Fourth Grade Math Pacing Guide - Home - Waynesboro ... Grade Math Pacing Guide Revised: May 2017...

FourthGradeMathPacingGuide

Revised:May2017

NumberandNumberSense

Virginia2016MathematicsStandardsofLearningCurriculumFramework

Introduction

WaynesboroPublicSchools

301PineAvenueWaynesboro,Virginia22980

www.waynesboro.k12.va.us

MathDiet

40%ConceptualLearning20%ProblemSolving20%Fluency20%ProceduralSkills

ImportantPacingGuideInformation:

ThispacingguiderepresentsWaynesboroPublicSchool’scurriculum,basedonthe2009and2016VirginiaMathematicsStandardsofLearningforVirginiaPublicSchoolsfoundintheStateCurriculumFramework.

Pacingguidesarealwaysaworkinprogress.Pleasekeepnotesregardingyourexperienceswiththepacingguidesandassociatedassessments.Thisinformationwillbeusedtoimprovethepacingguidesovertime.

ResourceshavebeendividedtomatchspecificSOLstrands.AnyresourceshighlightedinlightbluecorrelatewithmultipleSOLstrandswithinaunitandhavenotbeenseparatedbyspecificSOL.

Anyinformationfoundwithin[brackets]islistedasapartoftheSOLbutisnotexplicitlytaughtorassessedduringtheunit.

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The2016MathematicsStandardsof LearningCurriculumFramework, a companiondocument to the2016MathematicsStandardsof Learning, amplifies theMathematics Standards of Learningand further defines the content knowledge, skills, and understandings that aremeasured by the Standards of Learningassessments.ThestandardsandCurriculumFrameworkarenotintendedtoencompasstheentirecurriculumforagivengradelevelorcourse.SchooldivisionsareencouragedtoincorporatethestandardsandCurriculumFrameworkintoabroader,locallydesignedcurriculum.TheCurriculumFrameworkdelineatesingreaterspecificitytheminimumcontentthatallteachersshouldteachandallstudentsshouldlearn.Teachersareencouragedtogobeyondthestandardsaswellastoselectinstructionalstrategiesandassessmentmethodsappropriateforallstudents.

TheCurriculumFrameworkalsoservesasaguideforStandardsofLearningassessmentdevelopment.Studentsareexpectedtocontinuetoconnectandapplyknowledgeandskills fromStandardsofLearningpresentedinpreviousgradesastheydeepentheirmathematicalunderstanding. Assessment itemsmaynotandshouldnotbeaverbatimreflectionoftheinformationpresentedintheCurriculumFramework.

Eachtopicinthe2016MathematicsStandardsofLearningCurriculumFrameworkisdevelopedaroundtheStandardsofLearning.TheformatoftheCurriculumFramework facilitates teacherplanningby identifying the key concepts, knowledge, and skills that shouldbe the focusof instruction for each standard. TheCurriculumFrameworkisdividedintotwocolumns:UnderstandingtheStandardandEssentialKnowledgeandSkills.Thepurposeofeachcolumnisexplainedbelow.

UnderstandingtheStandardThis section includes mathematical content and key concepts that assist teachers in planning standards-focused instruction The statements may providedefinitions,explanations,examples,andinformationregardingconnectionswithinandbetweengradelevel(s)/course(s).

EssentialKnowledgeandSkillsThis sectionprovidesadetailedexpansionof themathematics knowledgeand skills thateach student shouldknowandbeable todemonstrate. This isnotmeanttobeanexhaustivelistofstudentexpectations.

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MathematicalProcessGoalsforStudents

The content of the mathematics standards is intended to support the following five process goals for students: becoming mathematical problem solvers,communicatingmathematically,reasoningmathematically,makingmathematicalconnections,andusingmathematicalrepresentationstomodelandinterpretpracticalsituations.Practicalsituationsincludereal-worldproblemsandproblemsthatmodelreal-worldsituations.

MathematicalProblemSolving

Studentswillapplymathematicalconceptsandskillsandtherelationshipsamongthemtosolveproblemsituationsofvaryingcomplexities.Studentsalsowillrecognize and create problems from real-world data and situations within and outside mathematics and then apply appropriate strategies to determineacceptablesolutions.Toaccomplishthisgoal,studentswillneedtodeveloparepertoireofskillsandstrategiesforsolvingavarietyofproblems.Amajorgoalofthemathematicsprogramistohelpstudentsapplymathematicsconceptsandskillstobecomemathematicalproblemsolvers.

MathematicalCommunication

Students will communicate thinking and reasoning using the language of mathematics, including specialized vocabulary and symbolic notation, to expressmathematicalideaswithprecision.Representing,discussing,justifying,conjecturing,reading,writing,presenting,andlisteningtomathematicswillhelpstudentsclarifytheirthinkinganddeepentheirunderstandingofthemathematicsbeingstudied.Mathematicalcommunicationbecomesvisiblewherelearninginvolvesparticipationinmathematicaldiscussions.

MathematicalReasoning

Studentswillrecognizereasoningandproofasfundamentalaspectsofmathematics.Studentswilllearnandapplyinductiveanddeductivereasoningskillstomake,test,andevaluatemathematicalstatementsandtojustifystepsinmathematicalprocedures.Studentswilluselogicalreasoningtoanalyzeanargumentandtodeterminewhetherconclusionsarevalid.Inaddition,studentswillusenumbersensetoapplyproportionalandspatialreasoningandtoreasonfromavarietyofrepresentations.

MathematicalConnections

Studentswillbuilduponpriorknowledgetorelateconceptsandprocedures fromdifferent topicswithinmathematicsandseemathematicsasan integratedfield of study. Through the practical application of content and process skills, students will make connections among different areas of mathematics andbetweenmathematicsandotherdisciplines,andtoreal-worldcontexts.Scienceandmathematicsteachersandcurriculumwritersareencouragedtodevelopmathematicsandsciencecurriculathatsupport,apply,andreinforceeachother.

MathematicalRepresentationsStudentswillrepresentanddescribemathematicalideas,generalizations,andrelationshipsusingavarietyofmethods.Studentswillunderstandthatrepresentationsofmathematicalideasareanessentialpartoflearning,doing,andcommunicatingmathematics.Studentsshouldmakeconnectionsamongdifferentrepresentations–physical,visual,symbolic,verbal,andcontextual–andrecognizethatrepresentationisbothaprocessandaproduct.

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InstructionalTechnologyThe use of appropriate technology and the interpretation of the results from applying technology toolsmust be an integral part of teaching, learning, andassessment.However,facilityintheuseoftechnologyshallnotberegardedasasubstituteforastudent’sunderstandingofquantitativeandalgebraicconceptsandrelationshipsorforproficiencyinbasiccomputations.Studentsmustlearntouseavarietyofmethodsandtoolstocompute, includingpaperandpencil,mentalarithmetic,estimation,andcalculators.Inaddition,graphingutilities,spreadsheets,calculators,dynamicapplications,andothertechnologicaltoolsarenowstandardformathematicalproblemsolvingandapplicationinscience,engineering,businessandindustry,government,andpracticalaffairs.

Calculatorsandgraphingutilitiesshouldbeusedbystudentsforexploringandvisualizingnumberpatternsandmathematicalrelationships,facilitatingreasoningandproblemsolving,andverifyingsolutions.However,accordingtotheNationalCouncilofTeachersofMathematics,“…theuseofcalculatorsdoesnotsupplantthe need for students to develop proficiency with efficient, accurate methods of mental and pencil-and-paper calculation and in making reasonableestimations.” State and local assessments may restrict the use of calculators in measuring specific student objectives that focus on number sense andcomputation.Onthegradethreestateassessment,allobjectivesareassessedwithouttheuseofacalculator.Onthestateassessmentsforgradesfourthroughseven,objectivesthatareassessedwithouttheuseofacalculatorareindicatedwithanasterisk(*).

ComputationalFluencyMathematicsinstructionmustdevelopstudents’conceptualunderstanding,computationalfluency,andproblem-solvingskills.Thedevelopmentofrelatedconceptualunderstandingandcomputationalskillsshouldbebalancedandintertwined,eachsupportingtheotherandreinforcinglearning.

Computationalfluencyreferstohavingflexible,efficient,andaccuratemethodsforcomputing.Studentsexhibitcomputationalfluencywhentheydemonstratestrategicthinkingandflexibilityinthecomputationalmethodstheychoose,understand,andcanexplain,andproduceaccurateanswersefficiently.

Thecomputationalmethodsusedbyastudentshouldbebasedonthemathematicalideasthatthestudentunderstands,includingthestructureofthebase-tennumbersystem,numberrelationships,meaningofoperations,andproperties.Computationalfluencywithwholenumbersisagoalofmathematicsinstructionintheelementarygrades.Studentsshouldbefluentwiththebasicnumbercombinationsforadditionandsubtractionto20bytheendofgradetwoandthoseformultiplicationanddivisionbytheendofgradefour.Studentsshouldbeencouragedtousecomputationalmethodsandtoolsthatareappropriateforthecontextandpurpose.AlgebraReadinessThesuccessfulmasteryofAlgebraIiswidelyconsideredtobethegatekeepertosuccessinthestudyofupper-levelmathematics.“Algebrareadiness”describesthemasteryof,andtheabilitytoapply,theMathematicsStandardsofLearning,includingtheMathematicalProcessGoalsforStudents,forkindergartenthroughgradeeight.ThestudyofalgebraicthinkingbeginsinkindergartenandisprogressivelyformalizedpriortothestudyofthealgebraiccontentfoundintheAlgebraIStandardsofLearning.Includedintheprogressionofalgebraiccontentispatterning,generalizationofarithmeticconcepts,proportionalreasoning,andrepresentingmathematicalrelationshipsusingtables,symbols,andgraphs.TheK-8MathematicsStandardsofLearningformaprogressionofcontentknowledgeanddevelopthereasoningnecessarytobewell-preparedformathematicscoursesbeyondAlgebraI,includingGeometryandStatistics.

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Equity

“Addressingequityandaccessincludesbothensuringthatallstudentsattainmathematicsproficiencyandincreasingthenumbersofstudentsfromallracial,ethnic,linguistic,gender,andsocioeconomicgroupswhoattainthehighestlevelsofmathematicsachievement.”

–NationalCouncilofTeachersofMathematics

Mathematics programs should have an expectation of equity by providing all students access to quality mathematics instruction and offerings that areresponsivetoandrespectfulofstudents’priorexperiences,talents,interests,andculturalperspectives.Successfulmathematicsprogramschallengestudentstomaximize their academic potential and provide consistentmonitoring, support, and encouragement to ensure success for all. Individual students should beencouragedtochoosemathematicalprogramsofstudythatchallenge,enhance,andextendtheirmathematicalknowledgeandfutureopportunities.

Studentengagementisanessentialcomponentofequityinmathematicsteachingandlearning.Mathematicsinstructionalstrategiesthatrequirestudentstothinkcritically,toreason,todevelopproblem-solvingstrategies,tocommunicatemathematically,andtousemultiplerepresentationsengagesstudentsbothmentallyandphysically.Studentengagementincreaseswithmathematicaltasksthatemploytheuseofrelevant,appliedcontextsandprovideanappropriatelevelofcognitivechallenge.All students, includingstudentswithdisabilities,gifted learners,andEnglish language learnersdeservehigh-qualitymathematicsinstructionthataddressesindividuallearningneeds,maximizingtheopportunitytolearn.

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Focus3-5 StrandIntroduction

STRAND:NUMBERANDNUMBERSENSEMathematics instruction in grades three through five should continue to foster the development of number sense,with greater emphasis on decimals andfractions. Studentswith good number sense understand themeaning of numbers, developmultiple relationships and representations among numbers, andrecognizetherelativemagnitudeofnumbers.Theyshouldlearntherelativeeffectofoperatingonwholenumbers,fractions,anddecimalsandlearnhowtousemathematicalsymbolsandlanguagetorepresentproblemsituations.Numberandoperationsensecontinuestobethecornerstoneofthecurriculum.

Thefocusofinstructioningradesthreethroughfiveallowsstudentstoinvestigateanddevelopanunderstandingofnumbersensebymodelingnumbers,usingdifferentrepresentations(e.g.,physicalmaterials,diagrams,mathematicalsymbols,andwordnames),andmakingconnectionsamongmathematicsconceptsaswell as to other content areas. Students should develop strategies for reading, writing, and judging the size of whole numbers, fractions, and decimals bycomparingthem,usingavarietyofmodelsandbenchmarksasreferents(e.g.,or0.5).Studentsshouldapplytheirknowledgeofnumberandnumbersensetoinvestigateandsolveavarietyofproblemtypes.STRAND:COMPUTATIONANDESTIMATIONComputationandestimationingradesthreethroughfiveshouldfocusondevelopingfluencyinmultiplicationanddivisionwithwholenumbersandshouldbegintoextendstudents’understandingoftheseoperationstoworkwithdecimals.Instructionshouldfocusoncomputationactivitiesthatenablestudentstomodel,explain, and develop proficiencywith basic facts and algorithms. These proficiencies are often developed as a result of investigations and opportunities todevelopalgorithms.Additionally,opportunitiestodevelopandusevisualmodels,benchmarks,andequivalents,toaddandsubtractfractions,andtodevelopcomputationalproceduresfortheadditionandsubtractionofdecimalsareapriorityforinstructioninthesegrades.Multiplicationanddivisionwithdecimalswillbeexploredingradefive.

Students should develop an understanding of howwhole numbers, fractions, and decimals are written andmodeled; an understanding of themeaning ofmultiplicationanddivision,includingmultiplerepresentations(e.g.,multiplicationasrepeatedadditionorasanarray);anabilitytoidentifyanduserelationshipsamongoperationstosolveproblems(e.g.,multiplicationastheinverseofdivision);andtheabilitytousepropertiesofoperationstosolveproblems(e.g.,7´28isequivalentto(7´20)+(7´8)).

Students shoulddevelop computational estimation strategiesbasedonanunderstandingofnumber concepts, properties, and relationships. Practice shouldincludeestimationofsumsanddifferencesofcommonfractionsanddecimals,usingbenchmarks(e.g.,+mustbelessthan1becausebothfractionsarelessthan).Usingestimation,studentsshoulddevelopstrategiestorecognizethereasonablenessoftheirsolutions.

Additionally,studentsshouldenhancetheirabilitytoselectanappropriateproblem-solvingmethodfromamongestimation,mentalmathematics,paper-and-pencilalgorithms,andtheuseofcalculatorsandcomputers.Withactivitiesthatchallengestudentstousethisknowledgeandtheseskillstosolveproblemsinmanycontexts,studentsdevelopthefoundationtoensuresuccessandachievementinhighermathematics.

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STRAND:MEASUREMENTANDGEOMETRYStudents in grades three through five should be actively involved inmeasurement activities that require a dynamic interaction between students and theirenvironment.Studentscanseetheusefulnessofmeasurementifclassroomexperiencesfocusonmeasuringobjectsandestimatingmeasurements.Textbookexperiencescannotsubstituteforactivitiesthatutilizemeasurementtoanswerquestionsaboutrealproblems.Theapproximatenatureofmeasurementdeservesrepeatedattentionatthislevel.Itisimportanttobegintoestablishsomebenchmarksbywhichtoestimateorjudgethesizeofobjects.Studentsusestandardandnonstandard,age-appropriatetoolstomeasureobjects.Studentsalsouseage-appropriatelanguageofmathematicstoverbalizethemeasurementsoflength,weight/mass,liquidvolume,area,perimeter,temperature,andtime.Thefocusof instructionshouldbeanactiveexplorationoftherealworld inordertoapplyconcepts fromthetwosystemsofmeasurement(metricandU.S.Customary), to measure length, weight/mass, liquid volume/capacity, area, perimeter, temperature, and time. Students’ understanding of measurementcontinuestobeenhancedthroughexperiencesusingappropriatetoolssuchasrulers,balances,clocks,andthermometers.Thestudyofgeometryhelpsstudentsrepresentandmakesenseoftheworld.Ingradesthreethroughfive,reasoningskillstypicallygrowrapidly,andtheseskillsenablestudentstoinvestigategeometricproblemsofincreasingcomplexityandtostudyhowgeometrictermsrelatetogeometricproperties.Studentsdevelopknowledgeabouthowgeometricfiguresrelatetoeachotherandbegintousemathematicalreasoningtoanalyzeandjustifypropertiesandrelationshipsamongfigures.Students discover these relationships by constructing, drawing, measuring, comparing, and classifying geometric figures. Investigations should includeexplorationswitheverydayobjectsandotherphysicalmaterials.Exercisesthataskstudentstovisualize,draw,andcomparefigureswillhelpthemnotonlytodevelopanunderstandingoftherelationships,buttodeveloptheirspatialsenseaswell. Intheprocess,definitionsbecomemeaningful,relationshipsamongfiguresareunderstood,andstudentsarepreparedtousetheseideastodevelopinformalarguments.Students investigate, identify, draw representations of, and describe the relationships among points, lines, line segments, rays, and angles. Students applygeneralizationsaboutlines,angles,andtrianglestodevelopunderstandingaboutcongruence;parallel,intersecting,andperpendicularlines;andclassificationoftriangles.ThevanHiele theoryofgeometricunderstandingdescribeshowstudents learngeometryandprovidesa framework forstructuringstudentexperiences thatshouldleadtoconceptualgrowthandunderstanding.

● Level 0: Pre-recognition. Geometric figures are not recognized. For example, students cannot differentiate between three-sided and four-sidedpolygons.

● Level1:Visualization.Geometricfiguresarerecognizedasentities,withoutanyawarenessofthepartsoffiguresorrelationshipsbetweencomponentsofafigure.Studentsshouldrecognizeandnamefiguresanddistinguishagivenfigurefromothersthatlooksomewhatthesame.(Thisistheexpectedlevelofstudentperformanceduringkindergartenandgradeone.)

● Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of geometric figures.

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(Studentsareexpectedtotransitiontothislevelduringgradestwoandthree.)● Level3:Abstraction.Definitionsaremeaningful,withrelationshipsbeingperceivedbetweenpropertiesandbetweenfigures.Logicalimplicationsand

classinclusionsareunderstood,buttheroleandsignificanceofdeductionisnotunderstood.(Studentsshouldtransitiontothislevelduringgradesfiveandsixandfullyattainitbeforetakingalgebra.)

STRAND:PROBABILITYANDSTATISTICSStudentsenteringgradesthreethroughfivehavebeguntoexploretheconceptofthemeasurementofchanceandareabletodeterminepossibleoutcomesofgiven events. Students have utilized a variety of random generator tools, including random number generators (number cubes), spinners, and two-sidedcounters. Ingamesituations,studentshavehadinitialexperiences inpredictingwhetheragameisfairornotfair.Furthermore,studentsareableto identifyeventsaslikelyorunlikelytohappen.Thusthefocusofinstructioningradesthreethroughfiveistodeepentheirunderstandingoftheconceptsofprobabilityby·offeringopportunitiestosetupmodelssimulatingpracticalevents;·engagingstudentsinactivitiestoenhancetheirunderstandingoffairness;and· engaging students in activities that instill a spirit of investigation and exploration and providing students with opportunities to usemanipulatives.Thefocusofstatisticsinstructionistoassiststudentswithfurtherdevelopmentandinvestigationofdatacollectionstrategies.Studentsshouldcontinuetofocuson:·posingquestions;·collectingdataandorganizingthisdataintomeaningfulgraphs,charts,anddiagramsbasedonissuesrelatingtopracticalexperiences;·interpretingthedatapresentedbythesegraphs;·answeringdescriptivequestions(“Howmany?”“Howmuch?”)fromthedatadisplays;·identifyingandjustifyingcomparisons(“Whichisthemost?Whichistheleast?”“Whichisthesame?Whichisdifferent?”)abouttheinformation;·comparingtheirinitialpredictionstotheactualresults;and·communicatingtootherstheirinterpretationofthedata.Throughastudyofprobabilityandstatistics,studentsdeveloparealappreciationofdataanalysismethodsaspowerfulmeansfordecisionmaking.

STRAND:PATTERNSFUNCTIONSANDALGEBRAStudentsenteringgradesthreethroughfivehavehadopportunitiestoidentifypatternswithinthecontextoftheschoolcurriculumandintheirdailylives,andtheycanmakepredictionsaboutthem.Theyhavehadopportunities touse informal languagetodescribethechangeswithinapatternandtocomparetwopatterns.Studentshavealsobeguntoworkwiththeconceptofavariablebydescribingmathematicalrelationshipswithinapattern.Thefocusofinstructionistohelpstudentsdevelopasoliduseofpatterningasaproblemsolvingtool.Atthislevel,patternsarerepresentedandmodeledinavarietyofways,includingnumeric,geometric,andalgebraicformats.Studentsdevelopstrategiesfororganizinginformationmoreeasilytounderstandvarioustypesofpatternsandfunctionalrelationships.Theyinterpretthestructureofpatternsbyexploringanddescribingpatternsthatinvolvechange,andtheybegintogeneralizethesepatterns.Byinterpretingmathematicalsituationsandmodels,studentsbegintorepresentthese,usingsymbolsandvariablestowrite“rules”forpatterns,todescriberelationshipsandalgebraicproperties,andtorepresentunknownquantities.

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IntroductiontoInvestigationsProgramandRoutines

Duringthefirstweeksofschool,studentsshouldlearnInvestigationsgamesandroutinesthatconnecttotheessentialknowledgeinthisfirstunit.Theyshouldalsobecomefamiliarandcomfortablewithgroupingandorganizationalstructures;suchascircle,seat,center.

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SettingRoutines:Multiplication&DivisionFacts3rdGradeReviewWeek1-2SOL3.53.4c(Review)3.5Thestudentwillrecallmultiplicationfactsthroughthetwelvestable,andthecorrespondingdivisionfacts.3.4cThestudentwilldemonstratefluencywithmultiplicationfactsof0,1,2,5,and10

Number&NumberSense:PlaceValue,Comparing&RoundingWeeks3-7SOL4.1,4.3a-c4.1Thestudentwill

a)identifyorallyandinwritingtheplacevalueforeachdigitinawholenumberexpressedthroughmillions;read,write,andidentifytheplaceandvalueofeachdigitinanine-digitwholenumber;b)compareandordertwowholenumbersexpressedthroughmillionsusingsymbols(>,<,or=);andc)roundwholenumbersexpressedthroughmillionstothenearestthousand,tenthousand,andhundredthousand.UNDERSTANDINGTHESTANDARD

ESSENTIAL

UNDERSTANDINGSESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES

(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)

• ThestructureoftheBase-10numbersystemisbaseduponasimplepatternoftens,inwhichthevalueofeachplaceistentimesthevalueoftheplacetoitsright.

• Placevaluereferstothevalueofeachdigitanddependsuponthepositionofthedigitinthenumber.Forexample,inthenumber7,864,352,theeightisinthehundredthousandsplace,andthevalueofthe8iseighthundredthousandor800,000.

• Wholenumbersmaybewritteninavarietyofformsformats:

• Standard:1,234,567• Written:onemillion,twohundredthirty-four

thousand,fivehundredsixty-seven• Expanded:(1×1,000,000)+(2×100,000)+(3×

10,000)+(4×1,000)+(5×100)+(6×10)+(7×1)• Numbersarearrangedintogroupsofthreeplaces

calledperiods(ones,thousands,millions,…).Placeswithintheperiodsrepeat(hundreds,tens,ones).Commasareusedtoseparatetheperiods.Knowingthe

Allstudentsshould• Understandthe

relationshipsintheplacevaluesysteminwhichthevalueofeachplaceistentimesthevalueoftheplacetoitsright.

• Usethepatternsintheplacevaluesystemtoreadandwritenumbers.

• Understandthatreadingplacevaluecorrectlyisessentialwhencomparingnumbers.

• Understandthatroundinggivesaclosenumbertousewhenexactnumbersarenotneededforthesituationathand.

• Developstrategiesfor

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto• Readnine-digitwholenumbers,

presentedinstandardformandrepresentthesamenumberinwrittenform.(a)

• Writenine-digitwholenumbersinstandardformwhenthenumbersarepresentedorallyorinwrittenform.(a)

• Identifyandcommunicate,bothorallyandinwrittenform,theplacedandvalueforeachdigitinanine-digitwholenumbersexpressedthroughtheonemillionsplace.(a)

• Readwholenumbersthroughtheonemillionsplacethatarepresentedinstandardformat,andselectthe

4.1a–changingfrom7digitsto9digits4.1b–includesordering(limitedtofournumbers)

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valueoftheplaceplacevalueandperiodofanumberhelpsstudentsdeterminefindvaluesofdigitsinanynumberaswellasreadandwritenumbers.Studentsatthislevelwillworkwithnumbersthroughthemillionsperiod(nine-digitnumbers).

• Readingandwritinglargenumbersshouldbemeaningfulforstudents.Experiencescanbeprovidedthatrelatepracticalsituations(e.g.,numbersfoundinthestudents’environmentincludingpopulation,numberofschoollunchessoldstatewideinaday,etc.).

• ConcretematerialssuchasBase-10blocksandbundlesofsticksmaybeusedtorepresentwholenumbersthroughthousands.Largernumbersmayberepresentedbydigitcardsandplacevaluecharts,oronnumberlines.

• Numberlinesareusefultoolswhendevelopingaconceptualunderstandingofroundingwithwholenumbers.Whengivenanumbertoround,locateitonthenumberline.Next,determinetheclosestmultiplesofthousand,ten-thousand,orhundred-thousanditisbetween.Then,identifytowhichitiscloser.

• Mathematicalsymbols(>,<)usedtocomparetwounequalnumbersarecalledinequalitysymbols.

• Aprocedureforcomparingtwonumbersbyexaminingplacevaluemayincludethefollowing:

• Comparethedigitsinthenumberstodeterminewhich

numberisgreater(orwhichisless).• Useanumberlinetoidentifytheappropriate

placementofthenumbersbasedontheplacevalueofthedigits.

• Usetheappropriatesymbol>or<orwordsgreaterthanorlessthantocomparethenumbersintheorderinwhichtheyarepresented.

• Ifbothnumbershavethesamevalue,usethesymbol=

rounding.

matchingnumberinwrittenformat.• Writewholenumbersthroughthe

onemillionsplaceinstandardformatwhenthenumbersarepresentedorallyorinwrittenformat.

• Identifyandusethesymbolsforgreaterthan,lessthan,andequalto.

• Comparetwowholenumbersexpressedthroughtheonemillions,usingthewordsgreaterthan,lessthan,equalto,andnotequaltoorusingthesymbols>,<,=,or≠.(b)

• Orderuptofourwholenumbersexpressedthroughmillions.(b)

• Roundwholenumbersexpressedthroughmillionstheonemillionsplacetothenearestthousand,tenthousand,andhundred-thousandplace.(c)

• Identifytherangeofnumbersthatroundtoagiventhousand,tenthousand,andhundredthousand.(c)

4,367,000?4,368,000

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orwordsequalto.• Astrategyforroundingnumberstothenearest

thousand,tenthousand,andhundredthousandisasfollows:

• Useanumberlinetodeterminetheroundednumber(e.g.,whenrounding4,367,925tothenearestthousand,identifythe‘thousands’thenumberwouldfallbetweenonthenumberline,thendeterminethethousandthatthenumberisclosestto):

• Lookoneplacetotherightofthedigittowhichyouwishtoround.

• Ifthedigitislessthan5,leavethedigitintheroundingplaceasitis,andchangethedigitstotherightoftheroundingplacetozero.

• Ifthedigitis5orgreater,add1tothedigitintheroundingplaceandchangethedigitstotherightoftheroundingplacetozero.

EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf

● Wholenumber● Digit● >● <● =● ≠● Ordering● Compare● Greaterthan

● Lessthan● Equalto● Equivalent● Round● Place● *Placevalue● Value● One(s)

● Ten● Hundred● Thousand● Tenthousand● Hundredthousand● Million● TenMillion● HundredMillion

● Period● Range● Standar

dform● Written

form● Expande

dform

RESOURCES

BenchmarkLiteracy:SportsMath–Unit9

4,367,000?4,368,000

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4.3 Thestudentwilla)read,write,represent,andidentifydecimalsexpressedthroughthousandths;b)rounddecimalstothenearestwholenumber;,tenth,andhundredth;c)compareandorderdecimals;andd)givenamodel,writethedecimalandfractionequivalents.**Onthestateassessment,itemsmeasuringthisobjectiveareassessedwithouttheuseofacalculator.

UNDERSTANDINGTHESTANDARD ESSENTIALUNDERSTANDINGS

ESSENTIALKNOWLEDGEANDSKILLS

IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe

2018-2019pacingguide)● Decimalnumbersexpandthesetofwholenumbersand,like

fractions,areawayofrepresentingpartofawhole.● Thestructureofthebase-tenBase-10numbersystemisbased

uponasimplepatternoftens,whereeachplaceistentimesthevalueoftheplacetoitsright.Thisisknownasaten-to-oneplacevaluerelationship(e.g.,in2.35,3isinthetenthsplacesinceittakestenone-tenthstomakeonewhole).Usebase-tenproportionalmanipulatives,suchasplacevaluemats/charts,decimalsquares,base-tenblocks,metersticks,aswellastheten-to-onenon-proportionalmodel,money,toinvestigatethisrelationship.

● Understandingthesystemoftensmeansthattententhsrepresentsonewhole,tenhundredthsrepresentsonetenth,tenthousandthsrepresentsonehundredth.

● AdecimalpointseparatesthewholenumberplacesfromtheplacesthatarelessthanonePlacevaluesextendinfinitelyintwodirectionsfromadecimalpoint.Anumbercontainingadecimalpointiscalledadecimalnumberorsimplyadecimal.

● Toreaddecimals,– readthewholenumbertotheleftofthedecimalpoint,if

thereisone;– readthedecimalpointas“and”;– readthedigitstotherightofthedecimalpointjustasyou

wouldreadawholenumber;and– saythenameoftheplacevalueofthedigitinthesmallest

place.● Anydecimallessthan1willincludealeadingzero(e.g.,0.125).For

example0.125whichcanbereadas“zeroandonehundredtwenty-fivethousandths”oras“onehundredtwenty-fivethousandths.”

Allstudentsshould● Understandthe

placevaluestructureofdecimalsandusethisstructuretoread,write,andcomparedecimals.

● Understandthatdecimalnumberscanberoundedtoanestimatewhenexactnumbersarenotneededforthesituationathand.

● Understandthatdecimalsareroundedinawaythatissimilartothewaywholenumbersarerounded.

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto

• Readandwritedecimalsexpressedthroughthousandths,usingBase-10manipulatives,drawings,andnumericalsymbols.(a)

• Representandidentifydecimalsexpressedthroughthousandths,usingBase-10manipulatives,pictorialrepresentations,andnumericalsymbols(e.g.,relatetheappropriatedrawingto0.05).(a)

● Investigatetheten-to-oneplacevaluerelationshipfordecimalsthroughthousandths,usingbase-tenBase-10manipulatives(e.g.,placevaluemats/charts,decimalsquares,base-ten

4.3distaughtduringweeks21-27.

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● Decimalsmaybewritteninavarietyofforms:– Standard:26.537– Written:twenty-sixandfivehundredthirty-seven

thousandths– Expanded:(2×10)+(6×1)+(5×0.1)+(3×0.01)+(7×

0.001).20+6+0.5+0.03+0.007.● Strategiesforroundingwholenumberscanbeappliedtorounding

decimals.● Numberlinesareusefultoolswhendevelopingaconceptual

understandingofroundingwithdecimals.Whengivenadecimaltoroundtothenearestwholeoronesplace,locateitonthenumberline.Next,determinethetwowholenumbersitisbetween.Then,identifytowhichitiscloser.

● Theprocedureforroundingdecimalnumbersissimilartotheprocedureforroundingwholenumbers.

● Astrategyforroundingdecimalnumberstothenearesttenthandhundredthisasfollows:– Lookoneplacetotherightofthedigityouwanttoroundto.– Ifthedigitis5orgreater,add1tothedigitintherounding

place,anddropthedigitstotherightoftheroundingplace.– Ifthedigitislessthan5,leavethedigitintheroundingplace

asitis,anddropthedigitstotherightoftheroundingplace.● Differentstrategiesforroundingdecimalsinclude:

– Useanumberlinetolocateadecimalbetweentwonumbers.Forexample,18.83iscloserto18.8thanto18.9.

– Comparethedigitsinthenumberstodeterminewhichnumberisgreater(orwhichisless).

– Comparethevalueofdecimals,usingthesymbols>,<,=(e.g.,0.83>0.8or0.19<0.2).

– Orderthevalueofdecimals,fromleasttogreatestandgreatesttoleast(e.g.,0.83,0.821,0.8).

Base-10blocks,money).(a)

● Identifyandcommunicate,bothorallyandinwrittenform,thepositionandvalueofadecimalthroughthousandthsForexample,in0.385(e.g.,given0.385,the8isinthehundredthsplaceandhasavalueof0.08).(a)

● Rounddecimalsexpressedthroughthousandthstothenearestwholenumber,tenth,andhundredth..(b)

● Comparetwodecimalsexpressedthroughthousandths,usingthesymbols(>,<,=,≠)and/orwords(greaterthan,lessthan.equalto,andnotequalto).(c)

● Orderasetofuptofourdecimals,expressedthroughthousandths,fromleasttogreatestorgreatesttoleast.(c)


● Decimal● Decimalpoint● Decimalnumber● Standard● Written● Expanded

● Tenth● Hundredth● Thousandth● Compare● Order● >

● <● =● ≠● Greaterthan● Lessthan● Equalto

● Notequalto● Equivalent● Fraction● Fractionalpart● Mixednumber

15

Addition&SubtractionwithEstimationWeeks8-11SOLs4.4,4.16Multiplication&DivisionwithEstimationWeeks12-18SOLs4.4,4.164.4Thestudentwill

a)estimatesums,differences,[products,andquotients]ofwholenumbers;b)add,subtract,[andmultiply]wholenumbers;c)[dividewholenumbers,findingquotientswithandwithoutremainders;and]d)solvesingle-stepandmultistepaddition,subtraction,[andmultiplication]problemswithwholenumbers.

a) demonstratefluencywithmultiplicationfactsthrough12×12,andthecorrespondingdivisionfacts;*b) estimateanddeterminesums,differences,andproductsofwholenumbers;*c) estimateanddeterminequotientsofwholenumbers,withandwithoutremainders;*andd) createandsolvesingle-stepandmultisteppracticalproblemsinvolvingaddition,subtraction,andmultiplication,andsingle-step

practicalproblemsinvolvingdivisionwithwholenumbers.

*Onthestateassessment,itemsmeasuringthisobjectiveareassessedwithouttheuseofacalculator.UNDERSTANDINGTHESTANDARD ESSENTIAL

UNDERSTANDINGSESSENTIALKNOWLEDGE

ANDSKILLSIMPORTANTCHANGES


● Asumistheresultofaddingtwoormorenumbers.● Adifferenceistheamountthatremainsafteronequantityis

subtractedfromanother.● Anestimateisanumberclosetoanexactsolution.An

estimatetellsabouthowmuchorabouthowmany.● Differentstrategiesforestimatingincludeusingcompatible

numberstoestimatesumsanddifferencesandusingfront-endestimationforsumsanddifferences.– Compatiblenumbersarenumbersthatareeasytowork

withmentally.Numberpairsthatareeasytoaddorsubtractarecompatible.Whenestimatingasum,replaceactualnumberswithcompatiblenumbers(e.g.,52+74canbeestimatedbyusingthecompatiblenumbers50+75).Whenestimatingadifference,usenumbersthatareclosetotheoriginalnumbers.Tensandhundredsareeasytosubtract(e.g.,83–38isclose

Allstudentsshould● Developandusestrategies

toestimatewholenumbersumsanddifferencesandtojudgethereasonablenessofsuchresults.

● Understandthatadditionandsubtractionareinverseoperations.

● Understandthatdivisionistheoperationofmakingequalgroupsorequalshares.Whentheoriginalamountandthenumberofsharesareknown,dividetofindthesizeofeachshare.Whentheoriginalamount

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Estimatewholenumber

sums,differences,products,andquotients.

● Refineestimatesbyadjustingthefinalamount,usingtermssuchascloserto,between,andalittlemorethan.

● Determinethesumordifferenceoftwowholenumbers,each999,999orless,inverticaland

TheUnderstandingtheStandardandtheEssentialKnowledgeandSkillscolumnsareVERYDIFFERENT!Newstandardsarelocatedbelowtheold,inaseparatetable!

16

to80–40).– Thefront-endstrategyforestimatingiscomputingwith

thefrontdigits.Front-endestimationforadditioncanbeusedevenwhentheaddendshaveadifferentnumberofdigits.Theprocedurerequirestheadditionofthevaluesofthedigitsinthegreatestofthesmallestnumber.Forexample:

2367 → 2300 243 → 200 +1186 → +1100 3600● Front-endorleading-digitestimationalwaysgivesasumless

thantheactualsum;however,theestimatecanbeadjustedorrefinedsothatitisclosertotheactualsum.

● Additionisthecombiningofquantities;itusesthefollowingterms:

addend → 45,623 addend → +37,846 sum → 83,469● Subtractionistheinverseofaddition;ityieldsthedifference

betweentwonumbersandusesthefollowingterms: minuend → 45,698 subtrahend → –32,741 difference → 12,957● Beforeaddingorsubtractingwithpaperandpencil,addition

andsubtractionproblemsinhorizontalformshouldberewritteninverticalformbylininguptheplacesvertically.

● UsingBase-10materialstomodelandstimulatediscussionaboutavarietyofproblemsituationshelpsstudentsunderstandregroupingandenablesthemtomovefromtheconcretetothepictorial,totheabstract.Regroupingisusedinadditionandsubtractionalgorithms.Inaddition,whenthesuminaplaceis10ormore,isusedtoregroupthesumssothatthereisonlyonedigitineachplace.Insubtraction,whenthenumber(minuend)inaplaceisnotenoughfromwhichtosubtract,regroupingisrequired.

● Acertainamountofpracticeisnecessarytodevelopfluencywithcomputationalstrategiesformultidigitnumbers;

andthesizeofeachshareareknown,dividetofindthenumberofshares.

● Understandthatmultiplicationanddivisionareinverseoperations.

● Understandvariousrepresentationsofdivisionandthetermsusedindivisionaredividend,divisor,andquotient.

dividend÷divisor=quotient quotient

divisor ● Understandhowtosolve

single-stepandmultistepproblemsusingwholenumberoperations.

horizontalformwithorwithoutregrouping,usingpaperandpencil,andusingacalculator.

● Estimateandfindtheproductsoftwowholenumberswhenonefactorhastwodigitsorfewerandtheotherfactorhasthreedigitsorfewer,usingpaperandpencilandcalculators.

● Estimateandfindthequotientoftwowholenumbers,givenaone-digitdivisorandatwo-orthree-digitdividend.

● Solvesingle-stepandmultistepproblemsusingwholenumberoperations.

● Verifythereasonablenessofsums,differences,products,andquotientsofwholenumbersusingestimation.

17

however,thepracticemustbemeaningful,motivating,andsystematicifstudentsaretodevelopfluencyincomputation,whethermentally,withmanipulativematerials,orwithpaperandpencil.

● Calculatorsareanappropriatetoolforcomputingsumsanddifferencesoflargenumbers,particularlywhenmasteryofthealgorithmhasbeendemonstrated.

● Thetermsassociatedwithmultiplicationare factor → 376 factor → ×23 product → 8,648● Onemodelofmultiplicationisrepeatedaddition.● Anothermodelofmultiplicationisthe“PartialProduct”

model. 24 ×3 12← Multiplytheones:3×4=12 +60← Multiplythetens:3×20=60 72● Anothermodelofmultiplicationisthe“AreaModel”(which

alsorepresentspartialproducts)andshouldbemodeledfirstwithBase-10blocks.(e.g.,23x68)

● Studentsshouldcontinuetodevelopfluencywithsingle-digitmultiplicationfactsandtheirrelateddivisionfacts.

● Calculatorsshouldbeusedtosolveproblemsthatrequiretediouscalculations.

● Estimationshouldbeusedtocheckthereasonablenessoftheproduct.Examplesofestimationstrategiesincludethefollowing:

Thefront-endmethod:multiplythefrontdigitsandthencompletetheproductbyrecordingthenumberofzerosfoundinthefactors.Itisimportanttodevelopunderstandingofthisprocessbeforeusingthestep-by-stepprocedure.

523 → 500 ×31 → ×30 15,000

18

Thisis3×5=15with3zeros.Compatiblenumbers:replacefactorswithcompatible

numbers,andthenmultiply.Opportunitiesforstudentstodiscoverpatternswith10andpowersof10shouldbeprovided.

64 → 64● ×11→ ×10

● Divisionistheoperationofmakingequalgroupsorequalshares.Whentheoriginalamountandthenumberofsharesareknown,dividetofindthesizeofeachshare.Whentheoriginalamountandthesizeofeachshareareknown,dividetofindthenumberofshares.BothsituationsmaybemodeledwithBase-10manipulatives.

● Multiplicationanddivisionareinverseoperations.● Termsusedindivisionaredividend,divisor,andquotient.dividend÷divisor=quotient quotient divisor )dividend● Opportunitiestoinventdivisionalgorithmshelpstudents

makesenseofthealgorithm.Teachersshouldteachdivisionbyvariousmethodssuchasrepeatedmultiplicationandsubtraction(partialquotients)beforeteachingthetraditionallongdivisionalgorithm.


● Closerto● Between● Alittlemorethan● Estimate

● Compatiblenumbers● Frontendestimation● +● Add

● Addition● Sum● -● Subtract

● Subtraction● Difference● Addend

19

4.4Thestudentwilla) demonstratefluencywithmultiplicationfactsthrough12×12,andthecorrespondingdivisionfacts;*b) estimateanddeterminesums,differences,andproductsofwholenumbers;*c) estimateanddeterminequotientsofwholenumbers,withandwithoutremainders;*andd) createandsolvesingle-stepandmultisteppracticalproblemsinvolvingaddition,subtraction,andmultiplication,andsingle-step

practicalproblemsinvolvingdivisionwithwholenumbers.

*Onthestateassessment,itemsmeasuringthisobjectiveareassessedwithouttheuseofacalculator.

UnderstandingtheStandard EssentialKnowledgeandSkills

• Computationalfluencyistheabilitytothinkflexiblyinordertochooseappropriatestrategiestosolveproblemsaccuratelyandefficiently.

• Thedevelopmentofcomputationalfluencyreliesonquickaccesstonumberfacts.Therearepatternsandrelationshipsthatexistinthefacts.Theserelationshipscanbeusedtolearnandretainthefacts.

• Acertainamountofpracticeisnecessarytodevelopfluencywithcomputationalstrategies;however,thepracticemustbemotivatingandsystematicifstudentsaretodevelopfluencyincomputation,whethermental,withmanipulativematerials,orwithpaperandpencil.

• Ingradethree,studentsdevelopedanunderstandingofthemeaningsofmultiplicationanddivisionofwholenumbersthroughactivitiesandpracticalproblemsinvolvingequal-sizedgroups,arrays,andlengthmodels.Inaddition,gradethreestudentshaveworkedonfluencyoffactsfor0,1,2,5,and10.

• Threemodelsusedtodevelopanunderstandingofmultiplicationinclude:

– Theequal-setsorequal-groupsmodellendsitselftosortingavarietyofconcreteobjectsintoequalgroupsandreinforcestheconceptofmultiplicationasawaytofindthetotalnumberofitemsinacollectionofgroups,withthesameamountineachgroup,andthetotalnumberofitemscanbefoundbyrepeatedadditionorskipcounting.

– Thearraymodel,consistingofrowsandcolumns(e.g.,threerowsoffourcolumnsfora3-by-4array),helpsbuildanunderstandingofthecommutativeproperty.


• Demonstratefluencywithmultiplicationthrough12×12,andthecorrespondingdivisionfacts.(a)

• Estimatewholenumbersums,differences,products,andquotients,withandwithoutcontext.(b,c)

• Applystrategies,includingplacevalueandthepropertiesofadditiontodeterminethesumordifferenceoftwowholenumbers,each999,999orless.(b)

• Applystrategies,includingplacevalueandthepropertiesofmultiplicationand/oraddition,todeterminetheproductoftwowholenumberswhenbothfactorshavetwodigitsorfewer.(b)

• Applystrategies,includingplacevalueandthepropertiesofmultiplicationand/oraddition,todeterminethequotientoftwowholenumbers,givenaone-digitdivisorandatwo-orthree-digitdividend,withandwithoutremainders.(c)

• Refineestimatesbyadjustingthefinalamount,usingtermssuchascloserto,between,andalittlemorethan.(b,c)

• Createandsolvesingle-stepandmultisteppracticalproblemsinvolvingaddition,subtraction,andmultiplicationwithwholenumbers.(d)

• Createandsolvesingle-steppracticalproblemsinvolvingdivisionwithwholenumbers.(d)

• Usethecontextinwhichapracticalproblemissituatedtointerpretthequotientandremainder.(d)

20

– Thelengthmodel(e.g.,anumberline)alsoreinforcesrepeatedadditionorskipcounting.

• Thereisaninverserelationshipbetweenmultiplicationanddivision.• Thenumberlinemodelcanbeusedtosolveamultiplicationproblemsuchas3×

6.Thisisrepresentedonthenumberlinebythreejumpsofsixorsixjumpsofthree,dependingonthecontextoftheproblem.

• Thenumberlinemodelcanbeusedtosolveadivisionproblemsuchas6÷3andisrepresentedonthenumberlinebynotinghowmanyjumpsofthreegofrom6to0.

• Thenumberlinemodelaboveshowstwojumpsofthreebetween6and0,answeringthequestionofhowmanyjumpsofthreegofrom6to0;therefore,6÷3=2.

● Inordertodevelopandusestrategiestolearnthemultiplicationfactsthroughthetwelvestable,studentsshoulduseconcretematerials,ahundredschart,andmentalmathematics.Strategiestolearnthemultiplicationfactsincludeanunderstandingofmultiples,propertiesofzeroandoneasfactors,commutativeproperty,andrelatedfacts.Investigatingarithmeticoperationswithwhole

numbershelpsstudentslearnaboutthedifferentpropertiesofarithmeticrelationships.Theserelationshipsremaintrueregardlessofthewholenumbers.

• Gradefourstudentsshouldexploreandapplythepropertiesofadditionandmultiplicationasstrategiesforsolvingaddition,subtraction,multiplication,anddivisionproblemsusingavarietyofrepresentations(e.g.,manipulatives,diagrams,andsymbols).

• Thepropertiesoftheoperationsare“rules”abouthownumbersworkandhowtheyrelatetooneanother.Studentsatthisleveldonotneedtousetheformaltermsforthesepropertiesbutshouldutilizethesepropertiestofurtherdevelopflexibilityandfluencyinsolvingproblems.Thefollowingpropertiesaremostappropriateforexplorationatthislevel:

0 1 2 3 4 5 6

21

– Theidentitypropertyofadditionstatesthatifzeroisaddedtoagivennumber,thesumisthesameasthegivennumber.Theidentitypropertyofmultiplicationstatesthatifagivennumberismultipliedbyone,theproductisthesameasthegivennumber.

– Thecommutativepropertyofadditionstatesthatchangingtheorderoftheaddendsdoesnotaffectthesum(e.g.,24+136=136+24).Similarly,thecommutativepropertyofmultiplicationstatesthatchangingtheorderofthefactorsdoesnotaffecttheproduct(e.g.,12×43=43×12).

– Theassociativepropertyofadditionstatesthatthesumstaysthesamewhenthegroupingofaddendsischanged(e.g.,15+(35+16)=(15+35)+16).Theassociativepropertyofmultiplicationstatesthattheproductstaysthesamewhenthegroupingoffactorsischanged[e.g.,16×(40×5)=(16×40)×5].

– Thedistributivepropertystatesthatmultiplyingasumbyanumbergivesthesameresultasmultiplyingeachaddendbythenumberandthenaddingtheproducts.Severalexamplesareshownbelow:– 3(9)=3(5+4)

3(5+4)=(3×5)+(3×4)

– 5×(3+7)=(5×3)+(5×7)

– (2×3)+(2×5)=2×(3+5)

– 9×239(20+3)180 27207

– 34×8

30 4

8

8×30=240

8×4=32

22

34

● Additionisthecombiningofquantities;itusesthefollowingterms:

addend → 45,623 addend → +37,846 sum → 83,469

● Subtractionistheinverseofaddition;ityieldsthedifferencebetweentwonumbersandusesthefollowingterms:

minuend → 45,698 subtrahend → –32,741 difference → 12,957

• Thetermsassociatedwithmultiplicationarelistedbelow:

factor → 76 factor → ×23 product → 1,748

• Inmultiplication,onefactorrepresentsthenumberofequalgroupsandtheotherfactorrepresentsthenumberinorsizeofeachgroup.Theproductisthetotalnumberinallofthegroups.

• Multiplicationcanalsorefertoamultiplicativecomparison,suchas:“GwenhassixtimesasmanystickersasPhillip”.Bothsituationsshouldbemodeledwithmanipulatives.

• Modelsofmultiplicationmayincluderepeatedadditionandcollectionsoflikesets,partialproducts,andareaorarraymodels.

• Divisionistheoperationofmakingequalgroupsorshares.Whentheoriginalamountandthenumberofsharesareknown,dividetodeterminethesizeofeachshare.Whentheoriginalamountandthesizeofeachshareareknown,dividetodeterminethenumberofshares.Bothsituationsmaybemodeledwithbase-tenmanipulatives.

• Divisionistheinverseofmultiplication.Termsusedindivisionaredividend,divisor,andquotient.

• Studentsbenefitfromexperienceswithvariousmethodsofdivision,suchas

23

repeatedsubtractionandpartialquotients.• Estimationcanbeusedtodeterminetheapproximationforandthentoverify

thereasonablenessofsums,differences,products,andquotientsofwholenumbers.Anestimateisanumberthatlieswithinarangeoftheexactsolution,andtheestimationstrategyusedinaparticularproblemdetermineshowclosethenumberistotheexactsolution.Anestimatetellsabouthowmuchorabouthowmany.

• Strategiessuchasroundingupordown,front-end,andcompatiblenumbersmaybeusedtoestimatesums,differences,products,andquotientsofwholenumbers.

• Theleastnumberofstepsnecessarytosolveasingle-stepproblemisone.• Theproblem-solvingprocessisenhancedwhenstudentscreateandsolvetheir

ownpracticalproblemsandmodelproblemsusingmanipulativesanddrawings.• Inproblemsolving,emphasisshouldbeplacedonthinkingandreasoningrather

thanonkeywords.Focusingonkey-wordssuchasinall,altogether,difference,etc.,encouragesstudentstoperformaparticularoperationratherthanmakesenseofthecontextoftheproblem.Akey-wordfocuspreparesstudentstosolvealimitedsetofproblemsandoftenleadstoincorrectsolutionsaswellaschallengesinupcominggradesandcourses.

• Extensiveresearchhasbeenundertakenoverthelastseveraldecadesregardingdifferentproblemtypes.Manyofthesestudieshavebeenpublishedinprofessionalmathematicseducationpublicationsusingdifferentlabelsandterminologytodescribethevariedproblemtypes.

• Studentsshouldexperienceavarietyofproblemtypesrelatedtomultiplicationanddivision.Someexamplesareincludedinthefollowingchart:à

24

• Studentsneedexposuretovarioustypesofpracticalproblemsinwhichtheymustinterpretthequotientandremainderbasedonthecontext.Thechartbelowincludesoneexampleofeachtypeofproblem.

• Studentswillsolveproblemsinvolvingthedivisionofdecimalsingradesfiveandsix.

4.16Thestudentwilla)recognizeanddemonstratethemeaningofequalityinanequation;andb)investigateanddescribetheassociativepropertyforaddition[andmultiplication.]

UNDERSTANDINGTHESTANDARD

ESSENTIALUNDERSTANDINGS

ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)

• Mathematicalrelationshipscanbeexpressedusingequations.

● Anexpressionisarepresentationofaquantity.Itismadeupofnumbers,variables,and/orcomputationalsymbols.Itdoesnothaveanequalsymbol(e.g.,8,15×12).

● Anequationrepresentstherelationshipbetweentwoexpressionsofequalvalue(e.g.,12×3=72÷2).

● Theequalsymbol(=)meansthatthevaluesoneithersideareequivalent(balanced).

Allstudentsshould● Understandthat

mathematicalrelationshipscanbeexpressedusingequations.

● Understandthatquantitiesonbothsidesofanequationmustbeequal.

● Understandthattheassociativepropertyfor

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Writeanequationtorepresentthe

relationshipbetweenequivalentmathematicalexpressions(e.g.,4´3=2´6;10+8=36÷2;12´4=60-12).

● Identifyandusetheappropriatesymboltodistinguishbetweenexpressionsthatareequalandexpressionsthatarenotequal,usingaddition,subtraction,

PleasenotepreviousSOL4.16b,isnowembeddedwithinnewSOL4.4EKS.

25

● Thenotequalsymbol(≠)meansthatthevaluesoneithersidearenotequivalent(notbalanced).

● Investigatingarithmeticoperationswithwholenumbershelpsstudentslearnaboutseveraldifferentpropertiesofarithmeticrelationships.Theserelationshipsremaintrueregardlessofthenumbers.

● Thecommutativepropertyforadditionstatesthatchangingtheorderoftheaddendsdoesnotaffectthesum(e.g.,4+3=3+4).Similarly,thecommutativepropertyformultiplicationstatesthatchangingtheorderofthefactorsdoesnotaffecttheproduct(e.g.,2×3=3×2).

● Theassociativepropertyforadditionstatesthatthesumstaysthesamewhenthegroupingofaddendsischanged[e.g.,15+(35+16)=(15+35)+16].Theassociativepropertyformultiplicationstatesthattheproductstaysthesamewhenthegroupingoffactorsischanged[e.g.,6×(3×5)=(6×3)×5].

additionmeansyoucanchangethegroupingsofthreeormoreaddendswithoutchangingthesum.

● Understandthattheassociativepropertyformultiplicationmeansyoucanchangethegroupingsofthreeormorefactorswithoutchangingtheproduct.

multiplication,anddivision(e.g.,4×12=8×6and64÷8≠8×8).

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Recognizeanddemonstratethatthe

equalssign(=)relatesequivalentquantitiesinanequation.

● Writeanequationtorepresentequivalentmathematicalrelationships(e.g.,4×3=2×6).

● Recognizeanddemonstrateappropriateuseoftheequalssigninanequation.

● Investigateanddescribetheassociativepropertyforadditionas(6+2)+3=6+(2+3).

● Investigateanddescribetheassociativepropertyformultiplicationas(3x2)x4=3x(2x4).


● =● ≠● Equal

● Equality● Equivalent

● Expression● Equation

● Variable● *AssociativeProperty

ofAddition

26

GraphingWeeks19-20SOLs4.144.14Thestudentwillcollect,organize,display,andinterpretdatafromavarietyofgraphs.

a) collect,organize,andrepresentdatainbargraphsandlinegraphs;b) interpretdatarepresentedinbargraphsandlinegraphs;andc) comparetwodifferentrepresentationsofthesamedata(e.g.,asetofdatadisplayedonachartandabargraph,achartandaline

graph,orapictographandabargraph).UNDERSTANDINGTHESTANDARD(BackgroundInformationforInstructorUseOnly)

ESSENTIALUNDERSTANDINGS ESSENTIALKNOWLEDGEANDSKILLS

IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)

● Dataanalysishelpsdescribedata,recognizepatternsortrends,andmakepredictions.

● Investigationsinvolvingpracticaldatashouldoccurfrequently;,anddatacanbecollectedthroughbriefclasssurveysorthroughmoreextendedprojectstakingmanydays.

● Studentsformulatequestions,predictanswerstoquestionsunderinvestigation,collectandrepresentinitialdata,andconsiderwhetherthedataanswerthequestions.

● Therearetwotypesofdata:categorical(e.g.,qualitative)andnumerical(e.g.,quantitative).Categoricaldataareobservationsaboutcharacteristicsthatcanbesortedintogroupsorcategories,whilenumericaldataarevaluesorobservationsthatcanbemeasured.Forexample,typesoffishcaughtwouldbecategoricaldatawhileweightsoffishcaughtwouldbenumericaldata.Whilestudentsneedtobeawareofthedifferences,theydonothavetoknowthetermsforeachtypeofdata.

● Bargraphsdisplaygroupeddatasuchascategoriesusingrectangularbarswhoselengthrepresentsthequantitythebarrepresents.Bargraphsshouldbeusedto

Allstudentsshould● Understandthedifferencebetween

representingcategoricaldataandrepresentingnumericaldata.

● Understandthatlinegraphsshowchangeovertime(numericaldata).

● Understandthatbargraphsshouldbeusedtocomparecountsofdifferentcategories(categoricaldata).

● Understandhowdatadisplayedinbarandlinegraphscanbeinterpretedsothatinformeddecisionscanbemade.

● Understandthatthetitleandlabelsofthegraphprovidethefoundationforinterpretingthedata.

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Collectdata,using,forexample,

observations,measurement,surveys,scientificexperiments,polls,orquestionnaires.(a)

● Organizedataintoachartortable.(a)

● ConstructanddisplayRepresentdatainbargraphs,labelingoneaxiswithequalwholenumberincrementsof1ormore(numericaldata)(e.g.,2,5,10,or100)andtheotheraxiswithcategoriesrelatedtothetitleofthegraph(categoricaldata)(e.g.,swimming,fishing,boating,andwaterskiingasthecategoriesof“FavoriteSummerSports”).(a)

● ConstructanddisplayRepresentdatainlinegraphs,labelingtheverticalaxiswithequalwholenumberincrementsof1ormore

27

comparecountsofdifferentcategories(categoricalorqualitativedata).Gridpapercanassiststudentsincreatinggraphswithgreateraccuracy.shouldbeusedtocomparecountsofdifferentcategories(categoricaldata).Usinggridpaperensuresmoreaccurategraphs. – Abargraphusesparallel,horizontalor

verticalbarstorepresentcountsforseveralcategories.Onebarisusedforeachcategory,withthelengthofthebarrepresentingthecountforthatcategory.

– Thereisspacebefore,between,andafterthebars.

– Theaxisthatdisplaysthescalerepresentingthecountforthecategoriesshouldbeginatzeroandextendoneincrementabovethegreatestrecordedpieceofdata.GradefourFourth-gradestudentsshouldcollectdatathatarerecordedinincrementsofwholenumbers,usuallymultiplesof1,2,5,10,or100.

– Eachaxisshouldbelabeled,andthegraphshouldbegivenatitle.

● Statementsrepresentingananalysisandinterpretationofthecharacteristicsofthedatainthegraph(e.g.,similaritiesanddifferences,leastandgreatest,thecategories,andtotalnumberofresponses)shouldbewritten.

● Linegraphsareusedtoshowhowtwocontinuousvariablesdatasets(numericalorquantitativedata)arerelated.Linegraphsmaybeusedtoshowhowonevariablechangesovertime(numericalorquantitativedata).Ifthisonevariableisnotcontinuous,thenabrokenlineisused.Bylookingatalinegraph,itcanbe

andthehorizontalaxiswithcontinuousdatacommonlyrelatedtotime(e.g.,hours,days,months,years,andage).Linegraphswillhavenomorethan10identifiedpointsalongacontinuumforcontinuousdata.Forexample,growthchartsshowingageversusheightplaceageonthehorizontalaxis(e.g.,1month,2months,3months,and4months).(a)

● Titlethegraphoridentifyanappropriatetitle.thetitleinagivengraphandLabeltheaxesoridentifytheappropriatelabels.theaxes.(a)

● Interpretdatabymakingobservationsfromsimplelineandbargraphsbydescribingthecharacteristicsofthedataandthedataasawhole(e.g.,thetimeperiodwhenthetemperatureincreasedthemost,thecategorywiththegreatest/least,categorieswiththesamenumberofresponses,similaritiesanddifferences,thetotalnumber).Onesetofdatawillberepresentedonagraph.Datapointswillbelimitedto30andcategoriesto8.(b)

● Interpretdatabymakinginferencesfrombargraphsandlinegraphs.(b)

● Interpretthedatatoanswerthequestionposed,andcomparetheanswertotheprediction(e.g.,“Thesummersportpreferredbymostisswimming,

28

determinedwhetherthevariablechangeinthedatasetisincreasing,decreasing,orstayingthesameovertime.– Thevaluesalongthehorizontalaxis

representcontinuousdataonagivenvariable,usuallysomemeasureoftime(e.g.,timeinyears,months,ordays).Thedatapresentedonalinegraphisreferredtoas“continuousdata,”asitrepresentsdatacollectedoveracontinuousperiodoftime.

– Thevaluesalongtheverticalaxisrepresenttherangeofvaluesinthecollecteddatasetatthegiventimeintervalonthehorizontalaxisarethescaleandrepresentthefrequencywithwhichthosevaluesoccurinthedataset.Thescalevaluesontheverticalaxisshouldrepresentequalincrementsofmultiplesofwholenumbers,fractions,ordecimals,dependinguponthedatabeingcollected.Thescaleshouldextendoneincrementabovethegreatestrecordedpieceofdata.

– Plotapointtorepresentthedatacollectedforeachtimeincrement,Uselinesegmentstoconnectthepointsinordermovinglefttoright.

– Eachaxisshouldbelabeled,andthegraphshouldbegivenatitle.

– Alinegraphtellswhethersomethinghasincreased,decreased,orstayedthesamewiththepassageoftime.Statementsrepresentingananalysisandinterpretationofthecharacteristicsofthedatainthegraphshouldbeincluded(e.g.,trendsofincreaseand/ordecrease,andleastandgreatest).

whichiswhatIpredictedbeforecollectingthedata.”).(b)

● Writeatleastonesentencetodescribetheanalysisandinterpretationofthedata,identifyingpartsofthedatathathavespecialcharacteristics,includingcategorieswiththegreatest,theleast,orthesame.(b)

● Comparetwodifferentrepresentationsofthesamedata(e.g.,asetofdatadisplayedonachartandabargraph;achartandalinegraph;apictographandabargraph).(c)

29


● Collect● Analyze● Interpret● Represent● Data● Bargraph

● *Linegraph● HorizontalNumberline● VerticalNumberLine● X-axis● Y-axis● Origin

● Increasing● Decreasing● Same● Continuousdata● Categoricaldata● Numericaldata

(Quantitativedata)

● Scale● *Frequency● Plot● Interval● Increment

RESOURCESBenchmarkLiteracy:SportsMath–Unit9

30

FractionsandDecimalsWeeks21-27SOLs4.2,4.3(d),4.5,4.64.2Thestudentwill

a)compareandorderfractionsandmixednumbers,withandwithoutmodels;*b)representequivalentfractions;*andc)identifythedivisionstatementthatrepresentsafraction,withmodelsandincontext.*Onthestateassessment,itemsmeasuringthisobjectiveareassessedwithouttheuseofacalculator.

UNDERSTANDINGTHESTANDARD(BackgroundInformationforInstructorUseOnly)



● Afractionisawayofrepresentingpartofawholeregion(i.e.,anareamodelasinaregion/areamodelorameasurementmodel),orpartofagroup(asini.e.,asetmodel),orpartofalength(i.e.,ameasurementmodel).Afractionisusedtonameapartofonethingorapartofacollectionofthings.

● Inthearea/regionandlength/measurementfractionmodels,thepartsmustbeequivalentequal.Inthesetmodel,theelementsofthesetdonothavetobeequal(i.e.,“Whatfractionoftheclassiswearingthecolorred?”).

● Inasetmodel,eachmemberofthesetisanequivalentpartoftheset.Insetmodels,thewholeneedstobedefined,butmembersofthesetmayhavedifferentsizesandshapes.Forinstance,ifawholeisdefinedasasetof10animals,theanimalswithinthesetmaybedifferent.Forexample,studentsshouldbeabletoidentifymonkeysasrepresenting!

!oftheanimalsinthefollowingset.

● Properfractions,improperfractions,andmixednumbers

aretermsoftenusedtodescribefractions.Aproperfractionisafractionwhosenumeratorislessthanthe

Allstudentsshould● Developan

understandingoffractionsaspartsofunitwholes,aspartsofacollection,andaslocationsonanumberline.

● Understandthatamixednumberisafractionthathastwoparts:awholenumberandaproperfraction.Themixednumberisthesumofthesetwoparts.

● Usemodels,benchmarks,andequivalentformstojudgethesizeoffractions.

● Recognizethatawholedividedintonineequalpartshassmallerpartsthanif

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Compareandordernomorethan

fourfractionshavinglikeandunlikedenominatorsof12orless,usingconcreteandpictorialmodels.manipulativemodelsanddrawings,suchasregion/areamodels.(a)

● Usebenchmarks(e.g.,0,!!or1)to

compareandordernomorethanfourfractionshavingunlikedenominatorsof12orless.(a)Compareandorderfractionshavingunlikedenominatorsof12orlessbycomparingthefractionstobenchmarks(e.g.,0,or1)todeterminetheirrelationshipstothebenchmarksorbyfindingacommondenominator.

● Compareandordernomorethanfourfractionswithlikedenominatorsof12orlessbycomparingnumberofparts(numerators)(e.g.,!

!<!

!).(a)

● Compareandordernomorethan

31

denominator.Animproperfractionisafractionwhosenumeratorisequaltoorgreaterthanthedenominator.Animproperfractionmaybeexpressedasamixednumber.Amixednumberiswrittenwithtwoparts:awholenumberandaproperfraction(e.g.,3!

!).

● Thevalueofafractionisdependentonboththenumberofequivalentpartsinawhole(denominator)andthenumberofthosepartsbeingconsidered(numerator).

● Themorepartsthewholeisdividedinto,thesmallertheparts(e.g.,!

!<!

!).

● Thedenominatortellshowmanyequalpartsareinthewholeorset.Thenumeratortellshowmanyofthosepartsarebeingcountedordescribed.

● Whenfractionshavethesamedenominator,theyaresaidtohave“commondenominators”or“likedenominators.”Comparingfractionswithlikedenominatorsinvolvescomparingonlythenumerators.

● Strategiesforcomparingfractionshavingunlikedenominatorsmayinclude:– comparingfractionstofamiliarbenchmarks(e.g.,0,!

!,

1);– findingdeterminingequivalentfractions,using

manipulativemodelssuchasfractionstrips,numberlines,fractioncircles,rods,patternblocks,cubes,Base-10base-tenblocks,tangrams,graphpaper,orpatternsinamultiplicationchartandpatterns;and

– findingdeterminingacommondenominatorbyfindingdeterminingtheleastcommonmultiple(LCM)ofbothdenominatorsandthenrewritingeachfractionasanequivalentfraction,usingtheLCMasthedenominator.

● Avarietyoffractionmodelsshouldbeusedtoexpandstudents’understandingoffractionsandmixednumbers:– Region/areamodels:asurfaceorareaissubdivided

intosmallerequalparts,andeachpartiscomparedwiththewhole(e.g.,fractioncircles,patternblocks,geoboards,gridpaper,colortiles).

– Setmodels:thewholeisunderstoodtobeasetofobjects,andsubsetsofthewholemakeupfractional

thewholehadbeendividedintofiveequalparts.

● Recognizeandgenerateequivalentformsofcommonlyusedfractionsanddecimals.

● Understandthedivisionstatementthatrepresentsafraction.

● Understandthatthemorepartsthewholeisdividedinto,thesmallertheparts(e.g.,

● <).

fourfractionswithlikenumeratorsandunlikedenominatorsof12orlessbycomparingthesizeoftheparts(e.g.,!

!<!!).(a)

● Compareandordernomorethanfourfractions(proper,orimproper),and/ormixednumbershavingdenominatorsof12orless.(a)

● Usethesymbols>,<,=,and≠tocomparethenumericalvalueoffractions(proper,orimproper),and/orandmixednumbershavingdenominatorsof12orless.(a)

● Representequivalentfractionsthroughtwelfths,usingregion/areamodels,setmodels,andmeasurement/lengthmodels.(b)

● Identifythedivisionstatementthatrepresentsafractionwithmodelsandincontext(e.g.,!

!meansthe

sameas3dividedby5!!or

representstheamountofmuffineachoffivechildrenwillreceivewhensharingthreemuffinsequally).(c)

32

parts(e.g.,counters,chips).– Measurementmodels:similartoareamodelsbut

lengthsinsteadofareasarecompared(e.g.,fractionstrips,rods,cubes,numberlines,rulers).

● Amixednumberhastwoparts:awholenumberandafraction.

● Equivalentfractionsnamethesameamount.Studentsshoulduseavarietyofrepresentationsandmodelstoidentifydifferentnamesforequivalentfractions.

● Whenpresentedwithafraction!!representingdivision,

thedivisionexpressionrepresentingthefractioniswrittenas3÷5.

● Thefraction!!maybeinterpretedastheamountofcake

eachpersonwillreceivewhen3cakesaredividedequallyamong4people.

● Studentsshouldfocusonfindingequivalentfractionsoffamiliarfractionssuchashalves,thirds,fourths,sixths,eighths,tenths,andtwelfths.

● Decimalsandfractionsrepresentthesamerelationships;however,theyarepresentedintwodifferentformats.Thedecimal0.25iswrittenas.Whenpresentedwiththefraction,thedivisionexpressionrepresentingafractioniswrittenas3dividedby5.


● Fraction● Fractionalpart● Partofawhole● Mixednumber● ImproperFraction

● ProperFraction● *Denominator● *Numerator● Commondenominator● Likedenominator

● Unlikedenominator● *Leastcommonmultiple● Leastcommondenominator● Compare

● Order● Equivalentfraction

4.3Thestudentwilld)givenamodel,writethedecimalandfractionequivalents.




● Decimalsandfractionsrepresentthesamerelationships;however,theyarepresentedin

Allstudentsshould● Understandthat

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematical

33

twodifferentformats.Thedecimal0.25iswrittenas!

!.Decimalnumbersareanother

wayofwritingfractions.Whenpresentedwiththefraction,thedivisionexpressionrepresentingafractioniswrittenas3dividedby5.TheBase-10modelsconcretelyrelatefractionstodecimals(e.g.,10-by-10grids,metersticks,numberlines,decimalsquares,money).

● Decimalnumbersareanotherwayofwritingfractions(halves,fourths,fifths,andtenths).TheBase-10Base-tenmodelsconcretelyrelatefractionstodecimals(e.g.,10-by-10grids,metersticks,numberlines,decimalsquares,decimalcirclesmoney).

● Provideafractionmodel(halves,fourths,fifths,andtenths)andaskstudentsforitsdecimalequivalent.

● Provideadecimalmodelandaskstudentsforitsfractionequivalent(halves,fourths,fifths,andtenths).

decimalsandfractionsrepresentthesamerelationship;however,theyarepresentedintwodifferentformats.

● Understandthatmodelsareusedtoshowdecimalandfractionequivalents.

reasoning,connections,andrepresentationsto● Representfractionsforhalves,fourths,

fifths,andtenthsasdecimalsthroughhundredths,usingconcreteobjects(e.g.,demonstratetherelationshipbetweenthefractionanditsdecimalequivalent0.25).(d)

● Relatefractionstodecimals,usingconcreteobjects(e.g.,10-by-10grids,metersticks,numberlines,decimalsquares,decimalcircles,money[coins]).(d)

● Writethedecimalandfractionequivalentforagivenmodel(e.g.,!

!=0.25or0.25

=!!;1.25=!

!or1!

!).(d)


● Decimal● Decimalpoint● Decimalnumber

● Tenth● Hundredth● Thousandth

● =● Fraction● Fractionalpart

● Mixednumber● Properfraction● Improperfraction

4.5 Thestudentwilla)determinecommonmultiplesandfactors,includingleastcommonmultipleandgreatestcommonfactor;b)addandsubtractfractionsandmixednumbershavinglikeandunlikedenominatorsthatarelimitedto2,3,4,5,6,8,10,

and12,andsimplifytheresultingfractions,usingcommonmultiplesandfactors;*andc)addandsubtractwithdecimals;andc)d)solvesingle-stepandmultisteppracticalproblemsinvolvingadditionandsubtractionwithfractionsandmixednumberswith

decimals.UNDERSTANDINGTHESTANDARD ESSENTIAL

UNDERSTANDINGSESSENTIALKNOWLEDGE

ANDSKILLSIMPORTANTCHANGES


● Afactorofawholenumberisawholenumberan Allstudentsshould Thestudentwilluseproblem PreviousSOL4.5cisnow4.6a

34

integerthatdividesevenlyintothatnumberwithnoremainderaremainderofzero.Afactorofanumberisadivisorofthenumber.

● Acommonfactoroftwoormorenumbersisadivisorthatallofthenumbersshare.

● Thegreatestcommonfactoroftwoormorenumbersisthelargestofthecommonfactorsthatallofthenumbersshare.

● Theproductofthenumberandanynaturalnumberisamultipleofthenumber.Amultipleofanumberistheproductofthenumberandanynaturalnumber.

● Commonmultiplesandcommonfactorscanbeusefulwhensimplifyingfractions.

● Theleastcommonmultipleoftwoormorenumbersisthelowestsmallestcommonmultipleofthegivennumbers.

● Estimationkeepsthefocusonthemeaningofthenumbersandoperations,encouragesreflectivethinking,andhelpsbuildinformalnumbersensewithfractions.Studentscanreasonwithbenchmarkstogetanestimatewithoutusinganalgorithm.

● Reasonableanswerstoproblemsinvolvingadditionandsubtractionoffractionscanbeestablishedbyusingbenchmarkssuchas0,!

!,and1.Forexample,!

!and!

!

areeachgreaterthan!!,sotheirsumisgreaterthan1.

● Studentsshouldinvestigateadditionandsubtractionwithfractions,usingavarietyofmodels(e.g.,fractioncircles,fractionstrips,linesrulers,linkingcubes,patternblocks).

● Whilethisstandardrequiresinstructioninsolvingproblemswithdenominatorsof2,3,4,5,6,8,10,and12,studentswouldbenefitfromexperienceswithotherdenominators.

● Whenstudentsusetheleastcommonmultipletodeterminecommondenominatorstoaddorsubtractfractionswithunlikedenominators,theleastcommonmultiplemaybegreaterthan12,butwillnotexceed60.

● Properfractions,improperfractions,andmixednumbersaretermsoftenusedtodescribefractions.A

• Understandandusecommonmultiplesandcommonfactorsforsimplifyingfractions.

● Developandusestrategiestoestimateadditionandsubtractioninvolvingfractionsanddecimals.

● Usevisualmodelstoaddandsubtractwithfractionsanddecimals.

solving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● FindDeterminecommon

multiplesandcommonfactorsofnumbers.(a)

● Determinetheleastcommonmultipleandgreatestcommonfactorofnomorethanthreenumbers.(a)

● Determineacommondenominatorforfractions,usingcommonmultiples.Commondenominatorsshouldnotexceed60.(b)Useleastcommonmultipleand/orgreatestcommonfactortofindacommondenominatorforfractions.

● Estimatethesumordifferenceoftwofractions.(b,c)

● Addandsubtractwithfractions(properorimproper)and/ormixednumbershavinglikeandunlikedenominatorswhosedenominatorsarelimitedto2,3,4,5,6,8,10,and12,andsimplifytheresultingfractionusingcommonmultiplesandfactors.(Subtractionwithfractionswillbelimitedtoproblemsthatdonotrequireregrouping).(b)

● Addandsubtractwithfractionshavingunlikedenominatorswhosedenominatorsarelimitedto2,

35

properfractionisafractionwhosenumeratorislessthanthedenominator.Animproperfractionisafractionwhosenumeratorisequaltoorgreaterthanthedenominator.Animproperfractionmaybeexpressedasamixednumber.Amixednumberiswrittenwithtwoparts:awholenumberandaproperfraction(e.g.,3!

!).

● Instructioninvolvingadditionandsubtractionoffractionsshouldincludeexperienceswithproperfractions,improperfractions,andmixednumbersasaddends,minuends,subtrahends,sums,anddifferences.

● Afractionisinsimplestformwhenitsnumeratoranddenominatorhavenocommonfactorsotherthanone.Thenumeratorcanbegreaterthanthedenominator.

● Theproblem-solvingprocessisenhancedwhenstudentscreateandsolvetheirownpracticalproblemsandmodelproblemsusingmanipulativesanddrawings.

● Inproblemsolving,emphasisshouldbeplacedonthinkingandreasoningratherthanonkeywords.Focusingonkeywordssuchasinall,altogether,difference,etc.encouragesstudentstoperformaparticularoperationratherthanmakesenseofthecontextoftheproblem.Itpreparesstudentstosolveaverylimitedsetofproblemsandoftenleadstoincorrectsolutions.

● Atthislevel,denominatorsoffractionsresultingfromsimplificationwillbelimitedto12orless.

● Whenaddingorsubtractingwithfractionshavinglikedenominators,addorsubtractthenumeratorsandusethesamedenominator.Writetheanswerinsimplestformusingcommonmultiplesandfactors.

● Whenaddingorsubtractingwithfractionshavingunlikedenominators,rewritethemasfractionswithacommondenominator.Theleastcommonmultiple(LCM)oftheunlikedenominatorsisacommondenominator(LCD).Writetheanswerinsimplestformusingcommonmultiplesandfactors.

● Additionandsubtractionofdecimalsmaybeexplored,usingavarietyofmodels(e.g.,10-by-10grids,number

3,4,5,6,8,10,and12,andsimplifytheresultingfractionusingcommonmultiplesandfactors.

● Solvesingle-steppracticalproblemsthatinvolveaddingandsubtractingaddingandsubtractingwithfractions(properorimproper)and/ormixednumbershavinglikeandunlikedenominatorswhosedenominatorsarelimitedto2,3,4,5,6,8,10,and12,andsimplifytheresultingfractionusingcommonmultiplesandfactors..(Subtractionwithfractionswillbelimitedtoproblemsthatdonotrequireregrouping).(c)

● Solvesingle-stepandmultistepproblemsthatinvolveaddingandsubtractingwithfractionsanddecimalsthroughthousandths.

● Addandsubtractwithdecimalsthroughthousandths,usingconcretematerials,pictorialrepresentations,andpaperandpencil.

36

lines,money).● Fordecimalcomputation,thesameideasdevelopedfor

wholenumbercomputationmaybeused,andtheseideasmaybeappliedtodecimals,givingcarefulattentiontotheplacementofthedecimalpointinthesolution.Lininguptenthstotenths,hundredthstohundredths,etc.helpstoestablishthecorrectplacementofthedecimal.

● Fractionsmayberelatedtodecimalsbyusingmodels(e.g.,10-by-10grids,decimalsquares,money).


● Multiple● Factor● Commonmultiple● Commonfactor● Leastcommonmultiple● Leastcommondenominator● Greatestcommonfactor● Simplify

● SimplestForm● Add● Addition● Addends● Subtract● Subtraction● Subtrahends● Minuends

● Sums● Differences● Product● Divisor● Fraction● Denominator● Numerator

● NaturalNumber● ProperFraction● ImproperFraction● MixedNumber● Simplify● Benchmark● Reasonableanswer

4.6 Thestudentwilla)addandsubtractdecimals;*andb)solvesingle-stepandmultisteppracticalproblemsinvolvingadditionandsubtractionwithdecimals.

*Onthestateassessment,itemsmeasuringthisobjectiveareassessedwithouttheuseofacalculator.UNDERSTANDINGTHESTANDARD ESSENTIAL



• Additionandsubtractionofdecimalsmaybeexplored,usingavarietyofmodels(e.g.,10-by-10grids,numberlines,money).

• Theproblem-solvingprocessisenhancedwhenstudentscreateandsolvetheirownpracticalproblemsandmodelproblemsusingmanipulativesanddrawings.

• Inproblemsolving,emphasisshouldbeplacedonthinkingandreasoningratherthanonkeywords.Focusingonkeywordssuchasinall,


• Estimatesumsanddifferencesofdecimals.(a)

• Addandsubtractdecimalsthroughthousandths,usingconcretematerials,

Pleasenoteallcontentof4.6hasbeenmovedfromprevious4.5candd.THISISNOTNEWThepreviousSOL4.6isnowembeddedinnewSOL4.8.

37

altogether,difference,etc.encouragesstudentstoperformaparticularoperationratherthanmakesenseofthecontextoftheproblem.Itpreparesstudentstosolveaverylimitedsetofproblemsandoftenleadstoincorrectsolutions.

• Theleastnumberofstepsnecessarytosolveasingle-stepproblemisone.

pictorialrepresentations,andpaperandpencil.(a)

• Solvesingle-stepandmultisteppracticalproblemsthatinvolveaddingandsubtractingwithdecimalsthroughthousandths.(b)

EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf● Addition● Add● Sum

● Subtraction● Subtract● Difference

● Decimal● Decimalpoint● Tenths

● Hundredths● Thousandths

38

MeasurementWeeks28-32SOLs4.6,4.7,4.84.7 ThestudentwillsolvepracticalproblemsthatinvolvedeterminingperimeterandareainU.S.Customaryandmetricunits.4.8 Thestudentwill

a)estimateandmeasurelengthanddescribetheresultinU.S.Customaryandmetricunits;b)estimateandmeasureweight/massanddescribetheresultinU.S.Customaryandmetricunits;c)giventheequivalentmeasureofoneunit,identifyequivalentmeasuresoflength,weight/mass,andliquidvolumebetweenunitswithin

theU.S.Customarysystem;andd)solvepracticalproblemsthatinvolvelength,weight/mass,andliquidvolumeinU.S.Customaryunits.



● Themeasurementoftheanobjectmustincludetheunitofmeasurealongwiththenumberofiterations.

● Lengthisthedistancebetweentwopointsalongalineorfigurefromonepointtoanother.

● U.S.Customaryunitsformeasurementoflengthincludeinches,feet,yards,andmiles.Appropriatemeasuringdevicesincluderulers,yardsticks,andtapemeasures.

● Metricunitsformeasurementoflengthincludemillimeters,centimeters,meters,andkilometers.Appropriatemeasuringdevicesincludecentimeterruler,meterstick,andtapemeasure.

● WhenmeasuringwithU.S.Customaryunits,studentsshouldbeabletomeasuretothenearestpartofaninch(!

!,!!,!!),inch,foot,or

yard.● Weightandmassaredifferent.Massisthe

amountofmatterinanobject.Weightisdeterminedbythepullofgravityonthemassofanobject.Themassofanobjectremainsthesameregardlessofitslocation.Theweightofanobjectchangesdependingonthegravitationalpullatitslocation.In

Allstudentsshould● Usebenchmarksto

estimateandmeasureweight/mass.

● IdentifyequivalentmeasuresbetweenunitswithintheU.S.Customaryandbetweenunitswithinthemetricmeasurements.

● Usebenchmarkstoestimateandmeasurelength.

● UnderstandhowtoconvertunitsoflengthbetweentheU.S.Customaryandmetricsystems,usingballparkcomparisons.

● UnderstandtherelationshipbetweenU.S.Customaryunitsandtherelationshipbetweenmetricunits.

● Usebenchmarkstoestimateandmeasure


• Determineanappropriateunitofmeasure(e.g.,inch,foot,yard,mile,millimeter,centimeter,andmeter)tousewhenmeasuringlengtheverydayobjectsinbothmetricandU.S.Customaryunits.(a)

• Estimatethelengthofeverydayobjects(e.g.,books,windows,tables)inbothmetricandU.S.Customaryunitsofmeasure.

• EstimateandmeasurethelengthofobjectsinbothmetricandU.S.Customaryunits,measuringtothenearestpartofaninch(!

!,!!,!!),and

tothenearestfoot,yard,mile,millimeter,centimeter,ormeter,andrecordthelengthincludingtheappropriateunitofmeasure(e.g.,24inches).(a)

• Compareestimatesofthelengthofobjectswiththeactual

PleasenotepreviousSOLs4.6,4.7,4.8havebeencombinedinthistablefornewSOL4.8.Additions,anddeletionsarerepresented.

39

everydaylife,mostpeopleareactuallyinterestedindetermininganobject’smass,althoughtheyusethetermweight(e.g.,“Howmuchdoesitweigh?”versus“Whatisitsmass?”).

● BalancesareappropriatemeasuringdevicestomeasureweightinU.S.Customaryunits(ounces,pounds)andmassinmetricunits(grams,kilograms).

● Practicalexperiencemeasuringthemassoffamiliarobjects(e.g.,foods,pencils,bookbags,shoes)helpstoestablishbenchmarksandfacilitatesthestudent’sabilitytoestimateweight/mass.

● Studentsshouldestimatethemass/weightofeverydayobjects(e.g.,foods,pencils,bookbags,shoes),usingappropriatemetricorU.S.Customaryunits.

● Practicalexperiencemeasuringthelengthoffamiliarobjectshelpstoestablishbenchmarksandfacilitatesthestudent’sabilitytoestimatelength.

● Studentsshouldestimatethelengthofeverydayobjects(e.g.,books,windows,tables)inbothmetricandU.S.Customaryunitsofmeasure.

● StudentsshouldmeasuretheliquidvolumeofeverydayobjectsinU.S.Customaryunits,includingcups,pints,quarts,gallons,andrecordthevolumeincludingtheappropriateunitofmeasure(e.g.,24gallons).

● StudentsatthislevelwillbegiventheequivalentmeasureofoneunitwhenaskedtodetermineequivalenciesbetweenunitsintheU.S.Customarysystem.– Forexample,studentswillbetoldone

gallonisequivalenttofourquartsandthenwillbeaskedtoapplythatrelationshiptodetermine:- thenumberofquartsinfivegallons;

volume.● Identifyequivalent

measurementsbetweenunitswithintheU.S.Customarysystem.

measurementofthelengthofobjects.(a)

● Determineanappropriateunitofmeasure(e.g.,ounce,pound,ton,gram,andkilogram)tousewhenmeasuringtheweight/massofeverydayobjectsinbothmetricandU.S.Customaryunits.(b)

● Estimateandmeasuretheweight/massofobjectsinbothmetricandU.S.Customaryunits(e.g.,ounce,pound,ton,gram,orkilogram)tothenearestappropriatemeasure,usingavarietyofmeasuringinstruments.(b)

● Recordtheweight/massofanobjectincludingtheappropriatewiththeunitofmeasure(e.g.,24grams).(b)

● Determineanappropriateunitofmeasure(cups,pints,quarts,gallons)tousewhenmeasuringliquidvolumeinU.S.Customaryunits.

● EstimatetheliquidvolumeofcontainersinU.S.Customaryunitsofmeasuretothenearestcup,pint,quart,andgallon.

● MeasuretheliquidvolumeofeverydayobjectsinU.S.Customaryunits,includingcups,pints,quarts,andgallons,andrecordthevolumeincludingtheappropriateunitofmeasure(e.g.,24gallons).

● Giventheequivalentmeasureofoneunit,identifyequivalentmeasuresbetweenunitswithintheU.S.Customarysystemfor:− length(inchesandfeet,feetand

yards,inchesandyards);yardsandmiles;

− weight/mass(ouncesand

40

- thenumberofgallonsequalto20quarts;

- Whenempty,Tim’s10-galloncontainercanholdhowmanyquarts?;or

- Mariahas20quartsoflemonade.Howmanyemptyone-galloncontainerswillshebeabletofill?

● Practicalexperiencemeasuringliquidvolumeoffamiliarobjectshelpstoestablishbenchmarksandfacilitatesthestudent’sabilitytoestimateliquidvolume.

● U.S.Customaryunitsformeasurementofliquidvolumeincludecups,pints,quarts,andgallons.

● StudentsshouldestimatetheliquidvolumeofcontainersinU.S.Customaryunitstothenearestcup,pint,quart,andgallon.

pounds);and− liquidvolume(cups,pints,

quarts,andgallons).(c)● Solvepracticalproblemsthatinvolve

length,weight/mass,andliquidvolumeinU.S.Customaryunits.(d)

● IdentifyequivalentmeasuresoflengthbetweenunitswithintheU.S.Customarymeasurementsandbetweenunitswithinthemetricmeasurements.

● IdentifyequivalentmeasuresofvolumebetweenunitswithintheU.S.Customarysystem.


● Weight● Weigh● Mass● *U.S.Customary● Metric● *MetricSystem● Ounce● Pound

● Gram● Kilogram● Equivalent● Measure● Estimate● Length● Inch● Feet

● Yard● Millimeter● Centimeter● Meter● Straightedge● Ruler● Liquid● Liquidvolume

● Volume● Cup● Pint● Quart● Gallon● Milliliter● Liter

RESOURCESBenchmarkLiteracy:MathtoMuchon–Unit4

4.7ThestudentwillsolvepracticalproblemsthatinvolvedeterminingperimeterandareainU.S.Customaryandmetricunits.UNDERSTANDINGTHESTANDARD



41

• Perimeteristhepathordistancearoundanyplanefigure.

• Todeterminetheperimeterofanypolygon,determinethesumofthelengthsofthesides.

• Areaisthesurfaceincludedwithinaplanefigure.Areaismeasuredbythenumberofsquareunitsneededtocoverasurfaceorplanefigure.

• Studentsshouldhaveopportunitiestoinvestigateanddiscover,usingmanipulatives,theformulasfortheareaofasquareandtheareaofarectangle.

- Areaofasquare=sidelength×sidelength

- Areaofrectangle=length×widthPerimeterandareashouldalwaysbelabeledwiththeappropriateunitofmeasure.


• Determinetheperimeterofapolygonwithnomorethaneightsides,whenthelengthsofthesidesaregiven,withdiagrams.

• Determinetheperimeterandareaofarectanglewhengiventhemeasureoftwoadjacentsides,withandwithoutdiagrams.

• Determinetheperimeterandareaofasquarewhenthemeasureofonesideisgiven,withandwithoutdiagrams.

• SolvepracticalproblemsthatinvolvedeterminingperimeterandareainU.S.Customaryandmetricunits.

PleasenotethisstandardisanADDITIONTOFOURTHGRADE.Itwaspreviously5.8.


● Area● SquareUnits● Perimeter● Units

● Length● Width● Distance● Polygon

● Adjacent● Surface● PlaneFigure

● Square● Rectangle● Formula

42

ElapsedTimeWeeks33SOL4.94.9Thestudentwillsolvepracticalproblemsrelatedtodetermineelapsedtimeinhoursandminuteswithina12-hourperiod.UNDERSTANDINGTHESTANDARD ESSENTIAL


(Thiscolumnwillnotbeincludedinthe2018-2019

pacingguide)● Elapsedtimeistheamountoftime

thathaspassedbetweentwogiventimes.

● Elapsedtimeshouldbemodeledanddemonstratedusinganalogclocksandtimelines.

● Elapsedtimecanbefoundbycountingonfromthebeginningtimeorcountingbackfromtheendingtime.tothefinishingtime.– Countthenumberofwholehours

betweenthebeginningtimeandthefinishingtime.

– Counttheremainingminutes.– Addthehoursandminutes.Forexample,tofindtheelapsedtimebetween10:15a.m.and1:25p.m.,count10minutes;andthen,add3hoursto10minutestofindthetotalelapsedtimeof3hoursand10minutes.

Allstudentsshould● Understanding

the“countingon”strategyfordeterminingelapsedtimeinhourandminuteincrementsovera12-hourperiodfroma.m.toa.m.orp.m.top.m.


• Solvepracticalproblemsrelatedtoelapsedtimeinhoursandminutes,withina12-hourperiod(withina.m.,withinp.m.,andacrossa.m.andp.m.):

–whengiventhebeginningtimeandtheendingtime,determinethetimethathaselapsed;

–whengiventhebeginningtimeandamountofelapsedtimeinhoursandminutes,determinetheendingtime;or

–whengiventheendingtimeandtheelapsedtimeinhoursandminutes,determinethebeginningtime.

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Determinetheelapsedtimeinhoursandminuteswithina12-hour

period(timescancrossbetweena.m.andp.m.).● Solvepracticalproblemsinrelationtotimethathaselapsed.

EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdfhttp://www.doe.virginia.gov/VDOE/Instruction/math_vocabulary/kindergarten-grade3.doc• Time• Hour

• Minute• TimePeriod

• *ElapsedTime• am

• pm

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GeometryandPatternsWeeks34-35SOLs4.10,4.11,4.12,4.154.10 Thestudentwill

a)identifyanddescriberepresentationsofpoints,lines,linesegments,rays,andangles,includingendpointsandvertices;andb)identifyanddescribeintersecting,parallel,andperpendicularlines.representationsoflinesthatillustrateintersection,

parallelism,andperpendicularity.UNDERSTANDINGTHESTANDARD ESSENTIAL



● Points,lines,linesegments,rays,andangles,includingendpointsandverticesarefundamentalcomponentsofnoncirculargeometricfigures.

● Apointisalocationinspace.Ithasnolength,width,orheight.Apointisusuallynamedwithacapitalletter.

● Theshortestdistancebetweentwopointsinaplane,aflatsurface,isalinesegment.

● Alineisacollectionofpointsextendinggoingonandoninfinitelyinbothdirections.Ithasnoendpoints.Whenalineisdrawn,atleasttwopointsonitcanbemarkedandgivencapitalletternames.Arrowsmustbedrawntoshowthatthelinegoesoninfinitelyinbothdirectionsinfinitely(e.g.,readas“thelineAB”).

● Alinesegmentispartofaline.Ithastwoendpointsandincludesallthepointsbetweenandincludingthethoseendpoints.Tonamealinesegment,nametheendpoints(e.g.,readas“thelinesegmentAB”).

● Arayispartofaline.Ithasoneendpointandextendscontinuesinfinitelyinonedirection.Tonamearay,saythenameofitsendpointfirstandthensaythenameofoneotherpointontheray(e.g.,readas“therayAB”).

● Tworaysthathavethesameendpointformanangle.Thisendpointiscalledthevertex.Anangleisformedbytworaysthatsharea

Allstudentsshould● Understandthatpoints,

lines,linesegments,rays,andangles,includingendpointsandverticesarefundamentalcomponentsofnoncirculargeometricfigures.

● Understandthattheshortestdistancebetweentwopointsonaflatsurfaceisalinesegment.

● Understandthatlinesinaplaneeitherintersectorareparallel.Perpendicularityisaspecialcaseofintersection.

● Identifypracticalsituationsthatillustrateparallel,intersecting,andperpendicularlines.

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Identifyanddescriberepresentationsof

points,lines,linesegments,rays,andangles,includingendpointsandvertices.(a)

● Usesymbolicnotationtonamepoints,lines,linesegments,rays,andangles.(a)

● Identifyparallel,perpendicular,andintersectinglinesegmentsinplaneandsolidfigures(b)

● Understandthatlinesinaplanecanintersectorareparallel.Perpendicularityisaspecialcaseofintersection.

● Identifypracticalsituationsthatillustrateparallel,intersecting,andperpendicularlines.(b)

● Usesymbolicnotationtodescribeparallellinesandperpendicularlines.(b)

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commonendpointcalledthevertex.Anglesarefoundwhereverlinesandlinesegmentsintersect.

● Ananglecanbenamedinthreedifferentwaysbyusing;– threeletterstoname,inthisorder:apoint

ononeray,thevertex,andapointontheotherray;

– oneletteratthevertex;or– anumberwritteninsidetheraysofthe

angle● Avertexisthepointatwhichtwolines,line

segments,orraysmeettoformanangle.Insolidfigures,avertexisthepointatwhichthreeormoreedgesmeet.

● Linesinaplaneeitherintersectorareparallel.Perpendicularityisaspecialcaseofintersection.

● Intersectinglineshaveonepointincommon.● Perpendicularlinesintersectatrightangles.

Thesymbol�isusedtoindicatethattwolinesareperpendicular.Forexample,thenotation𝐴𝐵⃡ ⊥ 𝐶𝐷⃡isreadas“lineABisperpendicularto lineCD.”

● Studentsneedexperiencesusinggeometricmarkingsinfigurestoindicatecongruenceofsidesandanglesandtoindicateparallelsides.

● Parallellineslieinthesameplaneandneverintersect.Parallellinesarealwaysthesamedistanceapartanddonotshareanypoints.Thesymbol�indicatesthattwoormorelinesareparallel.Forexample,thenotation𝐵𝐶 ̅⃡� 𝐹𝐺 ̅⃡isreadas“lineBCisparalleltoline

FG”.

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● Perpendicularlinesarespecialintersectinglinesthatformrightangleswheretheyintersect.

● Parallellinesarelinesthatlieinthesameplaceanddonotintersect.Parallellinesarealwaysthesamedistanceapartanddonotshareanypoints.

● Studentsshouldexploreintersection,parallelism,andperpendicularityinbothtwoandthreedimensions.Forexample,studentsshouldanalyzetherelationshipsbetweentheedgesofacube.Whichedgesareparallel?Whichareperpendicular?Whatplanecontainstheupperleftedgeandthelowerrightedgeofthecube?Studentscanvisualizethisbyusingtheclassroomitselftonoticethelinesformedbytheintersectionoftheceilingandwalls,ofthefloorandwall,andoftwowalls.


● Point● Line● Linesegment● Ray● Infinitely

● Endpoint● Vertex(vertices)● *Angle● Angleruler● *Intersectinglines

● *Perpendicularlines● Intersect● *Parallellines● Intersection

● Parallel● Parallelism● Perpendicular● Perpendicularity

4.11 Thestudentwilla)investigatecongruenceofplanefiguresaftergeometrictransformations,suchasreflection,translation,androtation,using

mirrors,paperfolding,andtracing;andb)recognizetheimagesoffiguresresultingfromgeometrictransformations,suchastranslation,reflection,androtation.

UNDERSTANDINGTHESTANDARD(BackgroundInformationforInstructorUseOnly)

ESSENTIALUNDERSTANDINGS ESSENTIALKNOWLEDGEANDSKILLS

IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)

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– ThevanHieletheoryofgeometricunderstandingdescribeshowstudentslearngeometryandprovidesaframeworkforstructuringstudentexperiencesthatshouldleadtoconceptualgrowthandunderstanding.– Level0:Pre-recognition.Geometric

figuresarenotrecognized.Forexample,studentscannotdifferentiatebetweenthree-sidedandfour-sidedpolygons.

– Level1:Visualization.Geometricfiguresarerecognizedasentities,withoutanyawarenessofpartsoffiguresorrelationshipsbetweencomponentsofafigure.Studentsshouldrecognizeandnamefiguresanddistinguishagivenfigurefromothersthatlooksomewhatthesame.(ThisistheexpectedlevelofstudentperformanceduringgradesKand1.)

– Level2:Analysis.Propertiesareperceivedbutareisolatedandunrelated.Studentsshouldrecognizeandnamepropertiesofgeometricfigures.(Studentsareexpectedtotransitiontothislevelduringgrades2and3.)

– Level3:Abstraction.Definitionsaremeaningful,withrelationshipsbeingperceivedbetweenpropertiesandbetweenfigures.Logicalimplicationsandclassinclusionareunderstood,buttheroleandsignificanceofdeductionisnotunderstood.(Studentsshouldtransitiontothislevelduringgrades5and6andfullyattainitbeforetakingalgebra.)

– Congruentfiguresarefigureshavingexactlythesamesizeandshape.Opportunitiesforexploringfiguresthatarecongruentand/or

Allstudentsshould– Understandthemeaningofthe

termcongruent.– Understandhowtoidentify

congruentfigures.– Understandthattheorientation

offiguresdoesnotaffectcongruencyornoncongruency.

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto– Recognizethecongruenceof

planefiguresresultingfromgeometrictransformationssuchastranslation,reflection,androtation,usingmirrors,paperfoldingandtracing.

PleasenotethatpreviousSOL4.11hasbeenmovedto5.14

NEWSOL4.11islocatedinatablebelowitwaspreviouslySOL3.14

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noncongruentcanbestbeaccomplishedbyusingphysicalmodels.

– Atranslationisatransformationinwhichanimageisformedbymovingeverypointonafigurethesamedistanceinthesamedirection.

– Areflectionisatransformationinwhichafigureisflippedoveralinecalledthelineofreflection.Allcorrespondingpointsintheimageandpreimageareequidistantfromthelineofreflection.

– Arotationisatransformationinwhichanimageisformedbyturningitspreimageaboutapoint.

– Theresultingfigureofatranslation,reflection,orrotationiscongruenttotheoriginalfigure.


● *Plane● Planefigure● FixedPoint● Congruent● Non-congruent● Transformations

● Translation● Reflection● Rotation● Flip● Slide● Turn

● Circle● Rectangle● Rhombus● Square● Triangle● Parallelogram

● 3-dimensional● Cube● Sphere● Rectangularprism● Rectangularsolid

4.11 Thestudentwillidentify,describe,compare,andcontrastplaneandsolidfiguresaccordingtotheircharacteristics(numberofangles,vertices,edges,andthenumberandshapeoffaces)usingconcretemodelsandpictorialrepresentations.

UNDERSTANDINGTHESTANDARD ESSENTIALKNOWLEDGEANDSKILLS

● Thestudyofgeometricfiguresmustbeactive,usingvisualimagesandconcretematerials(toolssuchasgraphpaper,patternblocks,geoboards,geometricsolids,andcomputersoftwaretools).

● Opportunitymustbeprovidedforbuildingandusinggeometricvocabularytodescribeplaneandsolidfigures.● Aplanefigureisanyclosed,two-dimensionalshape.● Asolidfigureisthree-dimensional,havinglength,width,andheight.● Afaceisanyflatsurfaceofasolidfigure.● Anangleisformedbytworayswithacommonendpointcalledthevertex.Anglesarefoundwhereverlines

and/orlinesegmentsintersect.● Anedgeisthelinesegmentwheretwofacesofasolidfigureintersect.


● Identifyconcretemodelsandpictorialrepresentationsofsolidfigures(cube,rectangularprism,squarepyramid,sphere,cone,andcylinder).● Identifyanddescribesolidfigures(cube,rectangularprism,squarepyramid,andsphere)

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● Avertexisthepointatwhichtwoormorelines,linesegments,orraysmeettoformanangle.Insolidfigures,avertexisthepointatwhichthreeormorefacesmeet.

● Acubeisasolidfigurewithsixcongruent,squarefaces.Alledgesarethesamelength.Acubehaseightverticesand12edges.

● Arectangularprismisasolidfigureinwhichallsixfacesarerectangles.Arectangularprismhaseightverticesand12edges.Acubeisaspecialcaseofarectangularprism.

● Asphereisasolidfigurewithallofitspointsthesamedistancefromitscenter.● Asquarepyramidisasolidfigurewithasquarebaseandfourfacesthataretriangleswithacommonvertex.A

squarepyramidhasfiveverticesandeightedges.● Characteristicsofsolidfiguresincludedatthisgradelevelaredefinedinthechartbelow:

accordingtotheircharacteristics(numberofangles,vertices,edges,andbythenumberandshapeoffaces).● Compareandcontrastplaneandsolidfigures(circle/sphere,square/cube,triangle/squarepyramid,andrectangle/rectangularprism)accordingtotheircharacteristics(numberofsides,angles,vertices,edges,andthenumberandshapeoffaces).


▪ PlaneGeometricFigure▪ *PlaneFigure▪ *Circle▪ *Square▪ *Rectangle▪ *Triangle▪ SolidGeometricFigure▪ *Cube

▪ *Rectangularsolid(prism)▪ *Squarepyramid▪ *Sphere▪ *Cone▪ *Cylinder▪ Side▪ Corner▪ Squarecorner

▪ *Edge▪ *Face▪ Angles▪ vertices▪ RightAngle▪ Endpoint▪ Shape▪ Symmetrical

▪ *LineofSymmetry▪ Transformation▪ Translation(slide)▪ Rotation(turn)▪ Reflection(flip)▪ Vertical▪ Horizontal▪ *Diagonal

4.12 Thestudentwilla)definepolygon;andb)identifypolygonswith10orfewersides.



ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedin

the2018-2019pacingguide)● Apolygonisaclosedplanegeometricfigure

composedofatleastthreelinesegmentsthatdonotcross.Noneofthesidesarecurved.

● Atriangleisapolygonwiththreeanglesandthreesides.

● Aquadrilateralisapolygonwithfoursides.

Allstudentsshould● Identifypolygonswith

10orfewersidesineverydaysituations.

● Identifypolygonswith10orfewersidesin

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connectionsandrepresentationto● Defineandidentifypropertiesof

polygonswith10orfewersides.

● PleasenotepreviousSOL4.12hasbeenmovedtoNEWSOL3.12–exceptforquadrilaterals

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● Arectangleisaquadrilateralwithfourrightangles.

● Asquareisarectanglewithfoursidesofequallength.

● Atrapezoidisaquadrilateralwithexactlyonepairofparallelsides.

● Aparallelogramisaquadrilateralwithbothpairsofoppositesidesparallel.

● Arhombusisaquadrilateralwith4congruentsides.

● Apentagonisa5-sidedpolygon.● Ahexagonisa6-sidedpolygon.● Aheptagonisa7-sidedpolygon.● Anoctagonisan8-sidedpolygon.● Anonagonisa9-sidedpolygon.● Adecagonisa10-sidedpolygon.

multipleorientations(rotations,reflections,andtranslationsofthepolygons).

● Identifypolygonsbynamewith10orfewersidesinmultipleorientations(rotations,reflections,andtranslationsofthepolygons).


● *Polygon–atwodimensionalsimpleclosedgeometricfigurethathaslinesegmentsassides

● Triangle● Quadrilateral● Rectangle● Square

● Trapezoid● Parallelogram● Rhombus● Pentagon● Hexagon

● Heptagon● Octagon● Nonagon● Decagon

4.12 Thestudentwillclassifyquadrilateralsasparallelograms,rectangles,squares,rhombi,and/ortrapezoids.

UnderstandingtheStandard EssentialKnowledgeandSkills

● Aquadrilateralisapolygonwithfoursides.● Aparallelogramisaquadrilateralwithbothpairsofoppositesidesparallelandcongruent.● Congruentfigureshavethesamesizeandshape.Congruentsidesarethesamelength.● Arectangleisaquadrilateralwithfourrightangles,and,oppositesidesthatareparallelandcongruent.● Thegeometricmarkingsshownontherectanglebelowindicateparallelsideswithanequalnumberofarrowsandcongruentsidesindicatedwithan


● Developdefinitionsforparallelograms,rectangles,squares,rhombi,andtrapezoids.● Identifypropertiesofquadrilateralsincludingparallel,perpendicular,andcongruentsides.● Classifyquadrilateralsasparallelograms,rectangles,squares,rhombi,and/ortrapezoids.

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equalnumberofhatch(hash)marks.

● Asquareisarectanglewithfourcongruentsidesandfourrightangles.● Atrapezoidisaquadrilateralwithexactlyonepairofparallelsides.● Arhombusisaquadrilateralwithfourcongruentsides.Propertiesofarhombusincludethefollowing:– oppositesidesarecongruent– oppositesidesareparallel– oppositeanglesarecongruent

● Compareandcontrastthepropertiesofquadrilaterals.● Identifyparallelsides,congruentsides,andrightanglesusinggeometricmarkingstodenotepropertiesofquadrilaterals.


▪ Quadrilateral▪ Rectangle

▪ Square▪ Trapezoid

▪ Parallelogram▪ Rhombus

▪ Pentagon

4.15 Thestudentwillidentify,describe,recognize,create,andextendpatternsfoundinobjects,pictures,numbers,andtables.numericalandgeometricpatterns.


ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe

2018-2019pacingguide)● Mostpatterningactivitiesshouldinvolvesome

formofconcretematerialstomakeupapattern.

● Studentswillidentifyandextendawidevarietyofpatterns,includingrhythmic,geometric,graphic,numerical,andalgebraic.Thepatternswillincludebothgrowingandrepeatingpatterns.

● Patternsandfunctionscanberepresentedinmanywaysanddescribedusingwords,tables,graphs,andsymbols.

● Patterningactivitiesshouldinvolvemakingconnectionsbetweenconcretematerialsandnumericalrepresentations(e.g.,numbersequence,table,description).Numericpatterns,atthislevel,willincludebothgrowingandrepeatingpatterns(limitedtoaddition,subtraction,andmultiplicationof

Allstudentsshould● Understandthatpatterns

andfunctionscanberepresentedinmanywaysanddescribedusingwords,tables,graphs,andsymbols.

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Describegeometricandnumerical

patterns,usingtables,symbols,orwords.● Createpatternsusingobjects,pictures,

numbers,andtables.Creategeometricandnumericalpatterns,usingconcretematerials,numberlines,tables,andwords.

● Extendpatterns,usingobjects,pictures,numbers,andtables.Extendgeometricandnumericalpatterns,usingconcretematerials,numberlines,tables,andwords.

● Solvepracticalproblemsthatinvolveidentifying,describing,andextending

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wholenumbersandadditionandsubtractionoffractionswithlikedenominatorsof12orless).

● Studentsneedexperienceswithgrowingpatternsusingconcretematerialsandcalculators.

● Reproductionofagivenpatterninadifferentrepresentation,usingsymbolsandobjects,laysthefoundationforwritingtherelationshipsymbolicallyoralgebraically.

● Tablesofvaluesshouldbeanalyzedforapatterntodeterminethenextvalue

● Samplegrowingpatternsthatare,orcanbe,representedasnumerical(arithmetic)growingpatternsinclude:

– 2,4,8,16,…;

– 8,10,13,17,…;

–!!,!!,1!

!,1!

!…;and

–

● Studentsingradethreehadexperiencesworkingwithinput/outputtables.Atthislevel,input/outputtablesshouldbeanalyzedforapatterntodetermineanunknownvalueordescribetherulethatexplainshowtofindtheoutputwhengiventheinput.Determiningandapplyingrulesbuildsthefoundationforfunctionalthinking.Sampleinput/outputtablesthatrequiredeterminationoftheruleormissingtermscanbefoundbelow:

single-operationinputandoutputrules,limitedtoaddition,subtraction,andmultiplicationofwholenumbersandadditionandsubtractionoffractionswithlikedenominatorsof12orless.

● Identifytheruleinasingle-operationnumericalpatternfoundinalistortable,limitedtoaddition,subtraction,andmultiplicationofwholenumbers.

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● Pattern ● Geometricpattern ● Numericalpattern

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ProbabilityandStatisticsWeek36SOLs4.134.13 Thestudentwill

a)predictdeterminethelikelihoodofanoutcomeofasimpleevent;andb)representprobabilityasanumberbetween0and1,inclusive;andc)createamodelorpracticalproblemtorepresentagivenprobability.



● Aspiritofinvestigationandexperimentationshouldpermeateprobabilityinstruction,wherestudentsareactivelyengagedinexplorationsandhaveopportunitiestousemanipulatives.

● Probabilityisthemeasureoflikelihoodthatchanceofaneventwilloccuroccurring.Aneventisacollectionofoutcomesfromaninvestigationorexperiment.

● Theprobabilityofaneventoccurringistheratioofdesiredoutcomestothetotalnumberofpossibleoutcomes.Ifalltheoutcomesofaneventareequallylikelytooccur,theprobabilityoftheevent=numberoffavorableoutcomestotalnumberofpossibleoutcomes.

● Thetermscertain,likely,equallylikely,unlikely,andimpossiblecanbeusedtodescribethelikelihoodofanevent.Ifalloutcomesofaneventareequallylikely,theprobabilityofaneventcanbeexpressedasafraction,wherethenumeratorrepresentsthenumberoffavorableoutcomesandthedenominatorrepresentsthetotalnumberofpossibleoutcomes.Ifalltheoutcomesofaneventareequallylikelytooccur,theprobabilityoftheeventisequalto:

numberoffavorableoutcomestotalnumberofpossibleoutcomes.

Allstudentsshould● Understandandapply

basicconceptsofprobability.

● Describeeventsaslikelyorunlikelyanddiscussthedegreeoflikelihood,usingthetermscertain,likely,equallylikely,unlikely,andimpossible.

● Predictthelikelihoodofanoutcomeofasimpleeventandtesttheprediction.

● Understandthatthemeasureoftheprobabilityofaneventcanberepresentedbyanumberbetween0and1,inclusive.

Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto● Modelanddetermineallpossible

outcomesofagivensimpleeventwheretherearenomorethan24possibleoutcomes,usingavarietyofmanipulatives,suchas(e.g,coins,numbercubes,andspinners).(a)

● Determinetheoutcomeofaneventthatisleastlikelytooccur(lessthanhalf)ormostlikelytooccurwheretherearenomorethan24possibleoutcomes.(greaterthanhalf)whenthenumberofpossibleoutcomesis24orless.(a)

● Writetheprobabilityofagivensimpleeventasafraction,wheretherearenomorethan24possibleoutcomes.wherethetotalnumberofpossibleoutcomesis24orfewer.(a)

● IdentifyDeterminethelikelihoodofaneventoccurringandrelateittoitswholenumberorfractionalrepresentation(e.g.,impossibleorzero;equallylikely;certainorone).

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Probabilityisquantifiedasanumberbetween0and1.Aneventis“impossible”ifithasaprobabilityof0(e.g.,ifeightballsareinabag,fouryellowandfourblue,thereiszeroprobabilitythataredballcouldbe

selected).Aneventis“certain”ifithasaprobabilityofone(e.g.,theprobabilitythatif10coins,allpennies,

areinabagthatitiscertainapennycouldbeselected).Theprobabilityofaneventoccurringis

representedbyaratiobetween0and1.Aneventis“impossible”ifithasaprobabilityof0(e.g.,the

probabilitythatthemonthofAprilwillhave31days).Aneventis“certain”ifithasaprobabilityof1(e.g.,theprobabilitythatthesunwillrisetomorrowmorning).● Conductexperimentstodeterminetheprobability

ofaneventoccurringforagivennumberoftrials(nomorethan25trials),usingmanipulatives(e.g.,thenumberoftimes“heads”occurswhenflippingacoin10times;thechancethatwhenthenamesof12classmatesareputinashoebox,anamethatbeginswithDwillbedrawn).

● Studentsshouldhaveopportunitiestodescribeininformalterms(i.e.,impossible,unlikely,aslikelyasunlikely,equallylikely,likely,andcertain)thedegreeoflikelihoodofaneventoccurring.

● Activitiesshouldincludepracticalexamples.● Foraneventsuchasflippingacoin,theequally

likelythingsthatcanhappenarecalledoutcomes.Forexample,therearetwoequallylikelyoutcomeswhenflippingacoin:thecoincanlandheadsup,orthecoincanlandtailsup.Thetwopossibleoutcomes,headsuportailsup,areequallylikely.

● Foranothereventsuchasspinningaspinnerthatisone-thirdredandtwo-thirdsblue,thetwooutcomes,redandblue,arenotequallylikely.Thisisanunfairspinner(sinceitisnotdividedequally),therefore,theoutcomesarenotequallylikely.

(a,b)● Createamodelorpracticalproblem

torepresentagivenprobability.(c)● Representprobabilityasapoint

between0and1,inclusively,onanumberline.

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Equallylikelyeventscanberepresentedwithfractionsofequivalentvalue.Forexample,onaspinnerwitheightsectionsofequalsize,wherethreeofthesectionsarelabeledG(green)andthreearelabeledB(blue),th

● echancesoflandingongreenoronblueareequallylikely;theprobabilityofeachoftheseeventsisthesame,or!

!.

● Studentsneedopportunitiestocreateamodelorpracticalproblemthatrepresentsagivenprobability.Forexample,ifaskedtocreateaboxofmarbleswheretheprobabilityofselectingablackmarbleis!

!,sampleresponsesmight

include:

● Whenaprobabilityexperimenthasveryfewtrials,

theresultscanbemisleading.Themoretimesanexperimentisdone,theclosertheexperimentalprobabilitycomestothetheoreticalprobability(e.g.,acoinlandsheadsuphalfofthetime).


● Probability● Event● Outcomes● Possibleoutcomes

● Certain● *Likely/likelihoodofanevent● Unlikely

● Impossible● Predict● Aslikely

● Asunlikely● *Equallylikely● Dataanalysis

Fourth Grade Math Pacing Guide - Home - Waynesboro ... Grade Math Pacing Guide Revised: May 2017...

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