WESTFIELDPUBLICSCHOOLS

Westfield, New Jersey

Officeof Instruction

Courseof Study

MATH 6

School...... Edison &RooseveltSchools

Department...... Mathematics

Length of Course...... Fullyear

GradeLevel...... 6

Prerequisite...... None

Date......

I. RATIONALE, DESCRIPTION ANDPURPOSE

The Math 6 curriculum links to and builds on students’ elementary mathematics experiences, providesfortheacquisitionandmastery of essentialskills,andintroducesnew conceptsorrevisits familiaronesingreaterdepth.Thereisastrongemphasis on understandingof and computing with rationalnumbers,rates and ratios, basicalgebraicexpressionsandequations,statisticalthinking,areaandvolume. It isimportantforallstudentstohaveaccesstoarich curriculumthatenablesthemtoacquireasolid knowledgeofmathematics,strongproblem-solvingskills,andtheabilitytoreasoneffectively,so that they will be well prepared for seventh grade math and beyond. In order to develop the foundation needed for the formal study of algebra and geometry, the sixth grade curriculum challenges students to think about mathematics, work collaboratively, and solve meaningful problems.

II. OBJECTIVES

This curriculumfulfillsWestfield Board ofEducation expectations for student achievement. Courseobjectives are aligned with the New Jersey Student Learning Standards for Mathematics, English LanguageArts, Science, Technology,and21stCentury Lifeand Careers.

1

Math6

CIP 3/17/2017

Students:

A. Demonstrate understandingofand applyratio concepts to solvemathematical and real-world problems

NJ Student Learning Standards for Mathematics6.RP

NJ Student Learning Standards for EnglishLanguageArts A.R7

NJ Student Learning Standards for Science P5

NJ Student Learning Standards for Technology 8.1

NJ Student Learning Standards for 21stCenturyLife andCareers9.1

B. Extend knowledgeof thenumbersystem to include rational numbers

NJ Student Learning Standards for Mathematics6.NS

NJ Student Learning Standards for Technology 8.1

C. Compute fluentlywith multi-digitwhole numbers, fractions and decimals

NJ Student LearningStandards for Mathematics6.NS

NJ Student Learning Standards for Science P5

NJ Student Learning Standards for Technology 8.1

D. Interpret,modeland solve real-world problems with single-variable expressions, equations and inequalities

NJ Student Learning Standards for Mathematics6.EE

NJ Student Learning Standards for EnglishLanguageArtsA.R7

NJ Student LearningStandards for Science P2

NJ Student Learning Standards for Technology 8.1

NJ Student Learning Standards for 21stCenturyLife andCareers 9.1

E. Solve problems involvingarea, surfacearea and volume

NJ Student Learning Standards for Mathematics6.G

NJ Student Learning Standards for Science P5

NJ Student Learning Standards for Technology 8.1

NJ Student Learning Standards for 21stCenturyLife andCareers9.1

F. Demonstrate understandingofand applyprinciples of statistical variability

NJ Student Learning Standards for Mathematics 6.SP

NJ Student Learning Standards for EnglishLanguageArts A.R7,A.W1,A.SL2,A.SL4

NJ Student Learning Standards for Science P3, P4, P7, P8

NJ Student Learning Standards for Technology 8.1

NJ Student Learning Standards for 21stCenturyLife andCareers9.1

G. Display, describeand summarizedata

NJ Student Learning Standards for Mathematics 6.SP

NJ Student Learning Standards for EnglishLanguageArts A.R7,A.W1,A.SL2,A.SL4

NJ Student Learning Standards for Science P3, P4, P7, P8

NJ Student Learning Standards for Technology 8.1

NJ Student Learning Standards for 21stCenturyLife andCareers9.1

H. Develop practicesand dispositions that lead to mathematical proficiency.

NJ Student Learning Standards for MathematicsSMP1 – SMP8

NJ Student Learning Standards forEnglishLanguageArts A.R7,A.R10,A.W1,A.SL1, A.SL3,A.SL4,A.SL5

NJ Student Learning Standards for ScienceP1 – P8

NJ Student Learning Standards for Technology8.1

NJ Student Learning Standards for21stCenturyLife andCareers 9.1

III.CONTENT, SCOPE AND SEQUENCE

Theimportanceofmathematics inthedevelopmentof all civilizations and cultures anditsrelevance tostudents’successregardlessof careerpathis addressedthroughouttheintermediatemathematics program. Emphasisisplacedonthedevelopmentof criticalthinking and problem-solvingskills, particularlythrough theuseofeverydaycontextsand real-world applications.

A. Ratios and proportional relationships

1. Usingratios and unitrates

2. Solvingword problems involvingratios and rates,such as percent and measurement conversions

B. Thenumbersystem

1. Computation

a. Multiplying and dividingfractions

b. Dividingmulti-digitnumbers usingthestandardalgorithm

c. Adding, subtracting, multiplying and dividingdecimals usingstandardalgorithms d. Finding common factorsand multiples

2. Rational numbers

a. Graphingnumbers onanumberline

b. Plottingpoints in the coordinate plane

c. Orderingandfindingtheabsolute value of numbers

d. Modelingand solvingreal-world problems involvingrational numbers

C. Expressions and equations

1. Numerical andalgebraicexpressions

a. Writingand evaluatingexpressions

b. Applyingorder ofoperationsto simplifyandgenerate equivalentexpressions

2. Single-variable equations and inequalities

a. Solvingsimple equations containingnon-negativerational numbers

b. Usingsimple equations to solve real-world and mathematical problems c. Usingsimple inequalitiesto representconstraints

d. Representingsolutions of simpleinequalities on thenumberline

3. Usingdependent and independent variables to describeandanalyze relationships

D. Geometry

1. Calculatingareas of triangles, quadrilaterals and polygons

2. Calculatingvolumeofprisms

3. Constructingpolygons in the coordinate plane

4. Usingnetsto determinesurface areaof prisms andpyramids

5. Solvingreal-world andmathematical problems involvinggeometric shapes

E. Statistics and probability

1. Data displays

a. Constructingdot plots, histogramsand boxplots b. Analyzingdata displays

2. Single-data distributions

a. Findingmeasures of center (median, mean)

b. Findingmeasures of variability(inter-quartile range, mean absolute deviation)

IV. INSTRUCTIONAL TECHNIQUES

A varietyof instructionalapproaches isemployedtoengage allstudents in the learningprocess and accommodate differences in readiness levels, interestsand learningstyles. Typical teaching techniques include, but arenot limited to, the following:

  1. Teacher-directedwholegroup instruction and modelingof procedures
  2. Mini-lessons or individualized instruction for reinforcement or re-teaching of concepts
  3. Guided investigations/explorations
  4. Problem-based learning
  5. Modeling with manipulatives
  6. Flexible grouping
  7. Differentiated tasks
  8. Spiral review
  9. Independent practice
  10. Use of technology
  11. Integration of mathematics with other disciplines.

V. EVALUATION

Multiple techniques are employed to assess student understanding of mathematical concepts, skills, and thinking processes. These may include, but are not limited to:

  1. Written tests and quizzes, including baseline and benchmark assessments
  2. Cumulative tests
  3. Standardized tests
  4. Electronic data-gathering and tasks
  5. Homework
  6. Independent or group projects
  7. Presentations.

VI. PROFESSIONAL DEVELOPMENT

The following recommended activities support this curriculum:

  1. Opportunities to learn from and share ideas about teaching and learning mathematics with colleagues through meetings and peer observations, including collaborations between intermediate and high school teachers
  2. Collaboration with colleagues and department supervisor to discuss and reflect upon unit plans, homework, and assessment practices
  3. Planning time to develop and discuss the results of implementing differentiated lessons and incorporating technology to enhance student learning
  4. Attendance at workshops,conferencesandcoursesthatfocusonrelevantmathematics content, pedagogy,alternate assessment techniques or technology.

APPENDIX I

New Jersey Student Learning StandardsforMathematicalPractice

The Standards for Mathematical Practice describe varieties of expertise thatmathematics educators at all levelsshouldseek todevelopintheirstudents.Thesepracticesrestonimportant“processesand proficiencies”withlongstanding importanceinmathematicseducation. The firstofthesearetheNCTM processstandardsofproblem solving,reasoningandproof,communication,representation,and connections. The secondarethestrandsofmathematicalproficiency specified in theNationalResearch Council’sreportAdding ItUp:adaptivereasoning,strategiccompetence,conceptualunderstanding (comprehensionofmathematicalconcepts,operationsandrelations),proceduralfluency (skillincarrying outproceduresflexibly,accurately,efficiently andappropriately),andproductivedisposition (habitual inclination to see mathematicsassensible, useful,and worthwhile,coupledwith a belief indiligence and one’sown efficacy).

SMP1 – Make sense ofproblems and persevere insolvingthem.

Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemand lookingforentrypointstoitssolution. Theyanalyzegivens,constraints,relationships,andgoals. They makeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthan simply jumping intoasolutionattempt. Theyconsideranalogousproblems,andtry specialcasesand simplerformsoftheoriginalproblem inordertogain insightintoitssolution. Theymonitorandevaluate theirprogressandchange course if necessary.Older studentsmight,depending onthe contextofthe problem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatorto gettheinformationthey need.Mathematically proficientstudentscanexplaincorrespondencesbetween equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships,graphdata,andsearch for regularity ortrends.Youngerstudentsmightrely onusing concrete objects orpicturestohelp conceptualize and solve aproblem. Mathematicallyproficientstudents checktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Does thismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsand identifycorrespondencesbetweendifferentapproaches.

SMP2 – Reason abstractlyand quantitatively.

Mathematically proficientstudentsmakesenseofthequantitiesand theirrelationshipsinproblem situations. Students bring two complementary abilities to bear on problems involving quantitative

relationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolically

andmanipulate the representing symbolsasif they havealife of theirown, withoutnecessarily attending totheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessin

order toprobe intothereferentsfor the symbols involved. Quantitative reasoningentails habits ofcreating

acoherent representationof the problemathand;considering theunits involved;attending to themeaning ofquantities, not justhow tocompute them;and knowing andflexibly using differentproperties of operationsand objects.

SMP3 – Constructviable arguments and critiquethe reasoningofothers.

Mathematically proficientstudentsunderstandandusestated assumptions, definitions, andpreviously establishedresults inconstructing arguments.Theymakeconjecturesandbuild alogicalprogressionof statements toexplore the truthoftheirconjectures. Theyareable toanalyzesituationsby breaking them into cases, and canrecognize and usecounterexamples. Theyjustifytheirconclusions, communicate them toothers,and respond to theargumentsofothers.They reasoninductively aboutdata,making plausible arguments that take intoaccount the context fromwhich the data arose. Mathematicallyproficient students are also abletocompare the effectiveness of two plausible arguments, distinguish correct logicor reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughthey arenot generalizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichan argumentapplies.Studentsatallgradescanlistenor readtheargumentsofothers,decidewhetherthey make sense, and askusefulquestions toclarifyorimprove the arguments.

SMP4 – Modelwith mathematics.

Mathematically proficientstudentscanapply themathematicsthey know tosolveproblemsarising in everyday life,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswriting anaddition equation todescribe asituation.Inmiddlegrades,a studentmightapply proportional reasoning toplan a schooleventoranalyzeaproblem inthecommunity.Byhighschool,astudentmightusegeometry to solveadesignproblem oruseafunction todescribehowonequantityofinterestdependsonanother. Mathematically proficientstudentswho can apply what theyknowarecomfortablemaking assumptions andapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater. They areabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuch toolsasdiagrams,two-waytables,graphs,flowcharts andformulas. Theycananalyzethoserelationships mathematically todraw conclusions.They routinely interprettheirmathematicalresultsin thecontextof thesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnot served its purpose.

SMP5 – Useappropriatetools strategically.

Mathematically proficientstudentsconsider theavailable toolswhensolving amathematicalproblem. Thesetoolsmightinclude pencilandpaper,concretemodels,aruler,aprotractor,acalculator,a spreadsheet, a computeralgebrasystem,a statisticalpackage,ordynamicgeometry software. Proficient studentsaresufficiently familiarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisions aboutwheneachofthese toolsmightbehelpful, recognizing boththeinsight tobegained andtheir limitations.Forexample, mathematically proficienthighschoolstudentsanalyzegraphsoffunctionsand solutionsgeneratedusing agraphing calculator. They detectpossibleerrorsby strategically using estimation andothermathematicalknowledge.Whenmakingmathematicalmodels,they know that technologycanenablethem tovisualizetheresultsofvaryingassumptions,exploreconsequences,and comparepredictionswith data.Mathematically proficientstudentsatvariousgrade levelsareableto identify relevantexternalmathematical resources, suchasdigitalcontentlocatedonawebsite,and use them toposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheir understandingofconcepts.

SMP6 – Attendto precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitionsindiscussionwithothersandin theirown reasoning. Theystate themeaning ofthesymbols theychoose,including using theequalsignconsistently andappropriately.Theyarecarefulabout specifying unitsofmeasure,and labeling axes toclarify thecorrespondence with quantitiesin aproblem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriatefortheproblem context.Intheelementarygrades,studentsgivecarefully formulated explanationsto eachother. By thetimethey reachhighschool they havelearnedtoexamineclaimsand make explicituse ofdefinitions.

SMP7 – Lookforand make useofstructure.

Mathematically proficientstudentslook closely todiscern apatternorstructure.Young students, for example, mightnotice thatthree andseven more is thesame amountasseven andthree more, ortheymay sortacollectionofshapes according tohowmany sidestheshapeshave.Later, studentswillsee7× 8 equalsthewell-remembered7×5+7×3,inpreparationforlearningaboutthedistributiveproperty.In theexpressionx2+9x+14,olderstudentscanseethe14as2×7andthe9as2+7.Theyrecognizethe

significanceofanexisting lineinageometricfigureandcanusethestrategy ofdrawinganauxiliary line forsolving problems.They alsocanstep back foranoverviewandshiftperspective. Theycansee complicatedthings,suchassomealgebraic expressions,assingleobjectsorasbeing composedofseveral objects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethat to realize thatits valuecannotbe morethan5 foranyrealnumbersxandy.

SMP8 – Lookforandexpressregularityin repeatedreasoning.

Mathematically proficientstudentsnoticeifcalculationsare repeated,and look both forgeneralmethods andforshortcuts.Upperelementary studentsmightnoticewhendividing 25by 11thattheyarerepeating thesamecalculationsoverandoveragain,andconcludetheyhavearepeating decimal.Bypaying attention to thecalculation ofslope asthey repeatedly check whetherpointsare ontheline through(1,2) with slope 3, middle school students might abstract the equation (y–2)/(x– 1)=3. Noticing theregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–1)(x2+x+1),and(x–1)(x3+x2+ x+1)mightleadthem tothegeneralformulaforthesum ofageometricseries.Astheywork tosolvea problem,mathematically proficientstudentsmaintain oversightoftheprocess, whileattending tothe details.Theycontinuallyevaluatethereasonableness of their intermediateresults.

New Jersey Student Learning Standardsfor MathematicalContent

In Grade6, instructionaltime should focuson fourcriticalareas: (1)connecting ratio andrateto whole numbermultiplication anddivision andusingconceptsofratio and rateto solve problems;(2)completing understandingofdivision of fractions andextendingthe notion ofnumberto the systemof rational numbers, whichincludesnegative numbers;(3)writing, interpreting, andusingexpressions and

equations;and (4)developingunderstandingofstatisticalthinking.

(1)Studentsusereasoningaboutmultiplication and division to solve ratioandrate problems about quantities. Byviewingequivalentratios andratesasderivingfrom,andextending, pairsofrows(or columns)inthemultiplicationtable,andbyanalyzingsimpledrawingsthatindicatetherelativesizeof quantities,studentsconnecttheirunderstandingofmultiplicationanddivisionwithratiosandrates.Thus

studentsexpandthescopeofproblemsforwhichtheycanusemultiplicationanddivisiontosolveproblems,

andtheyconnectratiosandfractions.Studentssolveawidevarietyofproblemsinvolvingratiosandrates. (2)Studentsusethemeaningof fractions,the meanings ofmultiplication and division, and the

relationshipbetween multiplication and divisionto understandand explain whythe proceduresfor dividingfractions make sense. Students usethese operations tosolve problems. Studentsextendtheir previousunderstandings ofnumberand the orderingofnumbers tothefullsystemof rationalnumbers, whichincludes negative rationalnumbers,and in particularnegative integers.Theyreason abouttheorder and absolute value ofrationalnumbersand aboutthelocation ofpoints inallfour quadrantsofthe

coordinateplane.

(3)Studentsunderstandthe use ofvariablesin mathematicalexpressions.Theywrite expressions and equations thatcorrespondto given situations,evaluateexpressions,and useexpressions andformulasto solve problems. Students understand thatexpressions in different forms can beequivalent, and theyuse the properties ofoperationsto rewriteexpressions in equivalentforms. Students knowthatthe solutions ofanequation are the values ofthe variablesthatmakethe equation true. Studentsusepropertiesof

operationsand theideaofmaintainingtheequalityofboth sides ofan equationtosolve simple one-step equations. Students constructand analyze tables, suchastablesofquantitiesthatare in equivalentratios,

andtheyuseequations (such as3x= y) to describe relationships between quantities.

(4)Buildingon andreinforcingtheirunderstandingofnumber, students beginto develop theirabilityto thinkstatistically. Studentsrecognize thata data distribution maynothave adefinite centerand that different ways to measure centeryield differentvalues.The median measures centerin the sensethatitis roughlythe middle value.The mean measurescenterinthe sense thatitisthe value thateach datapoint would take on if the totalofthe data valueswereredistributedequally, andalso inthe sense thatitisa balancepoint. Studentsrecognize thata measureofvariability(interquartilerangeormean absolute

deviation)canalso be useful forsummarizingdata because two verydifferentsetsofdata can havethe samemean and median yetbedistinguished bytheirvariability. Studentslearnto describeandsummarize numericaldata sets, identifyingclusters, peaks, gaps, and symmetry, consideringthe contextin which the data werecollected.

Studentsin Grade6also build on theirworkwithareain elementaryschoolbyreasoningabout relationships amongshapesto determine area, surfacearea,and volume. Theyfind areasofright triangles,othertriangles, and specialquadrilaterals bydecomposingtheseshapes,rearrangingor removingpieces, andrelatingthe shapesto rectangles.Usingthese methods,studentsdiscuss, develop,

andjustifyformulas forareasoftriangles andparallelograms. Studentsfind areasofpolygons and surface areas ofprisms and pyramids bydecomposingtheminto pieces whose areathey can determine.They

reason aboutrightrectangularprisms withfractionalside lengths to extendformulas forthe volume ofa rightrectangularprismto fractionalsidelengths.Theyprepareforworkon scale drawings and constructionsinGrade7bydrawingpolygons in thecoordinate plane.

Ratios andProportionalRelationships6.RP

Understandratioconcepts and useratio reasoningto solve problems.

1.Understandtheconceptofa ratioand useratiolanguage to describe aratiorelationshipbetweentwo quantities.Forexample, “The ratioofwingsto beaksin the bird house atthe zoowas 2:1,because for every 2 wingsthere was1 beak.” “For every vote candidate Areceived, candidate Creceived nearly three votes.”

2.Understandtheconceptofa unitrate a/b associated with aratioa:bwith b0, and userate language in the contextofaratiorelationship.Forexample, “This recipe has a ratio of3 cups offlourto 4 cupsofsugar, so thereis 3/4 cup offlourfor each cupofsugar.” “We paid $75 for 15 hamburgers, whichis arate of$5 per hamburger.” 1

3.Use ratio andrate reasoningto solve real-world and mathematicalproblems, e.g., byreasoningabout tablesofequivalentratios, tapediagrams, doublenumberlinediagrams, orequations.

a. Maketables ofequivalentratios relatingquantities with whole-numbermeasurements,find missing values inthetables, and plotthepairs ofvalues on the coordinate plane.Use tables to compare ratios.

b.Solve unitrateproblems includingthose involvingunitpricingand constantspeed.For example, ifittook7 hoursto mow4 lawns, then at thatrate, howmany lawns could bemowed in 35 hours? Atwhatrate werelawns being mowed?

c. Find apercentofa quantityasarate per100(e.g., 30%ofa quantitymeans30/100times the quantity);solve problems involvingfindingthe whole, given apartandthe percent.

d.Use ratio reasoning to convertmeasurementunits;manipulate andtransformunitsappropriately when multiplyingordividingquantities.

TheNumberSystem6.NS

Apply and extend previous understandings ofmultiplication and divisionto dividefractions by fractions.

1.Interpret andcompute quotientsoffractions,andsolveword problems involvingdivision of fractionsbyfractions,e.g., byusing visualfraction modelsand equations to representthe problem. For example,createa storycontextfor (2/3)÷(3/4)and useavisual fraction modelto showthe quotient;usethe relationship between multiplication and divisionto explainthat(2/3)÷(3/4)= 8/9 because3/4 of8/9is2/3.(In general,(a/b)÷(c/d)= ad/bc.)How muchchocolatewilleach person

1Expectations for unitrates in this grade are limited tonon-complexfractions.

getif3peopleshare1/2lbofchocolateequally? Howmany3/4-cupservings arein2/3 ofacup of yogurt? How wide is a rectangular strip oflandwith length 3/4 miand area1/2squaremi?

Computefluentlywithmulti-digitnumbersandfind commonfactorsand multiples.

2.Fluentlydivide multi-digitnumbers usingthe standardalgorithm.

3.Fluentlyadd,subtract, multiply, and divide multi-digitdecimals usingthe standardalgorithmfor each operation.

4.Find the greatestcommon factorof two whole numbers lessthan orequalto 100 and theleast commonmultiple oftwo whole numbers lessthanorequalto 12.Usethe distributive propertyto expressasumof two wholenumbers 1–100 witha common factorasamultiple ofa sumof two whole numbers withno common factor.For example,express 36 + 8 as 4(9 + 2).

Apply and extend previous understandings ofnumbers to the systemofrationalnumbers.

5.Understandthatpositive and negative numbers areused togetherto describequantitieshaving oppositedirections orvalues (e.g., temperature above/belowzero,elevationabove/belowsealevel, credits/debits,positive/negative electric charge);use positive and negative numbers torepresent quantities inreal-world contexts, explainingthe meaningof0 in eachsituation.

6.Understanda rationalnumberasa pointonthe numberline. Extend numberline diagrams and coordinateaxesfamiliarfromprevious gradesto representpoints ontheline and inthe plane with negative numbercoordinates.

a. Recognize opposite signs ofnumbers as indicatinglocations onopposite sides of0 on the number line;recognizethatthe opposite oftheopposite ofanumber isthenumberitself,e.g.,– (–3)= 3, and that0 isits own opposite.

b.Understandsigns ofnumbers in ordered pairs asindicatinglocationsinquadrantsof the coordinate plane;recognize thatwhentwo ordered pairs differonlybysigns, thelocationsofthe points arerelated byreflectionsacrossone orboth axes.

c. Findand positionintegers and otherrationalnumbers on a horizontalorverticalnumber line diagram; find andpositionpairs of integersand otherrationalnumbers ona coordinateplane.

7.Understandorderingand absolute value ofrationalnumbers.

a. Interpretstatements ofinequalityasstatements about the relative positionoftwonumbers on a number line diagram.For example,interpret–3 > –7 as a statementthat–3islocatedtothe rightof–7 ona number line oriented from leftto right.

b.Write,interpret, and explain statementsoforder forrationalnumbersin real-world contexts.For example, write–3 oC> –7 oCto express the factthat–3 oCis warmer than –7 oC.

c. Understandtheabsolute value ofa rationalnumberas its distance from0 on the number line; interpretabsolute value asmagnitudefora positive ornegative quantityin areal-worldsituation. For example,foran accountbalanceof–30 dollars, write |–30| = 30to describethe sizeofthe debtin dollars.

d.Distinguishcomparisons ofabsolute valuefromstatementsaboutorder.For example, recognize thatan accountbalance less than–30 dollarsrepresents a debtgreater than 30 dollars.

8.Solve real-world andmathematicalproblems by graphingpointsin all fourquadrants ofthe coordinate plane. Include useofcoordinatesand absolute value tofind distancesbetween points withthesame firstcoordinate or the same second coordinate.

Expressions and Equations6.EE

Apply and extend previous understandings ofarithmeticto algebraic expressions.

1.Writeand evaluate numericalexpressionsinvolvingwhole-numberexponents.

2.Write,read, and evaluate expressionsin which lettersstandfornumbers.

a. Writeexpressions thatrecord operations with numbersand with letters standingfornumbers.

For example,expressthe calculation “Subtractyfrom5” as 5–y.

b. Identifyparts ofan expression usingmathematicalterms (sum, term, product,factor, quotient, coefficient);view one or more parts ofan expression as asingle entity.For example, describe the expression 2(8+ 7)asa productof twofactors;view (8+ 7)as botha singleentityand a sum oftwo terms.

c.Evaluateexpressions atspecific valuesoftheirvariables. Include expressionsthatarisefrom formulasusedin real-world problems. Performarithmeticoperations, includingthoseinvolving whole-numberexponents, in theconventionalorder when there are no parentheses tospecifya particularorder(OrderofOperations).For example, use theformulasV= s3and A= 6 s2to

findthevolume andsurfacearea ofa cube withsidesof lengths= 1/2.

3.Applythepropertiesofoperationsto generate equivalentexpressions.For example, apply the distributiveproperty to theexpression 3 (2 + x)to producethe equivalentexpression 6 +3x;apply the distributiveproperty tothe expression 24x + 18y toproduce the equivalentexpression 6 (4x+

3y);apply propertiesofoperationsto y+ y +y toproducethe equivalentexpression 3y.

4.Identifywhen two expressions are equivalent(i.e., when the twoexpressionsname thesame number regardlessofwhich value issubstituted intothem).Forexample, the expressionsy+ y + y and 3y areequivalentbecausetheyname the samenumberregardlessofwhich numberystands for.