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Glossary
Additionandsubtractionwithin5,10,20,100,or1000.Additionorsubtractionoftwowhole
numberswithwholenumberanswers,andwithsumorminuendintherange0Ͳ5,0Ͳ10,0Ͳ20,or0Ͳ
100,respectively.Example:8+2=10isanadditionwithin10,14–5=9isasubtractionwithin20,
and55–18=37isasubtractionwithin100.
Additiveinverses.Twonumberswhosesumis0areadditiveinversesofoneanother.Example:3/4
and–3/4areadditiveinversesofoneanotherbecause3/4+(–3/4)=(–3/4)+3/4=0.
Associativepropertyofaddition.SeeTable3inthisGlossary.
Associativepropertyofmultiplication.SeeTable3inthisGlossary.
Bivariatedata.Pairsoflinkednumericalobservations.Example:alistofheightsandweightsforeach
playeronafootballteam.
Boxplot.Amethodofvisuallydisplayingadistributionofdatavaluesbyusingthemedian,quartiles,
andextremesofthedataset.Aboxshowsthemiddle50%ofthedata.1
Commutativeproperty.SeeTable3inthisGlossary.
Complexfraction.AfractionA/BwhereAand/orBarefractions(Bnonzero).
Computationalgorithm.Asetofpredefinedstepsapplicabletoaclassofproblemsthatgivesthe
correctresultineverycasewhenthestepsarecarriedoutcorrectly.Seealso:computationstrategy.
Computationstrategy.Purposefulmanipulationsthatmaybechosenforspecificproblems,maynot
haveafixedorder,andmaybeaimedatconvertingoneproblemintoanother.Seealso:computation
algorithm.
Congruent.Twoplaneorsolidfiguresarecongruentifonecanbeobtainedfromtheotherbyrigid
motion(asequenceofrotations,reflections,andtranslations).
Countingon.Astrategyforfindingthenumberofobjectsinagroupwithouthavingtocountevery
memberofthegroup.Forexample,ifastackofbooksisknowntohave8booksand3morebooks
areaddedtothetop,itisnotnecessarytocountthestackalloveragain.Onecanfindthetotalby
1
AdaptedfromWisconsinDepartmentofPublicInstruction,http://dpi.wi.gov/standards/mathglos.html,accessedMarch
2,2010.
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GLOSSARY
countingon—pointingtothetopbookandsaying“eight,”followingthiswith“nine,ten,eleven.
Thereareelevenbooksnow.”
Dotplot.See:lineplot.
Dilation.Atransformationthatmoveseachpointalongtheraythroughthepointemanatingfroma
fixedcenter,andmultipliesdistancesfromthecenterbyacommonscalefactor.
Expandedform.AmultiͲdigitnumberisexpressedinexpandedformwhenitiswrittenasasumof
singleͲdigitmultiplesofpowersoften.Forexample,643=600+40+3.
Expectedvalue.Forarandomvariable,theweightedaverageofitspossiblevalues,withweights
givenbytheirrespectiveprobabilities.
Firstquartile.ForadatasetwithmedianM,thefirstquartileisthemedianofthedatavaluesless
thanM.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},thefirstquartileis6.2Seealso:
median,thirdquartile,interquartilerange.
Fraction.Anumberexpressibleintheforma/bwhereaisawholenumberandbisapositivewhole
number.(ThewordfractioninthesestandardsalwaysreferstoanonͲnegativenumber.)Seealso:
rationalnumber.
Identitypropertyof0.SeeTable3inthisGlossary.
Independentlycombinedprobabilitymodels.Twoprobabilitymodelsaresaidtobecombined
independentlyiftheprobabilityofeachorderedpairinthecombinedmodelequalstheproductof
theoriginalprobabilitiesofthetwoindividualoutcomesintheorderedpair.
Integer.Anumberexpressibleintheformaor–aforsomewholenumbera.
InterquartileRange.Ameasureofvariationinasetofnumericaldata,theinterquartilerangeisthe
distancebetweenthefirstandthirdquartilesofthedataset.Example:Forthedataset{1,3,6,7,10,
12,14,15,22,120},theinterquartilerangeis15–6=9.Seealso:firstquartile,thirdquartile.
Lineplot.Amethodofvisuallydisplayingadistributionofdatavalueswhereeachdatavalueisshown
asadotormarkaboveanumberline.Alsoknownasadotplot.3
2
Manydifferentmethodsforcomputingquartilesareinuse.ThemethoddefinedhereissometimescalledtheMoore
andMcCabemethod.SeeLangford,E.,“QuartilesinElementaryStatistics,”JournalofStatisticsEducationVolume14,
Number3(2006).
3
AdaptedfromWisconsinDepartmentofPublicInstruction,op.cit.
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GLOSSARY
Mean.Ameasureofcenterinasetofnumericaldata,computedbyaddingthevaluesinalistand
thendividingbythenumberofvaluesinthelist.4Example:Forthedataset{1,3,6,7,10,12,14,15,
22,120},themeanis21.
Meanabsolutedeviation.Ameasureofvariationinasetofnumericaldata,computedbyaddingthe
distancesbetweeneachdatavalueandthemean,thendividingbythenumberofdatavalues.
Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},themeanabsolutedeviationis20.
Median.Ameasureofcenterinasetofnumericaldata.Themedianofalistofvaluesisthevalue
appearingatthecenterofasortedversionofthelist—orthemeanofthetwocentralvalues,ifthe
listcontainsanevennumberofvalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,90},
themedianis11.
Midline.Inthegraphofatrigonometricfunction,thehorizontallinehalfwaybetweenitsmaximum
andminimumvalues.
Multiplicationanddivisionwithin100.Multiplicationordivisionoftwowholenumberswithwhole
numberanswers,andwithproductordividendintherange0Ͳ100.Example:72÷8=9.
Multiplicativeinverses.Twonumberswhoseproductis1aremultiplicativeinversesofoneanother.
Example:3/4and4/3aremultiplicativeinversesofoneanotherbecause3/4×4/3=4/3×3/4=1.
Numberlinediagram.Adiagramofthenumberlineusedtorepresentnumbersandsupport
reasoningaboutthem.Inanumberlinediagramformeasurementquantities,theintervalfrom0to1
onthediagramrepresentstheunitofmeasureforthequantity.
Percentrateofchange.Arateofchangeexpressedasapercent.Example:ifapopulationgrowsfrom
50to55inayear,itgrowsby5/50=10%peryear.
Probabilitydistribution.Thesetofpossiblevaluesofarandomvariablewithaprobabilityassignedto
each.
Propertiesofoperations.SeeTable3inthisGlossary.
Propertiesofequality.SeeTable4inthisGlossary.
Propertiesofinequality.SeeTable5inthisGlossary.
Propertiesofoperations.SeeTable3inthisGlossary.
4
Tobemoreprecise,thisdefinesthearithmeticmean.
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GLOSSARY
Probability.Anumberbetween0and1usedtoquantifylikelihoodforprocessesthathaveuncertain
outcomes(suchastossingacoin,selectingapersonatrandomfromagroupofpeople,tossingaball
atatarget,ortestingforamedicalcondition).
Probabilitymodel.Aprobabilitymodelisusedtoassignprobabilitiestooutcomesofachance
processbyexaminingthenatureoftheprocess.Thesetofalloutcomesiscalledthesamplespace,
andtheirprobabilitiessumto1.Seealso:uniformprobabilitymodel.
Randomvariable.Anassignmentofanumericalvaluetoeachoutcomeinasamplespace.
Rationalexpression.AquotientoftwopolynomialswithanonͲzerodenominator.
Rationalnumber.Anumberexpressibleintheforma/bor–a/bforsomefractiona/b.Therational
numbersincludetheintegers.
Rectilinearfigure.Apolygonallanglesofwhicharerightangles.
Rigidmotion.Atransformationofpointsinspaceconsistingofasequenceofoneormore
translations,reflections,and/orrotations.Rigidmotionsarehereassumedtopreservedistancesand
anglemeasures.
Repeatingdecimal.Thedecimalformofarationalnumber.Seealso:terminatingdecimal.
Samplespace.Inaprobabilitymodelforarandomprocess,alistoftheindividualoutcomesthatare
tobeconsidered.
Scatterplot.Agraphinthecoordinateplanerepresentingasetofbivariatedata.Forexample,the
heightsandweightsofagroupofpeoplecouldbedisplayedonascatterplot.5
Similaritytransformation.Arigidmotionfollowedbyadilation.
Tapediagram.Adrawingthatlookslikeasegmentoftape,usedtoillustratenumberrelationships.
Alsoknownasastripdiagram,barmodel,fractionstrip,orlengthmodel.
Terminatingdecimal.Adecimaliscalledterminatingifitsrepeatingdigitis0.
Thirdquartile.ForadatasetwithmedianM,thethirdquartileisthemedianofthedatavalues
greaterthanM.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},thethirdquartileis15.
Seealso:median,firstquartile,interquartilerange.
5
AdaptedfromWisconsinDepartmentofPublicInstruction,op.cit.
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GLOSSARY
Transitivityprincipleforindirectmeasurement.IfthelengthofobjectAisgreaterthanthelengthof
objectB,andthelengthofobjectBisgreaterthanthelengthofobjectC,thenthelengthofobjectA
isgreaterthanthelengthofobjectC.Thisprincipleappliestomeasurementofotherquantitiesas
well.
Uniformprobabilitymodel.Aprobabilitymodelwhichassignsequalprobabilitytoalloutcomes.See
also:probabilitymodel.
Vector.Aquantitywithmagnitudeanddirectionintheplaneorinspace,definedbyanorderedpair
ortripleofrealnumbers.
Visualfractionmodel.Atapediagram,numberlinediagram,orareamodel.
Wholenumbers.Thenumbers0,1,2,3,….
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Table 1. Common addition and subtraction situations.6
GLOSSARY
Result Unknown
Add to
Take from
Twobunniesweresittingonthe
Twobunniessatonthegrass.
grass.Somemorebunnies
Threemorebunnieshopped
there.Howmanybunniesareon hoppedthere.Thentherewere
fivebunnies.Howmanybunnies
thegrassnow?
hoppedovertothefirsttwo?
2+3=?
2+?=5
Compare9
Somebunniesweresittingon
thegrass.Threemorebunnies
hoppedthere.Thentherewere
fivebunnies.Howmanybunnies
wereonthegrassbefore?
?+3=5
Fiveappleswereonthetable.I
atesomeapples.Thenthere
werethreeapples.Howmany
applesdidIeat?
5–?=3
Someappleswereonthetable.I
atetwoapples.Thentherewere
threeapples.Howmanyapples
wereonthetablebefore?
?–2=3
Total Unknown
Threeredapplesandtwogreen
applesareonthetable.How
Put Together/ manyapplesareonthetable?
Take Apart8 3+2=?
Start Unknown
Fiveappleswereonthetable.I
atetwoapples.Howmany
applesareonthetablenow?
5–2=?
Change Unknown
Difference Unknown
(“Howmanymore?”version):
Lucyhastwoapples.Juliehas
fiveapples.Howmanymore
applesdoesJuliehavethan
Lucy?
(“Howmanyfewer?”version):
Lucyhastwoapples.Juliehas
fiveapples.Howmanyfewer
applesdoesLucyhavethan
Julie?
2+?=5,5–2=?
Both Addends Unknown7
Addend Unknown
Fiveapplesareonthetable.
Threeareredandtherestare
green.Howmanyapplesare
green?
3+?=5,5–3=?
Grandmahasfiveflowers.How
manycansheputinherredvase
andhowmanyinherbluevase?
5=0+5,5=5+0
5=1+4,5=4+1
5=2+3,5=3+2
Bigger Unknown
Smaller Unknown
(Versionwith“more”):
Juliehasthreemoreapplesthan
Lucy.Lucyhastwoapples.How
manyapplesdoesJuliehave?
(Versionwith“fewer”):
Lucyhas3fewerapplesthan
Julie.Lucyhastwoapples.How
manyapplesdoesJuliehave?
2+3=?,3+2=?
(Versionwith“more”):
Juliehasthreemoreapplesthan
Lucy.Juliehasfiveapples.How
manyapplesdoesLucyhave?
(Versionwith“fewer”):
Lucyhas3fewerapplesthan
Julie.Juliehasfiveapples.How
manyapplesdoesLucyhave?
5–3=?,?+3=5
6
AdaptedfromBoxes2–4ofMathematicsLearninginEarlyChildhood,NationalResearchCouncil(2009,pp.32–33).
Thesetakeapartsituationscanbeusedtoshowallthedecompositionsofagivennumber.Theassociatedequations,whichhavethe
totalontheleftoftheequalsign,helpchildrenunderstandthatthe=signdoesnotalwaysmeanmakesorresultsinbutalwaysdoes
meanisthesamenumberas.
8
Eitheraddendcanbeunknown,sotherearethreevariationsoftheseproblemsituations.BothAddendsUnknownisaproductive
extensionofthisbasicsituation,especiallyforsmallnumberslessthanorequalto10.
9
FortheBiggerUnknownorSmallerUnknownsituations,oneversiondirectsthecorrectoperation(theversionusingmoreforthe
biggerunknownandusinglessforthesmallerunknown).Theotherversionsaremoredifficult.
7
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Table 2. Common multiplication and division situations.10
GLOSSARY
Equal Groups
Arrays,11
Area12
Compare
General
Unknown Product
Group Size Unknown
(“Howmanyineachgroup?”
Division)
Number of Groups
Unknown
(“Howmanygroups?”
Division)
3u6=?
3 u ? = 18 and 18 ÷ 3 = ?
? u 6 = 18 and 18 ÷ 6 = ?
Thereare3bagswith6
plumsineachbag.How
manyplumsarethereinall?
Measurementexample.You
need3lengthsofstring,each
6incheslong.Howmuch
stringwillyouneed
altogether?
If18plumsareshared
equallyinto3bags,thenhow
manyplumswillbeineach
bag?
Measurementexample.You
have18inchesofstring,
whichyouwillcutinto3
equalpieces.Howlongwill
eachpieceofstringbe?
If18plumsaretobepacked
6toabag,thenhowmany
bagsareneeded?
Measurementexample.You
have18inchesofstring,
whichyouwillcutintopieces
thatare6incheslong.How
manypiecesofstringwillyou
have?
Thereare3rowsofapples
with6applesineachrow.
Howmanyapplesarethere?
Areaexample.Whatisthe
areaofa3cmby6cm
rectangle?
If18applesarearrangedinto
3equalrows,howmany
appleswillbeineachrow?
Areaexample.Arectangle
hasarea18square
centimeters.Ifonesideis3
cmlong,howlongisaside
nexttoit?
If18applesarearrangedinto
equalrowsof6apples,how
manyrowswilltherebe?
Areaexample.Arectangle
hasarea18square
centimeters.Ifonesideis6
cmlong,howlongisaside
nexttoit?
Abluehatcosts$6.Aredhat
costs3timesasmuchasthe
bluehat.Howmuchdoesthe
redhatcost?
Measurementexample.A
rubberbandis6cmlong.
Howlongwilltherubber
bandbewhenitisstretched
tobe3timesaslong?
Aredhatcosts$18andthat
is3timesasmuchasablue
hatcosts.Howmuchdoesa
bluehatcost?
Measurementexample.A
rubberbandisstretchedto
be18cmlongandthatis3
timesaslongasitwasat
first.Howlongwasthe
rubberbandatfirst?
Aredhatcosts$18anda
bluehatcosts$6.Howmany
timesasmuchdoesthered
hatcostasthebluehat?
Measurementexample.A
rubberbandwas6cmlongat
first.Nowitisstretchedtobe
18cmlong.Howmanytimes
aslongistherubberband
nowasitwasatfirst?
aub=?
au?=pandpya=?
?ub=pandpyb=?
10
Thefirstexamplesineachcellareexamplesofdiscretethings.Theseareeasierforstudentsandshouldbegivenbefore
themeasurementexamples.
11
Thelanguageinthearrayexamplesshowstheeasiestformofarrayproblems.Aharderformistousethetermsrows
andcolumns:Theapplesinthegrocerywindowarein3rowsand6columns.Howmanyapplesareinthere?Bothforms
arevaluable.
12
Areainvolvesarraysofsquaresthathavebeenpushedtogethersothattherearenogapsoroverlaps,soarray
problemsincludetheseespeciallyimportantmeasurementsituations.
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GLOSSARY
Table 3. The properties of operations.
Herea,bandcstandforarbitrarynumbersinagivennumbersystem.Thepropertiesofoperations
applytotherationalnumbersystem,therealnumbersystem,andthecomplexnumbersystem.
Associativepropertyofaddition (a+b)+c=a+(b+c)
Commutativepropertyofaddition a+b=b+a
Additiveidentitypropertyof0 a+0=0+a=a
Existenceofadditiveinverses Foreveryathereexists–asothata+(–a)=(–a)+a=0.
Associativepropertyofmultiplication (aub)uc=au(buc)
Commutativepropertyofmultiplication aub=bua
Multiplicativeidentitypropertyof1 au1=1ua=a
Existenceofmultiplicativeinverses Foreveryaz0thereexists1/asothatau1/a=1/aua=1.
Distributivepropertyofmultiplicationover
au(b+c)=aub+auc
addition
Table 4. The properties of equality.
Herea,b,andcstandforarbitrarynumbersintherational,real,orcomplexnumbersystems.
Reflexivepropertyofequality a=a
Symmetricpropertyofequality Ifa=b,thenb=a.
Transitivepropertyofequality Ifa=bandb=c,thena=c.
Additionpropertyofequality Ifa=b,thena+c=b+c.
Subtractionpropertyofequality Ifa=b,thena–c=b–c.
Multiplicationpropertyofequality Ifa=b,thenauc=buc.
Divisionpropertyofequality Ifa=bandcz0,thenayc=byc.
Substitutionpropertyofequality Ifa=b,thenbmaybesubstitutedforainanyexpressioncontaininga.
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GLOSSARY
Table 5. The properties of inequality.
Herea,b,andcstandforarbitrarynumbersintherationalorrealnumbersystems.
Exactlyoneofthefollowingistrue:a<b,a=b,a>b.
Ifa>bandb>cthena>c.
Ifa>b,thenb<a.
Ifa>b,then–a<–b.
Ifa>b,thena±c>b±c.
Ifa>bandc>0,thenauc>buc.
Ifa>bandc<0,thenauc<buc.
Ifa>bandc>0,thenayc>byc.
Ifa>bandc<0,thenayc<byc.
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