a third of,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.
4SeeGlossary,Table1.5Sizesarecompareddirectlyorvisually,notcomparedbymeasuring.
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1
Mathematics|Grade3
InGrade3,instructionaltimeshouldfocusonfourcriticalareas:(1)
developingunderstandingofmultiplicationanddivisionandstrategies
formultiplicationanddivisionwithin100;(2)developingunderstanding
offractions,especiallyunitfractions(fractionswithnumerator1);(3)
developingunderstandingofthestructureofrectangulararraysandof
area;and(4)describingandanalyzingtwo-dimensionalshapes.
(1)Studentsdevelopanunderstandingofthemeaningsofmultiplication
anddivisionofwholenumbersthroughactivitiesandproblemsinvolving
equal-sizedgroups,arrays,andareamodels;multiplicationisfinding
anunknownproduct,anddivisionisfindinganunknownfactorinthese
situations.Forequal-sizedgroupsituations,divisioncanrequirefinding
theunknownnumberofgroupsortheunknowngroupsize.Studentsuse
propertiesofoperationstocalculateproductsofwholenumbers,using
increasinglysophisticatedstrategiesbasedonthesepropertiestosolve
multiplicationanddivisionproblemsinvolvingsingle-digitfactors.By
comparingavarietyofsolutionstrategies,studentslearntherelationship
betweenmultiplicationanddivision.
(2)Studentsdevelopanunderstandingoffractions,beginningwith
unitfractions.Studentsviewfractionsingeneralasbeingbuiltoutof
unitfractions,andtheyusefractionsalongwithvisualfractionmodels
torepresentpartsofawhole.Studentsunderstandthatthesizeofa
fractionalpartisrelativetothesizeofthewhole.Forexample,1/2ofthe
paintinasmallbucketcouldbelesspaintthan1/3ofthepaintinalarger
bucket,but1/3ofaribbonislongerthan1/5ofthesameribbonbecause
whentheribbonisdividedinto3equalparts,thepartsarelongerthan
whentheribbonisdividedinto5equalparts.Studentsareabletouse
fractionstorepresentnumbersequalto,lessthan,andgreaterthanone.
Theysolveproblemsthatinvolvecomparingfractionsbyusingvisual
fractionmodelsandstrategiesbasedonnoticingequalnumeratorsor
denominators.
(3)Studentsrecognizeareaasanattributeoftwo-dimensionalregions.
Theymeasuretheareaofashapebyfindingthetotalnumberofsame-
sizeunitsofarearequiredtocovertheshapewithoutgapsoroverlaps,
asquarewithsidesofunitlengthbeingthestandardunitformeasuring
area.Studentsunderstandthatrectangulararrayscanbedecomposedinto
identicalrowsorintoidenticalcolumns.Bydecomposingrectanglesinto
rectangulararraysofsquares,studentsconnectareatomultiplication,and
justifyusingmultiplicationtodeterminetheareaofarectangle.
(4)Studentsdescribe,analyze,andcomparepropertiesoftwo-
dimensionalshapes.Theycompareandclassifyshapesbytheirsidesand
angles,andconnectthesewithdefinitionsofshapes.Studentsalsorelate
theirfractionworktogeometrybyexpressingtheareaofpartofashape
asaunitfractionofthewhole.
Common Core State StandardS for matHematICSG
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3 | 2
2
operations and algebraic thinking
• represent and solve problems involving multiplication and division.
• Understand properties of multiplication and the relationship between multiplication and division.
• multiply and divide within 100.
• Solve problems involving the four operations, and identify and explain patterns in arithmetic.
number and operations in Base ten
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
number and operations—fractions
• develop understanding of fractions as numbers.
measurement and data
• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
• represent and interpret data.
• Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
• Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Geometry
• reason with shapes and their attributes.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 3 overviewmathematical Practices
1. Makesenseofproblemsandperseverein
solvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritique
thereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeated
reasoning.
Common Core State StandardS for matHematICSG
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3 | 2
3
operations and algebraic thinking 3.oa
Represent and solve problems involving multiplication and division.
1. Interpretproductsofwholenumbers,e.g.,interpret5×7asthetotalnumberofobjectsin5groupsof7objectseach.For example, describe a context in which a total number of objects can be expressed as 5 × 7.
2. Interpretwhole-numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3. Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.1
4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?.
Understand properties of multiplication and the relationship between multiplication and division.
5. Applypropertiesofoperationsasstrategiestomultiplyanddivide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
6. Understanddivisionasanunknown-factorproblem.For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Multiply and divide within 100.
7. Fluentlymultiplyanddividewithin100,usingstrategiessuchastherelationshipbetweenmultiplicationanddivision(e.g.,knowingthat8×5=40,oneknows40÷5=8)orpropertiesofoperations.BytheendofGrade3,knowfrommemoryallproductsoftwoone-digitnumbers.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
8. Solvetwo-stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.3
9. Identifyarithmeticpatterns(includingpatternsintheadditiontableormultiplicationtable),andexplainthemusingpropertiesofoperations.For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
1SeeGlossary,Table2.2Studentsneednotuseformaltermsfortheseproperties.3Thisstandardislimitedtoproblemsposedwithwholenumbersandhavingwhole-numberanswers;studentsshouldknowhowtoperformoperationsintheconven-tionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).
Common Core State StandardS for matHematICSg
ra
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3 | 24
Number and Operations in Base Ten 3.NBT
Use place value understanding and properties of operations to perform multi-digit arithmetic.4
1. Useplacevalueunderstandingtoroundwholenumberstothenearest10or100.
2. Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction.
3. Multiplyone-digitwholenumbersbymultiplesof10intherange10–90(e.g.,9×80,5×60)usingstrategiesbasedonplacevalueandpropertiesofoperations.
Number and Operations—Fractions5 3.NF
Develop understanding of fractions as numbers.1. Understandafraction1/basthequantityformedby1partwhena
wholeispartitionedintob equalparts;understandafractiona/basthequantityformedbyapartsofsize1/b.
2. Understandafractionasanumberonthenumberline;representfractionsonanumberlinediagram.
a. Representafraction1/bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1/bandthattheendpointofthepartbasedat0locatesthenumber1/bonthenumberline.
b. Representafractiona/bonanumberlinediagrambymarkingoffalengths1/bfrom0.Recognizethattheresultingintervalhassizea/bandthatitsendpointlocatesthenumbera/bonthenumberline.
3. Explainequivalenceoffractionsinspecialcases,andcomparefractionsbyreasoningabouttheirsize.
a. Understandtwofractionsasequivalent(equal)iftheyarethesamesize,orthesamepointonanumberline.
b. Recognizeandgeneratesimpleequivalentfractions,e.g.,1/2=2/4,4/6=2/3.Explainwhythefractionsareequivalent,e.g.,byusingavisualfractionmodel.
c. Expresswholenumbersasfractions,andrecognizefractionsthatareequivalenttowholenumbers.Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
d. Comparetwofractionswiththesamenumeratororthesamedenominatorbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.
Measurement and Data 3.MD
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
1. Tellandwritetimetothenearestminuteandmeasuretimeintervalsinminutes.Solvewordproblemsinvolvingadditionandsubtractionoftimeintervalsinminutes,e.g.,byrepresentingtheproblemonanumberlinediagram.
4Arangeofalgorithmsmaybeused.5Grade3expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,6,and8.
Common Core State StandardS for matHematICSG
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3 | 2
5
2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardunitsofgrams(g),kilograms(kg),andliters(l).6Add,subtract,multiply,ordividetosolveone-stepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.7
Represent and interpret data.
3. Drawascaledpicturegraphandascaledbargraphtorepresentadatasetwithseveralcategories.Solveone-andtwo-step“howmanymore”and“howmanyless”problemsusinginformationpresentedinscaledbargraphs.For example, draw a bar graph in which each square in the bar graph might represent 5 pets.
4. Generatemeasurementdatabymeasuringlengthsusingrulersmarkedwithhalvesandfourthsofaninch.Showthedatabymakingalineplot,wherethehorizontalscaleismarkedoffinappropriateunits—wholenumbers,halves,orquarters.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
5. Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement.
a. Asquarewithsidelength1unit,called“aunitsquare,”issaidtohave“onesquareunit”ofarea,andcanbeusedtomeasurearea.
b. Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.
6. Measureareasbycountingunitsquares(squarecm,squarem,squarein,squareft,andimprovisedunits).
7. Relateareatotheoperationsofmultiplicationandaddition.
a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.
b. Multiplysidelengthstofindareasofrectangleswithwhole-numbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.
c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.
d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposingthemintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
8. Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.
6Excludescompoundunitssuchascm3andfindingthegeometricvolumeofacontainer.7Excludesmultiplicativecomparisonproblems(problemsinvolvingnotionsof“timesasmuch”;seeGlossary,Table2).
Common Core State StandardS for matHematICSG
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3 | 2
6
Geometry 3.G
Reason with shapes and their attributes.
1. Understandthatshapesindifferentcategories(e.g.,rhombuses,rectangles,andothers)mayshareattributes(e.g.,havingfoursides),andthatthesharedattributescandefinealargercategory(e.g.,quadrilaterals).Recognizerhombuses,rectangles,andsquaresasexamplesofquadrilaterals,anddrawexamplesofquadrilateralsthatdonotbelongtoanyofthesesubcategories.
2. Partitionshapesintopartswithequalareas.Expresstheareaofeachpartasaunitfractionofthewhole.For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
Common Core State StandardS for matHematICSG
ra
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4 | 2
7
mathematics | Grade 4InGrade4,instructionaltimeshouldfocusonthreecriticalareas:(1)
developingunderstandingandfluencywithmulti-digitmultiplication,
anddevelopingunderstandingofdividingtofindquotientsinvolving
multi-digitdividends;(2)developinganunderstandingoffraction
equivalence,additionandsubtractionoffractionswithlikedenominators,
andmultiplicationoffractionsbywholenumbers;(3)understanding
thatgeometricfigurescanbeanalyzedandclassifiedbasedontheir
properties,suchashavingparallelsides,perpendicularsides,particular
anglemeasures,andsymmetry.
(1)Studentsgeneralizetheirunderstandingofplacevalueto1,000,000,
understandingtherelativesizesofnumbersineachplace.Theyapplytheir
understandingofmodelsformultiplication(equal-sizedgroups,arrays,
areamodels),placevalue,andpropertiesofoperations,inparticularthe
distributiveproperty,astheydevelop,discuss,anduseefficient,accurate,
andgeneralizablemethodstocomputeproductsofmulti-digitwhole
numbers.Dependingonthenumbersandthecontext,theyselectand
accuratelyapplyappropriatemethodstoestimateormentallycalculate
products.Theydevelopfluencywithefficientproceduresformultiplying
wholenumbers;understandandexplainwhytheproceduresworkbasedon
placevalueandpropertiesofoperations;andusethemtosolveproblems.
Studentsapplytheirunderstandingofmodelsfordivision,placevalue,
propertiesofoperations,andtherelationshipofdivisiontomultiplication
astheydevelop,discuss,anduseefficient,accurate,andgeneralizable
procedurestofindquotientsinvolvingmulti-digitdividends.Theyselect
andaccuratelyapplyappropriatemethodstoestimateandmentally
calculatequotients,andinterpretremaindersbaseduponthecontext.
(2)Studentsdevelopunderstandingoffractionequivalenceand
operationswithfractions.Theyrecognizethattwodifferentfractionscan
beequal(e.g.,15/9=5/3),andtheydevelopmethodsforgeneratingand
recognizingequivalentfractions.Studentsextendpreviousunderstandings
abouthowfractionsarebuiltfromunitfractions,composingfractions
fromunitfractions,decomposingfractionsintounitfractions,andusing
themeaningoffractionsandthemeaningofmultiplicationtomultiplya
fractionbyawholenumber.
(3)Studentsdescribe,analyze,compare,andclassifytwo-dimensional
shapes.Throughbuilding,drawing,andanalyzingtwo-dimensionalshapes,
studentsdeepentheirunderstandingofpropertiesoftwo-dimensional
objectsandtheuseofthemtosolveproblemsinvolvingsymmetry.
Common Core State StandardS for matHematICSG
ra
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4 | 2
8
Grade 4 overviewoperations and algebraic thinking
• Use the four operations with whole numbers to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.
number and operations in Base ten
• Generalize place value understanding for multi-digit whole numbers.
• Use place value understanding and properties of operations to perform multi-digit arithmetic.
number and operations—fractions
• extend understanding of fraction equivalence and ordering.
• Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
• Understand decimal notation for fractions, and compare decimal fractions.
measurement and data
• Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
• represent and interpret data.
• Geometric measurement: understand concepts of angle and measure angles.
Geometry
• draw and identify lines and angles, and classify shapes by properties of their lines and angles.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Common Core State StandardS for matHematICSG
ra
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4 | 2
9
operations and algebraic thinking 4.oa
Use the four operations with whole numbers to solve problems.
1. Interpretamultiplicationequationasacomparison,e.g.,interpret35=5×7asastatementthat35is5timesasmanyas7and7timesasmanyas5.Representverbalstatementsofmultiplicativecomparisonsasmultiplicationequations.
2. Multiplyordividetosolvewordproblemsinvolvingmultiplicativecomparison,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem,distinguishingmultiplicativecomparisonfromadditivecomparison.1
3. Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole-numberanswersusingthefouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.
Gain familiarity with factors and multiples.
4. Findallfactorpairsforawholenumberintherange1–100.Recognizethatawholenumberisamultipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1–100isamultipleofagivenone-digitnumber.Determinewhetheragivenwholenumberintherange1–100isprimeorcomposite.
Generate and analyze patterns.
5. Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
number and operations in Base ten2 4.nBt
Generalize place value understanding for multi-digit whole numbers.
1. Recognizethatinamulti-digitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
2. Readandwritemulti-digitwholenumbersusingbase-tennumerals,numbernames,andexpandedform.Comparetwomulti-digitnumbersbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.
3. Useplacevalueunderstandingtoroundmulti-digitwholenumberstoanyplace.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
4. Fluentlyaddandsubtractmulti-digitwholenumbersusingthestandardalgorithm.
5. Multiplyawholenumberofuptofourdigitsbyaone-digitwholenumber,andmultiplytwotwo-digitnumbers,usingstrategiesbasedonplacevalueandthepropertiesofoperations.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.
1SeeGlossary,Table2.2Grade4expectationsinthisdomainarelimitedtowholenumberslessthanorequalto1,000,000.
Common Core State StandardS for matHematICSG
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4 | 3
0
6. Findwhole-numberquotientsandremainderswithuptofour-digitdividendsandone-digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.
number and operations—fractions3 4.nf
Extend understanding of fraction equivalence and ordering.
1. Explainwhyafractiona/bisequivalenttoafraction(n×a)/(n×b)byusingvisualfractionmodels,withattentiontohowthenumberandsizeofthepartsdiffereventhoughthetwofractionsthemselvesarethesamesize.Usethisprincipletorecognizeandgenerateequivalentfractions.
2. Comparetwofractionswithdifferentnumeratorsanddifferentdenominators,e.g.,bycreatingcommondenominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas1/2.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswithsymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understandafractiona/bwitha>1asasumoffractions1/b.
a. Understandadditionandsubtractionoffractionsasjoiningandseparatingpartsreferringtothesamewhole.
b. Decomposeafractionintoasumoffractionswiththesamedenominatorinmorethanoneway,recordingeachdecompositionbyanequation.Justifydecompositions,e.g.,byusingavisualfractionmodel.Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Addandsubtractmixednumberswithlikedenominators,e.g.,byreplacingeachmixednumberwithanequivalentfraction,and/orbyusingpropertiesofoperationsandtherelationshipbetweenadditionandsubtraction.
d. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.
4. Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionbyawholenumber.
a. Understandafractiona/basamultipleof1/b.For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
b. Understandamultipleofa/basamultipleof1/b,andusethisunderstandingtomultiplyafractionbyawholenumber.For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solvewordproblemsinvolvingmultiplicationofafractionbyawholenumber,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
3Grade4expectationsinthisdomainarelimitedtofractionswithdenominators2,3,4,5,6,8,10,12,and100.
Common Core State StandardS for matHematICSG
ra
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4 | 3
1
Understand decimal notation for fractions, and compare decimal fractions.
5. Expressafractionwithdenominator10asanequivalentfractionwithdenominator100,andusethistechniquetoaddtwofractionswithrespectivedenominators10and100.4For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
6. Usedecimalnotationforfractionswithdenominators10or100.For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
7. Comparetwodecimalstohundredthsbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwodecimalsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualmodel.
measurement and data 4.md
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
1. Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz.;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.Recordmeasurementequivalentsinatwo-columntable.For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
2. Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.Representmeasurementquantitiesusingdiagramssuchasnumberlinediagramsthatfeatureameasurementscale.
3. Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
Represent and interpret data.
4. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
Geometric measurement: understand concepts of angle and measure angles.
5. Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:
a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.
b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.
4Studentswhocangenerateequivalentfractionscandevelopstrategiesforaddingfractionswithunlikedenominatorsingeneral.Butadditionandsubtractionwithun-likedenominatorsingeneralisnotarequirementatthisgrade.
Common Core State StandardS for matHematICSG
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de
4 | 3
2
6. Measureanglesinwhole-numberdegreesusingaprotractor.Sketchanglesofspecifiedmeasure.
7. Recognizeanglemeasureasadditive.Whenanangleisdecomposedintonon-overlappingparts,theanglemeasureofthewholeisthesumoftheanglemeasuresoftheparts.Solveadditionandsubtractionproblemstofindunknownanglesonadiagraminrealworldandmathematicalproblems,e.g.,byusinganequationwithasymbolfortheunknownanglemeasure.
Geometry 4.G
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
1. Drawpoints,lines,linesegments,rays,angles(right,acute,obtuse),andperpendicularandparallellines.Identifytheseintwo-dimensionalfigures.
2. Classifytwo-dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.
3. Recognizealineofsymmetryforatwo-dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline-symmetricfiguresanddrawlinesofsymmetry.
Common Core State StandardS for matHematICSG
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5 | 33
mathematics | Grade 5InGrade5,instructionaltimeshouldfocusonthreecriticalareas:(1)
developingfluencywithadditionandsubtractionoffractions,and
developingunderstandingofthemultiplicationoffractionsandofdivision
offractionsinlimitedcases(unitfractionsdividedbywholenumbersand
wholenumbersdividedbyunitfractions);(2)extendingdivisionto2-digit
divisors,integratingdecimalfractionsintotheplacevaluesystemand
developingunderstandingofoperationswithdecimalstohundredths,and
developingfluencywithwholenumberanddecimaloperations;and(3)
developingunderstandingofvolume.
(1)Studentsapplytheirunderstandingoffractionsandfractionmodelsto
representtheadditionandsubtractionoffractionswithunlikedenominators
asequivalentcalculationswithlikedenominators.Theydevelopfluency
incalculatingsumsanddifferencesoffractions,andmakereasonable
estimatesofthem.Studentsalsousethemeaningoffractions,of
multiplicationanddivision,andtherelationshipbetweenmultiplicationand
divisiontounderstandandexplainwhytheproceduresformultiplyingand
dividingfractionsmakesense.(Note:thisislimitedtothecaseofdividing
unitfractionsbywholenumbersandwholenumbersbyunitfractions.)
(2)Studentsdevelopunderstandingofwhydivisionprocedureswork
basedonthemeaningofbase-tennumeralsandpropertiesofoperations.
Theyfinalizefluencywithmulti-digitaddition,subtraction,multiplication,
anddivision.Theyapplytheirunderstandingsofmodelsfordecimals,
decimalnotation,andpropertiesofoperationstoaddandsubtract
decimalstohundredths.Theydevelopfluencyinthesecomputations,and
makereasonableestimatesoftheirresults.Studentsusetherelationship
betweendecimalsandfractions,aswellastherelationshipbetween
finitedecimalsandwholenumbers(i.e.,afinitedecimalmultipliedbyan
appropriatepowerof10isawholenumber),tounderstandandexplain
whytheproceduresformultiplyinganddividingfinitedecimalsmake
sense.Theycomputeproductsandquotientsofdecimalstohundredths
efficientlyandaccurately.
(3)Studentsrecognizevolumeasanattributeofthree-dimensional
space.Theyunderstandthatvolumecanbemeasuredbyfindingthetotal
numberofsame-sizeunitsofvolumerequiredtofillthespacewithout
gapsoroverlaps.Theyunderstandthata1-unitby1-unitby1-unitcube
isthestandardunitformeasuringvolume.Theyselectappropriateunits,
strategies,andtoolsforsolvingproblemsthatinvolveestimatingand
measuringvolume.Theydecomposethree-dimensionalshapesandfind
volumesofrightrectangularprismsbyviewingthemasdecomposedinto
layersofarraysofcubes.Theymeasurenecessaryattributesofshapesin
ordertodeterminevolumestosolverealworldandmathematicalproblems.
Common Core State StandardS for matHematICSG
ra
de
5 | 34
operations and algebraic thinking
• Write and interpret numerical expressions.
• analyze patterns and relationships.
number and operations in Base ten
• Understand the place value system.
• Perform operations with multi-digit whole numbers and with decimals to hundredths.
number and operations—fractions
• Use equivalent fractions as a strategy to add and subtract fractions.
• apply and extend previous understandings of multiplication and division to multiply and divide fractions.
measurement and data
• Convert like measurement units within a given measurement system.
• represent and interpret data.
• Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Geometry
• Graph points on the coordinate plane to solve real-world and mathematical problems.
• Classify two-dimensional figures into categories based on their properties.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 5 overview
Common Core State StandardS for matHematICSG
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operations and algebraic thinking 5.oa
Write and interpret numerical expressions.
1. Useparentheses,brackets,orbracesinnumericalexpressions,andevaluateexpressionswiththesesymbols.
2. Writesimpleexpressionsthatrecordcalculationswithnumbers,andinterpretnumericalexpressionswithoutevaluatingthem.For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Analyze patterns and relationships.
3. Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationshipsbetweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwopatterns,andgraphtheorderedpairsonacoordinateplane.For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
number and operations in Base ten 5.nBt
Understand the place value system.
1. Recognizethatinamulti-digitnumber,adigitinoneplacerepresents10timesasmuchasitrepresentsintheplacetoitsrightand1/10ofwhatitrepresentsintheplacetoitsleft.
2. Explainpatternsinthenumberofzerosoftheproductwhenmultiplyinganumberbypowersof10,andexplainpatternsintheplacementofthedecimalpointwhenadecimalismultipliedordividedbyapowerof10.Usewhole-numberexponentstodenotepowersof10.
3. Read,write,andcomparedecimalstothousandths.
a. Readandwritedecimalstothousandthsusingbase-tennumerals,numbernames,andexpandedform,e.g.,347.392=3×100+4×10+7×1+3×(1/10)+9×(1/100)+2×(1/1000).
b. Comparetwodecimalstothousandthsbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofcomparisons.
4. Useplacevalueunderstandingtorounddecimalstoanyplace.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluentlymultiplymulti-digitwholenumbersusingthestandardalgorithm.
6. Findwhole-numberquotientsofwholenumberswithuptofour-digitdividendsandtwo-digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.
7. Add,subtract,multiply,anddividedecimalstohundredths,usingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionandsubtraction;relatethestrategytoawrittenmethodandexplainthereasoningused.
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number and operations—fractions 5.nf
Use equivalent fractions as a strategy to add and subtract fractions.
1. Addandsubtractfractionswithunlikedenominators(includingmixednumbers)byreplacinggivenfractionswithequivalentfractionsinsuchawayastoproduceanequivalentsumordifferenceoffractionswithlikedenominators.For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
2. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewhole,includingcasesofunlikedenominators,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.Usebenchmarkfractionsandnumbersenseoffractionstoestimatementallyandassessthereasonablenessofanswers.For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3. Interpretafractionasdivisionofthenumeratorbythedenominator(a/b=a÷b).Solvewordproblemsinvolvingdivisionofwholenumbersleadingtoanswersintheformoffractionsormixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
4. Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionorwholenumberbyafraction.
a. Interprettheproduct(a/b)×qasapartsofapartitionofqintobequalparts;equivalently,astheresultofasequenceofoperationsa×q÷b.For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Findtheareaofarectanglewithfractionalsidelengthsbytilingitwithunitsquaresoftheappropriateunitfractionsidelengths,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.Multiplyfractionalsidelengthstofindareasofrectangles,andrepresentfractionproductsasrectangularareas.
5. Interpretmultiplicationasscaling(resizing),by:
a. Comparingthesizeofaproducttothesizeofonefactoronthebasisofthesizeoftheotherfactor,withoutperformingtheindicatedmultiplication.
b. Explainingwhymultiplyingagivennumberbyafractiongreaterthan1resultsinaproductgreaterthanthegivennumber(recognizingmultiplicationbywholenumbersgreaterthan1asafamiliarcase);explainingwhymultiplyingagivennumberbyafractionlessthan1resultsinaproductsmallerthanthegivennumber;andrelatingtheprincipleoffractionequivalencea/b=(n×a)/(n×b)totheeffectofmultiplyinga/bby1.
6. Solverealworldproblemsinvolvingmultiplicationoffractionsandmixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.
7. Applyandextendpreviousunderstandingsofdivisiontodivideunitfractionsbywholenumbersandwholenumbersbyunitfractions.1
a. Interpretdivisionofaunitfractionbyanon-zerowholenumber,
1Studentsabletomultiplyfractionsingeneralcandevelopstrategiestodividefrac-tionsingeneral,byreasoningabouttherelationshipbetweenmultiplicationanddivision.Butdivisionofafractionbyafractionisnotarequirementatthisgrade.
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andcomputesuchquotients.For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpretdivisionofawholenumberbyaunitfraction,andcomputesuchquotients.For example, create a story context 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 × (1/5) = 4.
c. Solverealworldproblemsinvolvingdivisionofunitfractionsbynon-zerowholenumbersanddivisionofwholenumbersbyunitfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
measurement and data 5.md
Convert like measurement units within a given measurement system.
1. Convertamongdifferent-sizedstandardmeasurementunitswithinagivenmeasurementsystem(e.g.,convert5cmto0.05m),andusetheseconversionsinsolvingmulti-step,realworldproblems.
Represent and interpret data.
2. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
3. Recognizevolumeasanattributeofsolidfiguresandunderstandconceptsofvolumemeasurement.
a. Acubewithsidelength1unit,calleda“unitcube,”issaidtohave“onecubicunit”ofvolume,andcanbeusedtomeasurevolume.
b. Asolidfigurewhichcanbepackedwithoutgapsoroverlapsusingnunitcubesissaidtohaveavolumeofncubicunits.
4. Measurevolumesbycountingunitcubes,usingcubiccm,cubicin,cubicft,andimprovisedunits.
5. Relatevolumetotheoperationsofmultiplicationandadditionandsolverealworldandmathematicalproblemsinvolvingvolume.
a. Findthevolumeofarightrectangularprismwithwhole-numbersidelengthsbypackingitwithunitcubes,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengths,equivalentlybymultiplyingtheheightbytheareaofthebase.Representthreefoldwhole-numberproductsasvolumes,e.g.,torepresenttheassociativepropertyofmultiplication.
b. ApplytheformulasV=l×w×handV=b×hforrectangularprismstofindvolumesofrightrectangularprismswithwhole-numberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.
c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwonon-overlappingrightrectangularprismsbyaddingthevolumesofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
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Geometry 5.G
Graph points on the coordinate plane to solve real-world and mathematical problems.
1. Useapairofperpendicularnumberlines,calledaxes,todefineacoordinatesystem,withtheintersectionofthelines(theorigin)arrangedtocoincidewiththe0oneachlineandagivenpointintheplanelocatedbyusinganorderedpairofnumbers,calleditscoordinates.Understandthatthefirstnumberindicateshowfartotravelfromtheorigininthedirectionofoneaxis,andthesecondnumberindicateshowfartotravelinthedirectionofthesecondaxis,withtheconventionthatthenamesofthetwoaxesandthecoordinatescorrespond(e.g.,x-axisandx-coordinate,y-axisandy-coordinate).
2. Representrealworldandmathematicalproblemsbygraphingpointsinthefirstquadrantofthecoordinateplane,andinterpretcoordinatevaluesofpointsinthecontextofthesituation.
Classify two-dimensional figures into categories based on their properties.
3. Understandthatattributesbelongingtoacategoryoftwo-dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
4. Classifytwo-dimensionalfiguresinahierarchybasedonproperties.
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mathematics | Grade 6InGrade6,instructionaltimeshouldfocusonfourcriticalareas:(1)
connectingratioandratetowholenumbermultiplicationanddivision
andusingconceptsofratioandratetosolveproblems;(2)completing
understandingofdivisionoffractionsandextendingthenotionofnumber
tothesystemofrationalnumbers,whichincludesnegativenumbers;
(3)writing,interpreting,andusingexpressionsandequations;and(4)
developingunderstandingofstatisticalthinking.
(1)Studentsusereasoningaboutmultiplicationanddivisiontosolve
ratioandrateproblemsaboutquantities.Byviewingequivalentratios
andratesasderivingfrom,andextending,pairsofrows(orcolumns)in
themultiplicationtable,andbyanalyzingsimpledrawingsthatindicate
therelativesizeofquantities,studentsconnecttheirunderstandingof
multiplicationanddivisionwithratiosandrates.Thusstudentsexpandthe
scopeofproblemsforwhichtheycanusemultiplicationanddivisionto
solveproblems,andtheyconnectratiosandfractions.Studentssolvea
widevarietyofproblemsinvolvingratiosandrates.
(2)Studentsusethemeaningoffractions,themeaningsofmultiplication
anddivision,andtherelationshipbetweenmultiplicationanddivisionto
understandandexplainwhytheproceduresfordividingfractionsmake
sense.Studentsusetheseoperationstosolveproblems.Studentsextend
theirpreviousunderstandingsofnumberandtheorderingofnumbers
tothefullsystemofrationalnumbers,whichincludesnegativerational
numbers,andinparticularnegativeintegers.Theyreasonabouttheorder
andabsolutevalueofrationalnumbersandaboutthelocationofpointsin
allfourquadrantsofthecoordinateplane.
(3)Studentsunderstandtheuseofvariablesinmathematicalexpressions.
Theywriteexpressionsandequationsthatcorrespondtogivensituations,
evaluateexpressions,anduseexpressionsandformulastosolveproblems.
Studentsunderstandthatexpressionsindifferentformscanbeequivalent,
andtheyusethepropertiesofoperationstorewriteexpressionsin
equivalentforms.Studentsknowthatthesolutionsofanequationarethe
valuesofthevariablesthatmaketheequationtrue.Studentsuseproperties
ofoperationsandtheideaofmaintainingtheequalityofbothsidesof
anequationtosolvesimpleone-stepequations.Studentsconstructand
analyzetables,suchastablesofquantitiesthatareinequivalentratios,
andtheyuseequations(suchas3x=y)todescriberelationshipsbetween
quantities.
(4)Buildingonandreinforcingtheirunderstandingofnumber,students
begintodeveloptheirabilitytothinkstatistically.Studentsrecognizethata
datadistributionmaynothaveadefinitecenterandthatdifferentwaysto
measurecenteryielddifferentvalues.Themedianmeasurescenterinthe
sensethatitisroughlythemiddlevalue.Themeanmeasurescenterinthe
sensethatitisthevaluethateachdatapointwouldtakeonifthetotalof
thedatavalueswereredistributedequally,andalsointhesensethatitisa
balancepoint.Studentsrecognizethatameasureofvariability(interquartile
rangeormeanabsolutedeviation)canalsobeusefulforsummarizing
databecausetwoverydifferentsetsofdatacanhavethesamemeanand
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medianyetbedistinguishedbytheirvariability.Studentslearntodescribe
andsummarizenumericaldatasets,identifyingclusters,peaks,gaps,and
symmetry,consideringthecontextinwhichthedatawerecollected.
StudentsinGrade6alsobuildontheirworkwithareainelementary
schoolbyreasoningaboutrelationshipsamongshapestodeterminearea,
surfacearea,andvolume.Theyfindareasofrighttriangles,othertriangles,
andspecialquadrilateralsbydecomposingtheseshapes,rearranging
orremovingpieces,andrelatingtheshapestorectangles.Usingthese
methods,studentsdiscuss,develop,andjustifyformulasforareasof
trianglesandparallelograms.Studentsfindareasofpolygonsandsurface
areasofprismsandpyramidsbydecomposingthemintopieceswhose
areatheycandetermine.Theyreasonaboutrightrectangularprisms
withfractionalsidelengthstoextendformulasforthevolumeofaright
rectangularprismtofractionalsidelengths.Theyprepareforworkon
scaledrawingsandconstructionsinGrade7bydrawingpolygonsinthe
coordinateplane.
Common Core State StandardS for matHematICS
ratios and Proportional relationships
• Understand ratio concepts and use ratio reasoning to solve problems.
the number System
• apply and extend previous understandings of multiplication and division to divide fractions by fractions.
• Compute fluently with multi-digit numbers and find common factors and multiples.
• apply and extend previous understandings of numbers to the system of rational numbers.
expressions and equations
• apply and extend previous understandings of arithmetic to algebraic expressions.
• reason about and solve one-variable equations and inequalities.
• represent and analyze quantitative relationships between dependent and independent variables.
Geometry
• Solve real-world and mathematical problems involving area, surface area, and volume.
Statistics and Probability
• develop understanding of statistical variability.
• Summarize and describe distributions.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 6 overview
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ratios and Proportional relationships 6.rP
Understand ratio concepts and use ratio reasoning to solve problems.
1. Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities.For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
2. Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb≠0,anduseratelanguageinthecontextofaratiorelationship.For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1
3. Useratioandratereasoningtosolvereal-worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.
a. Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.
b. Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.
d. Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.
the number System 6.nS
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
1. Interpretandcomputequotientsoffractions,andsolvewordproblemsinvolvingdivisionoffractionsbyfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, create a story context for (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 3/4 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 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Compute fluently with multi-digit numbers and find common factors and multiples.
2. Fluentlydividemulti-digitnumbersusingthestandardalgorithm.
3. Fluentlyadd,subtract,multiply,anddividemulti-digitdecimalsusingthestandardalgorithmforeachoperation.
4. Findthegreatestcommonfactoroftwowholenumberslessthanorequalto100andtheleastcommonmultipleoftwowholenumberslessthanorequalto12.Usethedistributivepropertytoexpressasumoftwowholenumbers1–100withacommonfactorasamultipleofasumoftwowholenumberswithnocommonfactor.For example, express 36 + 8 as 4 (9 + 2).
1Expectationsforunitratesinthisgradearelimitedtonon-complexfractions.
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Apply and extend previous understandings of numbers to the system of rational numbers.
5. Understandthatpositiveandnegativenumbersareusedtogethertodescribequantitieshavingoppositedirectionsorvalues(e.g.,temperatureabove/belowzero,elevationabove/belowsealevel,credits/debits,positive/negativeelectriccharge);usepositiveandnegativenumberstorepresentquantitiesinreal-worldcontexts,explainingthemeaningof0ineachsituation.
6. Understandarationalnumberasapointonthenumberline.Extendnumberlinediagramsandcoordinateaxesfamiliarfrompreviousgradestorepresentpointsonthelineandintheplanewithnegativenumbercoordinates.
a. Recognizeoppositesignsofnumbersasindicatinglocationsonoppositesidesof0onthenumberline;recognizethattheoppositeoftheoppositeofanumberisthenumberitself,e.g.,–(–3)=3,andthat0isitsownopposite.
b. Understandsignsofnumbersinorderedpairsasindicatinglocationsinquadrantsofthecoordinateplane;recognizethatwhentwoorderedpairsdifferonlybysigns,thelocationsofthepointsarerelatedbyreflectionsacrossoneorbothaxes.
c. Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram;findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane.
7. Understandorderingandabsolutevalueofrationalnumbers.
a. Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram.For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
b. Write,interpret,andexplainstatementsoforderforrationalnumbersinreal-worldcontexts.For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.
c. Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal-worldsituation.For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
d. Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder.For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
8. Solvereal-worldandmathematicalproblemsbygraphingpointsinallfourquadrantsofthecoordinateplane.Includeuseofcoordinatesandabsolutevaluetofinddistancesbetweenpointswiththesamefirstcoordinateorthesamesecondcoordinate.
expressions and equations 6.ee
Apply and extend previous understandings of arithmetic to algebraic expressions.
1. Writeandevaluatenumericalexpressionsinvolvingwhole-numberexponents.
2. Write,read,andevaluateexpressionsinwhichlettersstandfornumbers.
a. Writeexpressionsthatrecordoperationswithnumbersandwithlettersstandingfornumbers.For example, express the calculation “Subtract y from 5” as 5 – y.
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b. Identifypartsofanexpressionusingmathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofanexpressionasasingleentity.For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
c. Evaluateexpressionsatspecificvaluesoftheirvariables.Includeexpressionsthatarisefromformulasusedinreal-worldproblems.Performarithmeticoperations,includingthoseinvolvingwhole-numberexponents,intheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.
3. Applythepropertiesofoperationstogenerateequivalentexpressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
4. Identifywhentwoexpressionsareequivalent(i.e.,whenthetwoexpressionsnamethesamenumberregardlessofwhichvalueissubstitutedintothem).For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Reason about and solve one-variable equations and inequalities.
5. Understandsolvinganequationorinequalityasaprocessofansweringaquestion:whichvaluesfromaspecifiedset,ifany,maketheequationorinequalitytrue?Usesubstitutiontodeterminewhetheragivennumberinaspecifiedsetmakesanequationorinequalitytrue.
6. Usevariablestorepresentnumbersandwriteexpressionswhensolvingareal-worldormathematicalproblem;understandthatavariablecanrepresentanunknownnumber,or,dependingonthepurposeathand,anynumberinaspecifiedset.
7. Solvereal-worldandmathematicalproblemsbywritingandsolvingequationsoftheformx+p=qandpx=qforcasesinwhichp,qandxareallnonnegativerationalnumbers.
8. Writeaninequalityoftheformx>corx <c torepresentaconstraintorconditioninareal-worldormathematicalproblem.Recognizethatinequalitiesoftheformx>corx<chaveinfinitelymanysolutions;representsolutionsofsuchinequalitiesonnumberlinediagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
9. Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable,intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariablesusinggraphsandtables,andrelatethesetotheequation.For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Geometry 6.G
Solve real-world and mathematical problems involving area, surface area, and volume.
1. Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
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2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV = l w h and V = b htofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems.
3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
4. Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
Statistics and Probability 6.SP
Develop understanding of statistical variability.
1. Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedatarelatedtothequestionandaccountsforitintheanswers.For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
2. Understandthatasetofdatacollectedtoanswerastatisticalquestionhasadistributionwhichcanbedescribedbyitscenter,spread,andoverallshape.
3. Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber,whileameasureofvariationdescribeshowitsvaluesvarywithasinglenumber.
Summarize and describe distributions.
4. Displaynumericaldatainplotsonanumberline,includingdotplots,histograms,andboxplots.
5. Summarizenumericaldatasetsinrelationtotheircontext,suchasby:
a. Reportingthenumberofobservations.
b. Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.
c. Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.
d. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.
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mathematics | Grade 7InGrade7,instructionaltimeshouldfocusonfourcriticalareas:(1)
developingunderstandingofandapplyingproportionalrelationships;
(2)developingunderstandingofoperationswithrationalnumbersand
workingwithexpressionsandlinearequations;(3)solvingproblems
involvingscaledrawingsandinformalgeometricconstructions,and
workingwithtwo-andthree-dimensionalshapestosolveproblems
involvingarea,surfacearea,andvolume;and(4)drawinginferencesabout
populationsbasedonsamples.
(1)Studentsextendtheirunderstandingofratiosanddevelop
understandingofproportionalitytosolvesingle-andmulti-stepproblems.
Studentsusetheirunderstandingofratiosandproportionalitytosolve
awidevarietyofpercentproblems,includingthoseinvolvingdiscounts,
interest,taxes,tips,andpercentincreaseordecrease.Studentssolve
problemsaboutscaledrawingsbyrelatingcorrespondinglengthsbetween
theobjectsorbyusingthefactthatrelationshipsoflengthswithinan
objectarepreservedinsimilarobjects.Studentsgraphproportional
relationshipsandunderstandtheunitrateinformallyasameasureofthe
steepnessoftherelatedline,calledtheslope.Theydistinguishproportional
relationshipsfromotherrelationships.
(2)Studentsdevelopaunifiedunderstandingofnumber,recognizing
fractions,decimals(thathaveafiniteorarepeatingdecimal
representation),andpercentsasdifferentrepresentationsofrational
numbers.Studentsextendaddition,subtraction,multiplication,anddivision
toallrationalnumbers,maintainingthepropertiesofoperationsandthe
relationshipsbetweenadditionandsubtraction,andmultiplicationand
division.Byapplyingtheseproperties,andbyviewingnegativenumbers
intermsofeverydaycontexts(e.g.,amountsowedortemperaturesbelow
zero),studentsexplainandinterprettherulesforadding,subtracting,
multiplying,anddividingwithnegativenumbers.Theyusethearithmetic
ofrationalnumbersastheyformulateexpressionsandequationsinone
variableandusetheseequationstosolveproblems.
(3)StudentscontinuetheirworkwithareafromGrade6,solvingproblems
involvingtheareaandcircumferenceofacircleandsurfaceareaofthree-
dimensionalobjects.Inpreparationforworkoncongruenceandsimilarity
inGrade8theyreasonaboutrelationshipsamongtwo-dimensionalfigures
usingscaledrawingsandinformalgeometricconstructions,andtheygain
familiaritywiththerelationshipsbetweenanglesformedbyintersecting
lines.Studentsworkwiththree-dimensionalfigures,relatingthemtotwo-
dimensionalfiguresbyexaminingcross-sections.Theysolvereal-world
andmathematicalproblemsinvolvingarea,surfacearea,andvolumeof
two-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,
polygons,cubesandrightprisms.
(4)Studentsbuildontheirpreviousworkwithsingledatadistributionsto
comparetwodatadistributionsandaddressquestionsaboutdifferences
betweenpopulations.Theybegininformalworkwithrandomsampling
togeneratedatasetsandlearnabouttheimportanceofrepresentative
samplesfordrawinginferences.
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ratios and Proportional relationships
• analyze proportional relationships and use them to solve real-world and mathematical problems.
the number System
• apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
expressions and equations
• Use properties of operations to generate equivalent expressions.
• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Geometry
• draw, construct and describe geometrical figures and describe the relationships between them.
• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Statistics and Probability
• Use random sampling to draw inferences about a population.
• draw informal comparative inferences about two populations.
• Investigate chance processes and develop, use, and evaluate probability models.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 7 overview
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ratios and Proportional relationships 7.rP
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
2. Recognizeandrepresentproportionalrelationshipsbetweenquantities.
a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.
b. Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.
c. Representproportionalrelationshipsbyequations.For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
d. Explainwhatapoint (x, y) onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentiontothepoints(0,0)and(1, r) where r istheunitrate.
3. Useproportionalrelationshipstosolvemultistepratioandpercentproblems.Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
the number System 7.nS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
1. Applyandextendpreviousunderstandingsofadditionandsubtractiontoaddandsubtractrationalnumbers;representadditionandsubtractiononahorizontalorverticalnumberlinediagram.
a. Describesituationsinwhichoppositequantitiescombinetomake0.For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
b. Understandp+qasthenumberlocatedadistance|q|fromp,inthepositiveornegativedirectiondependingonwhetherqispositiveornegative.Showthatanumberanditsoppositehaveasumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.
c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse,p–q=p+(–q).Showthatthedistancebetweentworationalnumbersonthenumberlineistheabsolutevalueoftheirdifference,andapplythisprincipleinreal-worldcontexts.
d. Applypropertiesofoperationsasstrategiestoaddandsubtractrationalnumbers.
2. Applyandextendpreviousunderstandingsofmultiplicationanddivisionandoffractionstomultiplyanddividerationalnumbers.
a. Understandthatmultiplicationisextendedfromfractionstorationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(–1)(–1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbersbydescribingreal-worldcontexts.
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b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zerodivisor)isarationalnumber.Ifpandqareintegers,then–(p/q)=(–p)/q=p/(–q).Interpretquotientsofrationalnumbersbydescribingreal-worldcontexts.
c. Applypropertiesofoperationsasstrategiestomultiplyanddividerationalnumbers.
d. Convertarationalnumbertoadecimalusinglongdivision;knowthatthedecimalformofarationalnumberterminatesin0soreventuallyrepeats.
3. Solvereal-worldandmathematicalproblemsinvolvingthefouroperationswithrationalnumbers.1
expressions and equations 7.ee
Use properties of operations to generate equivalent expressions.
1. Applypropertiesofoperationsasstrategiestoadd,subtract,factor,andexpandlinearexpressionswithrationalcoefficients.
2. Understandthatrewritinganexpressionindifferentformsinaproblemcontextcanshedlightontheproblemandhowthequantitiesinitarerelated.For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solvemulti-stepreal-lifeandmathematicalproblemsposedwithpositiveandnegativerationalnumbersinanyform(wholenumbers,fractions,anddecimals),usingtoolsstrategically.Applypropertiesofoperationstocalculatewithnumbersinanyform;convertbetweenformsasappropriate;andassessthereasonablenessofanswersusingmentalcomputationandestimationstrategies.For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
4. Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.
a. Solvewordproblemsleadingtoequationsoftheformpx+q=randp(x+q)=r,wherep,q,andrarespecificrationalnumbers.Solveequationsoftheseformsfluently.Compareanalgebraicsolutiontoanarithmeticsolution,identifyingthesequenceoftheoperationsusedineachapproach.For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solvewordproblemsleadingtoinequalitiesoftheformpx+q>rorpx+q<r,wherep,q,andrarespecificrationalnumbers.Graphthesolutionsetoftheinequalityandinterpretitinthecontextoftheproblem.For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Geometry 7.G
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solveproblemsinvolvingscaledrawingsofgeometricfigures,includingcomputingactuallengthsandareasfromascaledrawingandreproducingascaledrawingatadifferentscale.
1Computationswithrationalnumbersextendtherulesformanipulatingfractionstocomplexfractions.
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2. Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticingwhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.
3. Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems;giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle.
5. Usefactsaboutsupplementary,complementary,vertical,andadjacentanglesinamulti-stepproblemtowriteandsolvesimpleequationsforanunknownangleinafigure.
6. Solvereal-worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.
Statistics and Probability 7.SP
Use random sampling to draw inferences about a population.
1. Understandthatstatisticscanbeusedtogaininformationaboutapopulationbyexaminingasampleofthepopulation;generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation.Understandthatrandomsamplingtendstoproducerepresentativesamplesandsupportvalidinferences.
2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Draw informal comparative inferences about two populations.
3. Informallyassessthedegreeofvisualoverlapoftwonumericaldatadistributionswithsimilarvariabilities,measuringthedifferencebetweenthecentersbyexpressingitasamultipleofameasureofvariability.For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
4. Usemeasuresofcenterandmeasuresofvariabilityfornumericaldatafromrandomsamplestodrawinformalcomparativeinferencesabouttwopopulations.For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Investigate chance processes and develop, use, and evaluate probability models.
5. Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthelikelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynorlikely,andaprobabilitynear1indicatesalikelyevent.
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6. Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesitandobservingitslong-runrelativefrequency,andpredicttheapproximaterelativefrequencygiventheprobability.For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7. Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy.
a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilitiesofevents.For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Developaprobabilitymodel(whichmaynotbeuniform)byobservingfrequenciesindatageneratedfromachanceprocess.For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
8. Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.
a. Understandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs.
b. Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespacewhichcomposetheevent.
c. Designanduseasimulationtogeneratefrequenciesforcompoundevents.For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
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mathematics | Grade 8InGrade8,instructionaltimeshouldfocusonthreecriticalareas:(1)formulating
andreasoningaboutexpressionsandequations,includingmodelinganassociation
inbivariatedatawithalinearequation,andsolvinglinearequationsandsystems
oflinearequations;(2)graspingtheconceptofafunctionandusingfunctions
todescribequantitativerelationships;(3)analyzingtwo-andthree-dimensional
spaceandfiguresusingdistance,angle,similarity,andcongruence,and
understandingandapplyingthePythagoreanTheorem.
(1)Studentsuselinearequationsandsystemsoflinearequationstorepresent,
analyze,andsolveavarietyofproblems.Studentsrecognizeequationsfor
proportions(y/x=mory=mx)asspeciallinearequations(y=mx+b),
understandingthattheconstantofproportionality(m)istheslope,andthegraphs
arelinesthroughtheorigin.Theyunderstandthattheslope(m)ofalineisa
constantrateofchange,sothatiftheinputorx-coordinatechangesbyanamount
A,theoutputory-coordinatechangesbytheamountm·A.Studentsalsousealinear
equationtodescribetheassociationbetweentwoquantitiesinbivariatedata(such
asarmspanvs.heightforstudentsinaclassroom).Atthisgrade,fittingthemodel,
andassessingitsfittothedataaredoneinformally.Interpretingthemodelinthe
contextofthedatarequiresstudentstoexpressarelationshipbetweenthetwo
quantitiesinquestionandtointerpretcomponentsoftherelationship(suchasslope
andy-intercept)intermsofthesituation.
Studentsstrategicallychooseandefficientlyimplementprocedurestosolvelinear
equationsinonevariable,understandingthatwhentheyusethepropertiesof
equalityandtheconceptoflogicalequivalence,theymaintainthesolutionsofthe
originalequation.Studentssolvesystemsoftwolinearequationsintwovariables
andrelatethesystemstopairsoflinesintheplane;theseintersect,areparallel,or
arethesameline.Studentsuselinearequations,systemsoflinearequations,linear
functions,andtheirunderstandingofslopeofalinetoanalyzesituationsandsolve
problems.
(2)Studentsgrasptheconceptofafunctionasarulethatassignstoeachinput
exactlyoneoutput.Theyunderstandthatfunctionsdescribesituationswhereone
quantitydeterminesanother.Theycantranslateamongrepresentationsandpartial
representationsoffunctions(notingthattabularandgraphicalrepresentations
maybepartialrepresentations),andtheydescribehowaspectsofthefunctionare
reflectedinthedifferentrepresentations.
(3)Studentsuseideasaboutdistanceandangles,howtheybehaveunder
translations,rotations,reflections,anddilations,andideasaboutcongruenceand
similaritytodescribeandanalyzetwo-dimensionalfiguresandtosolveproblems.
Studentsshowthatthesumoftheanglesinatriangleistheangleformedbya
straightline,andthatvariousconfigurationsoflinesgiverisetosimilartriangles
becauseoftheanglescreatedwhenatransversalcutsparallellines.Students
understandthestatementofthePythagoreanTheoremanditsconverse,andcan
explainwhythePythagoreanTheoremholds,forexample,bydecomposinga
squareintwodifferentways.TheyapplythePythagoreanTheoremtofinddistances
betweenpointsonthecoordinateplane,tofindlengths,andtoanalyzepolygons.
Studentscompletetheirworkonvolumebysolvingproblemsinvolvingcones,
cylinders,andspheres.
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the number System
• Know that there are numbers that are not rational, and approximate them by rational numbers.
expressions and equations
• Work with radicals and integer exponents.
• Understand the connections between proportional relationships, lines, and linear equations.
• analyze and solve linear equations and pairs of simultaneous linear equations.
functions
• define, evaluate, and compare functions.
• Use functions to model relationships between quantities.
Geometry
• Understand congruence and similarity using physical models, transparencies, or geometry software.
• Understand and apply the Pythagorean theorem.
• Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.
Statistics and Probability
• Investigate patterns of association in bivariate data.
mathematical Practices
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
Grade 8 overview
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the number System 8.nS
Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Knowthatnumbersthatarenotrationalarecalledirrational.Understandinformallythateverynumberhasadecimalexpansion;forrationalnumbersshowthatthedecimalexpansionrepeatseventually,andconvertadecimalexpansionwhichrepeatseventuallyintoarationalnumber.
2. Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2).For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
expressions and equations 8.ee
Work with radicals and integer exponents.
1. Knowandapplythepropertiesofintegerexponentstogenerateequivalentnumericalexpressions.For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
2. Usesquarerootandcuberootsymbolstorepresentsolutionstoequationsoftheformx2=pandx3=p,wherepisapositiverationalnumber.Evaluatesquarerootsofsmallperfectsquaresandcuberootsofsmallperfectcubes.Knowthat√2isirrational.
3. Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
4. Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothdecimalandscientificnotationareused.Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities(e.g.,usemillimetersperyearforseafloorspreading).Interpretscientificnotationthathasbeengeneratedbytechnology.
Understand the connections between proportional relationships, lines, and linear equations.
5. Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Usesimilartrianglestoexplainwhytheslopemisthesamebetweenanytwodistinctpointsonanon-verticallineinthecoordinateplane;derivetheequationy=mxforalinethroughtheoriginandtheequationy=mx+bforalineinterceptingtheverticalaxisatb.
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solvelinearequationsinonevariable.
a. Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).
b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.
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8. Analyzeandsolvepairsofsimultaneouslinearequations.
a. Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.
b. Solvesystemsoftwolinearequationsintwovariablesalgebraically,andestimatesolutionsbygraphingtheequations.Solvesimplecasesbyinspection.For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solvereal-worldandmathematicalproblemsleadingtotwolinearequationsintwovariables.For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
functions 8.f
Define, evaluate, and compare functions.
1. Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.Thegraphofafunctionisthesetoforderedpairsconsistingofaninputandthecorrespondingoutput.1
2. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
3. Interprettheequationy=mx+basdefiningalinearfunction,whosegraphisastraightline;giveexamplesoffunctionsthatarenotlinear.For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.
4. Constructafunctiontomodelalinearrelationshipbetweentwoquantities.Determinetherateofchangeandinitialvalueofthefunctionfromadescriptionofarelationshiporfromtwo(x,y)values,includingreadingthesefromatableorfromagraph.Interprettherateofchangeandinitialvalueofalinearfunctionintermsofthesituationitmodels,andintermsofitsgraphoratableofvalues.
5. Describequalitativelythefunctionalrelationshipbetweentwoquantitiesbyanalyzingagraph(e.g.,wherethefunctionisincreasingordecreasing,linearornonlinear).Sketchagraphthatexhibitsthequalitativefeaturesofafunctionthathasbeendescribedverbally.
Geometry 8.G
Understand congruence and similarity using physical models, trans-parencies, or geometry software.
1. Verifyexperimentallythepropertiesofrotations,reflections,andtranslations:
a. Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.
b. Anglesaretakentoanglesofthesamemeasure.
c. Parallellinesaretakentoparallellines.
2. Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.
1FunctionnotationisnotrequiredinGrade8.
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3. Describetheeffectofdilations,translations,rotations,andreflectionsontwo-dimensionalfiguresusingcoordinates.
4. Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.
5. Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Understand and apply the Pythagorean Theorem.
6. ExplainaproofofthePythagoreanTheoremanditsconverse.
7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.
8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.
Statistics and Probability 8.SP
Investigate patterns of association in bivariate data.
1. Constructandinterpretscatterplotsforbivariatemeasurementdatatoinvestigatepatternsofassociationbetweentwoquantities.Describepatternssuchasclustering,outliers,positiveornegativeassociation,linearassociation,andnonlinearassociation.
2. Knowthatstraightlinesarewidelyusedtomodelrelationshipsbetweentwoquantitativevariables.Forscatterplotsthatsuggestalinearassociation,informallyfitastraightline,andinformallyassessthemodelfitbyjudgingtheclosenessofthedatapointstotheline.
3. Usetheequationofalinearmodeltosolveproblemsinthecontextofbivariatemeasurementdata,interpretingtheslopeandintercept.For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
4. Understandthatpatternsofassociationcanalsobeseeninbivariatecategoricaldatabydisplayingfrequenciesandrelativefrequenciesinatwo-waytable.Constructandinterpretatwo-waytablesummarizingdataontwocategoricalvariablescollectedfromthesamesubjects.Userelativefrequenciescalculatedforrowsorcolumnstodescribepossibleassociationbetweenthetwovariables.For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?