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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)
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● Decimal● Decimalpoint● Decimalnumber● Standard● Written● Expanded
● Tenth● Hundredth● Thousandth● Compare● Order● >
● <● =● ≠● Greaterthan● Lessthan● Equalto
● Notequalto● Equivalent● Fraction● Fractionalpart● Mixednumber
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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
(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
● 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!
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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.
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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
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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.
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● Closerto● Between● Alittlemorethan● Estimate
● Compatiblenumbers● Frontendestimation● +● Add
● Addition● Sum● -● Subtract
● Subtraction● Difference● Addend
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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.
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto
• 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)
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– 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
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– 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
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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
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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
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• 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.
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● 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).
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● =● ≠● Equal
● Equality● Equivalent
● Expression● Equation
● Variable● *AssociativeProperty
ofAddition
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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
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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,
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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)
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EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● 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
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FractionsandDecimalsWeeks21-27SOLs4.2,4.3(d),4.5,4.64.2Thestudentwill
a)compareandorderfractionsandmixednumbers,withandwithoutmodels;*b)representequivalentfractions;*andc)identifythedivisionstatementthatrepresentsafraction,withmodelsandincontext.*Onthestateassessment,itemsmeasuringthisobjectiveareassessedwithouttheuseofacalculator.
UNDERSTANDINGTHESTANDARD(BackgroundInformationforInstructorUseOnly)
ESSENTIALUNDERSTANDINGS
ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
● 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
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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)
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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.
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● Fraction● Fractionalpart● Partofawhole● Mixednumber● ImproperFraction
● ProperFraction● *Denominator● *Numerator● Commondenominator● Likedenominator
● Unlikedenominator● *Leastcommonmultiple● Leastcommondenominator● Compare
● Order● Equivalentfraction
4.3Thestudentwilld)givenamodel,writethedecimalandfractionequivalents.
UNDERSTANDINGTHESTANDARD
ESSENTIALUNDERSTANDINGS
ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
● Decimalsandfractionsrepresentthesamerelationships;however,theyarepresentedin
Allstudentsshould● Understandthat
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematical
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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)
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● 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
(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
● Afactorofawholenumberisawholenumberan Allstudentsshould Thestudentwilluseproblem PreviousSOL4.5cisnow4.6a
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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,
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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.
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lines,money).● Fordecimalcomputation,thesameideasdevelopedfor
wholenumbercomputationmaybeused,andtheseideasmaybeappliedtodecimals,givingcarefulattentiontotheplacementofthedecimalpointinthesolution.Lininguptenthstotenths,hundredthstohundredths,etc.helpstoestablishthecorrectplacementofthedecimal.
● Fractionsmayberelatedtodecimalsbyusingmodels(e.g.,10-by-10grids,decimalsquares,money).
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● 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
UNDERSTANDINGSESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES
(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
• Additionandsubtractionofdecimalsmaybeexplored,usingavarietyofmodels(e.g.,10-by-10grids,numberlines,money).
• Theproblem-solvingprocessisenhancedwhenstudentscreateandsolvetheirownpracticalproblemsandmodelproblemsusingmanipulativesanddrawings.
• Inproblemsolving,emphasisshouldbeplacedonthinkingandreasoningratherthanonkeywords.Focusingonkeywordssuchasinall,
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto
• Estimatesumsanddifferencesofdecimals.(a)
• Addandsubtractdecimalsthroughthousandths,usingconcretematerials,
Pleasenoteallcontentof4.6hasbeenmovedfromprevious4.5candd.THISISNOTNEWThepreviousSOL4.6isnowembeddedinnewSOL4.8.
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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
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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.
UNDERSTANDINGTHESTANDARD ESSENTIALUNDERSTANDINGS
ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
● 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
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto
• 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.
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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
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- 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.
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● 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
ESSENTIALUNDERSTANDINGS
ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
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• Perimeteristhepathordistancearoundanyplanefigure.
• Todeterminetheperimeterofanypolygon,determinethesumofthelengthsofthesides.
• Areaisthesurfaceincludedwithinaplanefigure.Areaismeasuredbythenumberofsquareunitsneededtocoverasurfaceorplanefigure.
• Studentsshouldhaveopportunitiestoinvestigateanddiscover,usingmanipulatives,theformulasfortheareaofasquareandtheareaofarectangle.
- Areaofasquare=sidelength×sidelength
- Areaofrectangle=length×widthPerimeterandareashouldalwaysbelabeledwiththeappropriateunitofmeasure.
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto
• Determinetheperimeterofapolygonwithnomorethaneightsides,whenthelengthsofthesidesaregiven,withdiagrams.
• Determinetheperimeterandareaofarectanglewhengiventhemeasureoftwoadjacentsides,withandwithoutdiagrams.
• Determinetheperimeterandareaofasquarewhenthemeasureofonesideisgiven,withandwithoutdiagrams.
• SolvepracticalproblemsthatinvolvedeterminingperimeterandareainU.S.Customaryandmetricunits.
PleasenotethisstandardisanADDITIONTOFOURTHGRADE.Itwaspreviously5.8.
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● Area● SquareUnits● Perimeter● Units
● Length● Width● Distance● Polygon
● Adjacent● Surface● PlaneFigure
● Square● Rectangle● Formula
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ElapsedTimeWeeks33SOL4.94.9Thestudentwillsolvepracticalproblemsrelatedtodetermineelapsedtimeinhoursandminuteswithina12-hourperiod.UNDERSTANDINGTHESTANDARD ESSENTIAL
UNDERSTANDINGSESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES
(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.
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto
• 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
UNDERSTANDINGSESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES
(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
● 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.
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● 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.
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● *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.
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto
● 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).
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
▪ 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.
UNDERSTANDINGTHESTANDARD
ESSENTIALUNDERSTANDINGS
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).
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● *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
Thestudentwilluseproblemsolving,mathematicalcommunication,mathematicalreasoning,connections,andrepresentationsto
● 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.
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
▪ Quadrilateral▪ Rectangle
▪ Square▪ Trapezoid
▪ Parallelogram▪ Rhombus
▪ Pentagon
4.15 Thestudentwillidentify,describe,recognize,create,andextendpatternsfoundinobjects,pictures,numbers,andtables.numericalandgeometricpatterns.
UNDERSTANDINGTHESTANDARD ESSENTIALUNDERSTANDINGS
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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EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● Pattern ● Geometricpattern ● Numericalpattern
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ProbabilityandStatisticsWeek36SOLs4.134.13 Thestudentwill
a)predictdeterminethelikelihoodofanoutcomeofasimpleevent;andb)representprobabilityasanumberbetween0and1,inclusive;andc)createamodelorpracticalproblemtorepresentagivenprobability.
UNDERSTANDINGTHESTANDARD ESSENTIALUNDERSTANDINGS
ESSENTIALKNOWLEDGEANDSKILLS IMPORTANTCHANGES(Thiscolumnwillnotbeincludedinthe2018-2019pacingguide)
● 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).
EssentialVocabulary:Thesearethetermsthatneedtobeexplicitlytaught.Thosethathavea*nexttothemcanbefoundinflashcardformat(somecardscontainpictures)whichcanbefoundontheVirginiaDepartmentofEducationwebsiteorbycopyingandpastingthefollowingwebaddressintoyourinternetbrowser:http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/grade4-5.pdf
● Probability● Event● Outcomes● Possibleoutcomes
● Certain● *Likely/likelihoodofanevent● Unlikely
● Impossible● Predict● Aslikely
● Asunlikely● *Equallylikely● Dataanalysis