Grade 8– Crosswalk (Summary of Revisions): 2016 · PDF file · 2017-07-12Grade 8– Crosswalk (Summary of Revisions): 2016 Mathematics Standards of Learning and Curriculum.....

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Grade8–Crosswalk(SummaryofRevisions):2016MathematicsStandardsofLearningandCurriculumFrameworkAdditions(2016SOL) DeletionsfromGrade8(2009SOL)

• 8.1EKS–Userationalapproximationsofirrationalnumberstocompareandorderrealnumbers• 8.5EKS–Usetherelationshipbetweenpairsofanglesthatarevertical,adjacent,supplementary,and

complementarytodeterminethemeasureofanunknownangle• 8.12a,b,c–Representdatainboxplots,makeinferencesandcomparedata[MovedfromA.10]• 8.14bEKS–Simplifyalgebraicexpressionsinonevariable;representalgebraicexpressionsusing

concretematerials• 8.16a,b,d,ande–Slopeofaline(positive/negative/zero);identifyslopeandy-interceptofalinear

function;graphalinearfunction;makeconnectionsbetweenrepresentations• 8.17EKS–Writeverbalexpressionsandsentencesasalgebraicexpressionsandequations;write

algebraicexpressionsandequationsasverbalexpressionsandsentences• 8.18–Solvemultisteplinearinequalitiesinonevariableononeorbothsidesoftheinequalitysymbol,

includingpracticalproblems,andgraphthesolutiononanumberline

• 8.5a–Determinewhetheragivennumberisaperfectsquare[Includedin7.1dEKS]

• 8.6b–Measureanglesoflessthan360°• 8.7a–Investigateandsolveproblemsinvolvingvolumeand

surfaceareaofrectangularprisms,cylinders[Includedin7.4a],triangular-basedpyramids

• 8.7EKS–Describethetwodimensionalfiguresthatresultfromslicingathree-dimensionalfigureparalleltothebase

• 8.8aEKS–Rotateafigure180°,270°,and360°;dilateapolygonfromafixedpoint(nottheorigin)[IncludedinG.3]

• 8.15b–Solvetwo-steplinearinequalitiesandgraphtheresultsonanumberline[Movedto7.13]

ParameterChanges/Clarifications(2016SOL) MoveswithinGrade8(2009SOLto2016SOL)• 8.1EKS–Compareandordernomorethanfiverealnumbersexpressedasintegers,fractions,mixed

numbers,decimals,percents,numberswritteninscientificnotation,radicals(includespositiveandnegativesquareroots),andpi

• 8.3a–Estimateanddeterminethetwoconsecutiveintegersbetweenwhichasquarerootlies(expandedbeyondwholenumbers)

• 8.4EKS–Limitcomputingsimpleinterestgiventheprincipal,interestrate,andtime(years)• 8.6bEKS–Describevolumeofarectangularprismwhenoneattributeismultipliedbyafactorof!

!,!!,

!!,2,3or4;describeitssurfaceareawhenoneattributeismultipliedbyfactorof!

!or2

• 8.7aEKS–Restrictdilationstorighttrianglesorrectanglestoascalefactorof!!, !!, 2,3,or4usingthe

originascenterofdilation;bothtranslateandthenreflectapolygonoverthex-ory-axisorreflectapolygonoverthex-ory-axisandthentranslate;andtranslateapolygonbothhorizontallyandvertically

• 8.11a–Compareandcontrastprobabilityoftwoindependent/dependentevents[Movedfrom6.16]• 8.13–Makeobservationsaboutdatadisplayedlimitedtoscatterplotsonly• 8.14EKS–Evaluatealgebraicexpressionslimitedtowholenumberexponentsandintegerbases;limit

numberofreplacementsto3;representexpressionsusingconcrete/pictorialrepresentations• 8.15–Determinewhetheragivenrelationisafunctionusingdiscretepoints(orderedpairs,tables,and

graphs);setsofdataarelimitedtonomorethan10orderedpairs• 8.17EKS–Solvemultistepequations,uptofoursteps;coefficientsandnumerictermswillberational;

equationsmaycontainexpressionsthatneedexpansionusingthedistributivepropertyorrequirecombiningliketerms

• 8.14,8.17,and8.18EKSandUS-applypropertiesofrealnumbersandpropertiesofequality/inequality

• 8.1a–Simplifynumericalexpressions[Combinedwith8.14a]• 8.3–Practicalproblemswithrationalnumbers[Movedto8.4]• 8.4–Evaluatealgebraicexpressions[Includedin8.14a]• 8.5b–Determinetwoconsecutivewholenumbersbetween

whichasquarerootlies[Includedin8.3a]• 8.5EKS–Positiveandnegativesquareroots[Movedto8.3b]• 8.6a–[Movedto8.5]• 8.7–[Movedto8.6]• 8.8–[Movedto8.7]• 8.9–[Movedto8.8]• 8.10–[Movedto8.9]• 8.11–[Movedto8.10]• 8.12–[Movedto8.11]• 8.14–[Includedin8.15aEKSand8.16e]• 8.15a–Solvemultisteplinearequations[Movedto8.17]• 8.15c–Identifyproperties[Incorporatedinto8.14,8.17and8.18

EKSandUS]• 8.16–Graphalinearequationintwovariables[Includedin

8.16d]• 8.17–Domainandrange[Movedto8.15b];dependentand

independentvariable[Movedto8.16c]

EKS=EssentialKnowledgeandSkills,referringtothecolumnontherightsideoftheCurriculumFrameworkUS=UnderstandingtheStandard,referringtothecolumnontheleftsideoftheCurriculumFramework

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ComparisonofMathematicsStandardsofLearning–2009to20162009SOL 2016SOL

NumberandNumberSense8.1 Thestudentwill

a) simplifynumericalexpressionsinvolvingpositiveexponents,usingrationalnumbers,orderofoperations,andpropertiesofoperationswithrealnumbers;and[Combinedwith8.14a]

b) compareandorderdecimals,fractions,percents,andnumberswritteninscientificnotation.[MovedtoEKS]

8.1 Thestudentwillcompareandorderrealnumbers.

8.2 Thestudentwilldescribeorallyandinwritingtherelationshipsbetweenthesubsetsoftherealnumbersystem.

8.2 Thestudentwilldescribetherelationshipsbetweenthesubsetsoftherealnumbersystem.

8.3 Thestudentwilla) estimateanddeterminethetwoconsecutiveintegersbetweenwhicha

squarerootlies;and[Movedfrom8.5b]b) determineboththepositiveandnegativesquarerootsofagivenperfect

square.[Movedfrom8.5EKS]ComputationandEstimation

8.3 Thestudentwilla) solvepracticalproblemsinvolvingrationalnumbers,percents,ratios,and

proportions;andb) determinethepercentincreaseordecreaseforagivensituation.

8.4 Thestudentwillsolvepracticalproblemsinvolvingconsumerapplications.

8.4 Thestudentwillapplytheorderofoperationstoevaluatealgebraicexpressionsforgivenreplacementvaluesofthevariables.[Movedto8.14a]

8.5 Thestudentwilla) determinewhetheragivennumberisaperfectsquare;and[Includedin7.1]b) findthetwoconsecutivewholenumbersbetweenwhichasquarerootlies.

[Includedin8.3a]

MeasurementandGeometry8.6 Thestudentwill

a) verifybymeasuringanddescribetherelationshipsamongverticalangles,adjacentangles,supplementaryangles,andcomplementaryangles;and

b) measureanglesoflessthan360°.

8.5 Thestudentwillusetherelationshipsamongpairsofanglesthatareverticalangles,adjacentangles,supplementaryangles,andcomplementaryanglestodeterminethemeasureofunknownangles.

8.7 Thestudentwilla) investigateandsolvepracticalproblemsinvolvingvolumeandsurfaceareaof

prisms,cylinders,cones,andpyramids;andb) describehowchangingonemeasuredattributeofafigureaffectsthevolume

andsurfacearea.

8.6 Thestudentwilla) solveproblems,includingpracticalproblems,involvingvolumeandsurface

areaofconesandsquare-basedpyramids;andb) describehowchangingonemeasuredattributeofarectangularprismaffects

thevolumeandsurfacearea.8.8 Thestudentwill

a) applytransformationstoplanefigures;andb) identifyapplicationsoftransformations.

8.7 Thestudentwilla) givenapolygon,applytransformations,toincludetranslations,reflections,

anddilations,inthecoordinateplane;andb) identifypracticalapplicationsoftransformations.

8.9 Thestudentwillconstructathree-dimensionalmodel,giventhetoporbottom, 8.8 Thestudentwillconstructathree-dimensionalmodel,giventhetoporbottom,

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2009SOL 2016SOLside,andfrontviews. side,andfrontviews.

8.10 Thestudentwill a) verifythePythagoreanTheorem;andb) applythePythagoreanTheorem.

8.9 Thestudentwilla) verifythePythagoreanTheorem;andb) applythePythagoreanTheorem.

8.11 Thestudentwillsolvepracticalareaandperimeterproblemsinvolvingcompositeplanefigures.

8.10 Thestudentwillsolveareaandperimeterproblems,includingpracticalproblems,involvingcompositeplanefigures.

ProbabilityandStatistics8.12 Thestudentwilldeterminetheprobabilityofindependentanddependentevents

withandwithoutreplacement.

8.11 Thestudentwilla) compareandcontrasttheprobabilityofindependentanddependentevents;

and[Movedfrom6.16]b) determineprobabilitiesforindependentanddependentevents.

8.12 Thestudentwilla) representnumericaldatainboxplots;b) makeobservationsandinferencesaboutdatarepresentedinboxplots;andc) compareandanalyzetwodatasetsusingboxplots.[MovedfromA.10]

8.13 Thestudentwilla) makecomparisons,predictions,andinferences,usinginformationdisplayed

ingraphs;andb) constructandanalyzescatterplots.

8.13 Thestudentwilla) representdatainscatterplots;b) makeobservationsaboutdatarepresentedinscatterplots;andc) useadrawingtoestimatethelineofbestfitfordatarepresentedina

scatterplot.Patterns,Functions,andAlgebra

8.14 Thestudentwilla) evaluateanalgebraicexpressionforgivenreplacementvaluesofthe

variables;and[Movedfrom8.4ComputationandEstimationand8.1aNumberandNumberSense]

b) simplifyalgebraicexpressionsinonevariable. 8.15Thestudentwill

a) determinewhetheragivenrelationisafunction;andb) determinethedomainandrangeofafunction.[Movedfrom8.17]

8.14 Thestudentwillmakeconnectionsbetweenanytworepresentations(tables,graphs,words,andrules)ofagivenrelationship.[Includedin8.15aEKSand8.16e]

8.16Thestudentwilla) recognizeanddescribethegraphofalinearfunctionwithaslopethatis

positive,negative,orzero;b) identifytheslopeandy-interceptofalinearfunctiongivenatableof

values,agraph,oranequationiny=mx+bform;c) determinetheindependentanddependentvariable,givenapractical

situationmodeledbyalinearfunction;d) graphalinearfunctiongiventheequationiny=mx+bform;ande) makeconnectionsbetweenandamongrepresentationsofalinearfunction

usingverbaldescriptions,tables,equations,andgraphs.8.15 Thestudentwill

a) solvemultisteplinearequationsinonevariablewiththevariableononeand8.17 Thestudentwillsolvemultisteplinearequationsinonevariableononeorboth

sidesoftheequation,includingpracticalproblemsthatrequirethesolutionofa

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2009SOL 2016SOLtwosidesoftheequation;

b) solvetwo-steplinearinequalitiesandgraphtheresultsonanumberline;and[Movedto7.13]

c) identifypropertiesofoperationsusedtosolveanequation.[IncorporatedintoEKSandUSof8.14,8.17and8.18]

multisteplinearequationinonevariable.

8.18 Thestudentwillsolvemultisteplinearinequalitiesinonevariablewiththevariableononeorbothsidesoftheinequalitysymbol,includingpracticalproblems,andgraphthesolutiononanumberline.

8.16 Thestudentwillgraphalinearequationintwovariables.[Incorporatedinto8.16d] 8.17 Thestudentwillidentifythedomain,range,[Movedto8.15b]independent

variable,ordependentvariable[Movedto8.16c]inagivensituation.

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SOL Reporting Category: Number and Number Sense—The Real Number System - Dates: 8/14-8/18 Current Standard and

Essential Knowledge and Skills NEW Standard(s)

Essential Knowledge and Skills NEW: Understanding the Standard(s)

8.2Thestudentwilldescribeorallyandinwritingtherelationshipsbetweenthesubsetsoftherealnumbersystem.•Describeorallyandinwritingtherelationshipsamongthesetsofnaturalorcountingnumbers,wholenumbers,integers,rationalnumbers,irrationalnumbers,andrealnumbers.•IllustratetherelationshipsamongthesubsetsoftherealnumbersystembyusinggraphicorganizerssuchasVenndiagrams.Subsetsincluderationalnumbers,irrationalnumbers,integers,wholenumbers,andnaturalorcountingnumbers.•Identifythesubsetsoftherealnumbersystemtowhichagivennumberbelongs.•Determinewhetheragivennumberisamemberofaparticularsubsetoftherealnumbersystem,andexplainwhy.•Describeeachsubsetofthesetofrealnumbersandincludeexamplesandnonexamples.•Recognizethatthesumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.†Vertical Articulation: 5.3ab

8.2Thestudentwilldescribetherelationshipsbetweenthesubsetsoftherealnumbersystem.•Describeandillustratetherelationshipsamongthesubsetsoftherealnumbersystembyusingrepresentations(graphicorganizers,numberlines,etc.).Subsetsincluderationalnumbers,irrationalnumbers,integers,wholenumbers,andnaturalnumbers.•Classifyagivennumberasamemberofaparticularsubsetorsubsetsoftherealnumbersystem,andexplainwhy.•Describeeachsubsetofthesetofrealnumbersandincludeexamplesandnon-examples.•Recognizethatthesumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.

•Thesubsetsofrealnumbersincludenaturalnumbers(countingnumbers),wholenumbers,integers,rationalandirrationalnumbers.

•Somenumberscanbelongtomorethanonesubsetoftherealnumbers(e.g.,4isanaturalnumber,awholenumber,aninteger,andarationalnumber).Theattributesofonesubsetcanbecontainedinwholeorinpartinanothersubset.Therelationshipsbetweenthesubsetsoftherealnumbersystemcanbeillustratedusinggraphicorganizers(thatmayinclude,butnotbelimitedto,Venndiagrams),numberlines,andotherrepresentations.

•Thesetofnaturalnumbersisthesetofcountingnumbers{1,2,3,4...}.

•Thesetofwholenumbersincludesthesetofallthenaturalnumbersandzero{0,1,2,3…}.

•Thesetofintegersincludesthesetofwholenumbersandtheiropposites{…-2,-1,0,1,2…}.Zerohasnooppositeandisneitherpositivenornegative.

•Thesetofrationalnumbersincludesthesetofallnumbersthatcanbeexpressedasfractionsintheform𝑎𝑏whereaandbareintegersandbdoesnotequalzero.Thedecimalformofarationalnumbercanbeexpressedasaterminatingorrepeatingdecimal.Afewexamplesofrationalnumbersare√25,14,-2.3,75%,and4.59̅̅̅̅.

•Thesetofirrationalnumbersisthesetofallnonrepeating,nonterminatingdecimals.Anirrationalnumbercannotbewritteninfractionform(e.g.,π,√2,1.232332333…).

•Therealnumbersystemiscomprisedofallrationalandirrationalnumbers.

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Key Vocabulary: real numbers, rational, irrational, natural, counting numbers, whole, integers, sets, subsets, nonzero, sum, product, representations, number lines, graphic organizers, example, non-example, terminating, non-terminating, positive, negative, simplify

DOE Lessons/Resources: Ø ESS Lessons Ø ARI Companion Ø Henrico Website

Teacher Notes and Elaborations: Studentsneedadditionalpracticeillustratingtherelationshipsamongthesubsetsoftherealnumbersystem.Examples:WhichareNOTintegers?

WhichfallONLYintherationalsubset?Whichsubset(s)does4fallwithin?

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SOL Reporting Category: Number and Number Sense—Comparing & Ordering Numbers- Dates: 8/21-8/25 Current Standard and

Essential Knowledge and Skills

NEW Standard(s) Essential Knowledge and Skills

NEW: Understanding the Standard(s)

8.1Thestudentwillb)compareandorderdecimals,fractions,percents,andnumberswritteninscientificnotation.•Compareandordernomorethanfivefractions,decimals,percents,andnumberswritteninscientificnotationusingpositiveandnegativeexponents.Orderingmaybeinascendingordescendingorder.Vertical Articulation: 6.2bd and 7.1bc

8.1Thestudentwillcompareandorderrealnumbers.•Compareandordernomorethanfiverealnumbersexpressedasintegers,fractions(properorimproper),decimals,mixednumbers,percents,numberswritteninscientificnotation,radicals,andπ.Radicalsmayincludebothpositiveandnegativesquarerootsofvaluesfrom0to400.Orderingmaybeinascendingordescendingorder.•Userationalapproximations(tothenearesthundredth)ofirrationalnumberstocompareandorder,locatingvaluesonanumberline.Radicalsmayincludebothpositiveandnegativesquarerootsofvaluesfrom0to400yieldinganirrationalnumber.

• Realnumberscanberepresentedasintegers,fractions(properorimproper),decimals,percents,numberswritteninscientificnotation,radicals,andπ.Itisoftenusefultoconvertnumberstobecomparedand/ororderedtoonerepresentation(e.g.,fractions,decimalsorpercents).

•Properfractions,improperfractions,andmixednumbersaretermsoftenusedtodescribefractions.Aproperfractionisafractionwhosenumeratorislessthanthedenominator.Animproperfractionisafractionwhosenumeratorisequaltoorgreaterthanthedenominator.Animproperfractionmaybeexpressedasamixednumber.Amixednumberiswrittenwithtwoparts:awholenumberandaproperfraction(e.g.,358).Fractionscanhaveapositiveornegativevalue.

•Thedensitypropertystatesthatbetweenanytworealnumbersliesanotherrealnumber.Forexample,between3and5wecanfind4;between4.0and4.2wecanfind4.16;between4.16and4.17wecanfind4.165;between4.165and4.166wecanfind4.1655,etc.Thus,wecanalwaysfindanothernumberbetweentwonumbers.Studentsarenotexpectedtoknowthetermdensitypropertybuttheconceptallowsforadeeperunderstandingofthesetofrealnumbers.

•Scientificnotationisusedtorepresentverylargeorverysmallnumbers.•Anumberwritteninscientificnotationistheproductoftwofactors:a

decimalgreaterthanorequaltoonebutlessthan10multipliedbyapowerof10(e.g.,3.1×105=310,000and3.1×10–5=0.000031).

•Anyrealnumberraisedtothezeropoweris1.Theonlyexceptiontothisruleiszeroitself.Zeroraisedtothezeropowerisundefined.

Key Vocabulary: fractions (proper/improper, mixed), numerator, denominator, decimals, percents, scientific notation, base, power of, exponent, zero power, rounding, pi, positive, negative, integers, radicals, square roots, irrational numbers


Teacher Notes and Elaborations:

Whatdoyouknowabout:50%,0.5,10050

,5x10 1−

Selecteachnumberthatisbetween431 and195%.

Understand: ascending, descending

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SOL Reporting Category: Number and Number Sense--Expressions - Dates: 8/28-9/1 9/5-9/8 Current Standard and



8.1Thestudentwilla)simplifynumericalexpressionsinvolvingpositiveexponents,usingrationalnumbers,orderofoperations,andpropertiesofoperationswithrealnumbers;and•Simplifynumericalexpressionscontaining:1)exponents(wherethebaseisarationalnumberandtheexponentisapositivewholenumber);2)fractions,decimals,integersandsquarerootsofperfectsquares;and3)groupingsymbols(nomorethan2embeddedgroupingsymbols).Orderofoperationsandpropertiesofoperationswithrealnumbersshouldbeused.8.4 The student will apply the order

of operations to evaluate algebraic expressions for given replacement values of the variables.

• Substitute numbers for variables in algebraic expressions and simplify the expressions by using the order of operations. Exponents are positive and limited to whole numbers less than 4. Square roots are limited to perfect squares.

• Apply the order of operations to evaluate formulas. Problems will be limited to positive exponents. Square roots may be included in the expressions but limited to perfect squares.

8.14 Thestudentwilla) evaluateanalgebraic

expressionforgivenreplacementvaluesofthevariables;and

b) simplifyalgebraicexpressionsinonevariable.

• Usetheorderofoperationsandapplythepropertiesofrealnumberstoevaluatealgebraicexpressionsforthegivenreplacementvaluesofthevariables.Exponentsarelimitedtowholenumbersandbasesarelimitedtointegers.Squarerootsarelimitedtoperfectsquares.Limitthenumberofreplacementstonomorethanthreeperexpression.(a)

• Representalgebraicexpressionsusingconcretematerialsandpictorialrepresentations.Concretematerialsmayincludecoloredchipsoralgebratiles.(a)

• Simplifyalgebraicexpressionsinonevariable.Expressionsmayneedtobeexpanded(usingthedistributiveproperty)orrequirecombiningliketermstosimplify.Expressionswillincludeonlylinearandnumericterms.Coefficientsandnumerictermsmayberational.(b)

• Anexpressionisarepresentationofaquantity.Itmaycontainnumbers,variables,

and/oroperationsymbols.Itdoesnothavean“equalsign(=)”(e.g., !!,5x,140-

38.2,-18·21,(5+2x)·4).Anexpressioncannotbesolved.• Anumericalexpressioncontainsonlynumbers,theoperationssymbols,and

groupingsymbols.• Expressionsaresimplifiedusingtheorderofoperations.• Simplifyinganalgebraicexpressionmeanstowritetheexpressionasamore

compactandequivalentexpression.Thisusuallyinvolvescombiningliketerms.• Liketermsaretermsthathavethesamevariablesandexponents.Thecoefficients

donotneedtomatch(e.g.,12xand-5x;45and-5!!;9y,-51yand

!!y.)

• Liketermsmaybeaddedorsubtractedusingthedistributiveandotherproperties.Forexample,

- 2(x-!! )+5x=2x–1+5x=2x+5x–1=7x-1

- w+w–2w=(1+1)w–2w=2w–2w=(2–2)w=0w=0• Theorderofoperationsisasfollows:- First,completealloperationswithingroupingsymbols*.Iftherearegrouping

symbolswithinothergroupingsymbols,dotheinnermostoperationfirst.- Second,evaluateallexponentialexpressions.- Third,multiplyand/ordivideinorderfromlefttoright.- Fourth,addand/orsubtractinorderfromlefttoright.*Parentheses(),brackets[],braces{},absolutevalue

(i.e., 3(−5 + 2) –7),andthedivisionbar(i.e.,!!!!!!

)shouldbetreatedasgroupingsymbols.

• Propertiesofrealnumberscanbeusedtoexpresssimplification.Studentsshouldusethefollowingproperties,whereappropriate,tofurtherdevelopflexibilityandfluencyinproblemsolving(limitationsmayexistforthevaluesofa,b,orcinthisstandard):

- Commutativepropertyofaddition:𝑎 + 𝑏 = 𝑏 + 𝑎.- Commutativepropertyofmultiplication:𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎.- Associativepropertyofaddition: 𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐).- Associativepropertyofmultiplication: 𝑎 ∙ 𝑏 ∙ 𝑐 = 𝑎 ∙ (𝑏 ∙ 𝑐).- Subtractionanddivisionareneithercommutativenorassociative.- Distributiveproperty(overaddition/subtraction):𝑎 ∙ 𝑏 + 𝑐 = 𝑎 ∙ 𝑏 + 𝑎 ∙ 𝑐 𝑎𝑛𝑑 𝑎 ∙ 𝑏 − 𝑐 = 𝑎 ∙ 𝑏 − 𝑎 ∙ 𝑐.

- Theadditiveidentityiszero(0)becauseanynumberaddedtozeroisthenumber.Themultiplicativeidentityisone(1)becauseanynumbermultipliedbyoneisthe

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Vertical Articulation: 6.8, 7.13ab, and A.1

number.Therearenoidentityelementsforsubtractionanddivision.- Identitypropertyofaddition(additiveidentityproperty):𝑎 + 0 = 𝑎 𝑎𝑛𝑑 0 + 𝑎 =𝑎.

- Identitypropertyofmultiplication(multiplicativeidentityproperty):𝑎 ∙ 1 =𝑎 𝑎𝑛𝑑 1 ∙ 𝑎 = 𝑎.

- Inversesarenumbersthatcombinewithothernumbersandresultinidentityelements

[e.g.,5+(–5)=0;15·5=1].

- Inversepropertyofaddition(additiveinverseproperty):𝑎 + −𝑎 = 0 𝑎𝑛𝑑 −𝑎 + 𝑎 = 0.

- Inversepropertyofmultiplication(multiplicativeinverseproperty):𝑎 ∙ !!=

1 𝑎𝑛𝑑 !!∙ 𝑎 = 1.

- Zerohasnomultiplicativeinverse.- Multiplicativepropertyofzero:𝑎 ∙ 0 = 0 𝑎𝑛𝑑 0 ∙ 𝑎 = 0.- Divisionbyzeroisnotapossiblemathematicaloperation.Itisundefined.- Substitutionproperty:If𝑎 = 𝑏,thenbcanbesubstitutedforainanyexpression,

equation,orinequality.• Apowerofanumberrepresentsrepeatedmultiplicationofthenumber.For

example,(–5)4means(–5)·(–5)·(–5)·(−5).Thebaseisthenumberthatismultiplied,andtheexponentrepresentsthenumberoftimesthebaseisusedasafactor.Inthisexample,(–5)isthebase,and4istheexponent.Theproductis625.Noticethatthebaseappearsinsidethegroupingsymbols.Themeaningchangeswiththeremovalofthegroupingsymbols.Forexample,–54means5·5·5·5negatedwhichresultsinaproductof-625.Theexpression–(5)4meanstotaketheoppositeof5·5·5·5whichis-625.Studentsshouldbeexposedtoallthreerepresentations.

• Analgebraicexpressionisanexpressionthatcontainsvariablesandnumbers.• Algebraicexpressionsareevaluatedbysubstitutingnumbersforvariablesand

applyingtheorderofoperationstosimplifytheresultingnumericexpression. Key Vocabulary: expression, algebraic, order of operations, grouping (parentheses, brackets, braces, absolute value), exponent, multiplication, division, addition, subtraction, coefficient, numeric terms, like terms, properties (commutative +/x, associative +/x, distributive, +/x identity, +/x inverse, multiplication property of zero), power of, with replacement


Teacher Notes and Elaborations: Practicewritinganswerasanimproperfraction.

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SOL Reporting Category: Computation and Estimation—Perfect Squares - Dates: 9/11-9/15 Current Standard and



8.5 The student will a) determine whether a given number is a

perfect square; and b) find the two consecutive whole numbers

between which a square root lies. • Identify the perfect squares from 0 to 400. • Identify the two consecutive whole numbers

between which the square root of a given whole number from 0 to 400 lies (e.g., 57 lies between 7 and 8 since 72 = 49 and 82 = 64).

• Define a perfect square. • Find the positive or positive and negative square

roots of a given whole number from 0 to 400. (Use the symbol to ask for the positive root and − when asking for the negative root.)

Vertical Articulation: 6.5, 7.1ad, and A.3

8.3Thestudentwilla) estimateanddeterminethetwo

consecutiveintegersbetweenwhichasquarerootlies;and

b) determineboththepositiveandnegativesquarerootsofagivenperfectsquare.

• Estimateandidentifythetwoconsecutiveintegersbetweenwhichthepositiveornegativesquarerootofagivennumberlies.Numbersarelimitedtonaturalnumbersfrom1to400.(a)

• Determinethepositiveornegativesquarerootofagivenperfectsquarefrom1to400.(b)

• Aperfectsquareisawholenumberwhosesquarerootisaninteger.

• Thesquarerootofagivennumberisanynumberwhich,whenmultipliedtimesitself,equalsthegivennumber.

• Boththepositiveandnegativerootsofwholenumbers,exceptzero,canbedetermined.Thesquarerootofzeroiszero.Thevalueisneitherpositivenornegative.Zero(awholenumber)isaperfectsquare.

• Thepositiveandnegativesquarerootofanywholenumberotherthanaperfectsquareliesbetweentwoconsecutiveintegers(e.g., 57liesbetween7and8since72=49and82=64;− 11liesbetween-4and-3since(-4)2=16and(-3)2=9).

• Thesymbol maybeusedtorepresentapositive(principal)rootand- maybeusedtorepresentanegativeroot.

• Thesquarerootofawholenumberthatisnotaperfectsquareisanirrationalnumber(e.g., 2 isanirrationalnumber).Anirrationalnumbercannotbeexpressedexactlyasafraction!

!

wherebdoesnotequal0.• Squarerootsymbolsmaybeusedtorepresentsolutionsto

equationsoftheformx2=p.Examplesmayinclude:- Ifx2=36,thenxis 36=6or− 36=-6.- Ifx2=5,thenxis 5or− 5.

• Studentscanusegridpaperandestimationtodeterminewhatisneededtobuildaperfectsquare.Thesquarerootofapositivenumberisusuallydefinedasthesidelengthofasquarewiththeareaequaltothegivennumber.Ifitisnotaperfectsquare,theareaprovidesameansforestimation.

Key Vocabulary: square, square root, perfect square, consecutive, positive, negative, estimate, whole number, , − , rational, irrational


Teacher Notes and Elaborations: Graph the square root of 45 on number line as a way to show the two consecutive integers the square root lies between.

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SOL Reporting Category: Computation and Estimation—Proportional Reasoning - Dates: 9/18-10/13 Current Standard and



8.3 The student will a) solve practical problems involving rational

numbers, percents, ratios, and proportions; and

b) determine the percent increase or decrease for a given situation.

• Write a proportion given the relationship of equality between two ratios.

• Solve practical problems by using computation procedures for whole numbers, integers, fractions, percents, ratios, and proportions. Some problems may require the application of a formula.

• Maintain a checkbook and check registry for five or fewer transactions.

• Compute a discount or markup and the resulting sale price for one discount or markup.

• Compute the percent increase or decrease for a one-step equation found in a real life situation.

• Compute the sales tax or tip and resulting total. • Substitute values for variables in given formulas.

For example, use the simple interest formula I prt= to determine the value of any missing variable when given specific information.

• Compute the simple interest and new balance earned in an investment or on a loan for a given number of years.

Vertical Articulation: 6.7 and 7.4

8.4 Thestudentwillsolvepracticalproblemsinvolvingconsumerapplications.

• Solvepracticalproblemsinvolvingconsumerapplicationsbyusingproportionalreasoningandcomputationproceduresforrationalnumbers.

• Reconcileanaccountbalancegivenastatementwithfiveorfewertransactions.

• Computeadiscountormarkupandtheresultingsalepriceforonediscountormarkup.

• Computethesalestaxortipandresultingtotal.

• Computethesimpleinterestandnewbalanceearnedinaninvestmentoronaloangiventheprincipalamount,interestrate,andtimeperiodinyears.

• Computethepercentincreaseordecreasefoundinapracticalsituation.

• Rationalnumbersmaybeexpressedaswholenumbers,integers,fractions,percents,andnumberswritteninscientificnotation.

• Practicalproblemsmayinclude,butarenotlimitedto,thoserelatedtoeconomics,sports,science,socialscience,transportation,andhealth.Someexamplesincludeproblemsinvolvingtheamountofapaycheckpermonth,commissions,fees,thediscountpriceonaproduct,temperature,simpleinterest,salestaxandinstallmentbuying.

• Apercentisaratiowithadenominatorof100.• Reconcilinganaccountisaprocessusedtoverifythattwosets

ofrecords(usuallythebalancesoftwoaccounts)areinagreement.Reconciliationisusedtoensurethatthebalanceofanaccountmatchestheactualamountofmoneydepositedand/orwithdrawnfromtheaccount.

• Adiscountisapercentoftheoriginalprice.Thediscountpriceistheoriginalpriceminusthediscount.

• Simpleinterest(I)foranumberofyearsisdeterminedbyfindingtheproductoftheprincipal(p),theannualrateofinterest(r),andthenumberofyears(t)oftheloanorinvestmentusingtheformulaI=prt.

• Thetotalvalueofaninvestmentisequaltothesumoftheoriginalinvestmentandtheinterestearned.

• Thetotalcostofaloanisequaltothesumoftheoriginalcostandtheinterestpaid.

• Percentincreaseandpercentdecreasearebothpercentsofchangemeasuringthepercentaquantityincreasesordecreases.

Key Vocabulary: Rational numbers, percents, rate, ratio, proportion, percent increase, percent decrease, equality, checkbook, checkbook register, transaction, deposit, withdraw, debit, credit, tax, tip, discount, simple interest, balance, mark-up, investment, loan, interest


Teacher Notes and Elaborations: ¾’sof48cookiesarechocolatechip.Eachcookiecosts$0.45.Howmuchwillonlythechocolatechipcookiescost?

Acamerasellsfor$326.Youmakeadownpaymentof$23.Ifyoupaytheremainingamountinsixequalmonthlypayments,whatamountwillyoupayeachmonth?

12

SOL Reporting Category: Probability and Statistics--Probability - Dates: 10/19-10/27 Current Standard and Essential Knowledge

and Skills

NEW Standard(s) Essential Knowledge

and Skills


8.12 The student will determine the probability of independent and dependent events with and without replacement. • Determinethe

probabilityofnomorethanthreeindependentevents.

• Determinetheprobabilityofnomorethantwodependenteventswithoutreplacement.

• Comparetheoutcomesofeventswithandwithoutreplacement.

Vertical Articulation: 6.16ab and 7.9 & 7.10

8.11Thestudentwilla) compareand

contrasttheprobabilityofindependentanddependentevents;and

b) determineprobabilitiesforindependentanddependentevents.

• Determinewhethertwoeventsareindependentordependent.(a)

• Compareandcontrasttheprobabilityofindependentanddependentevents.(a)

• Determinetheprobabilityoftwoindependentevents.(b)

• Determinetheprobabilityoftwodependentevents.(b)

• Asimpleeventisoneevent(e.g.,pullingonesockoutofadrawerandexaminingtheprobabilityofgettingonecolor).

• Ifalloutcomesofaneventareequallylikely,thetheoreticalprobabilityofaneventoccurringisequaltotheratioofdesiredoutcomestothetotalnumberofpossibleoutcomesinthesamplespace.

• Theprobabilityofaneventoccurringcanberepresentedasaratioortheequivalentfraction,decimal,orpercent.

• Theprobabilityofaneventoccurringisaratiobetween0and1.Aprobabilityofzeromeanstheeventwillneveroccur.Aprobabilityofonemeanstheeventwillalwaysoccur.

• Twoeventsareeitherdependentorindependent.• Iftheoutcomeofoneeventdoesnotinfluencetheoccurrenceoftheotherevent,theyarecalledindependent.Iftwoeventsareindependent,thentheprobabilityofthesecondeventdoesnotchangeregardlessofwhetherthefirstoccurs.Forexample,thefirstrollofanumbercubedoesnotinfluencethesecondrollofthenumbercube.Otherexamplesofindependenteventsare,butnotlimitedto:flippingtwocoins;spinningaspinnerandrollinganumbercube;flippingacoinandselectingacard;andchoosingacardfromadeck,replacingthecardandselectingagain.

• Theprobabilityoftwoindependenteventsisfoundbyusingthefollowingformula:P(AandB)=P(A)·P(B)

- Example:Whenrollingasix-sidednumbercubeandflippingacoin,simultaneously,whatistheprobabilityofrollinga3onthecubeandgettingaheadsonthecoin?P(3andheads)=!

!∙ !!= !

!"

• Iftheoutcomeofoneeventhasanimpactontheoutcomeoftheotherevent,theeventsarecalleddependent.Ifeventsaredependentthenthesecondeventisconsideredonlyifthefirsteventhasalreadyoccurred.Forexample,ifyouchooseabluecardfromasetofninedifferentcoloredcardsthathasatotaloffourbluecardsandyoudonotplacethatbluecardbackinthesetbeforeselectingasecondcard,thechanceofselectingabluecardthesecondtimeisdiminishedbecausetherearenowonlythreebluecardsremainingintheset.Otherexamplesofdependenteventsinclude,butarenotlimitedto:choosingtwomarblesfromabagbutnotreplacingthefirstafterselectingit;determiningtheprobabilitythatitwillsnowandthatschoolwillbecancelled.

• Theprobabilityoftwodependenteventsisfoundbyusingthefollowingformula:P(AandB)=P(A)·P(BafterA)- Example:Youhaveabagholdingablueball,aredball,andayellowball.Whatistheprobabilityofpickingablueballoutofthebagonthefirstpickthenwithoutreplacingtheblueballinthebag,pickingaredballonthesecondpick?P(blueandred)=P(blue)·P(redafterblue)=!

!∙ !!= !

!

Key Vocabulary: probability, independent, dependent, replacement, ratio, equivalent, simple event, theoretical


Teacher Notes and Elaborations: Understandvariouswaysindependenteventscanbedescribed(onedie,twodice,onespinner,abagofmarbleswithreplacement).

13

Dependentprobability–the2ndeventiseffectedbythe1stevent(withoutreplacement,setaside,etc.)

Examples:20%chanceofrainMonday,40%chanceofrainTuesday.WhatisprobabilityitwillNOTrainMondayandTuesday.

GivenaDependentprobabilityscenario,thenaskingwhatistheprobabilityofthe2ndeventoccurring.Graphtheanswertoaprobabilityquestiononanumberline.

SOL Reporting Category: Probability and Statistics--Statistics - Dates: 10/30-11/3

Current Standard and Essential Knowledge

and Skills

NEW Standard(s) Essential Knowledge and

Skills


8.13Thestudentwilla) makecomparisons,

predictions,andinferences,usinginformationdisplayedingraphs;and

b) constructandanalyzescatterplots.

• Collect,organize,andinterpretadatasetofnomorethan20itemsusingscatterplots.Predictfromthetrendanestimateofthelineofbestfitwithadrawing.

• Interpretasetofdatapointsinascatterplotashavingapositive

8.12Thestudentwilla) representnumericaldata

inboxplots;b) makeobservationsand

inferencesaboutdatarepresentedinboxplots;and

c) compareandanalyzetwodatasetsusingboxplots.

• Collectanddisplayanumericdatasetofnomorethan20items,usingboxplots.(a)

• Makeobservationsandinferencesaboutdatarepresentedinaboxplot.(b)

• Givenadatasetrepresentedinaboxplot,

8.12• Aboxplot(box-and-whiskerplot)isaconvenientandinformativewaytorepresentsingle-variable

(univariate)data.• Boxplotsareeffectiveatgivinganoverallimpressionoftheshape,center,andspreadofthedata.It

doesnotshowadistributioninasmuchdetailasastemandleafplotorahistogram.• Aboxplotwillallowyoutoquicklyanalyzeasetofdatabyidentifyingkeystatisticalmeasures(median

andrange)andmajorconcentrationsofdata.• Aboxplotusesarectangletorepresentthemiddlehalfofasetofdataandlines(whiskers)atboth

endstorepresenttheremainderofthedata.Themedianismarkedbyaverticallineinsidetherectangle.

• Thefivecriticalpointsinaboxplot,commonlyreferredtoasthefive-numbersummary,arelowerextreme(minimum),lowerquartile,median,upperquartile,andupperextreme(maximum).Eachofthesepointsrepresentstheboundsforthefourquartiles.Intheexamplebelow,thelowerextremeis15,thelowerquartileis19,themedianis21.5,theupperquartileis25,andtheupperextremeis29.

• Therangeisthedifferencebetweentheupperextremeandthelowerextreme.Theinterquartile

range(IQR)isthedifferencebetweentheupperquartileandthelowerquartile.Usingtheexampleabove,therangeis14or29-15.Theinterquartilerangeis6or25–19.

14

relationship,anegativerelationship,ornorelationship.

Vertical Articulation: 6.14abc, 7.11ab, and A.11

identifyanddescribethelowerextreme(minimum),upperextreme(maximum),median,upperquartile,lowerquartile,range,andinterquartilerange.(b)

• Compareandanalyzetwodatasetsrepresentedinboxplots.(c)

8.13 Thestudentwilla) representdatain

scatterplots;b) makeobservations

aboutdatarepresentedinscatterplots;and

c)useadrawingtoestimatethelineofbestfitfordatarepresentedinascatterplot.

• Collect,organize,andrepresentadatasetofnomorethan20itemsusingscatterplots.(a)

• Makeobservationsaboutasetofdatapointsinascatterplotashavingapositivelinearrelationship,anegativelinearrelationship,ornorelationship.(b)

• Estimatethelineofbestfitwithadrawingfordatarepresentedinascatterplot.(c)

• Whenthereareanoddnumberofdatavaluesinasetofdata,themedianwillnotbeconsideredwhencalculatingthelowerandupperquartiles.- Example:Calculatethemedian,lowerquartile,andupperquartileforthefollowingdatavalues:

356789111313Median:8;LowerQuartile:5.5;UpperQuartile:12

• Inthepulserateexample,shownbelow,manystudentsincorrectlyinterpretthatlongersectionscontainmoredataandshorteronescontainless.Itisimportanttorememberthatroughlythe same amount of data is in each section.Thenumbersintheleftwhisker(lowestofthedata)arespreadlesswidelythanthoseintherightwhisker.

• Boxplotsareusefulwhencomparinginformationabouttwodatasets.Thisexamplecomparesthetest

scoresforacollegeclassofferedattwodifferenttimes.

Usingtheseboxplots,comparisonscouldbemadeaboutthetwosetsofdata,suchascomparingthemedianscoreofeachclassortheInterquartileRange(IQR)ofeachclass.8.13• Ascatterplotillustratestherelationshipbetweentwosetsofnumericaldatarepresentedbytwo

variables(bivariatedata).Ascatterplotconsistsofpointsonthecoordinateplane.Thecoordinatesofthepointrepresentthemeasuresofthetwoattributesofthepoint.

• Inascatterplot,eachpointmayrepresentanindependentanddependentvariable.Theindependentvariableisgraphedonthehorizontalaxisandthedependentisgraphedontheverticalaxis.

• Scatterplotscanbeusedtopredictlineartrendsandestimatealineofbestfit.• Alineofbestfithelpsinmakinginterpretationsandpredictionsaboutthesituationmodeledinthe

dataset.LinesandcurvesofbestfitareexploredmoreinAlgebraItomakeinterpretationsandpredictions.

• Ascatterplotcansuggestvariouskindsoflinearrelationshipsbetweenvariables.Forexample,weightandheight,whereweightwouldbeony-axisandheightwouldbeonthex-axis.Linearrelationshipsmaybepositive(rising)ornegative(falling).Ifthepatternofpointsslopesfromlowerlefttoupperright,itindicatesapositivelinearrelationshipbetweenthevariablesbeingstudied.Ifthepatternofpointsslopesfromupperlefttolowerright,itindicatesanegativelinearrelationship.• Forexample:Thefollowingscatterplotsillustratehowpatternsindatavaluesmayindicatelinear

relationships.

NorelationshipPositiverelationshipNegativerelationship

15

• Alinearrelationshipbetweenvariablesdoesnotnecessarilyimplycausation.Forexample,asthe

temperatureatthebeachincreases,thesalesatanicecreamstoreincrease.Ifdatawerecollectedforthesetwovariables,apositivelinearrelationshipwouldexist,however,thereisnocausalrelationshipbetweenthevariables(i.e.,thetemperatureoutsidedoesnotcauseicecreamsalestoincrease,butthereisarelationshipbetweenthetwo).

• Therelationshipbetweenvariablesisnotalwayslinear,andmaybemodeledbyothertypesoffunctionsthatarestudiedinhighschoolandcollegelevelmathematics.

Key Vocabulary: scatterplot, positive/negative/no relationship, linear, boxplot, variables, median, quartile-lower extreme, lower, median, upper, upper extreme


Teacher Notes and Elaborations: Scatterplots–As“x”increases,whatdoes“y”do?Thex-axishastheIndependentvariable,they-axishasthedependentvariable.

Studentsinterpretthemeaningoftherelationshipshownonthescatterplot.(Ex.Tallermotherstendtohavetallerdaughters)Studentsmakepredictionsusingalineofbestfit.

16

SOL Reporting Category: Patterns, Functions, & Algebra—Equations and Inequalities - Dates: 11/9-12/18 Current Standard and




8.15 Thestudentwilla) solvemultisteplinear

equationsinonevariableononeandtwosidesoftheequation;

b) solvetwo-steplinearinequalitiesandgraphtheresultsonanumberline;and

c) identifypropertiesofoperationsusedtosolveanequation.

• Solve two- to four-step linear equations in one variable using concrete materials, pictorial representations, and paper and pencil illustrating the steps performed.

• Solve two-step inequalities in one variable by showing the steps and using algebraic sentences.

• Graph solutions to two-step linear inequalities on a number line.

• Identify properties of operations used to solve an equation from among:

- the commutative properties of addition and multiplication;

- theassociativepropertiesofadditionandmultiplication;

- thedistributiveproperty;- theidentitypropertiesof

additionandmultiplication;- thezeropropertyof

multiplication;- theadditiveinverseproperty;

8.17Thestudentwillsolvemultisteplinearequationsinonevariablewiththevariableononeorbothsidesoftheequation,includingpracticalproblemsthatrequirethesolutionofamultisteplinearequationinonevariable.

• Representandsolvemultisteplinearequationsinonevariablewiththevariableononeorbothsidesoftheequation(uptofoursteps)usingavarietyofconcretematerialsandpictorialrepresentations.Applypropertiesofrealnumbersandpropertiesofequalitytosolvemultisteplinearequationsinonevariable(uptofoursteps).Coefficientsandnumerictermswillberational.Equationsmaycontainexpressionsthatneedtobeexpanded(usingthedistributiveproperty)orrequirecollectingliketermstosolve.

• Writeverbalexpressionsandsentencesasalgebraicexpressionsandequations.

• Writealgebraicexpressionsandequationsasverbalexpressionsandsentences.

• Solvepracticalproblemsthatrequirethesolutionofamultisteplinearequation.

• Confirmalgebraicsolutionstolinearequationsinonevariable.

• Amultistepequationmayinclude,butnotbelimitedtoequationssuchasthefollowing:2𝑥 + 1=!!

!;−3 2𝑥 + 7 = !

!𝑥;2𝑥 + 7 − 5𝑥 = 27;−5𝑥 − 𝑥 + 3 = −12.

• Anexpressionisarepresentationofquantity.Itmaycontainnumbers,variables,and/oroperationsymbols.Itdoesnothavean“equalsign(=)”(e.g., !

!,5x,140−38.2,18·21,5

+x.)• Anexpressionthatcontainsavariableisavariableexpression.Avariableexpressionis

likeaphrase:asaphrasedoesnothaveaverb,soanexpressiondoesnothavean“equalsign(=)”.Anexpressioncannotbesolved.

• Averbalexpressioncanberepresentedbyavariableexpression.Numbersareusedwhentheyareknown;variablesareusedwhenthenumbersareunknown.Forexample,theverbalexpression“anumbermultipliedbyfive”couldberepresentedbythevariableexpression“n·5”or“5n”.

• Analgebraicexpressionisavariableexpressionthatcontainsatleastonevariable(e.g.,2x–3).

• Averbalsentenceisacompletewordstatement(e.g.,“Thesumoftwoconsecutiveintegersisthirty-five.”couldberepresentedby“n+(n+1)=35”).

• Analgebraicequationisamathematicalstatementthatsaysthattwoexpressionsareequal(e.g.,2x+3=-4x+1).

• Inanequation,the“equalsign(=)”indicatesthatthevalueoftheexpressionontheleftisequivalenttothevalueoftheexpressionontheright.

• Liketermsaretermsthathavethesamevariablesandexponents.Thecoefficientsdonotneedtomatch(e.g.,12xand−5x;45and−5!

!;9y,−51yand!

!y.)

• Liketermsmaybeaddedorsubtractedusingthedistributiveandotherproperties.Forexample,- 4.6y–5y=(−4.6–5)y=−9.6y- w+w–2w=(1+1)w–2w=2w–2w=(2–2)w=0∙w=0

• Real-worldproblemscanbeinterpreted,represented,andsolvedusinglinearequationsinonevariable.

• Propertiesofrealnumbersandpropertiesofequalitycanbeusedtosolveequations,justifysolutionsandexpresssimplification.Studentsshouldusethefollowingproperties,whereappropriate,tofurtherdevelopflexibilityandfluencyinproblemsolving(limitationsmayexistforthevaluesofa,b,orcinthisstandard):

- Commutativepropertyofaddition:𝑎 + 𝑏 = 𝑏 + 𝑎.- Commutativepropertyofmultiplication:𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎.

17

and- themultiplicativeinverse

property.Vertical Articulation: 6.18, 7.14ab & 7.15ab, and A.2, A.4, A.5, & A.6

8.18 Thestudentwillsolvemultisteplinearinequalitiesinonevariablewiththevariableononeorbothsidesoftheinequalitysymbol,includingpracticalproblems,andgraphthesolutiononanumberline.

• Applypropertiesofrealnumbersandpropertiesofinequalitytosolvemultisteplinearinequalities(uptofoursteps)inonevariablewiththevariableononeorbothsidesoftheinequality.Coefficientsandnumerictermswillberational.Inequalitiesmaycontainexpressionsthatneedtobeexpanded(usingthedistributiveproperty)orrequirecollectingliketermstosolve.

• Graphsolutionstomultisteplinearinequalitiesonanumberline.

• Writeverbalexpressionsandsentencesasalgebraicexpressionsandinequalities.

• Writealgebraicexpressionsandinequalitiesasverbalexpressionsandsentences.

• Solvepracticalproblemsthatrequirethesolutionofamultisteplinearinequalityinonevariable.

• Identifyanumericalvalue(s)thatispartofthesolutionsetofagiveninequality.

- Associativepropertyofaddition: 𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐).- Associativepropertyofmultiplication: 𝑎 ∙ 𝑏 ∙ 𝑐 = 𝑎 ∙ (𝑏 ∙ 𝑐).- Subtractionanddivisionareneithercommutativenorassociative.- Distributiveproperty(overaddition/subtraction):𝑎 ∙ 𝑏 + 𝑐 = 𝑎 ∙ 𝑏 + 𝑎 ∙ 𝑐 𝑎𝑛𝑑

𝑎 ∙ 𝑏 − 𝑐 = 𝑎 ∙ 𝑏 − 𝑎 ∙ 𝑐.- Theadditiveidentityiszero(0)becauseanynumberaddedtozeroisthenumber.The

multiplicativeidentityisone(1)becauseanynumbermultipliedbyoneisthenumber.Therearenoidentityelementsforsubtractionanddivision.

- Identitypropertyofaddition(additiveidentityproperty):𝑎 + 0 = 𝑎 𝑎𝑛𝑑 0 + 𝑎 = 𝑎.- Identitypropertyofmultiplication(multiplicativeidentityproperty):𝑎 ∙ 1 = 𝑎 𝑎𝑛𝑑 1 ∙

𝑎 = 𝑎.- Inversesarenumbersthatcombinewithothernumbersandresultinidentityelements

(e.g.,5+(–5)=0;15·5=1).

- Inversepropertyofaddition(additiveinverseproperty):𝑎 + −𝑎 = 0 𝑎𝑛𝑑 −𝑎 +𝑎 = 0.

- Inversepropertyofmultiplication(multiplicativeinverseproperty):𝑎 ∙ !!= 1 𝑎𝑛𝑑 !

!∙

𝑎 = 1.- Zerohasnomultiplicativeinverse.- Multiplicativepropertyofzero:𝑎 ∙ 0 = 0 𝑎𝑛𝑑 0 ∙ 𝑎 = 0.- Divisionbyzeroisnotapossiblemathematicaloperation.Itisundefined.8.18• Amultistepinequalitymayinclude,butnotbelimitedtoinequalitiessuchasthe

following: 2𝑥 + 1>!!

!;−3 2𝑥 + 7 ≤ !

!𝑥;2𝑥 + 7 − 5𝑥 < 27;−5𝑥 − 𝑥 + 3 > −12.

• Whenbothexpressionsofaninequalityaremultipliedordividedbyanegativenumber,theinequalitysignreverses.

• Asolutiontoaninequalityisthevalueorsetofvaluesthatcanbesubstitutedtomaketheinequalitytrue.

• Inaninequality,therecanbemorethanonevalueforthevariablethatmakestheinequalitytrue.Therecanbemanysolutions.(i.e.,x+4>−3thenthesolutionsisx>−7.Thismeansthatxcanbeanynumbergreaterthan−7.Afewsolutionsmightbe−6.5,−3,0,4,25,etc.)

• Real-worldproblemscanbemodeledandsolvedusinglinearinequalities.• Thepropertiesofrealnumbersandpropertiesofinequalitycanbeusedtosolve

inequalities,justifysolutions,andexpresssimplification.Studentsshouldusethefollowingproperties,whereappropriate,tofurtherdevelopflexibilityandfluencyinproblemsolving(limitationsmayexistforthevaluesofa,b,orcinthisstandard).

- Commutativepropertyofaddition:𝑎 + 𝑏 = 𝑏 + 𝑎.- Commutativepropertyofmultiplication:𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎.- Associativepropertyofaddition: 𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐).

18

- Associativepropertyofmultiplication: 𝑎 ∙ 𝑏 ∙ 𝑐 = 𝑎 ∙ (𝑏 ∙ 𝑐).- Subtractionanddivisionareneithercommutativenorassociative.- Distributiveproperty(overaddition/subtraction):𝑎 ∙ 𝑏 + 𝑐 = 𝑎 ∙ 𝑏 + 𝑎 ∙ 𝑐 𝑎𝑛𝑑

𝑎 ∙ 𝑏 − 𝑐 = 𝑎 ∙ 𝑏 − 𝑎 ∙ 𝑐.- Theadditiveidentityiszero(0)becauseanynumberaddedtozeroisthenumber.The

multiplicativeidentityisone(1)becauseanynumbermultipliedbyoneisthenumber.Therearenoidentityelementsforsubtractionanddivision.

- Identitypropertyofaddition(additiveidentityproperty):𝑎 + 0 = 𝑎 𝑎𝑛𝑑 0 + 𝑎 = 𝑎.- Identitypropertyofmultiplication(multiplicativeidentityproperty):𝑎 ∙ 1 = 𝑎 𝑎𝑛𝑑 1 ∙

𝑎 = 𝑎.- Inversesarenumbersthatcombinewithothernumbersandresultinidentityelements

(e.g.,5+(–5)=0;15·5=1).

- Inversepropertyofaddition(additiveinverseproperty):𝑎 + −𝑎 = 0 𝑎𝑛𝑑 −𝑎 + 𝑎 =0.

- Inversepropertyofmultiplication(multiplicativeinverseproperty):𝑎 ∙ !!= 1 𝑎𝑛𝑑 !

!∙

𝑎 = 1.- Zerohasnomultiplicativeinverse.- Multiplicativepropertyofzero:𝑎 ∙ 0 = 0 𝑎𝑛𝑑 0 ∙ 𝑎 = 0.- Divisionbyzeroisnotapossiblemathematicaloperation.Itisundefined.- Substitutionproperty:If𝑎 = 𝑏,thenbcanbesubstitutedforainanyexpression,

equation,orinequality.- Additionpropertyofinequality:If𝑎 < 𝑏,then 𝑎 + 𝑐 < 𝑏 + 𝑐;if𝑎 > 𝑏,then𝑎 + 𝑐 >

𝑏 + 𝑐.- Subtractionpropertyofinequality:If𝑎 < 𝑏,then 𝑎 − 𝑐 < 𝑏 − 𝑐;if𝑎 > 𝑏,then𝑎 − 𝑐 >

𝑏 − 𝑐.- Multiplicationpropertyofinequality:If𝑎 < 𝑏 𝑎𝑛𝑑 𝑐 > 0,then𝑎 ∙ 𝑐 < 𝑏 ∙ 𝑐;if𝑎 >

𝑏 𝑎𝑛𝑑 𝑐 > 0,then𝑎 ∙ 𝑐 > 𝑏 ∙ 𝑐.

- Multiplicationpropertyofinequality(multiplicationbyanegativenumber):If𝑎 <𝑏 𝑎𝑛𝑑 𝑐 < 0,then𝑎 ∙ 𝑐 > 𝑏 ∙ 𝑐;if𝑎 > 𝑏 𝑎𝑛𝑑 𝑐 < 0,then𝑎 ∙ 𝑐 < 𝑏 ∙ 𝑐.

- Divisionpropertyofinequality:If𝑎 < 𝑏 𝑎𝑛𝑑 𝑐 > 0,then!!< !

!;if𝑎 > 𝑏 𝑎𝑛𝑑 𝑐 > 0,then

!!> !

!.

- Divisionpropertyofinequality(divisionbyanegativenumber):If𝑎 < 𝑏 𝑎𝑛𝑑 𝑐 < 0,then!!> !

!;if𝑎 > 𝑏 𝑎𝑛𝑑 𝑐 < 0,then!

!< !

!.

Key Vocabulary: equation, inequality, linear, variable, coefficient, constant, graph, number line, verbal, algebraic, properties, like terms


Teacher Notes and Elaborations: Forchallenge,usefractioncoefficientsandconstants.Closedcircleforgraphingvs.Opencircleforgraphing.Whichdirectionforarrowifvariableisontheright.6<x

19

SOL Reporting Category: Patterns, Functions, & Algebra --Functions - Dates: 1/3-1/19 Current Standard and




8.14 Thestudentwillmakeconnectionsbetweenanytworepresentations(tables,graphs,words,andrules)ofagivenrelationship.

• Graph in a coordinate plane ordered pairs that represent a relation.

• Describe and represent relations and functions, using tables, graphs, words, and rules. Given one representation, students will be able to represent the relation in another form.

• Relate and compare different representations for the same relation.

8.16 Thestudentwillgrapha

linearequationintwovariables.

• Construct a table of ordered pairs by substituting values for x in a linear equation to find values for y.

• Plot in the coordinate plane ordered pairs (x, y) from a table.

• Connect the ordered pairs to form a straight line (a continuous function).

• Interpret the unit rate of the proportional relationship graphed as the slope of the graph, and compare two different proportional relationships represented in different ways.†

8.17 Thestudentwillidentify

8.15Thestudentwilla) determinewhetheragiven

relationisafunction;andb) determinethedomainand

rangeofafunction.• Determinewhetherarelation,

representedbyasetoforderedpairs,atable,oragraphofdiscretepointsisafunction.Setsarelimitedtonomorethan10orderedpairs.(a)

• Identifythedomainandrangeofafunctionrepresentedasasetoforderedpairs,atable,oragraphofdiscretepoints.(b)

8.16 Thestudentwilla) recognizeanddescribethe

graphofalinearfunctionwithaslopethatispositive,negative,orzero;

b) identifytheslopeandy-interceptofalinearfunctiongivenatableofvalues,agraph,oranequationiny=mx+bform;

c) determinetheindependentanddependentvariable,givenapracticalsituationmodeledbyalinearfunction;

d) graphalinearfunctiongiventheequationiny=mx+bform;and

e) makeconnectionsbetweenandamongrepresentationsofalinearfunctionusingverbal

• Arelationisanysetoforderedpairs.Foreachfirstmember,theremaybemanysecondmembers.

• Afunctionisarelationbetweenasetofinputs,calledthedomain,andasetofoutputs,calledtherange,withthepropertythateachinputisrelatedtoexactlyoneoutput.

• Asatableofvalues,afunctionhasauniquevalueassignedtothesecondvariableforeachvalueofthefirstvariable.Inthe“notafunction”example,theinputvalue

“1”hastwodifferentoutputvalues,5and-3,assignedtoit,sotheexampleisnotafunction.

• Asasetoforderedpairs,afunctionhasauniqueordifferenty-valueassignedtoeachx-value.Forexample,thesetoforderedpairs,{(1,2),(2,4),(3,2),(4,8)}isafunction.Thissetoforderedpairs,{(1,2),(2,4),(3,2),(2,3)},isnotafunctionbecausethex-valueof“2”hastwodifferenty-values.

• Asagraphofdiscretepoints,arelationisafunctionwhen,foranyvalueofx,averticallinepassesthroughnomorethanonepointonthegraph.

• Somerelationsarefunctions;allfunctionsarerelations.• Graphsoffunctionscanbediscreteorcontinuous.• Inadiscretefunctiongraphthereareseparate,distinctpoints.Youwouldnotusea

linetoconnectthesepointsonagraph.Thepointsbetweentheplottedpointshavenomeaningandcannotbeinterpreted.Forexample,thenumberofpetsperhouseholdrepresentsadiscretefunctionbecauseyoucannothaveafractionofapet.

• Functionsmayberepresentedasorderedpairs,tables,graphs,equations,physicalmodels,orinwords.Anygivenrelationshipcanberepresentedusingmultiplerepresentations.

• Adiscussionaboutdeterminingwhetheracontinuousgraphofarelationisafunctionusingtheverticallinetestmayoccuringradeeight,butwillbeexploredfurtherinAlgebraI.

• Thedomainisthesetofalltheinputvaluesfortheindependentvariableorx-values(firstnumberinanorderedpair).

20

thedomain,range,independentvariableordependentvariableinagivensituation.

• Apply the following algebraic terms appropriately: domain, range, independent variable, and dependent variable.

• Identify examples of domain, range, independent variable, and dependent variable.

• Determine the domain of a function.

• Determine the range of a function.

• Determine the independent variable of a relationship.

• Determine the dependent variable of a relationship.

Vertical Articulation: 6.20, 7.12, and A.7

descriptions,tables,equations,andgraphs.

• Recognizeanddescribealinewithaslopethatispositive,negative,orzero(0).(a)

• Givenatableofvaluesforalinearfunction,identifytheslopeandy-intercept.Thetablewillincludethecoordinateofthey-intercept.(b)

• Givenalinearfunctionintheformy=mx+b,identifytheslopeandy-intercept.(b)

• Giventhegraphofalinearfunction,identifytheslopeandy-intercept.Thevalueofthey-interceptwillbelimitedtointegers.Thecoordinatesoftheorderedpairsshowninthegraphwillbelimitedtointegers.(b)

• Identifythedependentandindependentvariable,givenapracticalsituationmodeledbyalinearfunction.(c)

• Giventheequationofalinearfunctionintheformy=mx+b,graphthefunction.Thevalueofthey-interceptwillbelimitedtointegers.(d)

• Writetheequationofalinearfunctionintheformy=mx+bgivenvaluesfortheslope,m,andthey-interceptorgivenapracticalsituationinwhichtheslope,m,andy-interceptaredescribedverbally.(e)

• Makeconnectionsbetweenandamongrepresentationsofalinearfunctionusingverbaldescriptions,tables,equations,andgraphs.(e).

• Therangeisthesetofalltheoutputvaluesforthedependentvariableory-values(secondnumberinanorderedpair

• Ifafunctioniscomprisedofadiscretesetoforderedpairs,thenthedomainisthesetofallthex-coordinates,andtherangeisthesetofallthey-coordinates.Thesesetsofvaluescanbedeterminedgivendifferentrepresentationsofthefunction.- Example:Thedomainofafunctionis{−1,1,2,3}andtherangeis{−3,3,5}.The

followingarerepresentationsofthisfunction:o Thefunctionrepresentedasatableofvalues:

o Thefunctionrepresentedasasetoforderedpairs:{(-1,5),(1,-3),(2,3),(3,5)}o Thefunctionrepresentedasagraphonacoordinateplane:

8.16• Alinearfunctionisanequationintwovariableswhosegraphisastraightline,atype

ofcontinuousfunction.• Alinearfunctionrepresentsasituationwithaconstantrate.Forexample,when

drivingatarateof35mph,thedistanceincreasesasthetimeincreases,buttherateofspeedremainsthesame.

• Slope(m)representstherateofchangeinalinearfunctionorthe“steepness”oftheline.Theslopeofalineisarateofchange,aratiodescribingtheverticalchangetothehorizontalchange.

slope=!!!"#$ !" !!!!"#$ !" !

= !"#$%&'( !!!"#$!!"#$!%&'( !!!"#$

• Alineisincreasingifitrisesfromlefttoright.Theslopeispositive(i.e.,m>0).• Alineisdecreasingifitfallsfromlefttoright.Theslopeisnegative(i.e.,m<0).• Ahorizontallinehaszeroslope(i.e.,m=0).

21

• Adiscussionaboutlineswithundefinedslope(verticallines)shouldoccurwithstudentsingradeeightmathematicstocompareundefinedslopetolineswithadefinedslope.FurtherexplorationofthisconceptwilloccurinAlgebraI.

• Alinearfunctioncanbewrittenintheformy=mx+b,wheremrepresentstheslopeorrateofchangeinycomparedtox,andbrepresentsthey-interceptofthegraphofthelinearfunction.They-interceptisthepointatwhichthegraphofthefunctionintersectsthey-axisandmaybegivenasasinglevalue,b,orasthelocationofapoint(0,b).- Example:Giventheequationofthelinearfunctiony=−3x+2,theslopeis−3or

!!!andthe

y-interceptis2or(0,2).- Example:Thetableofvaluesrepresentsalinearfunction.

Inthetable,thepoint(0,2)representsthey-intercept.Theslopeisdeterminedbyobservingthechangeineachy-valuecomparedtothecorrespondingchangeinthex-value.

slope=m=!!!"#$ !" !!!"#$%!!!"#$ !" !!!"#$%

=!!!!

=−3

• Theslope,m,andy-interceptofalinearfunctioncanbedeterminedgiventhegraphofthefunction.- Example:Giventhegraphofthelinearfunction,determinetheslopeandy-

intercept.

Giventhegraphofalinearfunction,they-interceptisfoundbydeterminingwherethelineintersectsthey-axis.They-interceptwouldbe2orlocatedatthepoint(0,2).Theslopecanbefoundbydeterminingthechangeineachy-valuecomparedtothechangeineachx-value.Here,wecoulduseslopetrianglestohelpvisualizethis:slope=m=!!!"#$ !" !!!"#$%

!!!"#$ !" !!!"#$%=!!!!

=−3

• Graphingalinearfunctiongivenanequationcanbeaddressedusingdifferentmethods.Onemethodinvolvesdeterminingatableoforderedpairsbysubstitutingintotheequationvaluesforonevariableandsolvingfortheothervariable,plotting

22

theorderedpairsinthecoordinateplane,andconnectingthepointstoformastraightline.Anothermethodinvolvesusingslopetrianglestodeterminepointsontheline.

- Example:Graphthelinearfunctionwhoseequationisy=5x−1.Inordertographthelinearfunction,wecancreateatableofvaluesbysubstitutingarbitraryvaluesforxtodeterminingcoordinatingvaluesfory:

Thevaluescanthenbeplottedaspointsonagraph.Knowingtheequationofalinearfunctionwritteniny=mx+bprovidesinformationabouttheslopeandy-interceptofthefunction.Iftheequationisy=5x−1,thentheslope,m,ofthelineis5or!

!andthey-interceptis−1andcanbelocatedatthepoint

(0,−1).Wecangraphthelinebyfirstplottingthey-intercept.Wealsoknow,slope=m=!!!"#$ !" !!!"#$%

!!!"#$ !" !!!"#$%=!!!!

Otherpointscanbeplottedonthegraphusingtherelationshipbetweentheyandxvalues.Slopetrianglescanbeusedtohelplocatetheotherpointsasshowninthegraphbelow:

• Atableofvaluescanbeusedinconjunctionwithusingslopetrianglestoverifythe

graphofalinearfunction.They-interceptislocatedonthey-axiswhichiswherethex-coordinateis0.Thechangeineachy-valuecomparedtothecorrespondingx-valuecanbeverifiedbythepatternsinthetableofvalues.

• Theaxesofacoordinateplanearegenerallylabeledxandy;however,anylettersmaybeusedthatareappropriateforthefunction.

• Afunctionhasvaluesthatrepresenttheinput(x)andvaluesthatrepresenttheoutput(y).Theindependentvariableistheinputvalue.

• Thedependentvariabledependsontheindependentvariableandistheoutput

23

value.• Belowisatableofvaluesforfindingtheapproximatecircumferenceofcircles,C=

πd,wherethevalueofπisapproximatedas3.14.Diameter Circumference

1in. 3.14in.

2in. 6.28in.

3in. 9.42in.

4in. 12.56in.

- Theindependentvariable,orinput,isthediameterofthecircle.Thevaluesforthediametermakeupthedomain.

- Thedependentvariable,oroutput,isthecircumferenceofthecircle.Thesetofvaluesforthecircumferencemakesuptherange.

• Inagraphofacontinuousfunctioneverypointinthedomaincanbeinterpreted.Therefore,itispossibletoconnectthepointsonthegraphwithacontinuouslinebecauseeverypointonthelineanswerstheoriginalquestionbeingasked.

• Thecontextofaproblemmaydeterminewhetheritisappropriatefororderedpairsrepresentingalinearrelationshiptobeconnectedbyastraightline.Iftheindependentvariable(x)representsadiscretequantity(e.g.,numberofpeople,numberoftickets,etc.)thenitisnotappropriatetoconnecttheorderedpairswithastraightlinewhengraphing.Iftheindependentvariable(x)representsacontinuousquantity(e.g.,amountoftime,temperature,etc.),thenitisappropriatetoconnecttheorderedpairswithastraightlinewhengraphing.

- Example:Thefunctiony=7xrepresentsthecostindollars(y)forxticketstoanevent.Thedomainofthisfunctionwouldbediscreteandwouldberepresentedbydiscretepointsonagraph.Notallvaluesforxcouldberepresentedandconnectingthepointswouldnotbeappropriate.

- Example:Thefunctiony=−2.5x+20representsthenumberofgallonsofwater(y)remainingina20-gallontankbeingdrainedforxnumberofminutes.Thedomaininthisfunctionwouldbecontinuous.Therewouldbeanx-valuerepresentinganypointintimeuntilthetankisdrainedsoconnectingthepointstoformastraightlinewouldbeappropriate(Note:thecontextoftheproblemlimitsthevaluesthatxcanrepresenttopositivevalues,sincetimecannotbenegative.).

• Functionscanberepresentedasorderedpairs,tables,graphs,equations,physicalmodels,orinwords.Anygivenrelationshipcanberepresentedusingmultiplerepresentations.

• Theequationy=mx+bdefinesalinearfunctionwhosegraph(solution)isastraightline.Theequationofalinearfunctioncanbedeterminedgiventheslope,m,andthey-intercept,b.Verbaldescriptionsofpracticalsituationsthatcanbe

24

modeledbyalinearfunctioncanalsoberepresentedusinganequation.- Example:Writetheequationofalinearfunctionwhoseslopeis!

!andy-

interceptis−4,orlocatedatthepoint(0,−4).Theequationofthislinecanbefoundbysubstitutingthevaluesfortheslope,m=!!,andthey-intercept,b=−4,intothegeneralformofalinearfunctiony=mx

+b.Thus,theequationwouldbey=!!x–4.

- Example:Johnchargesa$30flatfeetotroubleshootapersonalwatercraftthatisnotworkingproperlyand$50perhourneededforanyrepairs.Writealinearfunctionthatrepresentsthetotalcost,yofapersonalwatercraftrepair,basedonthenumberofhours,x,neededtorepairit.Assumethatthereisnoadditionalchargeforparts.Inthispracticalsituation,they-intercept,b,wouldbe$30,torepresenttheinitialflatfeetotroubleshootthewatercraft.Theslope,m,wouldbe$50,sincethatwouldrepresenttherateperhour.Theequationtorepresentthissituationwouldbey=50x+30.

• Aproportionalrelationshipbetweentwovariablescanberepresentedbyalinearfunctiony=mxthatpassesthroughthepoint(0,0)andthushasay-interceptof0.Thevariableyresultsfromxbeingmultipliedbym,therateofchangeorslope.

• Thelinearfunctiony=x+brepresentsalinearfunctionthatisanon-proportionaladditiverelationship.Thevariableyresultsfromthevaluebbeingaddedtox.Inthislinearrelationship,thereisay-interceptofb,andtheconstantrateofchangeorslopewouldbe1.Inalinearfunctionwithaslopeotherthan1,thereisacoefficientinfrontofthexterm,whichrepresentstheconstantrateofchange,orslope.

• Proportionalrelationshipsandadditiverelationshipsbetweentwoquantitiesarespecialcasesoflinearfunctionsthatarediscussedingradesevenmathematics.

Key Vocabulary: relation, function, table, graph, rule, linear, coordinate plane, order pairs, x/y axis, discrete, continuous, domain, range, input, output, independent, dependent, slope, y-intercept, proportional


Teacher Notes and Elaborations: DIXI – Domain, Input, X-coordinate, Indep. Variable ROYD – Range, Ouput, Y-coordinate, Depend. Variable. Domain – the set of all x-coordinates. Range – the set of all y-coordinates. Make a mapping to determine if a relation is also a function. Repeated domain or range members listed only once.

25

SOL Reporting Category: Measurement and Geometry --Angles - Dates: 2/13-2/17 Current Standard and



8.6Thestudentwilla) verifybymeasuringanddescribethe

relationshipsamongverticalangles,adjacentangles,supplementaryangles,andcomplementaryangles;and

b) measureanglesoflessthan360°.• Measure angles of less than 360° to the nearest

degree, using appropriate tools. • Identify and describe the relationships between

angles formed by two intersecting lines. • Identify and describe the relationship between

pairs of angles that are vertical. • Identify and describe the relationship between

pairs of angles that are supplementary. • Identify and describe the relationship between

pairs of angles that are complementary. • Identify and describe the relationship between

pairs of angles that are adjacent. • Use the relationships among supplementary,

complementary, vertical, and adjacent angles to solve practical problems.†

Vertical Articulation:

6.13 and 7.7

8.5Thestudentwillusetherelationshipsamongpairsofanglesthatareverticalangles,adjacentangles,supplementaryangles,andcomplementaryanglestodeterminethemeasureofunknownangles.

• Identifyanddescribetherelationshipbetweenpairsofanglesthatarevertical,adjacent,supplementary,andcomplementary.

• Usetherelationshipsamongsupplementary,complementary,vertical,andadjacentanglestosolveproblems,includingpracticalproblems,involvingthemeasureofunknownangles.

• Verticalanglesareapairofnonadjacentanglesformedbytwointersectinglines.Verticalanglesarecongruentandshareacommonvertex.

• Complementaryanglesareanytwoanglessuchthatthesumoftheirmeasuresis90°.

• Supplementaryanglesareanytwoanglessuchthatthesumoftheirmeasuresis180°.

• Complementaryandsupplementaryanglesmayormaynotbeadjacent.

• Adjacentanglesareanytwonon-overlappinganglesthatshareacommonrayandacommonvertex.

Key Vocabulary: angles, degrees, vertical, adjacent, nonadjacent, supplementary, complementary, intersecting lines, common ray, common vertex


Teacher Notes and Elaborations: Angle A is 74 degrees. What is the measure of its complement? What is the measure of its supplement? Given a picture of intersecting lines, if Angle 3 is 34 degrees, find the measures of the remaining angles based upon angle relationships.

26

SOL Reporting Category: Measurement and Geometry —Pythagorean Theorem - Dates: 2/21-2/24

Current Standard and Essential Knowledge and Skills



8.10 Thestudentwilla) verifythePythagoreanTheorem;andb) applythePythagoreanTheorem.• Identify the parts of a right triangle (the

hypotenuse and the legs). • Verify a triangle is a right triangle given the

measures of its three sides. • Verify the Pythagorean Theorem, using

diagrams, concrete materials, and measurement. • Find the measure of a side of a right triangle,

given the measures of the other two sides. • Solve practical problems involving right

triangles by using the Pythagorean Theorem. 8.5 Connection Vertical Articulation: 6.12 and 7.6

8.9Thestudentwilla) verifythePythagorean

Theorem;andb) applythePythagorean

Theorem.• VerifythePythagorean

Theorem,usingdiagrams,concretematerials,andmeasurement.(a)

• Determinewhetheratriangleisarighttrianglegiventhemeasuresofitsthreesides.(b)

• Determinethemeasureofasideofarighttriangle,giventhemeasuresoftheothertwosides.(b)

• SolvepracticalproblemsinvolvingrighttrianglesbyusingthePythagoreanTheorem.(b)

• ThePythagoreanTheoremisessentialforsolvingproblemsinvolvingrighttriangles.

• Therelationshipbetweenthesidesandanglesofrighttrianglesareusefulinmanyappliedfields.

• Inarighttriangle,thesquareofthelengthofthehypotenuseequalsthesumofthesquaresofthelegs.ThisrelationshipisknownasthePythagoreanTheorem:a2+b2=c2.

• ThePythagoreanTheoremisusedtodeterminethemeasureofany

27

oneofthethreesidesofarighttriangleifthemeasuresoftheothertwosidesareknown.

• TheconverseofthePythagoreanTheoremstatesthatifthesquareofthelengthofthehypotenuseequalsthesumofthesquaresofthelegsinatriangle,thenthetriangleisarighttriangle.Thiscanbeusedtodeterminewhetheratriangleisarighttrianglegiventhemeasuresofitsthreesides.

• Wholenumbertriplesthatarethemeasuresofthesidesofrighttriangles,suchas(3,4,5),(6,8,10),(9,12,15),and(5,12,13),arecommonlyknownasPythagoreantriples.

• Thehypotenuseofarighttriangleisthesideoppositetherightangle.• Thehypotenuseofarighttriangleisalwaysthelongestsideoftheright

triangle.• Thelegsofarighttriangleformtherightangle.

Key Vocabulary: right triangle, right angle, legs, hypotenuse, Pythagorean triples


Teacher Notes and Elaborations: A scalene triangle has sides that measure 12 cm, 5 cm, and 15 cm. Could it also be a right angle?

SOL Reporting Category: Measurement and Geometry —Composite Figures - Dates: 2/12-3/2 Current Standard and



8.11 Thestudentwillsolvepracticalareaandperimeterproblemsinvolvingcompositeplanefigures.

• Subdivide a figure into triangles, rectangles, squares, trapezoids and semicircles. Estimate the area of subdivisions and combine to determine the area of the composite figure.

• Use the attributes of the subdivisions to determine the perimeter and circumference of a figure.

• Apply perimeter, circumference and area formulas to solve practical problems.

Vertical Articulation: 6.10abcd

8.10Thestudentwillsolveareaandperimeterproblems,includingpracticalproblems,involvingcompositeplanefigures.

• Subdivideaplanefigureintotriangles,rectangles,squares,trapezoids,parallelograms,andsemicircles.Determinetheareaofsubdivisionsandcombinetodeterminetheareaofthecompositeplanefigure.

• Subdivideaplanefigureintotriangles,rectangles,squares,trapezoids,parallelograms,andsemicircles.Usetheattributesofthesubdivisionstodeterminethe

• Aplanefigureisanytwo-dimensionalshapethatcanbedrawninaplane.

• Apolygonisaclosedplanefigurecomposedofatleastthreelinesegmentsthatdonotcross.

• Theperimeteristhepathordistancearoundanyplanefigure.Theperimeterofacircleiscalledthecircumference.

• Theareaofacompositefigurecanbefoundbysubdividingthefigureintotriangles,rectangles,squares,trapezoids,parallelograms,circles,andsemicircles,calculatingtheirareas,andcombiningtheareastogetherbyadditionand/orsubtractionbaseduponthegivencompositefigure.

• Theareaofarectangleiscomputedbymultiplyingthelengthsoftwoadjacentsides(A=lw).

• Theareaofatriangleiscomputedbymultiplyingthemeasureofitsbasebythemeasureofitsheightanddividingtheproductby2ormultiplyingby!

! (A= !!

!orA= !

!𝑏ℎ).

28

perimeterofthecompositeplanefigure.

• Applyperimeter,circumference,andareaformulastosolvepracticalproblemsinvolvingcompositeplanefigures.

• Theareaofaparallelogramiscomputedbymultiplyingthemeasureofitsbasebythemeasureofitsheight(A=bh).

• Theareaofatrapezoidiscomputedbytakingtheaverageofthemeasuresofthetwobasesandmultiplyingthisaveragebytheheight(𝐴 = !

!ℎ(𝑏! + 𝑏!)).

• Theareaofacircleiscomputedbymultiplyingpitimestheradiussquared(𝐴 = 𝜋𝑟!).

• Thecircumferenceofacircleisfoundbymultiplyingpibythediameterormultiplyingpiby2timestheradius(𝐶 = 𝜋𝑑or𝐶 = 2𝜋𝑟).

• Theareaofasemicircleishalftheareaofacirclewiththesamediameterorradius

Key Vocabulary: area, perimeter, circumference, formulas, composite figures, plane figure, subdivide, two-dimensional, polygon, triangles, square, trapezoids, parallelogram, semicircles,



Practice: Determine the shaded area of a shape inside of a larger shape. Legs make the base and height of a right triangle.

29

SOL Reporting Category: Measurement and Geometry—Orthographic Projections - Dates: 3/5-3/9 Current Standard and



8.9 Thestudentwillconstructathree-dimensionalmodel,giventhetoporbottom,side,andfrontviews.

• Construct three-dimensional models, given the top or bottom, side, and front views.

• Identify three-dimensional models given a two-dimensional perspective.

Vertical Articulation:

8.8 Thestudentwillconstructathree-dimensionalmodel,giventhetoporbottom,side,andfrontviews.

• Constructthree-dimensionalmodels,giventhetoporbottom,side,andfrontviews.

• Identifythree-dimensionalmodelsgivenatwo-dimensionalperspective.

• Identifythetwo-dimensionalperspectivefromthetoporbottom,side,andfrontview,givenathree-dimensionalmodel.

• Athree-dimensionalobjectcanberepresentedasatwo-dimensionalmodelwithviewsoftheobjectfromdifferentperspectives.

• Three-dimensionalmodelsofgeometricsolidscanbeusedtounderstandperspectiveandprovidetactileexperiencesindeterminingtwo-dimensionalperspectives.

• Three-dimensionalmodelsofgeometricsolidscanberepresentedonisometricpaper.

• Thetopviewisamirrorimageofthebottomview.

Key Vocabulary: three-dimensional, two-dimensional, geometric solid, top/side/bottom/front views, mirror image



30

SOL Reporting Category: Measurement and Geometry –Surface Area & Volume - Dates: 3/12-3/23 Current Standard and



8.7 Thestudentwilla) investigateandsolvepracticalproblems

involvingvolumeandsurfaceareaofprisms,cylinders,cones,andpyramids;and

b) describehowchangingonemeasuredattributeofthefigureaffectsthevolumeandsurfacearea.

• Distinguish between situations that are applications of surface area and those that are applications of volume.

• Investigate and compute the surface area of a square or triangular pyramid by finding the sum of the areas of the triangular faces and the base using concrete objects, nets, diagrams and formulas.

• Investigate and compute the surface area of a cone by calculating the sum of the areas of the side and the base, using concrete objects, nets, diagrams and formulas.

• Investigate and compute the surface area of a right cylinder using concrete objects, nets, diagrams and formulas.

• Investigate and compute the surface area of a rectangular prism using concrete objects, nets, diagrams and formulas.

• Investigate and compute the volume of prisms, cylinders, cones, and pyramids, using concrete objects, nets, diagrams, and formulas.

• Solve practical problems involving volume and surface area of prisms, cylinders, cones, and pyramids.

• Compare and contrast the volume and surface area of a prism with a given set of attributes with the volume of a prism where one of the attributes has been increased by a factor of 2, 3, 5 or 10.

• Describe the two-dimensional figures that result from slicing three-dimensional figures parallel to the base (e.g., as in plane sections of right rectangular prisms and right rectangular pyramids).†

8.6 Thestudentwilla) solveproblems,includingpractical

problems,involvingvolumeandsurfaceareaofconesandsquare-basedpyramids;and

b) describehowchangingonemeasuredattributeofarectangularprismaffectsthevolumeandsurfacearea.

• Distinguishbetweensituationsthatareapplicationsofsurfaceareaandthosethatareapplicationsofvolume.(a)

• Determinethesurfaceareaofconesandsquare-basedpyramidsbyusingconcreteobjects,nets,diagramsandformulas.(a)

• Determinethevolumeofconesandsquare-basedpyramids,usingconcreteobjects,diagrams,andformulas.(a)

• Solvepracticalproblemsinvolvingvolumeandsurfaceareaofconesandsquare-basedpyramids.(a)

• Describehowthevolumeofarectangularprismisaffectedwhenonemeasuredattributeismultipliedbyafactorof!

!,!!,!!,2,3,or4.(b)

• Describehowthesurfaceareaofarectangularprismisaffectedwhenonemeasuredattributeismultipliedbyafactorof!

!or2.(b)

• Apolyhedronisasolidfigurewhosefacesareallpolygons.• Netsaretwo-dimensionalrepresentationsofathree-dimensional

figurethatcanbefoldedintoamodelofthethree-dimensionalfigure.

• Surfaceareaofasolidfigureisthesumoftheareasofthesurfacesofthefigure.

• Volumeistheamountacontainerholds.• Arectangularprismisapolyhedronthathasacongruentpairof

parallelrectangularbasesandfourfacesthatarerectangles.Arectangularprismhaseightverticesandtwelveedges.Inthiscourse,prismsarelimitedtorightprismswithbasesthatarerectangles.

• Thesurfaceareaofarectangularprismisthesumoftheareasofthefacesandbases,foundbyusingtheformulaS.A.=2lw+2lh+2wh.Allsixfacesarerectangles.

• Thevolumeofarectangularprismiscalculatedbymultiplyingthelength,widthandheightoftheprismorbyusingtheformulaV=lwh.

• Acubeisarectangularprismwithsixcongruent,squarefaces.Alledgesarethesamelength.Acubehaseightverticesandtwelveedges.

• Aconeisasolidfigureformedbyafacecalledabasethatisjoinedtoavertex(apex)byacurvedsurface.Inthisgradelevel,conesarelimitedtorightcircularcones.

• Thesurfaceareaofarightcircularconeisfoundbyusingtheformula,S.A.=πr2+πrl,wherelrepresentstheslantheightofthecone.Theareaofthebaseofacircularconeisπr2.

• ThevolumeofaconeisfoundbyusingV=!!πr2h,wherehisthe

heightandπr2istheareaofthebase.• Asquare-basedpyramidisapolyhedronwithasquarebaseand

fourfacesthataretriangleswithacommonvertex(apex)abovethebase.Inthisgradelevel,pyramidsarelimitedtorightregularpyramidswithasquarebase.

• Thevolumeofapyramidis!!Bh,whereBistheareaofthebase

andhistheheight.• Thesurfaceareaofapyramidisthesumoftheareasofthe

triangularfacesandtheareaofthebase,foundbyusingtheformulaS.A.=!

!lp+Bwherelistheslantheight,pistheperimeter

31

Vertical Articulation: 6.9 and 7.5abc

ofthebaseandBistheareaofthebase.• ThevolumeofapyramidisfoundbyusingtheformulaV=!

!Bh,

whereBistheareaofthebaseandhistheheight.• Thevolumeofprismscanbefoundbydeterminingtheareaofthe

baseandmultiplyingthatbytheheight.• Theformulafordeterminingthevolumeofconesandcylindersare

similar.Forcones,youaredetermining!!ofthevolumeofthe

cylinderwiththesamesizebaseandheight.ThevolumeofaconeisfoundbyusingV=!

!πr2h.Thevolumeofacylinderistheareaof

thebaseofthecylindermultipliedbytheheight,foundbyusingtheformula,V=πr2h,wherehistheheightandπr2istheareaofthebase.

• Thecalculationofdeterminingsurfaceareaandvolumemayvarydependingupontheapproximationforpi.Commonapproximationsforπinclude3.14,!!

!,orthepibuttononthecalculator.

• Whenthemeasurementofoneattributeofarectangularprismischangedthroughmultiplicationordivisionthevolumeincreasesbythesamefactorbywhichtheattributeincreased.Forexample,ifaprismhasavolumeof2·3·4,thevolumeis24cubicunits.However,ifoneoftheattributesisdoubled,thevolumedoubles.Thatis,2·3·8,thevolumeis48cubicunitsor24doubled.

• Whenoneattributeofarectangularprismischangedthroughmultiplicationordivision,thesurfaceareaisaffecteddifferentlythanthevolume.Theformulaforsurfaceareaofarectangularprismis2(lw)+2(lh)+2(wh)whenthewidthisdoubledthenfourfacesareaffected.Forexample,arectangularprismwithlength=7in.,width=4in.,andheight=3in.wouldhaveasurfaceareaof2 7 ∙ 4 + 2 7 ∙ 3 + 2 4 ∙ 3 or122squareinches.Iftheheightisdoubledto6inchesthenthesurfaceareawouldbefoundby2 7 ∙ 4 + 2 7 ∙ 6 + 2(4 ∙ 6)or188squareinches.

Key Vocabulary: volume, surface area, prisms, cylinder, cones, square-based pyramid, attribute, two-dimensional, three dimensional, polyhedron, nets


Teacher Notes and Elaborations: Strategy: Write the entire formula then plug in replacement values below. Follow the order of operations.

32

Brian has two rectangular prisms. The height of one prism is 5 times the height of the other prism. Both prisms have the same length and width.

Which best describes the volume of the taller prism?

a) The volume is 5 times greater. b) The volume is 10 times greater. c) The volume is 20 times greater. d) The volume is 100 times greater.

Calc. Volume of each and compare results.

33

SOL Reporting Category: Measurement and Geometry --Transformations - Dates: 3/13-3/17 Current Standard and



8.8 Thestudentwilla) applytransformationstoplane

figures;andb) identifyapplicationsof

transformations.• Demonstrate the reflection of a

polygon over the vertical or horizontal axis on a coordinate grid.

• Demonstrate 90°, 180°, 270°, and 360°clockwise and counterclockwise rotations of a figure on a coordinate grid. The center of rotation will be limited to the origin.

• Demonstrate the translation of a polygon on a coordinate grid.

• Demonstrate the dilation of a polygon from a fixed point on a coordinate grid.

• Identify practical applications of transformations including, but not limited to, tiling, fabric, and wallpaper designs, art and scale drawings.

• Identify the type of transformation in a given example.

Vertical Articulation: 6.11ab and 7.8

8.7 Thestudentwilla) givenapolygon,applytransformations,to

includetranslations,reflections,anddilations,inthecoordinateplane;and

b) identifypracticalapplicationsoftransformations.

• Givenapreimageinthecoordinateplane,identifythecoordinateoftheimageofapolygonthathasbeentranslatedvertically,horizontally,oracombinationofboth.(a)

• Givenapreimageinthecoordinateplane,identifythecoordinatesoftheimageofapolygonthathasbeenreflectedoverthex-ory-axis.(a)

• Givenapreimageinthecoordinateplane,identifythecoordinatesoftheimageofarighttriangleorarectanglethathasbeendilated.Scalefactorsarelimitedto!

!,!!,2,3,or4.

Thecenterofthedilationwillbetheorigin.(a)• Givenapreimageinthecoordinateplane,identifythecoordinatesoftheimageofapolygonthathasbeentranslatedandreflectedoverthex-ory-axis,orreflectedoverthex-ory-axisandthentranslated.(a)

• Sketchtheimageofapolygonthathasbeentranslatedvertically,horizontally,oracombinationofboth.(a)

• Sketchtheimageofapolygonthathasbeenreflectedoverthex-ory-axis.(a)

• Sketchtheimageofadilationofarighttriangleorarectanglelimitedtoascalefactorof!

!,!!,2,3,or4.The

centerofthedilationwillbetheorigin.(a)• Sketchtheimageofapolygonthathasbeentranslatedandreflectedoverthex-ory-axis,orreflectedoverthex-ory-axisandthentranslated.(a)

• Identifythetypeoftranslationinagivenexample.(a,b)

• Identifypracticalapplicationsoftransformationsincluding,butnotlimitedto,tiling,fabric,wallpaperdesigns,art,andscaledrawings.(b)

• Translationsandreflectionsmaintaincongruencebetweenthepreimageandimagebutchangelocation.Dilationsbyascalefactorotherthan1produceanimagethatisnotcongruenttothepreimagebutissimilar.Reflectionschangetheorientationoftheimage.

• Atransformationofafigure,calledpreimage,changesthesize,shape,and/orpositionofthefiguretoanewfigure,calledtheimage.

• AtransformationofpreimagepointAcanbedenotedastheimageA’(readas“Aprime”).

• Areflectionisatransformationinwhichanimageisformedbyreflectingthepreimageoveralinecalledthelineofreflection.Eachpointontheimageisthesamedistancefromthelineofreflectionasthecorrespondingpointinthepreimage.

• Atranslationisatransformationinwhichanimageisformedbymovingeverypointonthepreimagethesamedistanceinthesamedirection.

• Adilationisatransformationinwhichanimageisformedbyenlargingorreducingthepreimageproportionallybyascalefactorfromthecenterofdilation(limitedtotheoriginingradeeight).Adilationofafigureandtheoriginalfigurearesimilar.Thecenterofdilationmayormaynotbeonthepreimage.

• Theresultoffirsttranslatingandthenreflectingoverthex-ory-axismaynotresultinthesametransformationofreflectingoverthex-ory-axisandthentranslating.

• Practicalapplicationsmayinclude,butarenotlimitedto,thefollowing:o Areflectionofaboatinwatershowsanimageoftheboatflippedupsidedownwiththewaterlinebeingthelineofreflection;

o Atranslationofafigureonawallpaperpatternshowsthesamefigureslidthesamedistanceinthesamedirection;and

o Adilationofamodelairplaneistheproductionmodeloftheairplane.

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Key Vocabulary: transformations, plane figures, reflection, rotation, translation, dilation, preimage, image, coordinates, polygon, scale factor, reducing, enlarging, prime (A’), similar


Teacher Notes and Elaborations: Studentsneedtopracticedemonstratingtransformationsinthecoordinateplaneandshowingknowledgeofthespecifictransformationrules.

Grade 8– Crosswalk (Summary of Revisions): 2016 · PDF file · 2017-07-12Grade 8– Crosswalk (Summary of Revisions): 2016 Mathematics Standards of Learning and Curriculum.....

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