TeachingMultiplicationandDivisionFactstotheWhole-to-Part,VisualLearner:

AGuidetoDevelopingFluencyWithMathFacts.

Foradditionalsupport,pleasegotothewebsitehttp://cwoodinmathfacts.tripod.com.

ChristopherL.Woodin,Ed.M.2008

Efficient Fact Learning • Fluency Involves Accuracy and Efficiency. Fluencyisachievedthroughtheaccuraterepetitionofknowledge,inthiscasemathfact“sentences.”It is very important to drill for fluency only after the student is able to produce a math fact accurately. The studentshouldneverbeplacedinasituationthatforcesthemtoguessandproduceafacterror.Verbalizinginaccuratefactsentencescompromisesthelearningprocessandleadstofrustration.Studentsshouldalwayshavetheabilitytoeitherrefertothecorrectanswerandreadit,usealearnedstrategytocreatetheanswer,orsay,“pass”andhavethecorrectanswersuppliedbeforeexpressingtheentirefactsentence.Agreatdealoftimeisspentinmathclassdevelopingstrategiesthatpromoteawarenessoffactfamilypatterns,andempower students to construct accurate fact sentences. Progress in efficiency cannot be achieved without accuracy.

• Provide Wait Time For Verbal Expression. Manystudentswithlanguage-basedlearningdisabilitiesneedampletimetoformulateandexpresstheirresponses.Learnedstrategiesthathelpyourchildmayinvolvevisualizingagraphicorganizer,orreferringtoapatterninherenttothefactfamily.Thisactivereasoningdemandsprocessingtime.Allowthestudenttomakearesponseorsay“pass”beforeattemptingtocueorpromptaresponse.Responsesmaytake15secondstoformulate.Trytobepatientandnotletyourimpatienceorfrustrationruboffonyourchild.Itdoesn’thelp-itjustaddstotheiranxietylevel.Timea15secondperiodofsilencepriortowork-ingwithyourchild.Itseemslikeaneternity,buttimespentprocessinginathoughtfulmannerwilleven-tually produce an accurate, and increasingly efficient response. In this manner, the efficient expression of mathfactsmaybeintegratedwiththevisual,kinesthetic,orsemantic(meaning-based)patternsthathavebeenassociatedwiththatfactfamilyduringmathclass.

• Multiplication Facts Should Be Practiced in Conjunction With Their Division Counterparts. Learningfactsinarelationalcontextthatpromptsthestudenttorecognizeandexpresstheminbothmultiplicationanddivisionformatsisveryimportant.Factsmaybeexpressedwithfourfactsentences:twomultiplication,andtwodivision(withtheexceptionofperfectsquares).Theentirefactfamilyismod-eledwitharectangularmatrixdiagramthatdovetailsvisuallywiththeareamodelofmultiplication,aswellastraditionaldivisionnotation(seeFigure1below).Gainingexposuretothedynamicexpressionoffact knowledge allows students to develop necessary flexibility with their knowledge base. Creating these relatedmathsentencesfromacompleteddiagramprovidesarelativelyerrorfreeopportunitytopractice.Italsoinvolvesprocessingthethreefactelementsratherthanmerelyrepeatingthem.Timeusedtoholdthefactinformationinshorttermmemorywhileformulatingtherelatedfactsmayfacilitatetransferofthefacttolongtermmemory.Aninitialstrategytofacilitatethisintegrationinvolvesthealterationoftraditionalmultiplicationsyntax(number/wordorder).Promptmultiplicationsentenceproductionwiththeproduct.Thisservestoactivateanimageofthefactasawhole,aswellasrelatedbackgroundsemanticknowledge.Forinstance,“Pictureasixpack.Howisitmade?”“6=2x3.”Figure1. RectangularMatrixDiagramAreaModelDivisionNotationRelatedFacts Initially:

Alteringthesyntaxfosterstheintegrationofmultiplicationanddivisionfactsasthethreeelementsofeachfactareactivatedsimultaneously,thenarticulatedandstoredasacompleteunit. From:ArithmeticfortheWholetoPartLearner(inpress)2008,ChristopherWoodin,Ed.M

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Order Of Fact Instruction

•Cadencedfacts:x2,x10,x5,x1 Thesefactsarethemostimportant,andmostfrequentlyrequiredfacts.Theyarecalledcadencedfactsastheproductsmaybeskipcountedwithameteredbeat-inthemannerofadrumcadence.Thesefourfactfamiliescomprise64%ofthe100multiplicationfacts.Theyarenecessaryformultidigitcomputation,tellingtime, developing base ten relationships, and fraction simplification. They also lend themselves to many seman-ticassociationsthatmaybemodeledandexpressedthroughwordproblems.Skipcountinghelpsfamiliarizethe student with the products; however, it is not a reliable or efficient means to produce facts. The process of defining factors of products by skip counting may place overwhelming demands on the student’s auditory pro-cessingabilities.Studentsoftenmiscountthenumberofbeatsofthecadence,orarriveatthecorrectproductwithouteverrehearsingtheentirefactsentence.Asaresult,theybecomerigidlydependentonthiscountingstrategyandfailtomakeprogresstowardrecognizingandretrievingtheentiremultiplicationfactsentence,oradivisioncorrelate. Time spent producing the cadenced facts in an effortless, fluent manner has a marked impact on math learning. These are the fact families that need regular drill and practice to develop fluency. Time spent at homeworkingonthesefactfamilieswillhavethegreatestpositiveimpactonyourchild.

DivisibilityRules Thex2,x10x5,andx1factfamilieshaveproductsthatareeasilyrecognizable.Asaresult,divisibil-ity rules have been developed to define them. These rules are helpful in that they provide an efficient means torecognizeandaccept,orexcludeandrejectaproductfromeachfamily.Priortopracticingamultiplica-tionfactfamily,itishelpfultoreviewitsaccompanyingdivisibilityrule.Therulewillhelptoconstrainthestudent’sproductstoapoolofacceptableanswers.SeeFigure2foravisualdepictionofthefollowingrules.Highlightingeachfactfamilyona1-100chartprovidesanopportunityforthestudenttoseethesepatterns.

Prior to working on a specific fact family, highlight the multiples. Work on one fact family at a time. Achieve fluency with that family before moving to the next. Examine the pattern. Cadenced facts will be represented onthe1-100chartascolumns.Guidethestudenttodetectthefollowingpatterns,thendevelopthefollowingdivisibilityrules.Findthe1-100chartonthefollowingpage.

•Highlightmultiplesoftwowithyellow.Theyhavea0,2,4,6,or8intheirone’splace.Evennumbers.

• Highlight multiples of 5 with red. Even multiples of five always end in zero while odd multiples end in five.

•Multiplesoftenwillbeorangeafterthe2xand5xfamilieshavebeenhighlighted..Thesemultiplesalwayshaveazerointheirone’splace.

Figure2 Highlightwithyellow. Highlightwithred. Thesebecomeorangeasaresult. Multiplesof2 Multiplesof5 Multiplesof102 4 6 8 10 5 10 10 12 14 16 18 20 15 20 20 25 30 30 35 40 40 45 50 50...

From:ArithmeticfortheWholetoPartLearner(inpress)2008,ChristopherWoodin,Ed.M

Use the 1 - 100 Chart to Explore Divisibility

Explore each fact family visually using the 1-100 chart. Prior to working on a specific fact family, highlight the multiples. Work on one fact family at a time. Achieve fluency with that family before moving to the next. Guidethestudenttotogeneratetheirdivisibilityrulesbasedonthesepatterns.Continuetousethesamechartasyouexploreeachfactfamilyinit’spropersequence.

Cadenced Facts:The2x,5x,and10xfamiliescreatecolumnsonthe1-100chart.Theseproductshavesimi-lar,recognizabledigitsintheirone’splace.

TwoTimesFamily:Colorallofthe2xfactproductsyellow.Thestudentmayskipcounttodothis.Makesurethatthestudentstartswith2!Pattern:The2xproductswillemergeas5(vertical)columnsofyellow.Divisibility rule to generate,thenwrite:Multiplesoftwohave0,2,4,6,or8intheirone’splace.

FiveTimesFamily:Colorallofthe5xfactproductsred.Pattern: The5xproductswillemergeas2(vertical)columnsofred.Divisibility rule: Multiples of five have 0 or 5 in their one’s place.

TenTimesFamily:The10xproductsbecomeorangeastheyhavealreadybeenhighlightedyellowandred.Pattern: The10xproductswillemergeas1(vertical)columnoforange.Divisibility rule:Multiplesoftenhavea0intheirone’splace.

Other Patterns: The9x,3x,6xfamiliescreatediagonalpatternsthatproducedivisibilityrules.

NineTimesFamily:Colorallofthe9xfactproductsdarkblue.Pattern: The9xproductswillemergeasadiagonalpattern.Divisibility rule: Multiplesofninehavedigitsthataddto9,oramultipleofnine(like18:1+8=9).

ThreeTimesFamily:Colorallofthe3xfactproductslightblue-somewillalreadybedarkblue.Pattern: The3xproductswillemergeasadiagonalpattern.Divisibility rule: Multiplesofthreehavedigitsthataddto3,6or9,oramultipleofthereof.Eachdiagonalwilladdtothesamesumof3,6or9.

SixTimesFamily:The6xproductsbecomegreenastheyhavealreadybeenhighlightedyellowandblue.Pattern: The6xproductswillemergeasa“checkerboard”diagonalpattern.Divisibility rule:Multiplesofsixareevenmultiplesofthree,therefore,theymustpassthedivisibilityrulesforboth2and3.

Time spent developing these divisibility rules will have a later benefit. These rules may be used to check multi-digitmultiplicationproducts,simplifyfractions,andfactornumbers.Timeshouldbespentondevelopingtheserelatedskillsinconjunctionwitheachmathfactfamily.

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MultiplicationTableAreaModel

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123412 8Usethemultiplicationtablelikeageoboardtoshowarea.

Masktherightsideandbottomofthemultiplicationtablewithtwo pieces of paper. This will define a rectangular area of the chartthatwillcorrespondtoamultiplicationordivisionfact.

For instance- note the eight square unit area defined by the 2x4gridshowntotheright.Thiswouldpromptthestudenttosay,“8=2x4,or8÷2=4.”Startusingterminologyrelat-ingtotheconceptofareatorelatefactswiththisapplication.Notethatthisrectanglealsorelatestotherectangularmatrixdiagramthatispicturedbelow.

MultiplicationTable

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Usethemultiplicationtableinconjunctionwiththe1-100chartac-tivitiesthathavealreadybeendescribed..Highlighttheappropriaterow and column of that specific fact family after identifying these multiplesonthe1-100chart.

Afterhighlightingtherowandcolumn,writetheproductsinpencilwhileworkingonthatfactfamily.Havethestudentinktheminwhentheycanbeexpressedindependently.

Highlightingthemultiplicationchartinthismannerprovidesagreatdeal of motivation for the student and justifies the suggested order of factinstruction.

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8 = 2 x 48 = 4 x 28 ÷ 2 = 48 ÷ 4 = 2Then...2 x 4 = 84 x 2 = 8

Eachfactshouldbeverbalized,and/orwritteninit’sentiretysothatitisstoredasanintactverbalstring.Usevisualcuestopromptverbalresponses.Useverbalcuestopromptthestudenttoproduceadiagram.Thiscross-modalprocessingfostersintegrationbetweenthe auditory, fine motor, and visual processing systems.

Avoidaskingthestudenttotraceorcopydiagrams,or“parrot”factsastheycandosowithminimalnovelprocessing.Activeprocess-ing-theactofbuildingstructureservestoprolongtheamountoftimethattheactisheldinshorttermmemory.Thisfacilitatesthelong-termretentionofthefact.

RectangularMatrixDiagram

RelatedFacts

MultiplicationGridWithHighlighted2xFacts

Learning the related facts of the most predictable fact families first provides a way for the student to learn many factsbelongingtolesspredictablefactfamilies.Forinstance,learning2x7=14withinthecontextofthetwotimesfamilywillalsoteachthestudent7x2=14.Soonmostofthechartbecomeshighlightedasfactsareas-similated.•Afterlearningthetwotimesfamily,19factsarelearned,81remainunfamiliar.• After learning the five times family, 17 additional facts are learned, 64 remain unfamiliar.•Afterlearningthetentimesfamily,15additionalfactsarelearned,49remainunfamiliar.Over1/2way!•Afterlearningtheonetimesfamily,13additionalfactsarelearned,36remainunfamiliar.•Afterlearningtheninetimesfamily,11additionalfactsarelearned,25remainunfamiliar.•Afterlearningthesixtimesfamily,9additionalfactsarelearned,only16remainunfamiliar.•Afterlearningtheperfectsquares,4additionalfactsarelearned,only12:6pairsarelefttomemorize.

UseHandstoModelTwo-TimesFactsandPlaceValueHave students hold out their left hand with fingers extended. * Make sure that their hand is within their visual field- so that they can see it.Teacher: “You are showing me five fingers, one time.”

Theteachershouldmodelthisbymirroringtheirimage-extendinghisrighthandashefacesthestudent. “High five” the student to make a “ clap.”Teacher: “ This is five, two times or one ten.”

Theteachershouldaskthestudenttokeep the student’s left fingers extended. Teacher: “Now show me seven fingers by sticking out two more fingers (onyourrighthand).”Teacher:“Youareshowingmesevenonetime.”Astheteacherfacesthestudent,theteachermodelsthisbymirroringtheirimage:The teacher extends the five fingers on his right hand, aswellastwofromhislefthand.Teacher:“Thisisfourteen.Itisseventwotimes.Twotimessevenisfourteen”

Thishandmodelrepresentstheproduct14withinthestudentsprimaryreferenceframe.Thisprovidesthestudentwithainternally-basedplace-valuetemplatetohelphimencodethedigitsintheirproperorder(14,not41).

ShootoutGameStart the game in the ready position: The teacher is extending all five of the fingers on his right hand. This hand should hang at belt level- like an old west gun fighter about to “draw” his sidearm. The student should assume a similar posture -have his five left hand fingers extended with his left hand hanging athisside.Theteachershouldcalloutanumberfrom5to9.Thatisthesignalto“draw.”Both teacher and student rush to display the quantity of fingers to match the number called-out- pointing their handsatoneanother.

The pairs of hands will create a two times fact that should be verbalized by the winner of this finger drawing contest.Theotherpersonshouldstatearelatedfact.For example: The teacher calls out, “seven.” Both teacher and student “shoot-out” seven fingers as pictured above. The teacher and student should move toward each other, clap their five-fingered hands, and identify the number of additional fingers that will comprise the one’s digit of the product (in this case 2+2=4) If the student gets his seven extended first, he must verbalize that 2x fact: “14 = 2 x 7, ” or “ 7 x 2 = 14.”Theteachershouldrespondwitharelateddivisionfact,“14÷2=7.”

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FactTwisterThisactivityreinforcesnumberfactsaswellasdirectionality.Itcanbeplayedindividuallyorinsmallgroups.Write the products of the fact family you are addressing on paper plates. Arrange the paper plates on the floor in frontofthestudent.Havethestudentsansweryourfactquestionsbytouchingthecorrectplatewiththeappend-ageyouname.•ExampleTeacher, “Put you right foot on the product of two times five.”Whenastudentcorrectlyperformsthetask,signalsuccessbyaskingfortheparrotedfact,2x5=10;thecommutedmultiplicationfact,5x2=10;oroneofthetworelateddivisionfacts,10÷5=2or10÷2=5.

Rad MinuteAfterthestudentwriteshisname,dateanddayontheRadMinutesheet,havehimfoldthepaperalongthedotted line. The first order of business is to have the student fill-in the products from top to bottom. This may beachievedbyskipcountingbytwos:“2,4,6...”etc.Next,havethestudentcheckalloftheproductsusingthe2xdivisibilityrule: “All of the products must have a 0,2,4,6 or 8 in their one’s place.”Theteachershouldalsochecktheproductsforaccuracywithineachfact.Forinstance,beonthelookoutforerrorslike2x7=18.Acheckfordivisibilitywillnotpickupthistypeoferror.Afterthat,havethestudentcopyeachmathfactinitsentirety.Eachfactshouldbewrittenfromlefttorightlikeasentence.Donotallowthestudent to write in columns: all of the first digits of the facts, then all of the “x” signs, etc. The object is to have thestudentrehearseeachfactasaunitsothatitmaybestored,thenlaterretrievedfromauditorymemory.SeetheRadMinutesheetonthefollowingpage.

FactBallAfter practicing a multiplication fact family using flash cards or another production task, play a game of fact ball.Writenumbersonasoft,lightball-perhapsatennisball.Tosstheballtothestudent.Thestudentshoulduse the first number that he sees on the ball to begin the fact sentence. Initially, write only those factors that are most familiar, perhaps 1,2,5,10. Add more factors as the student gains confidence with harder facts.Forexample,theteachertossestheballtoJohnnywhocatchesitwithonehand.HaveJohnnyreadthenumberfromtheballthatisclosesttohisthumb.“Six,”saysJohnny.“Ok,”saystheteacher,“saya2timessentencestartingwith6....”Johnnysays,“Sixtimestwois12.”Theteacherrespondswitharelatedfact,“Twotimessixis12.”Theteachermayalsochoosetomodeloneofthetworelateddivisionfacts:12÷2=6,or12÷6=2.Johnnythentossesorrollstheballtotheteacherandtherolesarereversed.

Makesurethatthestudenthastheabilitytolookupeachcorrectmathfactifnecessary.Thestudentshouldneverbeforcedtoguessatananswer.Ifunsureofafact,promptthestudenttoeitherwaitandthink,lookitup,orsay“pass”-andhavetheanswersuppliedbytheteacher.Thestudentshouldthenrehearsethefactbysayingitinitsentirety.

More Activities to Learn 2x Facts

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• Constant Time Delay This process involves showing the student a flash card, then having the student raise his hand or point to ananswerwhenheisable.Afterwaitingfourseconds,clapyourhandstopromptthestudenttorespondwiththemissingfactororproduct.Ifhisansweriscorrect,havehimthenreciteacompletemultiplicationsentenceandarelateddivisionsentence.Theimposeddelayencouragesthestudenttointernallyrecitethefact.Re-searchhasshownthisproceduretofacilitatelongtermretentionoffactsinsomechildren

©2008C.Woodin

UseGrossMotorKinestheticPracticetoDriveIntegratedFactProduction

Describethegrossmotorexercisehere

Many students have fine motor processing delays or impediments. Theseissuesareoftenevidencedbyrelativelylowscoreonthepro-cessingSpeedIndexscoreoftheW.I.S.C.Thesestudentsmayhavefine motor planning issues that may impact their handwriting output, or fine motor problems associated with their verbal articulation. Their slow production rate, exacerbated perhaps by inefficient sensory-mo-tor or visual feedback loops prevent them from refining their rate or qualityofproduction.Grossmotor-kinestheticpracticemayhaveanimmediatepositiveeffectontheirabilitytointegratevisualandaudi-toryprocessingwithmotorskills.

RAD MINUTE

201) Fill in blanks.2) Check with divisibility rules.3) Copy each number sentence.

Warm -up One Minute Quiz

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AddingwiththeIconSlideRule

•Cut-outthetwonumberlinestrips.•Tapestrip#1tothestudent’sdeskinahorizon-talformat.•Placethesecondnumberlinestripunderthetapedstrip.•Movethesecondstriprightorlefttoalignthetwoiconstobeadded.

Option#2-Makeaslideruleoutofthetwoiconnumberlinestrips.Reinforcethestripsandholderbyputtingtapeontheirbacksbeforecuttingthemout.Cutouttheholder,thenputaknifeslitinalongeachdarkline.Insertthestripsintothetachistoscope(holder)below.•MovethestripssothattheIconstobeaddedarealignedbetweentheslits.

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FactMosaicsThisjigsaw-typepuzzlereinforcesnumberfacts,thefollowingofsequentialdirections,directionality,prob-lem-solving,andperceptionofmultiplereferenceframes.Itisplayedeitherindividuallyorinsmallgroups.StartbyhavingeachstudentcompleteaRadMinuteforthefactfamilyyouwanttoaddress.RadMinutetem-platescanbefoundonthewebsitehttp://cwoodinmathfacts.tripod.com.Correcttheworksheetstoensurecorrectnumbersentences.Thesenumberfactsaretheonlycluesforreassemblingthepuzzleonceitiscutintopieces.Giveeachstudentasquarepieceofpaperdividedintonineequalsquares,likeatic-tac-toegrid.Drawthelinescarefullysotheninepiecesareidentical.Inthefuture,studentscanmakethegridsfollowingmicrounitedinstructions.UsingthecorrectedRadMinutesheets,havethestudentswritetheproduct(six)ononesideofagridlineandthepartialfactstatement(2x3)ontheother.Again,checktoseethattheproductsorquotientsareaccurate.Whenstudentshavelabeledallgridlineswithastatementandopposingproduct,turnthesheetover.Havestudentsdrawasimpledesignthatcoverstheentirebackofthesheet.Cutthepaperalongthegridlinestoproduceamosaicofninepieces.Ask students to reassemble this jigsaw-type puzzle using the front side. They can check the puzzle by flipping itovertothedesignside.Oncestudentsreassembletheirownmosaictheycantrademosaicswithanotherstudent.Thisactivityisagoodwaytoreinforceanytypeofshortquestionandanswerfacts.

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DiagramsthatGenerateMultiplicationandDivisionFacts•Completethediagrams.•Aftertheyarechecked,writefourrelatedmultiplicationanddivisionsentencesforeachdiagram.

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RAD MINUTE

201) Fill in blanks.2) Check with divisibility rules.3) Copy each number sentence.

Warm -up One Minute Quiz

1x = 3 x =

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RAD MINUTE

201) Fill in blanks.2) Check with divisibility rules.3) Copy each number sentence.

Warm -up One Minute Quiz

1x = 3 x =

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Iconic Grouping of Nickels and Multiplication

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Linkpairsofnickels(tens)withaline.

5¢ 5¢“Nickels”and5IconsfortheSkipCountingTemplate-Userealnickelsifpossible.

Directions for the Skip Counting Template Using Nickels ( or Five Icons)Add nickels, one by one. If the one’s place is vacant, have the student fill the empty space with a nickel, then finger-write the number of tens, and the number of ones in the appropriate place-value boxes, and verbalize thisnumber.Iftheone’splacealreadyhasanickelinit,havethestudentpickitupandcombineitwiththenew nickel to fill one of the vacant ten spots. Have the student finger-write the number of tens, and the num-berofonesintheappropriateplace-valueboxes,andverbalizethisnumber.Examples:• Give the student the first nickel. Ask the student what it is worth ( 5 ). Ask,“Howmanyones?”Havehimplacethenickelonthevacantone’sspot,thenwriteit’svalue(5)intheone’splacebox.•Givethestudentthesecondnickel.Thereisnospottoplacethenickelintheone’splace.The student must move the first nickel and combine it with the new nickel to fill one entire ten’s place. Ask,“Howmanytens?”Thereisoneten.Havethestudentwritethenumeral1intheten’splacebox.Asyoupointtothevacantone’sspot,ask,“Howmanyadditionalonesarethere?”Therearezeroadditionalones.Havethestudentwritethenumeral0intheone’splacebox.Havethestudentrewritethetwodigits:1tenand0ones.Nowhavethestudentverbalizethevalueof1tenand0ones(ten).•Givethestudentthethirdnickel.Havehimplacethenickelonthevacantone’sspot.Ask,“Howmanytens?”Thereisoneten.Havethestudentwritethenumeral1intheten’splacebox.Asyoupointtotherecently filled one’s spot, ask, “ How many additional ones are there?” There are five additional ones. Have the student write the numeral 5 in the one’s place box. Havethestudentrewritethetwodigits:1tenand5ones.Nowhavethestudentverbalizethevalueof1tenand 5 ones ( fifteen). •thisactivitycanbeextendedtomultiplicationbyaskingthestudenttoidentifythenumberof timesthathehas placed a nickel on the sheet, then creating a “five times...” sentence to match.

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NumberofOnes

“Write” the digits with your index finger.

SkipCountingTemplateUsingNickels

NumberofTens

GrossMotor-PrimaryReferenceFrame5Xfacts

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Thisstudentholdsacardwitha3ononesideanda15ontheother.

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Thisstudentholdsacardwitha9ononesideanda45ontheother.

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Traineestandshere-alwaysfac-ingthe12posi-tion!

Teacher

Levelone...Teachthe4benchmarklocationsof3,6,9,12ontheanalogclockdial.a)Usingacircle-the“round“KEY”portionofabasketballcourt,oracirculararrangementoficonspaintedontheground,haveyourstudentsformalinebehind,andslightlytotheleftofthe6position.Ifyouareconstructingthecircleoutside,havethe12pointtowardthesunoruphill,orboth.Ifinside,havealargeartifactatthe12position(basketballhoop-etc.).Thiswillhelpmaintainthecorrectreferenceframe.

Preparefour,“3x5”cards.Putalarge“3”onone,6,9,and12ontheotherthree.Placethe12 at the top of the circle. Hand one of the three remaining cards to the first person in line. Havethatpersongotothecenteroftheclockface.Askthemtoreadtheircard.Whilefacingthe12.E.g.,“3.”Theteachershouldagainask,lookatthe12.Theteachershouldmovetothe3positionandstamptheirfeet.Teacher,“I’monthe3.Pointtothe3.”Whenthestudentpointswiththeirrightarm,theteachersays,“Sidesteptothe3-keepfacingthe12.”Thestudentshouldgotothe3positionandstandthere.Askthenextstudenttostandatthecenter.Askthestudentstandingonthe3tostamp.Askthestudentatthecentertopointtothestompingnoise.Theteachershouldask,“Whereareyoupointing?”“3.”Teacher,“Side-steptothethreeandtakethatperson’scard.Repeatfortheremainderofthestudents-havethe first person go a second time. Then repeat the drill for the 6 position -backing into posi-tion,thenthe9position-sidesteppingtotheleft.

b)Onthenextday,repeatthesamedrill.Seehowquicklythestudentscangotothethreepositions.Then,ifthestudentsappeartobedoingwell,movetothenextactivity.Givethefirst three students in line a card -one 3, the next 6 , the last one : 9...

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Have the first student go to the center. Ask, “Where’s the 3?” Have the student point to the 3.Iftheyarepointingtothewrongspot-theteachergoestothatspotandstampsherfeettodrawattentiontoherposition.

Oncethestudentpointstothecorrectposition,askthemtogothere-maintainingeyecontactandbodyorientationtowardsthe12position.Repeatthisforthenexttwostudentswithcards.Thefourthpersoninlinewillnothaveacard.Havethestudenttoethelineinthecenter.

Ask,”Whereisthe3?”Havethestudentpointtothe3-ifunsure,askthepersonstandingonthe3positiontoprovideacuebystampingtheirfeet.Askthestudenttopointtoall3posi-tionstwice,thenonthelastrequest,askthestudenttopointtothe3.Oncethecenterstudentpointstothe3correctly,havethestudentside-stepintoposition,thentakethecardofthestudentstandingatthatposition.Continue,replacingthe6and9positionsaswell.Trytoruneachstudentthroughtwice.

Repeatthisdrilluntilallofthestudentscanpredictablypointtothe3,6,and9positions.

c)Learntheminutelocationsfor15,30and45minutes.Writethesetimesonthebacksofthecorresponding 3,6,9 cards. Run the students through the previous drill to fill the three posi-tions.Asthethreestudentstaketheirpositions,tellthemtoreadtheminutenumbersbacksoftheircards.Next,Havethe4thstudenttakethecenter.Havethestudentpointtothe3.Askthestudentthenumberofminutesforthatposition-havethepersonholdingthe3cardfacethe“15”towardthecenter.Havethe3,6,9peoplecuetheminutetimesasnecessary.Onthesecondtimethrough,interchangeclockpositionnumbersandthe15,30,45minutepositions.

d)Repeatdrillcuntilthestudentscanpointtotheminuteornumberpositionwithoutcueing.

e)Askthecenterpersontopointtoabenchmarklocationandhavethestudentidentifythecorrectminuteamountforthatlocation.Then,theteachergoestothatlocationandmovesone(noisydeliberate)clockwisestep.Teachershouldthenaskwhatnumberpositionthattheyarestanding-andthetimeinminutes.

f)Putitalltogether-make5xfacts.Studentsareaskedtopointtoanumberorminuteloca-tion,thensaythecorrespondingnumber.Thenpromptthestudentstosaythe5xfactthatrelatesthetwoquantities45=9x5.Haveastudentinlinesaythesamefact-startingwithadifferentnumber-Startwith9!9x5=45Startwith5!

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•Drawtheminutehandtoshowthegivenminutes.•Writetherelated5xfact. 9

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PredictableVisualPatterns: Theninetimesfamilyand,evenmultiplesof6addanadditional16%for80%ofthemultiplicationtable.Bothofthesefactfamilieshavedivisibilityrulesthatinvolveaddingthedigitsoftheproducts.Allofthex9productsonthemultiplicationtablearecomprisedofdigitsthataddtonine.Forinstance,9x2=18(thedigitsinthisproduct:1+8addto9).The9xfacts are highly predictible when flash cards present the division form of the fact. The missing factor will be one larger than the ten’s digit of the product (see figure 3). When students are presented with a division flash card based on the 9x fact family, cue the student to identify the digitinthetensplaceoftheproduct(ordividend).Oncetheydothis,theyneedonlytoaddonetoarriveatthemissingfactor.Havethemrecitethefactasamultiplicationsentencethatstartswiththismissingfactor,thenrecitetheotherthreerelatedfacts.

Figure3

92 7Add1totheten’sdigitoftheproduct:2+1=3.....“3x9=27”

Havethestudentreciteseveralrelatedfacts:

Prompts:“Givemeadivisionfact,startwith27.”“27÷9=3.”“Givemeadifferntdivisionfact.Divideby3.”“27÷3=9.”“Givemeamultiplicationfact.Startwith27.”“27=3x9.”“Givemeadifferntmultiplicationfact.Startwith9.”“9x3=27.“Givemeamultiplicationfact.Startwith3.”“3x9=27.

PerfectSquares32,42,72,82addanadditional4%foratotalof84%ofthetable.8factsremaintobememorized.Theyare:3x4,3x6,3x7,3x8,4x7,4x8,7x6,7x8

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When presenting the nine times family flash cards that present the factors and a missing prod-uct,useasimilarstrategy.Theten’sdigitinthemissingproductwillbeonesmallerthanthetopfactor.Thestudentshouldsubtractonefromthetopfactor,thensaytheresultingnumberasadecade.Thisisusuallyenoughtotriggertheseconddigitoftheproduct.Ifnot,itmaybecomputed by finding the missing addend to 9.

Start using these flash cards after the student has successfully practiced the flash cards that feature the product ( number in the rectangle). These flash cards are most productive after the ninetimesproductshavebecomefamiliarthroughoralrecitation.

Beginbyusingthistemplateasaworksheet.Havethestudentwritetheproductsinsideeachrectangle. The “trick” is to first write the tens digit by subtracting one from the number in the left circle. This ten’s digit should then said as a decade: In figure 4, the student writes 2, but is promptedtosay“twenty.”Itishopedthatsaying“twenty...”willprompttheremainderoftheproduct.Ifitdoesn’t,thestudentprobablyneedsmorepratice with the flash cards that supply the product.

Onecueingmethodinvolvesremindingthestudent thatthedigitsoftheproductmustaddto9. Thisisbestdonevisuallybysupplyinga9Icon -andseparatingtwodotsfromtherest.Usethestudent’s fingers to mask the top two dots, or separate them with a drawn line, piece of string, or pencil. See figure 5

Itshouldbenotedthatthisisnotreallyatrickatall. Ninetimesfactsaresimilar,yetalwayssmallerthan acomparabletentimesfact.Forinstance:10x3=30. 9x3=(alittlesmallerthan30)Twenty...7.

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2

5 == +

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Multiplyingby7.DistributiveProperty.

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Multiplyingby7.DistributiveProperty.

2(+)=___+___2(7)=_____

3(+)=___+___3(7)=

4(+)=___+___4(7)=_____

5(+)=___+___5(7)=

6(+)=___+___6(7)=_____

7(+)=___+___7(7)=

8(+)=___+___8(7)=_____

9(+)=___+___9(7)=