Diocese of Baton Rouge
Mathematics Standards
Grade 6
Introduction
In Grade6,instructionaltimeshould focuson fourcriticalareas: (1)connecting ratioand rateto wholenumbermultiplication and divisionand using conceptsofratio and rateto solveproblems;(2)completing understanding ofdivisionof fractionsand extending thenotionofnumber to thesystem of rationalnumbers,which includesnegativenumbers; (3)writing,interpreting,and using expressionsand equations; and (4)developing understandingof statisticalthinking.
(1)Studentsusereasoningaboutmultiplication and divisionto solveratioand rateproblemsaboutquantities.Byviewingequivalentratiosandratesasderivingfrom,andextending,pairsofrows(orcolumns)inthemultiplicationtable,andbyanalyzingsimpledrawingsthatindicatetherelativesizeofquantities,studentsconnecttheirunderstandingofmultiplicationand divisionwithratiosandrates.Thusstudentsexpandthescopeofproblemsforwhichtheycanusemultiplicationand divisiontosolveproblems,andtheyconnectratiosandfractions.Studentssolveawidevarietyofproblemsinvolvingratiosandrates.
(2)Studentsusethe meaningof fractions,themeanings ofmultiplication and division,and therelationshipbetweenmultiplication and divisiontounderstand and explainwhytheproceduresfordividing fractionsmakesense.Studentsusetheseoperationsto solveproblems.Studentsextend theirpreviousunderstandings of numberand theorderingofnumbersto thefullsystemof rationalnumbers,whichincludesnegativerationalnumbers,and in particularnegativeintegers.Theyreason abouttheorderand absolutevalueof rationalnumbers and aboutthelocationof pointsin allfourquadrantsof thecoordinateplane.
(3)Studentsunderstand theuseofvariablesin mathematicalexpressions.They writeexpressionsand equationsthatcorrespond togiven situations,evaluateexpressions,and useexpressionsand formulastosolveproblems.Studentsunderstand thatexpressionsin differentformscan beequivalent,andthey usethepropertiesofoperations to rewriteexpressionsinequivalentforms.Studentsknowthatthesolutionsof anequationarethevaluesof thevariablesthatmaketheequation true.Studentsusepropertiesofoperationsand theidea ofmaintaining theequality ofboth sidesofan equationto solvesimpleone-stepequations.Studentsconstructand analyzetables,suchastablesof quantitiesthatarein equivalentratios,and theyuseequations(suchas 3x=y)todescriberelationshipsbetweenquantities.
(4)Buildingon and reinforcing theirunderstanding ofnumber,studentsbegintodevelop theirability tothinkstatistically.Studentsrecognizethata datadistribution may nothavea definitecenterandthatdifferentwaystomeasurecenteryield differentvalues.Themedianmeasurescenterinthesensethatitisroughly themiddlevalue.Themeanmeasurescenterin thesensethatitisthevaluethateach datapointwould takeon ifthetotalof thedatavalueswereredistributed equally,and alsointhesensethatitis a balancepoint.Studentsrecognizethatameasureofvariability (interquartilerangeormean absolutedeviation)can alsobeusefulforsummarizing databecausetwo verydifferentsetsof datacan havethesame mean andmedian yetbedistinguished bytheirvariability.Studentslearn todescribeand summarizenumericaldatasets,identifying clusters,peaks,gaps,and symmetry,considering thecontext in which thedatawerecollected.
Studentsin Grade6 also build on theirworkwith areain elementaryschoolbyreasoning aboutrelationshipsamongshapestodeterminearea,surfacearea,andvolume.Theyfind areasof righttriangles,othertriangles,andspecialquadrilateralsbydecomposing theseshapes,rearrangingor removing pieces, and relating theshapesto rectangles.Using thesemethods,studentsdiscuss,develop,and justifyformulasforareasoftrianglesand parallelograms.Studentsfind areasof polygonsandsurfaceareasof prismsandpyramidsbydecomposingthem into pieceswhoseareatheycandetermine.Theyreason aboutrightrectangularprismswith fractionalsidelengths to extend formulasforthevolume of a rightrectangularprism to fractionalsidelengths.Theyprepareforworkon scaledrawingsand constructionsinGrade7 bydrawing polygonsin thecoordinateplane.
RatiosandProportionalRelationships DBR.6.RP
A.Understand ratioconceptsand use ratio reasoningto solve problems.
1.Understand theconceptofa ratio and useratio languageto describea ratio relationship betweentwoquantities.For example, “Theratioofwingsto beaks in thebird houseatthezoowas2:1,becauseforevery2wingstherewas1beak.” “Foreveryvotecandidate Areceived,candidateC received nearlythreevotes.”
2.Understand theconceptofa unitrate a/bassociatedwith a ratio a:b withb0,and useratelanguagein thecontextof aratiorelationship.Forexample,“Thisrecipehasa ratioof3cupsofflourto4cupsofsugar,so thereis3/4cup offlourforeachcup ofsugar.” “Wepaid $75for 15hamburgers,whichisarate of$5per hamburger.”1
3.Useratioand ratereasoning tosolvereal-world andmathematicalproblems,e.g.,byreasoning abouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.
a.Maketablesofequivalentratiosrelating quantitieswith whole-numbermeasurements,findmissing valuesin thetables,and plotthe pairs ofvaluesonthecoordinateplane.Usetablesto compareratios.
b.Solveunitrateproblemsincludingthoseinvolving unitpricing and constantspeed.Forexample,ifittook7hourstomow4lawns,then atthatrate,howmanylawnscould bemowed in35 hours?Atwhatunitratewerelawnsbeing mowed?
c.Find a percentof aquantityas arateper100(e.g.,30%ofa quantity means 30/100 timesthequantity);solveproblemsinvolving finding thewhole,given a partand thepercent.
d.Useratioreasoning to convertmeasurementunits; manipulateand transform unitsappropriately whenmultiplyingor dividing quantities.
TheNumberSystem DBR.6.NS
A.Apply and extend previousunderstandingsofmultiplication and division todivide fractionsby fractions.
1.Interpretandcomputequotientsof fractions,and solveword problemsinvolvingdivisionof fractionsbyfractions,e.g.,by using visualfractionmodelsand equationsto representtheproblem.For example,createastory contextfor (2/3)÷(3/4)andusea visualfractionmodel toshowthequotient;usetherelationship betweenmultiplication anddivisionto explain that(2/3)÷(3/4)=8/9 because3/4of8/9is2/3.(In general,(a/b) ÷(c/d)=ad/bc.)Howmuch
chocolatewilleachperson getif3peopleshare1/2lbofchocolateequally?Howmany3/4-cup servingsarein2/3ofacup ofyogurt?How wideisa rectangular stripoflandwith length3/4miandarea1/2squaremi?
B.Compute fluently withmulti-digitnumbersand find common factorsand multiples.
2.Fluentlydividemulti-digitnumbersusing thestandard algorithm.
3.Fluentlyadd,subtract,multiply,and dividemulti-digitdecimalsusing thestandard algorithmforeach operation.
1Expectationsfor unitratesinthisgrade are limited tonon-complexfractions
4.Find thegreatestcommonfactoroftwo wholenumberslessthan orequalto 100 and theleastcommonmultipleoftwowholenumberslessthanor equalto 12.Usethedistributiveproperty to express asum oftwowholenumbers1–100 witha common factoras amultipleof asum oftwo wholenumberswith no commonfactor.Forexample,express36+8as4(9+2).
C.Apply and extend previousunderstandingsofnumberstothe systemofrationalnumbers.
5.Understand thatpositive and negativenumbersareused togethertodescribequantitieshaving oppositedirections or values(e.g.,temperature above/below zero,elevation above/below sea level,credits/debits,positive/negativeelectriccharge);usepositiveand negativenumberstorepresentquantitiesin real-worldcontexts,explaining themeaning of0in each situation.
6.Understand a rationalnumberas a pointonthenumberline.Extend numberlinediagramsandcoordinateaxesfamiliarfrompreviousgradestorepresentpointsonthelineand in theplanewith negativenumbercoordinates.
a.Recognizeoppositesignsofnumbersasindicating locationsonoppositesides of0onthenumberline;recognizethatthe oppositeoftheoppositeof anumberisthenumberitself,e.g.,–(–3)=3,and that0isitsownopposite.
b.Understand signsof numbersinordered pairsasindicating locationsin quadrants ofthecoordinateplane;recognizethatwhen two ordered pairsdifferonlyby signs,thelocationsof thepointsarerelated byreflectionsacrossoneor both axes.
c.Find and position integersand otherrationalnumbers onahorizontalorverticalnumberlinediagram;findand position pairsof integersand otherrationalnumbersona coordinateplane.
7.Understand ordering and absolutevalueofrationalnumbers.
a.Interpretstatementsof inequalityasstatementsabouttherelativepositionof twonumbersona numberlinediagram.Forexample,interpret–3 –7asa statementthat–3islocated totherightof–7 on a numberlineorientedfromlefttoright.
b.Write,interpret,and explain statementsoforderforrationalnumbersin real-world contexts.Forexample,write –3oC–7oCto expressthefactthat–3oC iswarmer than–7oC.
c.Understand theabsolutevalueofa rationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudefora positiveor negativequantityin a real-world situation.For example,for anaccountbalanceof –30 dollars,write|–30|=30todescribethesizeofthedebtindollars.
d.Distinguish comparisonsofabsolutevaluefrom statementsaboutorder.For example,recognizethatanaccountbalancelessthan–30 dollarsrepresentsadebtgreater than30dollars.
8.Solvereal-world andmathematicalproblemsby graphing pointsin allfourquadrants ofthecoordinateplane.Includeuseof coordinatesand absolutevaluetofinddistancesbetween pointswith thesamefirstcoordinateorthesamesecond coordinate.
ExpressionsandEquations DBR.6.EE
A.Apply and extend previousunderstandingsofarithmetic to algebraicexpressions.
1.Writeandevaluatenumericalexpressionsinvolvingwhole-numberexponents.
2.Write,read,andevaluateexpressionsin which lettersstand for numbers.
a.Writeexpressionsthatrecord operationswith numbersand with lettersstanding for numbers.Forexample,expressthecalculation “Subtracty from5”as5–y.
b.Identifypartsofan expression using mathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofan expression asa singleentity.For example,describetheexpression2(8+7) asaproductoftwofactors; view(8+7) asboth asingleentityand a sumoftwoterms.
c.Evaluateexpressionsatspecificvaluesof theirvariables.Includeexpressionsthatarisefrom formulasused in real-world problems.Perform arithmeticoperations,including thoseinvolvingwhole-numberexponents,in theconventionalorderwhen thereareno parenthesesto specifya particularorder(OrderofOperations).Forexample,usetheformulasV =s3and A =6 s2to find thevolumeandsurfacearea of a cubewithsidesoflength s=1/2.
3.Applythepropertiesofoperationstogenerateequivalentexpressions.For example,applythedistributivepropertyto theexpression3(2+x) to producetheequivalentexpression6+3x; applythedistributiveproperty totheexpression24x+18y toproducetheequivalentexpression6(4x+3y); applypropertiesofoperationstoy +y+y to producetheequivalentexpression3y.
4.Identifywhen two expressionsareequivalent(i.e.,when thetwoexpressionsnamethesamenumberregardlessof whichvalueissubstituted intothem).For example,theexpressionsy +y +y and3y areequivalentbecausetheynamethesamenumber regardlessofwhich numberystandsfor.
B.Reason aboutand solveone-variable equationsand inequalities.
5.Understand solving anequationor inequalityasa processof answering a question:whichvaluesfromaspecified set,ifany,maketheequation orinequality true?Usesubstitutionto determinewhether agivennumberin aspecifiedsetmakesanequationor inequalitytrue.
6.Usevariablesto representnumbersandwriteexpressionswhen solving a real-world ormathematicalproblem;understand thatavariablecan representan unknownnumber,or, depending onthepurposeathand,anynumberin aspecifiedset.
7.Solvereal-world andmathematicalproblemsby writing and solvingequations and inequalitiesof theform x+p
=qand px=qforcasesinwhich p,qand xareallnonnegativerationalnumbers.Inequalitieswillinclude<,>, ≤,and ≥.
8.Writeaninequality oftheform xcor xcto representa constraintorcondition in a real-worldormathematicalproblem.Recognizethatinequalitiesoftheformxcor xchaveinfinitely many solutions;representsolutionsof such inequalitieson numberlinediagrams.
C.Representand analyzequantitativerelationships between dependentand independentvariables.
9.Usevariablesto representtwo quantitiesin a real-world problemthatchangeinrelationshiptoone another;writean equation toexpress onequantity,thoughtofasthedependentvariable,in termsoftheother quantity,thoughtof astheindependentvariable.Analyzetherelationship between thedependentand independentvariablesusing graphsandtables,and relatetheseto theequation.For example,in a probleminvolvingmotionatconstantspeed,listandgraphordered pairsofdistancesandtimes,andwritetheequation d=65ttorepresenttherelationshipbetween distanceandtime.
Geometry DBR.6.G
A.Solve real-world and mathematical problemsinvolvingarea,surface area,and volume.
1.Find theareaof righttriangles,othertriangles,specialquadrilaterals,and polygonsbycomposing into rectanglesor decomposinginto trianglesandothershapes;apply thesetechniquesin thecontextof solvingreal-world andmathematicalproblems.
2.Find thevolumeof a rightrectangularprismwith fractionaledgelengthsby packing itwith unitcubesof theappropriateunitfraction edgelengths,and showthatthevolumeisthesameas would befound by multiplyingtheedgelengthsof theprism.Apply theformulasV =lwhand V=bhto find volumesof rightrectangularprismswith fractionaledgelengthsin thecontextof solving real-world andmathematicalproblems.
3.Drawpolygonsinthecoordinateplanegiven coordinatesforthe vertices; usecoordinatestofindthelength of asidejoining pointswith thesamefirstcoordinateor thesamesecond coordinate.Applythesetechniquesin thecontextof solving real-world and mathematicalproblems.
4.Representthree-dimensionalfiguresusing netsmadeup of rectanglesandtriangles,and usethenetsto find thesurfaceareaof thesefigures.Applythesetechniquesin thecontextof solving real-world andmathematicalproblems.
StatisticsandProbability DBR.6.SP
A.Develop understanding ofstatistical variability.
1.Recognizea statisticalquestion asonethatanticipatesvariabilityin thedatarelated to thequestion andaccountsforitin theanswers.For example,“HowoldamI?”isnota statisticalquestion,but“Howold arethestudentsinmyschool?” isa statisticalquestion becauseoneanticipatesvariabilityin students’ages.
2.Understand thata setof datacollected toanswer astatisticalquestion has a distribution thatcan bedescribedbyitscenter,spread,and overall shape.
3.Recognizethatameasureof centerfor anumericaldatasetsummarizesallof itsvalueswitha singlenumber,whileameasureofvariation describeshowitsvaluesvary with asinglenumber.
B.Summarize and describe distributions.
4.Displaynumericaldatain plotsona numberline,includingdotplots,histograms,and box plots.
5.Summarizenumericaldatasets in relation totheircontext,such asby:
a.Reportingthenumberofobservations.
b.Describing thenatureof theattributeunderinvestigation,including howitwasmeasured and itsunitsofmeasurement.
c.Giving quantitativemeasures ofcenter(medianand/ormean)and variability (interquartilerange)aswellasdescribing any overall pattern and anystriking deviationsfrom theoverall pattern with referencetothecontextinwhich thedataweregathered.
d.Relating thechoice of measuresofcenterandvariabilityto theshape of thedatadistribution andthecontextinwhich thedataweregathered.
Diocese of Baton Rouge Mathematics Standards: Grade 6 July, 2017 Page 1