Homework Helper 1

MathHomeworkHelperNumberandOperationsinBaseTenMCC4.NBT.1Recognizethatinamulti‐digitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.MCC4.NBT.2Readandwritemulti‐digitwholenumbersusingbase‐tennumerals,numbernames,andexpandedform.Comparetwomulti‐digitnumbersbasedonmeaningsofthedigitsineachplace,using>,=,and<symbolstorecordtheresultsofthecomparison.BaseTenreferstothenumberingsystemincommonuse.Takeanumberlike475,basetenreferstotheposition,the5isintheone'splace,the7isintheten'splaceandthe4isinthehundred'splace.Eachnumberis10timesthevaluetotherightofit,hencetheterm‘baseten’.Base10blocksareoftenusedinclasstohelpstudentsgraspnumber.Basetenblockshave‘aunit’torepresentone,‘arod’torepresenttenand‘aflat’torepresent100.

millions’

hundred

thousands’

ten

thousands’

thousands’

hundreds’

tens’

ones’

tenths’

hundredths’

6, 4 7 2, 9 1 3. 5 8

Example:7place:tenthousands’place value:70,000 Example:9place:hundreds’place value:900

digit:number

expandedform:toshowanumberasthesumofthevalueofitsdigits Example:7,965inexpandedformis7,000+900+60+5 Example:onethousandtwelveinexpandedformis1,000+10+2

numeral:theusualwayofwritinganumber Example:234,908 Example:400+20+2is422 Example:Fifteenis15.

numbername:toshowanumberasword(s) Example:14,009:thenumbernameisfourteenthousand,nine Example:23.09:thenumbernameistwenty‐threeandninehundredths Example:2.9:thenumbernameistwoandninetenths

Think:Placeisaword(name)Valueisanumber(digits)

Homework Helper 2

ComparingWholeNumbersSymbolsareusedtoshowhowthesizeofonenumbercomparestoanother.Thesesymbolsare<(lessthan),>(greaterthan),and=(equals).Forexample,since2issmallerthan4and4islargerthan2,wecanwrite:2<4,whichsaysthesameas4>2andofcourse,4=4.

• Tocomparetwowholenumbers,firstputtheminnumeralform.• Thenumberwithmoredigitsisgreaterthantheother.• Iftheyhavethesamenumberofdigits,comparethedigitsinthehighestplacevalueposition(theleftmostdigitofeachnumber).Theonehavingthelargerdigitisgreaterthantheother.

• Ifthosedigitsarethesame,comparethenextpairofdigits(lookatthenumberstotheright).

• Repeatthisuntilthepairofdigitsisdifferent.Thenumberwiththelargerdigitisgreaterthantheother.

Example:402hasmoredigitsthan42,so402>42.Example:402and412havethesamenumberofdigits.Wecomparetheleftmostdigitofeachnumber:4ineachcase.Movingtotheright,wecomparethenexttwonumbers:0and1.Sincezeroislessthanone,402islessthan412or402<412.

RoundingMCC4.NBT.3Useplacevalueunderstandingtoroundmulti‐digitwholenumberstoanyplace.

RuleOne.Determinewhatyourroundingdigitisandunderlineit.Thenlooktotherightsideofit.Ifthedigitis0,1,2,3,or4donotchangethedigityouunderlined.Alldigitsthatareontherighthandsideoftherequestedroundingdigitwillbecome0.RuleTwo.Determinewhatyourroundingdigitisandunderlineit.Thenlooktotherightofit.Ifthedigitis5,6,7,8,or9,yourunderlinednumberupbyone.Alldigitsthatareontherighthandsideoftherequestedroundingdigitwillbecome0.

roundingtothenearestwholenumber(theones’place) 1.25to1 45.8to46 23.398to23

roundingtothenearestten 21to20 95to100 441 to440 1,125to1,130

roundingtothenearesthundred 236 to200 754 to800 1,992to2,000 1,232to1,200

roundingtothenearestthousand 2,363to2,000 7,541to8,000 14,227to14,000

roundingtothenearestmillion 2,908,674to3,000,000 14,643,960to15,000,000

Homework Helper 3

EstimationMCC4.OA.3Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.

estimatethesum:roundthenumbers,thenadd23 20 345 300 9,1229,000+89 +90 +496 +500 +5,631+6,000110 800 15,000estimatethedifference:roundthenumbers,thensubtract45 50 7457009,129 9,000‐21 ­20 ‐496 ­500 ‐5,631­6,00030 200 3,000estimatetheproduct:roundthenumbersandmultiply78 80 67 70 254 300x5x5 x18 x20 x349x300400 1,400 90,000estimatethequotient:roundthenumbersanddivide61÷2= 891÷32=60÷2=30 900÷30=30

Homework Helper 4

MultiplicationMCC4.NBT.5Multiplyawholenumberofuptofourdigitsbyaone‐digitnumber,andmultiplytwotwo‐digitnumbers,usingstrategiesbasedonplacevalueandthepropertiesofoperations.Illustrateandexplainthecalculationbyusingequations,rectanglearrays,and/orareamodels.PropertiesofMultiplicationcommutativepropertyofmultiplication:theorderofthefactorsdoesnotchangeaproduct. Example:2 4=4 28=8

associativepropertyofmultiplication:thewayfactorsaregroupeddoesnotchangetheproduct. Example:(2 3) 4=2 (3 4) 6 4=2 12 24=24

multiplicativeidentityproperty:anynumbermultipliedbyoneremainsthesamenumber. Example:5 1=5or1×5=5

distributivepropertyofmultiplication:multiplyingasumbyanumberisthesameasmultiplyingeachaddendbythenumberandthenaddingtheproblem. Example:2 (3+4)=(2 3)+(2 4) 2 7=6+814=14 Example:432x4=(400x4)+(30x4)+(2x4) 1,728=1,600+120+81,728=1,728 zeropropertyofmultiplication:anynumbermultipliedbyzerowillresultintheproductofzero Example:5 0=0or0×5=0

MultiplicationAlgorithm +4+2+1+1+312 23 432723 2125 x2 x7x2x6 x12x97 24 1618644,338 42175+210+2250 2522,425

Homework Helper 5

RectangleArrays

AreaModels

DivisionMCC4.NBT.6Findwhole‐numberquotientsandremainderswithuptofour‐digitdividendsandone‐digitdivisors,usingstrategiesbasedonplacevalue,thepropertiesofoperations,and/ortherelationshipbetweenmultiplicationanddivision.Illustrateandexplainthecalculationbyusingequations,rectangulararrays,and/orareamodels.

DivisionAlgorithm

9R292R8928R71049 ‐81‐81‐81‐82292503 ‐21‐18‐087939‐72‐32772 ‐72 0

Homework Helper 6

DivisibilityRules(forwholenumbers)Anumberisdivisibleby2ifthedigitinitsones’positioniseven,(either0,2,4,6,or8).Anumberisdivisibleby3ifthesumofallitsdigitsisdivisibleby3.Anumberisdivisibleby4ifthenumberformedbythelasttwodigitsisdivisibleby4.Anumberisdivisibleby5ifthedigitinitsones’positionis0or5.Anumberisdivisibleby6ifitisanevennumberanddivisibleby3.Anumberisdivisibleby8ifthenumberformedbythelastthreedigitsisdivisibleby8.Anumberisdivisibleby9ifthesumofallitsdigitsisdivisibleby9.Anumberisdivisibleby10ifthedigitinisones’placeiszero.

GeometryMCC4.G.1Drawpoints,lines,linesegments,rays,angles(acute,right,obtuse),andperpendicularandparallellines.Identifytheseintwo‐dimensionalobjects.MCC4.G.2Classifytwo‐dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.MCC4.G.3Recognizealineofsymmetryforatwo‐dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline‐symmetricfiguresanddrawlinesofsymmetry.MCC4.MD.3Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.

intersectinglines:twolinesthatcrossparallellines:linesthatneverintersectperpendicularlines:linesthatintersectandformrightanglesarea:tofigurearea,multiplythelengthbythewidth.5ft5ft×8ft=40squarefeet8ftarea=40squarefeet

8in.×8in.=64squareinches8in.area=64squareinches*Squarehasallequalsides.Sometimesonlyonelinesegmentislabeledwiththemeasurement.

9miles� area=72squaremiles�×9mi=72squaremi. 72sqmi.÷9mi.=8mi.

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perimeter:tofigureperimeter,alladdthesides.�+3m+4m+3m+4m=17m3m3m+4m+3m+4m=14m5ft5+5+8+8=26feet�4m17m‐14m=3mperimeter=26feetP=17m�=3m 8ft4m3m

Angle:tworaysthatshareacommonendpoint.

AcuteAngleAnglethatislessthan90°.

RightAngleAnglethatis90°.

ObtuseAngleAnglethatismorethan90°.

Triangle:three‐sidedpolygonwhoseangles’sumare180°.RightTriangle

onerightangle.perpendicularlines

AcuteTriangleallacuteangles.

ObtuseTriangleoneobtuseangle.

EquilateralTriangle

Allthreeequallengthsides

IsoscelesTriangleTwoequallengthsides

ScaleneTriangleNosidesofequallength

60°+90°+�=180° 60°60°180°+90°‐150°� 150°30°y=30°

Quadrilateralsfour‐sidedpolygon

TrapezoidParallelogramonesetofparallellines twosetsofparallellinesoppositesidesareequal

RhombusRectangletwosetsofparallellinestwosetsofparallellines

allsidesequal oppositesidesareequal allrightangles

(perpendicularlines)Square

twosetsofparallellinesallsidesareequalallrightangles(perpendicularlines)

Homework Helper 8

Symmetry• Lineofsymmetryisalinethatdividesafigureintotwocongruentparts,eachofwhichisthemirrorimageoftheother.

• Whenthefigurehavingalineofsymmetryisfoldedalongthelineofsymmetry,thetwopartsshouldmatchup.

AngleMeasurementMCC4.MD.5Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:

a) Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirclearcbetweenthepointswherethetworaysintersectthecircle.

b) Ananglethatturnsthrough1/360ofacircleiscalleda“one‐degreeangle”,andcanbeusedtomeasureangles.

MCC4.MD.6Measureanglesinwhole‐numberdegreesusingaprotractor.Sketchanglesofspecifiedmeasure.

Protractor:tooltomeasureangles.

ToMeasureanAngle• Findthecenterholeonthestraightedgeoftheprotractor.• Placetheholeoverthevertex,orpoint,oftheangleyouwishtomeasure.• Lineupthezeroonthestraightedgeoftheprotractorwithoneofthesidesoftheangle.

• Findthepointwherethesecondsideoftheanglecrossesthecurvededgeoftheprotractor.Readthenumberthatiswrittenontheprotractoratthispoint.Thisisthemeasureoftheangleindegrees.

ToConstructanAngle• Usethestraightedgeoftheprotractortodrawastraightline.Thislinewillformonesideofyourangle.

• Findthecenterholeonthestraightedgeoftheprotractor.Placetheholeoveroneendpointofthelineyouhavedrawn.(continuedonnextpage)

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• Lineupthezeroonthestraightedgeoftheprotractorwiththeline.• Makeamarkatthenumberonthecurvededgeoftheprotractorthatcorrespondstothedesiredmeasureofyourangle.Forexample,markat90fora90‐degreeangle.

• Usethestraightedgeoftheprotractortoconnectthemarktotheendpointofthefirstline,forminganangle.

MCC4.MD.7Recognizeanglemeasureasadditive.Whenanangleisdecomposedintonon‐overlappingparts,theanglemeasureofthewholeisthesumoftheanglemeasuresoftheparts.Solveadditionandsubtractionproblemstofindunknownanglesonadiagraminrealworldandmathematicalproblems.

CircleVocabularyCenter:thepointdirectlyinthemiddleofthecircle.Radius:alinesegmentthattravelsfromthecenterofthecircletotheoutsiderim.RadiusDiameter:alinesegmentthattravelscompletelyacrossthecirclepassingthroughthecenter.Diameter

Benchmarkangles:90°isarightangle

180°isastraightangle

MeasurementMCC4.MD.1Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;lb,oz;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.MCC4.MD.2Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.

metricunits customaryunitsweightmass

1kilogram(kg)=1,000grams(g) 1pound(lb)=16ounces(oz)

lengthdistance

1kilometer(km)=1,000meter(m)1meter(m)=100centimeters(cm)

1yard(yd)=3feet(ft)or36in.1foot(ft)=12inches(in.)

capacityliquidvolume

1liter(l)=1,000millimeters(ml) 1gallon(gal)=4quarts(qt)1quart(qt)=2pints(pt)1pint(pt)=2cups(c)

time1hour=60minutes1minute=60seconds

Toconvertalargerunittoasmallerunitofmeasure,youmultiply.Forexampleyouchange5meters(largeunit)tocentimeters(smallerunit),thereare100centimetersinonemeter.Multiply5×100=500cm.Therefore5m=500cm.

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Sometimesconversionsaremulti‐stepproblemssuchasconverting2gallonstopints.First,multiply2by4,whichis8quarts;thenmultiple8quartsby2whichequals16pints.

DataMCC.MD.4Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(12,14,18).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.Alineplotisagraphthatshowsfrequencyofdataalonganumberline.Itisbesttousealineplotwhencomparingfewerthan25numbers.Itisaquick,simplewaytoorganizedata.Alineplotconsistsofahorizontalnumberline,onwhicheachvalueofasetisdenotedbyanxoverthecorrespondingvalueonthenumberline.Thenumberofx'saboveeachscoreindicateshowmanytimeseachvalueoccurredinthedataset.dataset:

18,18,38,38,38,48,58,68,68,68,68,78,88,88,98,128,128

TheLengthofJamie’sEarthworms

NumberandOperations­FractionsGrade4expectationsinthisdomainarelimitedtofractionswithdenominatorsof2,3,4,5,6,8,10,12,and100.

Extendunderstandingoffractionequivalenceandordering.

MCC4.NF.1Explainwhyafraction��isequivalenttoafraction(��) (��) byusingvisualfractionmodels,withattentiontohowthenumberandsizeofthepartsdiffereventhoughthetwofractionsthemselvesarethesamesize.Usethisprincipletorecognizeandgenerateequivalentfractions.Equivalentfractionscanbecreatedbymultiplying(ordividing)boththenumeratoranddenominatorbythesamenumber.Wecandothisbecause,ifyoumultiplyboththenumeratoranddenominatorofafractionbythesamenumber,thefractionremainsunchangedinvalue.Inthemodelbelow,youcouldgetthefraction46bymultiplyingboththetopandbottomof23by2.

�� �� =����=����

���� �� =����=��=MCC4.NF.2Comparetwofractionswithdifferentnumeratorsanddifferentdenominators,e.g.,bycreatingcommondenominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas12.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordthe

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resultsofcomparisonswithsymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.Buildfractionsfromunitfractionsbyapplyingandextendingpreviousunderstandingsofoperationsonwholenumbers.Iftwofractionshavedifferentnumeratorsanddenominatorsitisdifficulttodeterminewhichfractionislarger.Itiseasiertodeterminewhichislargerifbothfractionshavethesamedenominator.Step1:Findacommondenominator.Theleastcommondenominatoroftwodenominatorsisactuallythesmallestnumberthatisdivisiblebyeachofthedenominators.

Tofindtheleastcommondenominator,simplylistthemultiplesofeachdenominator(multiplyby2,3,4,etc.outtoabout6orsevenusuallyworks)thenlookforthesmallestnumberthatappearsineachlist.Example:Supposewewantedtocompare512and13.Tofindtheleastcommondenominatorasfollows:• Firstwelistthemultiplesofeachdenominator.Multiplesof12are12,24,36,48,60,…Multiplesof3are3,6,9,12,15,18,21,,…Now,whenyoulookatthelistofmultiples,youcanseethat12isthesmallestnumberthatappearsineachlist.Therefore,theleastcommondenominatorof512and13is12.

Step2:Multiplythenumeratoranddenominatorofonefractionbythesamenumbersobothfractionswillhavethesamedenominator.

• Forexample,if512and13arebeingcompared,13shouldbemultipliedby44.Itdoesnotchangethevalueof13tobemultipliedby44(whichisequalto1)becauseanynumbermultipliedby1isstillthesamenumber.Afterthemultiplication�×� �×� =���thecomparisoncanbemadebetween512and412.So512isgreaterthan412or13.

• Youmayhavetomultiplybothfractionsbydifferentnumberstoproducethesamedenominatorforbothfractions.Forexampleif23and34arecompared,firstwelistthemultiplesofthetwodenominators.

Multiplesof3are3,6,9,12,15,18,…Multiplesof4are4,8,12,16,20,24,…

Next,weneedtomultiply23by44toget812andmultiply34by33toget912.Thefraction34whichisequalto912islargerthan23whichisequalto812.

MCC4.NF.3Understandafraction��witha>1asasumoffractions1/b.

a) Understandadditionandsubtractionoffractionsasjoiningandseparatingpartsreferringtothesamewhole.

b) Decomposeafractionintoasumoffractionswiththesamedenominatorinmorethanoneway,recordingeachdecompositionbyanequation.Justifydecompositions,e.g.,byusingavisualfractionmodel.

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DecompositionofFractionsExamples:38=18+18+18or38=18+28218=1+1+18or88+88+18=218.

c) Addandsubtractmixednumberswithlikedenominators,e.g.,byreplacingeachmixednumberwithanequivalentfraction,and/orbyusingpropertiesofoperationsandtherelationshipbetweenadditionandsubtraction.

Tochange2¾(amixednumber)intoanequivalentfraction:Multiply4x2,thenadd3=11,thisisthenumerator.Usethesamedenominator(4),andtheequivalentfractionis11/4.Ordecomposethemixednumber2¾to44+44+34=114(remember44=1)

d)Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.

MCC4.NF.4Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionbyawholenumber.

a.Understandafraction��asamultipleof1�.Forexample,useavisualfractionmodeltorepresent54astheproduct5×14,recordingtheconclusionbytheequation54=5×14.

b.Understandamultipleof��asamultipleof1�,andusethisunderstandingtomultiplyafractionbyawholenumber.

Forexample,useavisualfractionmodeltoexpress3×25as6×15recognizingthisproductas65.Ingeneral,n×��=(�×�) �.

c.Solvewordproblemsinvolvingmultiplicationofafractionbyawholenumber,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.

Decimals

Understanddecimalnotationforfractions,andcomparedecimalfractions.MCC4.NF.5Expressafractionwithdenominator10asanequivalentfractionwithdenominator100,andusethistechniquetoaddtwofractionswithrespectivedenominators10and100.Forexample,express310as30100,andadd310+4100=34100.

110=0.1or0.10 810=0.8or0.80 9100=0.0926100=0.26

MCC4.NF.6Usedecimalnotationforfractionswithdenominators10or100.

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Forexample,rewrite0.62as62100;describealengthas0.62meters;locate0.62onanumberlinediagram.MCC4.NF.7Comparetwodecimalstohundredthsbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwodecimalsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualmodel.Decimalsarecomparedexactlythesamewayaswholenumbers:bycomparingthedifferentplacevaluesfromlefttoright.Tohelpyou,writethetwonumbersintotheplacevaluetablesontopofeachother.Thencomparethedifferentplacevaluesinthetwonumbersfromlefttoright,startingfromthebiggestplacevalue.Withdecimalnumbers,youcannotassumethenumberwiththemostdigitsisthelargestnumber.Seebelow.

Above,thetwonumbershavethesamevalueintheones’place.Next,thedecimalsarecomparedstartingwithtenthsplaceandthenhundredthsplace(ifnecessary).Ifonedecimalhasahighernumberinthetenthsplacethenitislargerthanadecimalwithfewertenths.Ifthetenthsareequalcomparethehundredthsuntilonedecimalislargerortherearenomoreplacestocompare.Ifeachdecimalplacevalueisthesamethenthedecimalsareequal.OperationsandAlgebraicThinking

Usethefouroperationswithwholenumberstosolveproblems.MCC4.OA.1Interpretamultiplicationequationasacomparison,e.g.,interpret35=5×7asastatementthat35is5timesasmanyas7and7timesasmanyas5.Representverbalstatementsofmultiplicativecomparisonsasmultiplicationequations.SeeCommutativePropertyofMultiplication.MCC4.OA.2Multiplyordividetosolvewordproblemsinvolvingmultiplicativecomparison,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem,distinguishingmultiplicativecomparisonfromadditivecomparison.(Seechartonthelastpage)

1.3 1.031.3>1.03

1 .3 1 .0 3

tens’

ones’

tenths

hundredths

2.16 2.52.16<2.5

2 .1 6 2 .5

tens’

ones’

tenths

hundredths

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MCC4.OA.3Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole‐numberanswersusingthefouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.

Solvingforavariable(unknownquantity)inaproblem

AdditionProblems�+23=5422+�=5654–23=3156–22=34�=31�=34

SubtractionProblems�‐23=5456‐�=2254+23=7756–22=34�=77�=34

MultiplicationProblems�x5=657x�=9865÷5=1398÷7=14�=13�=14

DivisionProblems65÷�=598÷�=1465÷5=1398÷14=7�=13�=98

Gainfamiliaritywithfactorsandmultiples.MCC4.OA.4Findallfactorpairsforawholenumberintherange1–100.Recognizethatawholenumberisamultipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1–100isamultipleofagivenone‐digitnumber.Determinewhetheragivenwholenumberintherange1–100isprimeorcomposite.

Factorpairs:twowholenumbersthatmultipliedtogethertogetaproduct.

Multiples:aproductoftwogivenwholenumbers. multiplesof2:2,4,6,8,10,12,14,16,18,20,… multiplesof8:8,16,24,32,40,48,56,64,72,80,…

Primenumbers:numberswithonefactorpair. 17=1×17 11=1×11Compositenumbers:numberswithmorethanonefactorpairs. 6=1×6 16=1×16 48=1×48 6=2×3 16=2×8 48=2×24 16=4×4 48=4×12 48=6×8Generateandanalyzepatterns.MCC4.OA.5Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.

Forexample,giventherule“Add3”andthestartingnumber1,generatetermsintheresultingsequenceandobservethatthetermsappeartoalternatebetweenoddandevennumbers.Explaininformallywhythenumberswillcontinuetoalternateinthisway.

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StandardsforMathematicalPracticeStudentsareexpectedto:1.Makesenseofproblemsandpersevereinsolvingthem.

Studentsknowthatdoingmathematicsinvolvessolvingproblemsanddiscussinghowtheysolvedthem.

2.Reasonabstractlyandquantitatively.Whenproblemsolving,studentsshouldconnectthequantitytowrittensymbolsandcreatealogicalrepresentationoftheproblemathand,consideringboththeappropriateunitsinvolvedandthemeaningofquantities.

3.Constructviableargumentsandcritiquethereasoningofothers.Studentsshouldexplaintheirthinkingandmakeconnectionsbetweenmodelsandequations.Studentsshouldexplaintheirthinkingtoothersandrespondtoothers’thinking.

4.Modelwithmathematics.Studentsshouldexperimentwithrepresentingproblemsituationsinmultiplewaysincludingnumbers,words(mathematicallanguage),drawingpictures,usingobjects,makingachart,list,orgraph,creatinganequation,etc.

5.Useappropriatetoolsstrategically.Studentsshouldconsidertheavailabletoolswhensolvingaproblemanddecidewhencertaintoolsmightbehelpful.

6.Attendtoprecision.Studentsshoulduseclearandpreciselanguageinexplainingtheirreasoningwithothers.Theycarefulnotateunitsofmeasureandareabletostatethemeaningofsymbolstheychoose.

7.Lookforandmakeusofstructure.Studentslookcloselytodiscoverapatternorstructure.

8.Lookforandexpressregularityinrepeatedreasoning.Studentsshouldnoticerepetitiveactionsincomputationtomakegeneralizations.Studentsusemodelstoexplaincalculationandunderstandhowalgorithmswork.

Glossaryaddend:numbersthatareaddedtogethercompositenumber:anumberwithmorethanonefactorpair. 12=1×12 2×63×4denominatoristhebottomnumberofthefraction;showshowmanyequalpartstheitemisdividedinto;adivisordifference:answertoasubtractionproblemdigit:numberdividend:thenumberthatanothernumberisdividedinto

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Example:45÷5=9,45isthedividenddivisor:thenumberthatanothernumberisdividedby Example:45÷5=9,5isthedivisorequation:statetwothingsarethesameusingnumbersandmathsymbols.Anequalsignisused. Example:10=1+9 8×2=6+10equivalentfractions:havethesamevalue,butmaylookdifferent(12=24)even:anumberendingwith0,2,4,6,or8.Evennumberscanbeevenlydividedintotwogroups.expandedform:toshowanumberasthesumofthevalueofitsdigitsexpression:partofanumbersentencethathasnumbersandoperationsigns,butithasnoequalsignfactor:numbersthataremultipliedfactorpair:twowholenumbersthatmultipliedtogethertogetoneproductimproperfractions:haveanumeratorwiththehighestnumber,54mixednumber:isawholenumberandafraction,212multiple:aproductoftwogivenwholenumbers. multiplesof8:8,16,24,32,40,48,56,64,72,80,…numbername:toshowanumberasword(s)numeral:theusualwayofwritinganumbernumerator:isthetopnumberofthefraction;howmanyequalpartsyouhaveodd:anynumberthatendswith1,3,5,7,or9.Oddnumberscannotbeevenlydividedintotwogroups.parallellines:linesthatnevercrossperpendicularlines:linesthatmeetorcrossatrightangles(90°)primenumber:anumberwithonefactorpair Examples:3=1×3

17=1×17product:answertoamultiplicationproblem factor×factor=productproperfractions:haveadenominatorwiththehighestnumber,78quotient:answertoadivisionproblem dividend÷divisor=quotientExample:45÷5=9,9isthequotientremainder:theportionthatwillnotmakeacompletegroup Example:21÷5=4R1,1istheremaindersum:answertoanadditionproblem addend+addend=sumvariable:aletterorsymbolthatrepresentsanumberyoudon’tknow.

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