Arkansas Mathematics Standards

Grades 6-8

2016

Introduction to the Grades 6-8 Arkansas MathematicsStandards

When charged with the task of revising the previous mathematics standards, a group ofqualifiedindividualsfromacrossthestatecametogethertocraftstandardsthatwerespecificfortheschoolsandstudentsofArkansas.Theresultofthiswork,theArkansasMathematicsStandards,iscontainedinthisdocument.Thesestandardsreflectwhateducatorsacrossourstateknowtobebestforourstudents.

These standards retain the same structure as the previous standards in terms of organization.Thestandardsareorganizedbydomains,clusters,andstandards.Domainsrepresentthebigideasthataretobestudiedateachgradelevelandsometimesacrossgradebands.Thesebigideassupporteducatorsindeterminingtheproperamountoffocusandinstructionaltimetobegiventoeachofthesetopics.

Clustersrepresentcollectionsofstandardsthataregroupedtogethertohelpeducatorsunderstandthebuilding blocks of rich and meaningful instructional units. These units help students makeconnectionswithinclustersandavoidseeingmathematicsasadiscreetlistofskillsthattheymustmaster.Standardsrepresentthefoundationalbuildingblocksofmathinstruction.Thestandardsoutlinedinthisdocumentworktogethertoensurethatstudentsarecollegeandcareerreadyandontrackforsuccess.

Thereareadditionalsimilaritiessharedbythesenewstandardsandthepreviousstandards.Themainsimilarityisthestructureofthenomenclature.Theonlychangethatwasmadetothenamingsystemwas intended to reflect that these standards belong to Arkansas. However, educators may stillsearchforopeneducationresourcesbyusingthelastpartofthelabel,whichwilllinktotheresourcesforthepreviousstandards.Newstandardscanbefoundattheendofeachclusterinwhichanewstandardwasdeemednecessary.

Anothersimilaritytothepreviousstandardsistheuseofthesymbols(+)and(*)todistinguishcertainstandardsfromothers.Theplus(+)symbolisusedtodesignatestandardsthataretypicallybeyondthescopeofanAlgebraIIcourse.However,someoftheplus(+)standardsarenowincludedincoursesthatare not considered to be beyond Algebra II. Standards denoted with the asterisk (*) symbolrepresentthemodelingcomponentofthestandards.Thesestandardsshouldbepresentedinamodelingcontextwhere students are required to engage in the modeling process that is outlined in the StandardsforMathematicalPractice.

The revision committee opted to include some new elements in the Arkansas MathematicsStandardsthatrepresentanattemptatgreaterclarityandmoreconsistentimplementationacrossthestate.Manyof the revisions are a rewording of the original Common Core State Standards. The purpose oftherewording is often to help educators better understand the areas of emphasis and focus withintheexisting standard. Likewise, many of the standards are separated into a bulleted list of content.Thisdoes not mean that teachers should treat this content as a checklist of items that they must teachoneat a time. The content was bulleted out so that teachers can better understand all that is includedinsome of the broader standards.

Many of the examples that were included in the original standardswereeitherchangedforclarityorseparatedfromthebodyoftheactualstandard.Thecommitteewanted educators to understand that the examples included in the body of the standards document in nowayreflect all of the possible examples. Likewise, these examples do not mandate curriculum orproblemtypes.Localdistrictsarefreetoselectthecurriculumandinstructionalmethodstheythinkbestfortheirstudents.

Insomeinstances,notesofclarificationwereadded.Thesenoteswereintendedtoclarify,forteachers,whattheexpectationsareforthestudent.Likewise,thesenotesprovideinstructionalguidanceaswellas limitations so that teachers can better understand the scope of the standard. This will helptheeducatorsindeterminingwhatisdevelopmentallyappropriateforstudentswhentheyareworkingwithcertainstandards.

Finally,theArkansasMathematicsStandardswillbecomealivingdocument.ThestaffoftheArkansasDepartmentofEducationhopesthatthisdocumentportraysthehardworkoftheArkansaseducatorswho took part in the revision process and that it represents an improvement to the previous setofstandards.Asthesestandardsareimplementedacrossschoolsinthestate,theArkansasDepartmentofEducation welcomes further suggestions related to notes of clarification, examples,professionaldevelopment needs, and future revisions of thestandards.

Abbreviations:

Ratios and Proportional Relationships – RP

The Number System – NS

Expressions and Equations – EE

Geometry – G

Statistics and Probability – SP

Functions – F

Ratios andProportionalRelationships / Understand ratio concepts and use ratio reasoning to solve problems
AR.Math.Content.6.RP.A.1 / Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities
Forexample,"Theratioofwingstobeaksinthebirdhouseatthezoowas2:1,becauseforevery2wingstherewas1beak.""ForeveryvotecandidateAreceived,candidateCreceivednearlythreevotes."
AR.Math.Content.6.RP.A.2 / Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb≠0,anduseratelanguageinthecontextofaratiorelationship
Forexample,"Thisrecipehasaratioof3cupsofflourto4cupsofsugar,sothereis3/4cupofflourforeachcupofsugar.""Wepaid$75for15hamburgers,whichisarateof$5perhamburger."
Note:Expectationsforunitratesinthisgradearelimitedtonon-complexfractions.
AR.Math.Content.6.RP.A.3 / Useratioandratereasoningtosolvereal-worldandmathematicalproblems(e.g.,byreasoningabouttablesofequivalent ratios, tape diagrams, double number line diagrams, or equations):
  • Useandcreatetablestocompareequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane
  • Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed
Forexample:Ifittook7hourstomow4lawns,thenatthatrate,howmanylawnscouldbemowedin35hours?Atwhatrate were lawns beingmowed?
  • Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity)
  • Solveproblemsinvolvingfindingthewhole,givenapartandthepercent
  • Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhen multiplying or dividingquantities
Example:How many centimeters are in 7 feet, given that 1 inch ≈ 2.54 cm?

Note: Conversion factors will be given. Conversions can occur both between and across the metric and English system. Estimates are not expected.
The NumberSystem / Applyandextendpreviousunderstandingsofmultiplicationanddivisiontodividefractionsbyfractions
AR.Math.Content.6.NS.A.1 /
  • Interpret and compute quotients offractions
  • Solvewordproblemsinvolvingdivisionoffractionsbyfractions(e.g.,byusingvariousstrategies,includingbutnotlimitedto,visualfractionmodelsandequationstorepresenttheproblem)
Forexample: Createastorycontextfor(2/3)÷(3/4)anduseavisualfractionmodeltoshowthequotient;usetherelationshipbetweenmultiplicationanddivisiontoexplainthat(2/3)÷(3/4)=8/9because3/4of8/9is2/3.How many 3/4-cup servings are in 2/3 of a cup of yogurt?
Note: In general, (a/b) ÷ (c/d) = ad/bc.
The NumberSystem / Computefluentlywithmulti-digitnumbersandfindcommonfactorsandmultiples
AR.Math.Content.6.NS.B.2 / Usecomputationalfluencytodividemulti-digitnumbersusingastandardalgorithm
Note:Astandardalgorithmcanbeviewedas,butshouldnotbelimitedto,thetraditionalrecordingsystem. A standard algorithm denotes any valid base-tenstrategy.
AR.Math.Content.6.NS.B.3 / Usecomputationalfluencytoadd,subtract,multiply,anddividemulti-digitdecimalsandfractionsusingastandard algorithm for each operation
Note:Astandardalgorithmcanbeviewedas,butshouldnotbelimitedto,thetraditionalrecordingsystem. A standard algorithm denotes any valid base-tenstrategy.
AR.Math.Content.6.NS.B.4 /
  • Findthegreatestcommonfactoroftwowholenumberslessthanorequalto100usingprimefactorization as well as othermethods
  • Findtheleastcommonmultipleoftwowholenumberslessthanorequalto12usingprimefactorization as well as othermethods
  • Usethedistributivepropertytoexpressasumoftwowholenumbers1-100withacommonfactorasamultipleofasumoftwowholenumberswithnocommonfactor
Forexample,express36+8as4(9+2).
The NumberSystem / Applyandextendpreviousunderstandingsofnumberstothesystemofrationalnumbers
AR.Math.Content.6.NS.C.5 / Understandthatpositiveandnegativenumbersareusedtogethertodescribequantitieshavingoppositedirectionsorvalues,explainingthemeaningof0(e.g.,temperatureabove/belowzero,elevationabove/below sea level, credits/debits, positive/negative electric charge)
AR.Math.Content.6.NS.C.6 / Understandarationalnumberasapointonthenumberline
Extendnumberlinediagramsandcoordinateaxesfamiliarfrompreviousgradestorepresentpointsonthelineandintheplanewithnegativenumbercoordinates:
  • Recognizeoppositesignsofnumbersasindicatinglocationsonoppositesidesof0onthenumberline
  • Recognizethattheoppositeoftheoppositeofanumberisthenumberitself(e.g.,-(-3)=3,andthat0is its ownopposite)
  • Understandsignsofnumbersinorderedpairsasindicatinglocationsinquadrantsofthecoordinateplane
  • Recognizethatwhentwoorderedpairsdifferonlybysigns,thelocationsofthepointsarerelatedbyreflections across one or bothaxes
  • Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram
  • Findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane

AR.Math.Content.6.NS.C.7 / Understand ordering and absolute value of rational numbers:
  • Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram
Forexample,interpret-3-7asastatementthat-3islocatedtotherightof-7on a number line oriented from left toright.
  • Write,interpret,andexplainstatementsoforderforrationalnumbersinreal-worldcontexts
Forexample,write-3o C-7o Ctoexpressthefactthat-3o Ciswarmerthan-7o C.
  • Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline
  • Interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal-worldsituation
Forexample,foranaccountbalanceof-30dollars,write|-30|=30todescribethesizeofthedebtindollars.
  • Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder
Forexample,recognizethatanaccountbalancelessthan-30dollarsrepresentsadebtgreaterthan30dollars.
AR.Math.Content.6.NS.C.8 /
  • Solvereal-worldandmathematicalproblemsbygraphingpointsinallfourquadrantsofthecoordinateplane
  • Usecoordinatesandabsolutevaluetofinddistancesbetweenpointswiththesamefirstcoordinate or the same secondcoordinate

ExpressionsandEquations / Applyandextendpreviousunderstandingsofarithmetictoalgebraicexpressions
AR.Math.Content.6.EE.A.1 / Write and evaluate numerical expressions involving whole-number exponents
AR.Math.Content.6.EE.A.2 / Write,read,andevaluateexpressionsinwhichletters(variables)standfornumbers:
  • WriteexpressionsthatrecordoperationswithnumbersandwithlettersstandingfornumbersForexample,expressthecalculation‘subtractyfrom5’or‘ylessthan5’as5-y.
  • Identifypartsofanexpressionusingmathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofanexpressionasasingleentity
Forexample,describetheexpression2(8+7)asaproductoftwofactors;view(8+7)asbothasingleentityandasumoftwoterms.
  • Evaluateexpressionsatspecificvaluesoftheirvariables
  • Includeexpressionsthatarisefromformulasusedinreal-worldproblems
  • Performarithmeticoperations,includingthoseinvolvingwhole-numberexponents,intheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations)
Forexample,usetheformulasinvolvedinmeasurementsuchasV=s3andA=6s2tofindthevolumeandsurfaceareaofacubewithsidesoflengths=1/2.
AR.Math.Content.6.EE.A.3 / Apply the properties of operations to generate equivalent expressions
Forexample: Applythedistributivepropertytotheexpression3(2+x)toproducetheequivalentexpression6+3x;applythedistributivepropertytotheexpression24x+18ytoproducetheequivalentexpression 6(4x+3y);applypropertiesofoperationstoy+y+ytoproducetheequivalentexpression3y.
Note: Includes but not limited to the distributive property.
AR.Math.Content.6.EE.A.4 / Identifywhentwoexpressionsareequivalent(i.e.,whenthetwoexpressionsnamethesamenumberregardless of which value is substituted into them)
Forexample: Theexpressionsy+y+yand3yareequivalentbecausetheynamethesamenumberregardlessofwhich number y stands for.
Expressions andEquations / Reasonaboutandsolveone-variableequationsandinequalities
AR.Math.Content.6.EE.B.5 / Understandsolvinganequationorinequalityasaprocessofansweringaquestion:
  • Usingsubstitution,whichvaluesfromaspecifiedset,ifany,maketheequationorinequalitytrue?

AR.Math.Content.6.EE.B.6 /
  • Usevariablestorepresentnumbersandwriteexpressionswhensolvingareal-worldormathematicalproblem
  • Understandthatavariablecanrepresentanunknownnumberoranynumberinaspecifiedset

AR.Math.Content.6.EE.B.7 / Solvereal-worldandmathematicalproblemsbywritingandsolvingequationsoftheformx+p=qandpx=q
forcasesinwhichp,qandxareallnonnegativerationalnumbers
AR.Math.Content.6.EE.B.8 / For real world or mathematical problems:
  • Writeaninequalityoftheformxc,x≥c,xc,orx≤ctorepresentaconstraintorcondition
  • Recognizethatinequalitiesoftheformxcorxchaveinfinitelymanysolutions
  • Represent solutions of such inequalities on number linediagrams

Expressions andEquations / Representandanalyzequantitativerelationshipsbetweendependentandindependentvariables
AR.Math.Content.6.EE.C.9 / Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother:
  • Writeanequationtoexpressonequantity,thoughtofasthedependentvariable,intermsoftheotherquantity, thought of as the independentvariable
  • Analyzetherelationshipbetweenthedependentandindependentvariablesusinggraphsandtables,and relate these to theequation
Forexample: Inaprobleminvolvingmotionatconstantspeed,listandgraphorderedpairsofdistancesandtimes,andwritetheequationd=65ttorepresenttherelationshipbetweendistanceandtime.
Note:Theindependentvariableisthevariablethatcanbechanged;thedependentvariableisthevariablethatis affected by the change in the independent variable.
Geometry / Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume
AR.Math.Content.6.G.A.1 /
  • Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes
  • Applythesetechniquesinthecontextofsolvingreal-world and mathematicalproblems
Note:Trapezoidswillbedefinedtobeaquadrilateralwithatleastonepairofoppositesidesparallel,thereforeall parallelograms are trapezoids.
AR.Math.Content.6.G.A.2 /
  • Findthevolumeofarightrectangularprismincludingwholenumberandfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism
  • ApplytheformulasV=lwhandV=Bhtofindvolumesofrightrectangularprismsincludingfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems

AR.Math.Content.6.G.A.3 / Applythefollowingtechniquesinthecontextofsolvingreal-worldandmathematicalproblems:
  • Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices
  • Usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate

AR.Math.Content.6.G.A.4 / Applythefollowingtechniquesinthecontextofsolvingreal-worldandmathematicalproblems:
  • Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles
  • Use the nets to find the surface area of thesefigures

StatisticsandProbability / Developunderstandingofstatisticalvariability
AR.Math.Content.6.SP.A.1 / Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedatarelatedtothequestionandaccountsforitintheanswers
Forexample,‘HowoldamI?’isnotastatisticalquestion,but‘Howoldarethestudentsinmyschool?’isastatisticalquestionbecauseoneanticipatesvariabilityinstudents'ages.
Note:Statisticsisalsothenameforthescienceofcollecting,analyzingandinterpretingdata.Dataarethenumbers produced in response to a statistical question and are frequently collected from surveys orothersources (i.e.documents).
AR.Math.Content.6.SP.A.2 / Determine center, spread, and overall shape from a set of data
AR.Math.Content.6.SP.A.3 / Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber(mean,median,mode),whileameasureofvariation(interquartilerange,meanabsolutedeviation) describes how its values vary with a single number
Example:Ifthemeanheightofthestudentsintheclassis48”arethereanystudentsintheclasstallerthan48”?
StatisticsandProbability / Summarizeanddescribedistributions
AR.Math.Content.6.SP.B.4 / Displaynumericaldatainplotsonanumberline,includingdotplots,histograms,andboxplots
AR.Math.Content.6.SP.B.5 / Summarize numerical data sets in relation to their context, such as by:
  • Reporting the number ofobservations
  • Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement
  • Calculatequantitativemeasuresofcenter(includingbutnotlimitedtomedianandmean)andvariability(includingbutnotlimitedtointerquartilerangeandmeanabsolutedeviation)
  • Usethecalculationstodescribeanyoverallpatternandanystrikingdeviations(outliers)fromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered
Note: Instructional focus should be on summarizing and describing data distributions.
  • Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.Forexample,demonstrateinthecasewherethereare outliersinthedatamedianwouldbeabettermeasureofcenterthanthemean.

Ratios andProportionalRelationships / Analyzeproportionalrelationshipsandusethemtosolvereal-worldandmathematicalproblems
AR.Math.Content.7.RP.A.1 / Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areas,andotherquantitiesmeasured in like or different units
Forexample:Ifapersonwalks1/2mileineach1/4hour,computetheunitrateasthecomplexfraction1/2/1/4miles per hour, equivalently 2 miles per hour.
AR.Math.Content.7.RP.A.2 / Recognize and represent proportional relationships between quantities:
  • Decidewhethertwoquantitiesareinaproportionalrelationship(e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin)
  • Identifyunitrate(alsoknownastheconstantofproportionality)intables,graphs,equations,diagrams, and verbal descriptions of proportionalrelationships
  • Representproportionalrelationshipsbyequations(e.g.,iftotalcosttisproportionaltothenumbernofitemspurchasedataconstantpricep,therelationshipbetweenthetotalcostandthenumberofitems can be expressed as t =pn)
  • Explainwhatapoint(x,y)onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentiontothepoints(0,0)and(1,r)whereristheunitrate
Note: Unit rate connects to slope concept in 8thgrade.
AR.Math.Content.7.RP.A.3 / Use proportional relationships to solve multi-step ratio and percent problems
Note:Examplesincludebutarenotlimitedtosimpleinterest,tax,markupsandmarkdowns,gratuitiesandcommissions, fees, percent increase and decrease.
The NumberSystem / Apply and extend previous understandings of operations with fractions
AR.Math.Content.7.NS.A.1 / Applyandextendpreviousunderstandingsofadditionandsubtractiontoaddandsubtractrationalnumbers
Representadditionandsubtractiononahorizontalorverticalnumberlinediagram:
  • Describesituationsinwhichoppositequantitiescombinetomake0andshowthatanumberanditsoppositehaveasumof0(additiveinverses)(e.g.,Ahydrogenatomhas0chargebecauseitstwoconstituents are oppositelycharged.)
  • Understandp+qasanumberwherepisthestartingpointandqrepresentsadistancefrompinthepositiveornegativedirectiondependingonwhetherqispositiveornegative
  • Interpretsumsofrationalnumbersbydescribingreal-worldcontexts(e.g.,3+2meansbeginningat3,move2unitstotherightandendatthesumof5;3+(-2)meansbeginningat3,move2unitstotheleftandendatthesumof1;70+(-30)=40couldmeanafterearning$70,$30wasspentonanewvideogame, leaving a balance of$40)
  • Understandsubtractionofrationalnumbersasaddingtheadditiveinverse,p-q=p+(-q)
  • Showthatthedistancebetweentworationalnumbersonthenumberlineistheabsolutevalueoftheirdifferenceandapplythisprincipleinreal-worldcontexts(e.g.,thedistancebetween-5and6is11. -5 and 6 are 11 units apart on the number line)
  • Fluentlyaddandsubtractrationalnumbersbyapplyingpropertiesofoperationsasstrategies

AR.Math.Content.7.NS.A.2 / Applyandextendpreviousunderstandingsofmultiplicationanddivisionandoffractionstomultiplyanddividerationalnumbers:
  • Understandthatmultiplicationisextendedfromfractionstoallrationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythedistributiveproperty,andthe rules for multiplying signednumbers
  • Interpret products of rational numbers by describing real-worldcontexts
  • Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zerodivisor)isarationalnumber(e.g.,ifpandqareintegers,then-(p/q)=(-p)/q=p/(-q))
  • Interpret quotients of rational numbers by describing real-worldcontexts
  • Fluentlymultiplyanddividerationalnumbersbyapplyingpropertiesofoperationsasstrategies
  • Convert a fraction to a decimal using longdivision
  • Knowthatthedecimalformofafractionterminatesin0soreventuallyrepeats

AR.Math.Content.7.NS.A.3 / Solvereal-worldandmathematicalproblemsinvolvingthefouroperationswithrationalnumbers,includingbut not limited to complex fractions
Expressions andEquations / Use properties of operations to generate equivalent expressions
AR.Math.Content.7.EE.A.1 / Applypropertiesofoperationsasstrategiestoadd,subtract,expand,andfactorlinearexpressionswithrationalcoefficients
AR.Math.Content.7.EE.A.2 / Understandhowthequantitiesinaproblemarerelatedbyrewritinganexpressionindifferentforms
Forexample:a+0.05a=1.05ameansthat‘increaseby5%’isthesameas‘multiplyby1.05’ortheperimeterof a square with side length s can be written ass+s+s+sor 4s.
Expressions andEquations / Solvereal-lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations
AR.Math.Content.7.EE.B.3 / Solvemulti-step,real-life,andmathematicalproblemsposedwithpositiveandnegativerationalnumbersinany form using tools strategically:
  • Apply properties of operations to calculate with numbers in any form (e.g.,-(1/4)(n-4))
  • Convertbetweenformsasappropriate(e.g.,ifawomanmaking$25anhourgetsa10%raise,shewillmakeanadditional1/10ofhersalaryanhour,or$2.50,foranewsalaryof$27.50)
  • Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategies(e.g.,ifyouwanttoplaceatowelbar93/4incheslonginthecenterofadoorthatis271/2incheswide,youwillneedtoplacethebarabout9inchesfromeachedge;thisestimatecanbeusedasacheckon the exactcomputation)

AR.Math.Content.7.EE.B.4 /
  • Usevariablestorepresentquantitiesinareal-worldormathematicalproblem
  • Constructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities
  • Solvewordproblemsleadingtoequationsoftheseformspx+q=randp(x+q)=r,wherep,q,andrare specific rational numbers. Solve equations of these forms fluently
  • Writeanalgebraicsolutionidentifyingthesequenceoftheoperationsusedtomirrorthearithmeticsolution(e.g.,Theperimeterofarectangleis54cm.Itslengthis6cm.Whatisitswidth?Subtract2*6 from 54 and divide by 2; (2*6) + 2w =54)
  • Solvewordproblemsleadingtoinequalitiesoftheformpx+qrorpx+qr,wherep,q,andrarespecific rationalnumbers
  • Graphthesolutionsetoftheinequalityandinterpretitinthecontextoftheproblem(e.g.,Asasalesperson,youarepaid$50perweekplus$3persale.Thisweekyouwantyourpaytobeatleast$100.Writeaninequalityforthenumberofsalesyouneedtomake,anddescribethesolutions.)

Geometry / Drawconstruct,anddescribegeometricalfiguresanddescribetherelationshipsbetweenthem
AR.Math.Content.7.G.A.1 / Solveproblemsinvolvingscaledrawingsofgeometricfigures,includingcomputingactuallengthsandareasfromascaledrawingandreproducingascaledrawingatadifferentscale
Note: This concept ties into ratio and proportion.
AR.Math.Content.7.G.A.2 / Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions:
  • Giventhreemeasuresofanglesorsidesofatriangle,noticewhentheconditionsdetermineauniquetriangle, more than one triangle, or notriangle
  • Differentiate between regular and irregularpolygons

AR.Math.Content.7.G.A.3 / Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofright rectangular prisms and right rectangular pyramids
Geometry / Solvereal-lifeandmathematicalproblemsinvolvinganglemeasure,area,surfaceareaandvolume
AR.Math.Content.7.G.B.4 /
  • Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems.
  • Giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle

AR.Math.Content.7.G.B.5 / Usefactsaboutsupplementary,complementary,vertical,andadjacentanglesinamulti-stepproblemtowriteand solve simple equations for an unknown angle in a figure
AR.Math.Content.7.G.B.6 / Solvereal-worldandmathematicalproblemsinvolvingareaoftwo-dimensionalobjectsandvolumeandsurfaceareaofthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms
Statistics andProbability / Use random sampling to draw inferences about a population
AR.Math.Content.7.SP.A.1 / Understandthat:
  • Statisticscanbeusedtogaininformationaboutapopulationbyexaminingasampleofthepopulation
  • Generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation
  • Random sampling tends to produce representative samples and support validinferences

AR.Math.Content.7.SP.A.2 /
  • Usedatafromarandomsampletodrawinferencesaboutapopulationwithaspecificcharacteristic
  • Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions
Forexample:Estimatethemeanwordlengthinabookbyrandomlysamplingwordsfromthebook,orpredictthewinnerofaschoolelectionbasedonrandomlysampledsurveydata.Gaugehowfarofftheestimateorprediction mightbe.
Statistics andProbability / Draw informal comparative inferences about two populations
AR.Math.Content.7.SP.B.3 / Drawconclusionsaboutthedegreeofvisualoverlapoftwonumericaldatadistributionswithsimilarvariabilitysuchasinterquartilerangeormeanabsolutedeviation,expressingthedifferencebetweenthecentersasamultiple of a measure of variability such as mean, median, or mode
Forexample:Themeanheightofplayersonthebasketballteamis10cmgreaterthanthemeanheightofplayersonthesoccerteam,abouttwicethevariabilityoneitherteam;onadotplot,theseparationbetweenthe two distributions of heights is noticeable.
AR.Math.Content.7.SP.B.4 / Drawinformalcomparativeinferencesabouttwopopulationsusingmeasuresofcenterandmeasuresofvariability for numerical data from random samples
Forexample:Decidewhetherthewordsinachapterofaseventh-gradesciencebookaregenerallylongerthanthe words in a chapter of a fourth-grade science book.
Statistics andProbability / Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels
AR.Math.Content.7.SP.C.5 /
  • Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthelikelihoodoftheeventoccurring
  • Aprobabilitynear0indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynorlikely,andaprobabilitynear1indicatesalikelyevent

AR.Math.Content.7.SP.C.6 /
  • Collectdatatoapproximatetheprobabilityofachanceevent
  • Observe its long-run relativefrequency
  • Predict the approximate relative frequency given theprobability
Forexample:Whenrollinganumbercube600times,predictthata3or6wouldberolledroughly200times,but probably not exactly 200 times.
Note:Emphasisshouldbegiventotherelationshipbetweenexperimentalandtheoreticalprobability.
AR.Math.Content.7.SP.C.7 / Developaprobabilitymodelanduseittofindprobabilitiesofevents
Compareprobabilitiesfromamodeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy:
  • Developauniformprobabilitymodel,assigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilitiesofevents(e.g.,Ifastudentisselectedatrandomfromaclassof6girlsand4boys,theprobabilitythatJanewillbeselectedis.10andtheprobabilitythatagirlwillbeselectedis.60.)
  • Developaprobabilitymodel,whichmaynotbeuniform,byobservingfrequenciesindatageneratedfromachanceprocess(e.g.,Findtheapproximateprobabilitythataspinningpennywilllandheadsuporthatatossedpapercupwilllandopen-enddown.Dotheoutcomesforthespinningpennyappeartobe equally likely based on the observedfrequencies?)

AR.Math.Content.7.SP.C.8 / Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation:
  • Understandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs
  • Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams
  • Identifytheoutcomesinthesamplespacewhichcomposetheevent
Generatefrequenciesforcompoundeventsusingasimulation(e.g.,Whatisthefrequencyofpullingared card from a deck of cards and rolling a 5 on a die?)
The NumberSystem / Knowthattherearenumbersthatarenotrational,andapproximatethembyrationalnumbers
AR.Math.Content.8.NS.A.1 / Know that numbers that are not rational are called irrational:
  • Understand that every number has a decimalexpansion
For example:2=2.00…
  • Write a fraction a/b as a repeatingdecimal
  • Write a repeating decimal as afraction

AR.Math.Content.8.NS.A.2 / Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2)
Forexample:Bytruncatingthedecimalexpansionof√2,showthat√2isbetween1and2,thenbetween1.4and1.5,andexplainhowtocontinueontogetbetterapproximations.
Expressions andEquations / Work with radicals and integer exponents
AR.Math.Content.8.EE.A.1 / Knowandapplythepropertiesofintegerexponentstogenerateequivalentnumericalexpressionsusingproduct, quotient, power to a power, or expanded form
AR.Math.Content.8.EE.A.2 / Use square root and cube root symbols to represent solutions to equations:
  • Usesquarerootsymbolstorepresentsolutionstoequationsoftheformx2=p,wherepisapositiverational number
Evaluate square roots of small perfectsquares.
  • Usecuberootsymbolstorepresentsolutionstoequationsoftheformx3=p,wherepisarationalnumber.
Evaluatesquarerootsandcuberootsofsmallperfectcubes
AR.Math.Content.8.EE.A.3 / Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother
Forexample:EstimatethepopulationoftheUnitedStatesas3times108andthepopulationoftheworldas7times109,anddeterminethattheworldpopulationismorethan20timeslarger.
AR.Math.Content.8.EE.A.4 /
  • Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothstandard form and scientific notation areused
  • Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities(e.g.,usemillimetersperyearforseafloorspreading)
  • Interpret scientific notation that has been generated bytechnology

Expressions andEquations / Understandtheconnectionsbetweenproportionalrelationships,lines,andlinearequations
AR.Math.Content.8.EE.B.5 /
  • Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.
  • Comparetwodifferentproportionalrelationshipsrepresentedindifferentways(graphs,tables,equations)
Forexample:Compareadistance-timegraphtoadistance-timeequationtodeterminewhichoftwomovingobjects has greater speed.
AR.Math.Content.8.EE.B.6 /
  • Usinganon-verticalornon-horizontalline,showwhytheslopemisthesamebetweenanytwo distinct points by creating similar triangles
  • Write the equation y=mx + b for a line through the origin
  • Be able to write the equation y = mx + b for a line intercepting the vertical axis at b

Expressions andEquations / Analyzeandsolvelinearequationsandpairsofsimultaneouslinearequations
AR.Math.Content.8.EE.C.7 / Solve linear equations in one variable:
  • Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions
Note: Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbare differentnumbers)
  • Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms
Note: Students should solve equations with variables on both sides.
AR.Math.Content.8.EE.C.8 / Analyze and solve pairs of simultaneous linear equations:
  • Findsolutionstoasystemoftwolinearequationsintwovariablessotheycorrespondtopointsofintersection of theirgraphs
  • Solvesystemsofequationsintwovariablesalgebraicallyusingsimplesubstitutionandbyinspection(e.g.,3x+2y=5and3x+2y=6havenosolutionbecause3x+2ycannotsimultaneouslybe5and6)
  • Solvereal-worldmathematicalproblemsbyutilizingandcreatingtwolinearequationsintwovariables.
Forexample:Givencoordinatesfortwopairsofpoints,determinewhetherthelinethroughthefirstpairofpointsintersects the line through the second pair.
Functions / Define, evaluate, and comparefunctions
AR.Math.Content.8.F.A.1 /
  • Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput
  • Thegraphofafunctionistheset of ordered pairs consisting of an input and the corresponding output
Note: An informal discussion of function notation is needed; however, student assessment is not required.
AR.Math.Content.8.F.A.2 / Compareproperties(e.g.,y-intercept/initialvalue,slope/rateofchange)oftwofunctionseachrepresentedinadifferentway(e.g.,algebraically,graphically,numericallyintables,orbyverbaldescriptions)
Forexample:Givenalinearfunctionrepresentedbyatableofvaluesandalinearfunctionrepresentedbyanalgebraicexpression,determinewhichfunctionhasthegreaterrateofchange.
AR.Math.Content.8.F.A.3 / Identifytheuniquecharacteristicsoffunctions(e.g.,linear,quadratic,andexponential)bycomparingtheirgraphs, equations, and input/output tables
Functions / Use functions to model relationships between quantities
AR.Math.Content.8.F.B.4 / Construct a function to model a linear relationship between two quantities:
  • Determinetherateofchangeandinitialvalueofthefunctionfrom:
  • a verbal description of arelationship
  • two (x, y)values
  • atable
  • agraph
  • Interprettherateofchangeandinitialvalueofalinearfunctionintermsofthesituationitmodels, and in terms of its graph or a table ofvalues

AR.Math.Content.8.F.B.5 /
  • Describethefunctionalrelationshipbetweentwoquantitiesbyanalyzingagraph(e.g.,wherethefunction is increasing or decreasing, linear ornonlinear)
  • Sketchagraphthatexhibitsthefeaturesofafunctionthathasbeendescribedverbally

Geometry / Understandcongruenceandsimilarityusingphysicalmodels,transparencies,orgeometrysoftware
AR.Math.Content.8.G.A.1 / Verify experimentally the properties of rotations, reflections, and translations:
  • Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength
  • Angles are taken to angles of the samemeasure
  • Parallel lines are taken to parallellines

AR.Math.Content.8.G.A.2 /
  • Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthe first by a sequence of rotations, reflections, andtranslations
  • Giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem

AR.Math.Content.8.G.A.3 / Givenatwo-dimensionalfigureonacoordinateplane,identifyanddescribetheeffect(ruleornewcoordinates) of a transformation (dilation, translation, rotation, and reflection):
  • Image topre-image
  • Pre-image toimage

AR.Math.Content.8.G.A.4 /
  • Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations
  • Giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem

AR.Math.Content.8.G.A.5 / Use informal arguments to establish facts about:
  • The angle sum and exterior angle oftriangles
Forexample:Arrangethreecopiesofthesametrianglesothatthesumofthethreeanglesappearstoformaline.
  • Theanglescreatedwhenparallellinesarecutbyatransversal
Forexample:Giveanargumentintermsoftranslationsabouttheanglerelationships.
  • The angle-angle criterion for similarity oftriangles

Geometry / Understand and apply the Pythagorean Theorem
AR.Math.Content.8.G.B.6 / ModelorexplainaninformalproofofthePythagoreanTheoremanditsconverse
AR.Math.Content.8.G.B.7 / ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematical problems in two and three dimensions
AR.Math.Content.8.G.B.8 / ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem
Geometry / Solvereal-worldandmathematicalproblemsinvolvingvolumeofcylinders,cones,andspheres
AR.Math.Content.8.G.C.9 / Developandknowtheformulasforthevolumesand surface areas ofcones,cylinders,andspheresandusethemtosolvereal-world and mathematicalproblems
Statistics andProbability / Investigate patterns of association in bivariate data
AR.Math.Content.8.SP.A.1 /
  • Constructandinterpretscatterplotsforbivariatemeasurementdatatoinvestigatepatternsofassociation between twoquantities
  • Describepatternssuchasclustering,outliers,positiveornegativeassociation,linearassociation,andnonlinearassociation

AR.Math.Content.8.SP.A.2 /
  • Knowthatstraightlinesarewidelyusedtomodelrelationshipsbetweentwoquantitativevariables
  • Forscatterplotsthatsuggestalinearassociation,informallyfitastraightline,andinformallyassessthemodelfitbyjudgingthe closeness of the data points to the line
For example: Identify weak, strong, or no correlation.
AR.Math.Content.8.SP.A.3 / Usetheequationofalinearmodeltosolveproblemsinthecontextofbivariatemeasurementdata,interpreting the slope and intercepts
Forexample:Inalinearmodelforabiologyexperiment,interpretaslopeof1.5cm/hrasmeaningthatanadditionalhourofsunlighteachdayisassociatedwithanadditional1.5cminmatureplantheight.
AR.Math.Content.8.SP.A.4 /
  • Understandthatpatternsofassociationcanalsobeseeninbivariatecategoricaldatabydisplayingfrequencies and relative frequencies in a two-waytable
  • Constructandinterpretatwo-waytableontwocategoricalvariablescollectedfromthesamesubjects
  • Userelativefrequenciescalculatedforrowsorcolumnstodescribepossibleassociationbetweenthetwovariables
Example: Two-Way Frequency Table


Example: Two-Way Relative Frequency Table

For example: Students might be asked to interpret from the tables above, if they saw an SUV in the parking lot, would it be more likely to belong to a male or female?
Note: Suggested connections for instruction: Standard 8.NS.1. On the Two-Way Relative Frequency Table, it is not required to include the fractional representation for each value, this is simply provided as an example.

Glossary

Absolute Value / A numbers’ distance from 0 on the number line which gives its size, or magnitude, whether the number is positive or negative (in terms of functions, a piecewise defined function is the absolute value function)
Additive Inverses / Two numbers whose sum is 0 are additive inverses of one another
Bivariate Data / Data that has two variables
Complex Fraction / A fraction a/b where either a or b are fractions (bnonzero)
Coordinate Plane / A plane spanned by the x- and y-axis
Dependent Variable / A variable shows values depend on the values of another variable
Experimental Probability / The ratio of the number of times an event occurs to the total number of trials or times the activity is performed
Exponent / the power p in an expression of the form apused to show repeated multiplication
Expression / A mathematical phrase consisting of numbers, variables, and operations
First Quartile / For a data set with median M, the first quartile is the median of the data values less than M
Function / A rule or relationship in which there is exactly one output value for each input value
Function notation / A method of writing algebraic variables as function of other variables; for example f(x) = 3x is the same as y = 3x
Greatest Common Factor / The greatest factor that divides two numbers
Independent Variable / A variable whose values don’t depend on changes in other variables
Integer / A number expressible in the form of a or – a for some whole number a
Interquartile Range / A measure of variation in a set of numerical data; the interquartile range is the distance between the first and third quartiles of the data set
Irrational Number / A number that cannot be expressed as a fraction p/q for any integers p and q; have decimal expansions that neither terminate nor become periodic
Least Common Multiple / The smallest number that is exactly divisible by each member of a set of numbers
Mean / A measure of center in a set of numerical data, computed by adding the values in a slit then dividing by the number of values in the list
Mean Absolute Deviation / A measure of variation in a set of numerical data, computed by adding the distances between each data value and the mean, then dividing by the number of data values
Median / A measure of center in a set of numerical data; the median of a list of values is the value appearing at the center of a sorted version of the list – or the mean of the two central values, if the list contains an even number of values
Mode / A measure of center in a set of numerical data; the most common value in list of values
Order of Operations / A set of rules that define which procedures to perform first in order to evaluate a given expression
Probability / A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (tossing a coin, selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition)
Ratio / The ratio of two number r and s is written r/s, where r is the numerator and s is the denominator
Rational Number / A number that can be written as a ratio of two integers
Relation / A set of ordered pairs of data
Scale Drawing / A drawing with dimensions at a specific ratio relative to the actual size of the object
Scatter Plot / A graph in the coordinate plane representing a set of bivariate data
Standard Algorithm / Denotes any valid base-ten strategy
Theoretical Probability / The number of ways that the event can occur, divided by the total number of outcomes
Third Quartile / For a data set with median M, the third quartile is the median of the data values greater than M
Unit Rate / A comparison of two measurements in which one of the terms has a value of 1
Variable / A symbol used to represent an unknown or undetermined value in an expression or equation

Appendix