Speed math for kids achieve their full potential

CONTENTS
Preface
Introduction

Ahashare.com

Chapter1:Multiplication:GettingStarted
WhatisMultiplication?
TheSpeedMathematicsMethod
Chapter2:UsingaReferenceNumber
ReferenceNumbers
DoubleMultiplication
Chapter3:NumbersAbovetheReferenceNumber
MultiplyingNumbersinTheTeens
MultiplyingNumbersAbove100
SolvingProblemsinYourHead
DoubleMultiplication
Chapter4:MultiplyingAbove&BelowtheReferenceNumber
NumbersAboveandBelow
Chapter5:CheckingYourAnswers
SubstituteNumbers
Chapter6:MultiplicationUsingAnyReferenceNumber
Multiplicationbyfactors
Multiplyingnumbersbelow20
Multiplyingnumbersaboveandbelow20
Using50asareferencenumber
Multiplyinghighernumbers
Doublingandhalvingnumbers
Chapter7:MultiplyingLowerNumbers
Experimentingwithreferencenumbers

Chapter8:Multiplicationby11
Multiplyingatwo-digitnumberby11
Multiplyinglargernumbersby11
Multiplyingbymultiplesof11
Chapter9:MultiplyingDecimals


Multiplicationofdecimals
Chapter10:MultiplicationUsingTwoReferenceNumbers
Easymultiplicationby9
Usingfractionsasmultiples
Usingfactorsexpressedasdivision
Playingwithtworeferencenumbers
Usingdecimalfractionsasreferencenumbers
Chapter11:Addition
Addingfromlefttoright
Breakdownofnumbers
Checkingadditionbycastingoutnines
Chapter12:Subtraction
Numbersaround100
Easywrittensubtraction
Subtractionfromapowerof10
Checkingsubtractionbycastingnines
Chapter13:SimpleDivision
Simpledivision
Bonus:Shortcutfordivisionby9
Chapter14:LongDivisionbyFactors
WhatAreFactors?
Workingwithdecimals
Chapter15:StandardLongDivisionMadeEasy

Chapter16:DirectLongDivision
Estimatinganswers
Reversetechnique—roundingoffupwards
Chapter17:CheckingAnswers(Division)
Changingtomultiplication
Bonus:Castingtwos,tensandfives
Castingoutnineswithminussubstitutenumbers
Chapter18:FractionsMadeEasy
Workingwithfractions
Addingfractions


Subtractingfractions
Multiplyingfractions
Dividingfractions
Changingvulgarfractionstodecimals
Chapter19:DirectMultiplication
Multiplicationwithadifference
Directmultiplicationusingnegativenumbers
Chapter20:PuttingitAllintoPractice
HowDoIRememberAllofThis?
AdviceForGeniuses
Afterword
AppendixA:UsingtheMethodsintheClassroom
AppendixB:WorkingThroughaProblem
AppendixC:Learnthe13,14and15TimesTables
AppendixD:TestsforDivisibility
AppendixE:KeepingCount
AppendixF:PlusandMinusNumbers
AppendixG:Percentages

AppendixH:HintsforLearning
AppendixI:Estimating
AppendixJ:SquaringNumbersEndingin5
AppendixK:PracticeSheets
Index



Firstpublished2005byWrightbooks
animprintofJohnWiley&SonsAustralia,Ltd
42McDougallStreet,Milton,Qld4064
OfficealsoinMelbourne
©BillHandley2005
Themoralrightsoftheauthorhavebeenasserted.
NationalLibraryofAustraliaCataloguing-in-Publicationdata:
Handley,Bill.
Speedmathsforkids:Helpingchildrenachievetheirfullpotential.
Includesindex.
Forprimaryschoolstudents.
ISBN0731402278.
1.Mentalarithmetic.2.Mentalarithmetic–Studyandteaching(Primary).I.Title.
513.9
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or
transmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,
withoutthepriorpermissionofthepublisher.
CoverdesignbyRobCowpe


PREFACE
IcouldhavecalledthisbookFunWithSpeedMathematics.Itcontainssomeofthesamematerialasmy

other books and teaching materials. It also includes additional methods and applications based on the
strategiestaughtinSpeedMathematicsthat,Ihope,givemoreinsightintothemathematicalprinciples
andencouragecreativethought.Ihavewrittenthisbookforyoungerpeople,butIsuspectthatpeopleof
anyagewillenjoyit.Ihaveincludedsectionsthroughoutthebookforparentsandteachers.
AcommonresponseIhearfrompeoplewhohavereadmybooksorattendedaclassofmineis,‘Why
wasn’tItaughtthisatschool?’Peoplefeelthat,withthesemethods,mathematicswouldhavebeenso
much easier, and they could have achieved better results than they did, or they feel they would have
enjoyedmathematicsalotmore.Iwouldliketothinkthisbookwillhelponbothcounts.
IhavedefinitelynotintendedSpeedMathsforKidstobeaserioustextbookbutratherabooktobe
playedwithandenjoyed.IhavewrittenthisbookinthesamewaythatIspeaktoyoungstudents.Some
ofthelanguageandtermsIhaveusedaredefinitelynon-mathematical.Ihavetriedtowritethebook
primarilysoreaderswillunderstand.Alotofmyteachingintheclassroomhasjustbeenexplainingout
loudwhatgoesoninmyheadwhenIamworkingwithnumbersorsolvingaproblem.
I have been gratified to learn that many schools around the world are using my methods. I receive
emails every day from students and teachers who are becoming excited about mathematics. I have
produced a handbook for teachers with instructions for teaching these methods in the classroom and
withhandoutsheetsforphotocopying.Pleaseemailmeorvisitmywebsitefordetails.
BillHandley,Melbourne,2005

www.speedmathematics.com


INTRODUCTION
I have heard many people say they hate mathematics. I don’t believe them. They think they hate
mathematics. It’s not really maths they hate; they hate failure. If you continually fail at mathematics,
youwillhateit.No-onelikestofail.
Butifyousucceedandperformlikeageniusyouwilllovemathematics.Often,whenIvisitaschool,
students will ask their teacher, can we do maths for the rest of the day? The teacher can’t believe it.
Thesearekidswhohavealwayssaidtheyhatemaths.
Ifyouaregoodatmaths,peoplethinkyouaresmart.Peoplewilltreatyoulikeyouareagenius.Your

teachersandyourfriendswilltreatyoudifferently.Youwilleventhinkdifferentlyaboutyourself.And
thereisgoodreasonforit—ifyouaredoingthingsthatonlysmartpeoplecando,whatdoesthatmake
you?Smart!
Ihavehadparentsandteacherstellmesomethingveryinteresting.Someparentshavetoldmetheir
childjustwon’ttrywhenitcomestomathematics.Sometimestheytellmetheirchildislazy.Thenthe
childhasattendedoneofmyclassesorreadmybooks.Thechildnotonlydoesmuchbetterinmaths,
butalsoworksmuchharder.Whyisthis?Itissimplybecausethechildseesresultsforhisorherefforts.
Oftenparentsandteacherswilltellthechild,‘Justtry.Youarenottrying.’Ortheytellthechildtotry
harder.Thisjustcausesfrustration.Thechildwouldliketotryharderbutdoesn’tknowhow.Usually
childrenjustdon’tknowwheretostart.Sometimestheywillscrewuptheirfaceandhitthesideoftheir
head with their fist to show they are trying, but that is all they are doing. The only thing they
accomplishisaheadache.Bothchildandparentbecomefrustratedandangry.
I am going to teach you, with this book, not only what to do but how to do it. You can be a
mathematical genius. You have the ability to perform lightning calculations in your head that will
astonishyourfriends,yourfamilyandyourteachers.Thisbookisgoingtoteachyouhowtoperform
likeagenius—todothingsyourteacher,orevenyourprincipal,can’tdo.Howwouldyouliketobe
able to multiply big numbers or do long division in your head? While the other kids are writing the
problemsdownintheirbooks,youarealreadycallingouttheanswer.
Thekids(andadults)whoaregeniusesatmathematicsdon’thavebetterbrainsthanyou—theyhave
better methods. This book is going to teach you those methods. I haven’t written this book like a
schoolbook or textbook. This is a book to play with. You are going to learn easy ways of doing
calculations,andthenwearegoingtoplayandexperimentwiththem.Wewillevenshowofftofriends
andfamily.
WhenIwasinyearnineIhadamathematicsteacherwhoinspiredme.Hewouldtellusstoriesof
SherlockHolmesorofthrillermoviestoillustratehispoints.Hewouldoftensay,‘Iamnotsupposedto
beteachingyouthis,’or,‘Youarenotsupposedtolearnthisforanotheryearortwo.’OftenIcouldn’t
wait to get home from school to try more examples for myself. He didn’t teach mathematics like the
otherteachers.Hetoldstoriesandtaughtusshortcutsthatwouldhelpusbeattheotherclasses.Hemade
mathsexciting.Heinspiredmyloveofmathematics.
WhenIvisitaschoolIsometimesaskstudents,‘Whodoyouthinkisthesmartestkidinthisschool?’

ItellthemIdon’twanttoknowtheperson’sname.Ijustwantthemtothinkaboutwhothepersonis.
Then I ask, ‘Who thinks that the person you are thinking of has been told they are stupid?’ No-one
seemstothinkso.
Everyonehasbeentoldatonetimethattheyarestupid—butthatdoesn’tmakeittrue.Wealldo
stupid things. Even Einstein did stupid things, but he wasn’t a stupid person. But people make the
mistake of thinking that this means they are not smart. This is not true; highly intelligent people do


stupidthingsandmakestupidmistakes.Iamgoingtoprovetoyouasyoureadthisbookthatyouare
veryintelligent.Iamgoingtoshowyouhowtobecomeamathematicalgenius.

HowToReadThisBook
Readeachchapterandthenplayandexperimentwithwhatyoulearnbeforegoingtothenextchapter.
Dotheexercises—don’tleavethemforlater.Theproblemsarenotdifficult.Itisonlybysolvingthe
exercisesthatyouwillseehoweasythemethodsreallyare.Trytosolveeachprobleminyourhead.
Youcanwritedowntheanswerinanotebook.Findyourselfanotebooktowriteyouranswersandto
useasareference.Thiswillsaveyouwritinginthebookitself.Thatwayyoucanrepeattheexercises
severaltimesifnecessary.Iwouldalsousethenotebooktotryyourownproblems.
Remember,theemphasisinthisbookisonplayingwithmathematics.Enjoyit.Showoffwhatyou
learn.Usethemethodsasoftenasyoucan.Usethemethodsforcheckinganswerseverytimeyoumake
acalculation.Makethemethodspartofthewayyouthinkandpartofyourlife.
Now,goaheadandreadthebookandmakemathematicsyourfavouritesubject.


Chapter1
MULTIPLICATION:GETTINGSTARTED
Howwelldoyouknowyourmultiplicationtables?Doyouknowthemuptothe15or20timestables?
Do you know how to solve problems like 14 × 16, or even 94 × 97, without a calculator? Using the
speedmathematicsmethod,youwillbeabletosolvethesetypesofproblemsinyourhead.Iamgoing
toshowyouafun,fastandeasywaytomasteryourtablesandbasicmathematicsinminutes.I’mnot

goingtoshowyouhowtodoyourtablestheusualway.Theotherkidscandothat.
Using the speed mathematics method, it doesn’t matter if you forget one of your tables. Why?
Because if you don’t know an answer, you can simply do a lightning calculation to get an instant
solution.Forexample,aftershowingherthespeedmathematicsmethods,Iaskedeight-year-oldTrudy,
‘Whatis14times14?’Immediatelyshereplied,‘196.’
Iasked,‘Youknewthat?’
Shesaid,‘No,IworkeditoutwhileIwassayingit.’
Would you like to be able to do this? It may take five or ten minutes practice before you are fast
enoughtobeatyourfriendsevenwhentheyareusingacalculator.

WhatisMultiplication?
Howwouldyouaddthefollowingnumbers?
6+6+6+6+6+6+6+6=?
Youcouldkeepaddingsixesuntilyougettheanswer.Thistakestimeand,becausetherearesomany
numberstoadd,itiseasytomakeamistake.
Theeasymethodistocounthowmanysixestherearetoaddtogether,andthenusemultiplication
tablestogettheanswer.
Howmanysixesarethere?Countthem.
Thereareeight.
You have to find out what eight sixes added together would make. People often memorise the
answersoruseachart,butyouaregoingtolearnaveryeasymethodtocalculatetheanswer.
Asamultiplication,theproblemiswrittenlikethis:
8×6=
Thismeansthereareeightsixestobeadded.Thisiseasiertowritethan6+6+6+6+6+6+6+6
=.
Thesolutiontothisproblemis:
8×6=48

TheSpeedMathematicsMethod
Iamnowgoingtoshowyouthespeedmathematicswayofworkingthisout.Thefirststepistodraw

circlesundereachofthenumbers.Theproblemnowlookslikethis:

Wenowlookateachnumberandask,howmanymoredoweneedtomake10?


Westartwiththe8.Ifwehave8,howmanymoredoweneedtomake10?
Theansweris2.Eightplus2equals10.Wewrite2inthecirclebelowthe8.Ourequationnowlooks
likethis:

Wenowgotothe6.Howmanymoretomake10?Theansweris4.Wewrite4inthecirclebelowthe
6.
Thisishowtheproblemlooksnow:

Wenowtakeawaycrossways,ordiagonally.Weeithertake2from6or4from8.Itdoesn’tmatter
whichwaywesubtract,theanswerwillbethesame,sochoosethecalculationthatlookseasier.Two
from6is4,or4from8is4.Eitherwaytheansweris4.Youonlytakeawayonetime.Write4afterthe
equalssign.

Forthelastpartoftheanswer,you‘times’thenumbersinthecircles.Whatis2times4?Twotimes4
meanstwofoursaddedtogether.Twofoursare8.Writethe8asthelastpartoftheanswer.Theanswer
is48.

Easy, wasn’t it? This is much easier than repeating your multiplication tables every day until you
rememberthem.Andthisway,itdoesn’tmatterifyouforgettheanswer,becauseyoucansimplywork
itoutagain.
Doyouwanttotryanotherone?Let’stry7times8.Wewritetheproblemanddrawcirclesbelowthe
numberslikebefore:

Howmanymoredoweneedtomake10?Withthefirstnumber,7,weneed3,sowewrite3inthe
circlebelowthe7.Nowgotothe8.Howmanymoretomake10?Theansweris2,sowewrite2inthe

circlebelowthe8.
Ourproblemnowlookslikethis:

Now take away crossways. Either take 3 from 8 or 2 from 7. Whichever way we do it, we get the
sameanswer.Sevenminus2is5or8minus3is5.Fiveisouranswereitherway.Fiveisthefirstdigit
oftheanswer.Youonlydothiscalculationoncesochoosethewaythatlookseasier.
Thecalculationnowlookslikethis:

Forthefinaldigitoftheanswerwemultiplythenumbersinthecircles:3times2(or2times3)is6.
Writethe6astheseconddigitoftheanswer.


Hereisthefinishedcalculation:

Seveneightsare56.
Howwouldyousolvethisprobleminyourhead?Takebothnumbersfrom10toget3and2inthe
circles.Takeawaycrossways.Sevenminus2is5.Wedon’tsayfive,wesay,‘Fifty...’.Thenmultiply
thenumbersinthecircles.Threetimes2is6.Wewouldsay,‘Fifty...six.’
With a little practice you will be able to give an instant answer. And, after calculating 7 times 8 a
dozenorsotimes,youwillfindyouremembertheanswer,soyouarelearningyourtablesasyougo.

Testyourself
Herearesomeproblemstotrybyyourself.Doalloftheproblems,evenifyouknowyourtableswell.Thisisthebasicstrategywe
willuseforalmostallofourmultiplication.
a)9×9=
b)8×8=
c)7×7=
d)7×9=
e)8×9=
f)9×6=

g)5×9=
h)8×7=
Howdidyougo?Theanswersare:
a)81
b)64
c)49
d)63
e)72
f)54
g)45
h)56

Isn’tthistheeasiestwaytolearnyourtables?
Now, cover your answers and do them again in your head. Let’s look at 9 × 9 as an example. To
calculate9×9,youhave1below10eachtime.Nineminus1is8.Youwouldsay,‘Eighty...’.Then
youmultiply1times1togetthesecondhalfoftheanswer,1.Youwouldsay,‘Eighty...one.’
Ifyoudon’tknowyourtableswellitdoesn’tmatter.Youcancalculatetheanswersuntilyoudoknow
them,andno-onewilleverknow.

Multiplyingnumbersjustbelow100
Doesthismethodworkformultiplyinglargernumbers?Itcertainlydoes.Let’stryitfor96×97.
96×97=
Whatdowetakethesenumbersupto?Howmanymoretomakewhat?Howmanytomake100,so
wewrite4below96and3below97.

Whatdowedonow?Wetakeawaycrossways:96minus3or97minus4equals93.Writethatdown
as the first part of the answer. What do we do next? Multiply the numbers in the circles: 4 times 3
equals12.Writethisdownforthelastpartoftheanswer.Thefullansweris9,312.

Whichmethoddoyouthinkiseasier,thismethodortheoneyoulearntinschool?Idefinitelythink
thismethod;don’tyouagree?
Let’stryanother.Let’sdo98×95.
98×95=
Firstwedrawthecircles.

Howmanymoredoweneedtomake100?With98weneed2moreandwith95weneed5.Write2
and5inthecircles.

Nowtakeawaycrossways.Youcandoeither98minus5or95minus2.
98−5=93
or
95−2=93
Thefirstpartoftheansweris93.Wewrite93aftertheequalssign.

Nowmultiplythenumbersinthecircles.
2×5=10
Write10afterthe93togetananswerof9,310.

Easy.Withacoupleofminutespracticeyoushouldbeabletodotheseinyourhead.Let’stryone
now.
96×96=
Inyourhead,drawcirclesbelowthenumbers.
What goes in these imaginary circles? How many to make 100? Four and 4. Picture the equation
insideyourhead.Mentallywrite4and4inthecircles.
Now take away crossways. Either way you are taking 4 from 96. The result is 92. You would say,
‘Ninethousand,twohundred...’.Thisisthefirstpartoftheanswer.
Now multiply the numbers in the circles: 4 times 4 equals 16. Now you can complete the answer:
9,216.Youwouldsay,‘Ninethousand,twohundred...andsixteen.’
Thiswillbecomeveryeasywithpractice.

Tryitoutonyourfriends.Offertoracethemandletthemuseacalculator.Evenifyouaren’tfast
enoughtobeatthemyouwillstillearnareputationforbeingabrain.

Beatingthecalculator
Tobeatyourfriendswhentheyareusingacalculator,youonlyhavetostartcallingtheanswerbefore
they finish pushing the buttons. For instance, if you were calculating 96 times 96, you would ask


yourself how many to make 100, which is 4, and then take 4 from 96 to get 92. You can then start
saying, ‘Nine thousand, two hundred . . .’. While you are saying the first part of the answer you can
multiply4times4inyourhead,soyoucancontinuewithoutapause,‘...andsixteen.’
Youhavesuddenlybecomeamathsgenius!

Testyourself
Herearesomemoreproblemsforyoutodobyyourself.
a)96×96=
b)97×95=
c)95×95=
d)98×95=
e)98×94=
f)97×94=
g)98×92=
h)97×93=
Theanswersare:
a)9,216
b)9,215
c)9,025
d)9,310
e)9,212
f)9,118

g)9,016
h)9,021

Didyougetthemallright?Ifyoumadeamistake,gobackandfindwhereyouwentwrongandtry
again.Becausethemethodissodifferent,itisnotuncommontomakemistakesatfirst.
Areyouimpressed?
Now,dothelastexerciseagain,butthistime,doallofthecalculationsinyourhead.Youwillfindit
mucheasierthanyouimagine.Youneedtodoatleastthreeorfourcalculationsinyourheadbeforeit
reallybecomeseasy.So,tryitafewtimesbeforeyougiveupandsayitistoodifficult.
Ishowedthismethodtoaboyinfirstgradeandhewenthomeandshowedhisdadwhathecoulddo.
Hemultiplied96times98inhishead.Hisdadhadtogethiscalculatorouttocheckifhewasright!
Keepreading,andinthenextchaptersyouwilllearnhowtousethespeedmathsmethodtomultiply
anynumbers.


Chapter2
USINGAREFERENCENUMBER
Inthischapterwearegoingtolookatasmallchangetothemethodthatwillmakeiteasytomultiply
anynumbers.

ReferenceNumbers
Let’sgobackto7times8:
The10attheleftoftheproblemisourreferencenumber.Itisthenumberwesubtractthenumbers
wearemultiplyingfrom.
Thereferencenumberiswrittentotheleftoftheproblem.Wethenaskourselves,isthenumberwe
aremultiplyingaboveorbelowthereferencenumber?Inthiscase,bothnumbersarebelow,soweput
the circles below the numbers. How many below 10 are they? Three and 2. We write 3 and 2 in the
circles.Sevenis10minus3,soweputaminussigninfrontofthe3.Eightis10minus2,soweputa
minussigninfrontofthe2.

Wenowtakeawaycrossways:7minus2or8minus3is5.Wewrite5aftertheequalssign.

Now,hereisthepartthatisdifferent.Wemultiplythe5bythereferencenumber,10.Fivetimes10is
50,sowritea0afterthe5.(Howdowemultiplyby10?Simplyputa0attheendofthenumber.)Fifty
isoursubtotal.Hereishowourcalculationlooksnow:

Nowmultiplythenumbersinthecircles.threetimes2is6.Addthistothesubtotalof50forthe
finalanswerof56.thefullworkingoutlookslikethis:

Whyuseareferencenumber?
WhynotusethemethodweusedinChapter1?Wasn’tthateasier?Thatmethodused10and100as
referencenumbersaswell,wejustdidn’twritethemdown.
Usingareferencenumberallowsustocalculateproblemssuchas6×7,6×6,4×7and4×8.
Let’sseewhathappenswhenwetry6×7usingthemethodfromChapter1.
We draw the circles below the numbers and subtract the numbers we are multiplying from 10. We


write4and3inthecircles.Ourproblemlookslikethis:

Nowwesubtractcrossways:3from6or4from7is3.Wewrite3aftertheequalssign.

Fourtimes3is12,sowewrite12afterthe3forananswerof312.

Isthisthecorrectanswer?No,obviouslyitisn’t.
Let’sdothecalculationagain,thistimeusingthereferencenumber.

That’smorelikeit.
Youshouldsetoutthecalculationsasshownaboveuntilthemethodisfamiliartoyou,thenyoucan
simplyusethereferencenumberinyourhead.

Testyourself
Trytheseproblemsusingareferencenumberof10:
a)6×7=
b)7×5=
c)8×5=
d)8×4=
e)3×8=
f)6×5=
Theanswersare:
a)42
b)35
c)40
d)32
e)24
f)30

Using100asareferencenumber
Whatwasourreferencenumberfor96×97inChapter1?Onehundred,becauseweaskedhowmany
moredoweneedtomake100.
Theproblemworkedoutinfullwouldlooklikethis:

The technique I explained for doing the calculations in your head actually makes you use this
method.Let’smultiply98by98andyouwillseewhatImean.


If you take 98 and 98 from 100 you get answers of 2 and 2. Then take 2 from 98, which gives an
answer of 96. If you were saying the answer aloud, you would not say, ‘Ninety-six’, you would say,
‘Nine thousand, six hundred and . . .’. Nine thousand, six hundred is the answer you get when you
multiply96bythereferencenumber,100.
Now multiply the numbers in the circles: 2 times 2 is 4. You can now say the full answer: ‘Nine

thousand,sixhundredandfour.’Withoutusingthereferencenumberwemighthavejustwrittenthe4
after96.
Hereishowthecalculationlookswritteninfull:

Testyourself
Dotheseproblemsinyourhead:
a)96×96=
b)97×97=
c)99×99=
d)95×95=
e)98×97=
Youranswersshouldbe:
a)9,216
b)9,409
c)9,801
d)9,025
e)9,506

DoubleMultiplication
Whathappensifyoudon’tknowyourtablesverywell?Howwouldyoumultiply92times94?Aswe
haveseen,youwoulddrawthecirclesbelowthenumbersandwrite8and6inthecircles.Butifyou
don’tknowtheanswerto8times6youstillhaveaproblem.
Youcangetaroundthisbycombiningthemethods.Let’stryit.
Wewritetheproblemanddrawthecircles:

Wewrite8and6inthecircles.

Wesubtract(takeaway)crossways:either92minus6or94minus8.
Iwouldchoose94minus8becauseitiseasytosubtract8.Theeasywaytotake8fromanumberis
totake10andthenadd2.Ninety-fourminus10is84,plus2is86.Wewrite86aftertheequalssign.

Nowmultiply86bythereferencenumber,100,toget8,600.Thenwemustmultiplythenumbersin
thecircles:8times6.
If we don’t know the answer, we can draw two more circles below 8 and 6 and make another
calculation.Wesubtractthe8and6from10,givingus2and4.Wewrite2inthecirclebelowthe8,and
4inthecirclebelowthe6.
Thecalculationnowlookslikethis:

Wenowneedtocalculate8times6,usingourusualmethodofsubtractingdiagonally.Twofrom6is
4,whichbecomesthefirstdigitofthispartofouranswer.
Wethenmultiplythenumbersinthecircles.Thisis2times4,whichis8,thefinaldigit.Thisgives
us48.
Itiseasytoadd8,600and48.
8,600+48=8,648
Hereisthecalculationinfull.

Youcanalsousethenumbersinthebottomcirclestohelpyoursubtraction.Theeasywaytotake8
from94istotake10from94,whichis84,andaddthe2inthecircletoget86.Oryoucouldtake6
from92.Todothis,take10from92,whichis82,andaddthe4inthecircletoget86.
Withalittlepractice,youcandothesecalculationsentirelyinyourhead.

Notetoparentsandteachers
Peopleoftenaskme,‘Don’tyoubelieveinteachingmultiplicationtablestochildren?’
Myansweris,‘Yes,certainlyIdo.Thismethodistheeasiestwaytoteachthetables.Itisthefastestway,themostpainlesswayand
themostpleasantwaytolearntables.’
Andwhiletheyarelearningtheirtables,theyarealsolearningbasicnumberfacts,practisingadditionandsubtraction,memorising
combinations of numbers that add to 10, working with positive and negative numbers, and learning a whole approach to basic
mathematics.

Chapter3
NUMBERSABOVETHEREFERENCENUMBER
Whatifyouwanttomultiplynumbersabovethereferencenumber;above10or100?Doesthemethod
stillwork?Let’sfindout.

MultiplyingNumbersinTheTeens
Hereishowwemultiplynumbersintheteens.Wewilluse13×15asanexampleanduse10asour
referencenumber.
Both 13 and 15 are above the reference number, 10, so we draw the circles above the numbers,
insteadofbelowaswehavebeendoing.Howmuchabove10arethey?Threeand5,sowewrite3and
5inthecirclesabove13and15.Thirteenis10plus3,sowewriteaplussigninfrontofthe3;15is10
plus5,sowewriteaplussigninfrontofthe5.

Asbefore,wenowgocrossways.Thirteenplus5or15plus3is18.Wewrite18aftertheequalssign.

Wethenmultiplythe18bythereferencenumber,10,andget180.(Tomultiplyanumberby10we
add a 0 to the end of the number.) One hundred and eighty is our subtotal, so we write 180 after the
equalssign.

Forthelaststep,wemultiplythenumbersinthecircles.Threetimes5equals15.Add15to180and
wegetouranswerof195.Thisishowwewritetheprobleminfull:

If the number we are multiplying is above the reference number we put the circle above. If the
numberisbelowthereferencenumberweputthecirclebelow.
If the circled number is above we add diagonally. If the circled number is below we subtract
diagonally.
Thenumbersinthecirclesaboveareplusnumbersandthenumbersinthecirclesbelowareminus
numbers.

Let’stryanotherone.Howabout12×17?
The numbers are above 10 so we draw the circles above. How much above 10? Two and 7, so we
write2and7inthecircles.

What do we do now? Because the circles are above, the numbers are plus numbers so we add
crossways.Wecaneitherdo12plus7or17plus2.Let’sdo17plus2.
17+2=19
Wenowmultiply19by10(ourreferencenumber)toget190(wejustputa0afterthe19).Ourwork
nowlookslikethis:

Nowwemultiplythenumbersinthecircles.
2×7=14
Add14to190andwehaveouranswer.Fourteenis10plus4.Wecanaddthe10first(190+10=
200),thenthe4,toget204.
Hereisthefinishedproblem:

Testyourself
Nowtrytheseproblemsbyyourself.
a)12×15=
b)13×14=
c)12×12=
d)13×13=
e)12×14=
f)12×16=
g)14×14=
h)15×15=
i)12×18=
j)16×14=
Theanswersare:

a)180
b)182
c)144
d)169
e)168
f)192
g)196
h)225
i)216


j)224

Ifanyofyouranswerswerewrong,readthroughthissectionagain,findyourmistake,thentryagain.
Howwouldyousolve13×21?Let’stryit:
We still use a reference number of 10. Both numbers are above 10 so we put the circles above.
Thirteenis3above10,21is11above,sowewrite3and11inthecircles.
Twenty-oneplus3is24,times10is240.Threetimes11is33,addedto240makes273.Thisishow
thecompletedproblemlooks:

MultiplyingNumbersAbove100
Wecanuseourspeedmathsmethodtomultiplynumbersabove100aswell.Let’stry113times102.
Weuse100asourreferencenumber.

Addcrossways:
113+2=115
Multiplybythereferencenumber:
115×100=11,500
Nowmultiplythenumbersinthecircles:
2×13=26

Thisishowthecompletedproblemlooks:

SolvingProblemsinYourHead
Whenyouusethesestrategies,whatyousayinsideyourheadisveryimportant,andcanhelpyousolve
problemsmorequicklyandeasily.
Let’strymultiplying16by16.
ThisishowIwouldsolvethisprobleminmyhead:
16plus6(fromthesecond16)equals22,times10equals220
6times6is36
220plus30is250,plus6is256
Tryit.Seehowyougo.


Insideyourheadyouwouldsay:
16plus6...22...220...36...256
Withpractice,youcanleaveoutalotofthat.Youdon’thavetogothroughitstepbystep.Youwould
onlysaytoyourself:
220...256
Practisedoingthis.Sayingtherightthinginyourheadasyoudothecalculationcanbetterthanhalve
thetimeittakes.
Howwouldyoucalculate7×8inyourhead?Youwould‘see’3and2belowthe7and8.Youwould
take2fromthe7(or3fromthe8)andsay,‘Fifty’,multiplyingby10inthesamestep.Threetimes2is
6.Allyouwouldsayis,‘Fifty...six.’
Whatabout6×7?
Youwould‘see’4and3belowthe6and7.Sixminus3is3;yousay,‘Thirty’.Fourtimes3is12,
plus30is42.Youwouldjustsay,‘Thirty...forty-two.’
It’snotashardasitsounds,isit?Anditwillbecomeeasierthemoreyoudo.

DoubleMultiplication
Let’smultiply88by84.Weuse100asourreferencenumber.Bothnumbersarebelow100sowedraw

thecirclesbelow.Howmanybelowarethey?Twelveand16.Wewrite12and16inthecircles.Now
subtractcrossways:84minus12is72.(Subtract10,then2,tosubtract12.)
Multiplytheanswerof72bythereferencenumber,100,toget7,200.
Thecalculationsofarlookslikethis:

Wenowmultiply12times16tofinishthecalculation.

Thiscalculationcanbedonementally.
Nowaddthisanswertooursubtotalof7,200.
If you were doing the calculation in your head you would simply add 100 first, then 92, like this:
7,200plus100is7,300,plus92is7,392.Simple.
Youshouldeasilydothisinyourheadwithjustalittlepractice.

Testyourself
Trytheseproblems:
a)87×86=
b)88×88=
c)88×87=
d)88×85=
Theanswersare:
a)7,482
b)7,744
c)7,656


d)7,480

Combiningthemethodstaughtinthisbookcreatesendlesspossibilities.Experimentforyourself.

Notetoparentsandteachers

This chapter introduces the concept of positive and negative numbers. We will simply refer to them as plus and minus numbers
throughoutthebook.
Thesemethodsmakepositiveandnegativenumberstangible.Childrencaneasilyrelatetotheconceptbecauseitismadevisual.
Calculatingnumbersintheeightiesusingdoublemultiplicationdevelopsconcentration.Ifindmostchildrencandothecalculations
muchmoreeasilythanmostadultsthinktheyshouldbeableto.
Kidsloveshowingoff.Givethemtheopportunity.


Chapter4
MULTIPLYINGABOVE&BELOWTHEREFERENCENUMBER
Untilnow,wehavemultipliednumbersthatwerebothbelowthereferencenumberorbothabovethe
referencenumber.Howdowemultiplynumberswhenonenumberisabovethereferencenumberand
theotherisbelowthereferencenumber?

NumbersAboveandBelow
Wewillseehowthisworksbymultiplying97×125.Wewilluse100asourreferencenumber:
Ninety-seven is below the reference number, 100, so we put the circle below. How much below?
Three, so we write 3 in the circle. One hundred and twenty-five is above so we put the circle above.
Howmuchabove?Twenty-five,sowewrite25inthecircleabove.

Onehundredandtwenty-fiveis100plus25soweputaplussigninfrontofthe25.Ninety-sevenis
100minus3soweputaminussigninfrontofthe3.
Wenowcalculatecrossways.Either97plus25or125minus3.Onehundredandtwenty-fiveminus3
is 122. We write 122 after the equals sign. We now multiply 122 by the reference number, 100. One
hundredandtwenty-twotimes100is12,200.(Tomultiplyanynumberby100,wesimplyputtwozeros
afterthenumber.)Thisissimilartowhatwehavedoneinearlierchapters.
Thisishowtheproblemlookssofar:

Nowwemultiplythenumbersinthecircles.Threetimes25is75,butthatisnotreallytheproblem.
Wehavetomultiply25byminus3.Theansweris−75.

Nowourproblemlookslikethis:

Ashortcutforsubtraction
Let’stakeabreakfromthisproblemforamomenttohavealookatashortcutforthesubtractionswe
aredoing.
Whatistheeasiestwaytosubtract75?Letmeaskanotherquestion.Whatistheeasiestwaytotake9


from63inyourhead?
63−9=
Iamsureyougottherightanswer,buthowdidyougetit?Somewouldtake3from63toget60,then
takeanother6tomakeupthe9theyhavetotakeaway,andget54.
Somewouldtakeaway10from63andget53.Thentheywouldadd1backbecausetheytookaway1
toomany.Thiswouldalsogive54.
Somewoulddotheproblemthesamewaytheywouldwhenusingpencilandpaper.Thiswaythey
havetocarryandborrowintheirheads.Thisisprobablythemostdifficultwaytosolvetheproblem.
Remember,theeasiestwaytosolveaproblemisalsothefastest,withtheleastchanceofmakinga
mistake.
Most people find the easiest way to subtract 9 is to take away 10, then add 1 to the answer. The
easiestwaytosubtract8istotakeaway10,thenadd2totheanswer.Theeasiestwaytosubtract7isto
takeaway10,thenadd3totheanswer.
Whatistheeasiestwaytotake90fromanumber?Take100andgiveback10.
Whatistheeasiestwaytotake80fromanumber?Take100andgiveback20.
Whatistheeasiestwaytotake70fromanumber?Take100andgiveback30.
If we go back to the problem we were working on, how do we take 75 from 12,200? We can take
away100andgiveback25.Isthiseasy?Let’stryit.Twelvethousand,twohundredminus100?Twelve
thousand,onehundred.Plus25?Twelvethousand,onehundredandtwenty-five.
Sobacktoourexample.Thisishowthecompletedproblemlooks:

Withalittlepracticeyoushouldbeabletosolvetheseproblemsentirelyinyourhead.Practisewith

theproblemsbelow.

Testyourself
Trythese:
a)98×145=
b)98×125=
c)95×120=
d)96×125=
e)98×146=
f)9×15=
g)8×12=
h)7×12=
Howdidyougo?Theanswersare:
a)14,210
b)12,250
c)11,400
d)12,000
e)14,308
f)135
g)96
h)84