Woburn Lower School Calculationpolicy

Woburn Lower School CalculationPolicy

Rational

TheCalculationPolicyclarifiesprogressionincalculationwithexamplesthat are‘mathematicallytransparent’,inotherwords the waythecalculationworks is clearandsupportsthedevelopmentofmathematicalconcepts.

TheAimsofthecurriculum:

TheNationalCurriculum 2014formathematicsaimstoensurethatallpupils:

becomefluentinthefundamentalsofmathematics,includingthroughvariedandfrequentpracticewithincreasinglycomplexproblemsovertime,sothatpupilsdevelopconceptualunderstandingandtheabilityto recallandapplyknowledgerapidlyandaccurately.

reasonmathematicallybyfollowingalineof enquiry,conjecturingrelationshipsandgeneralisations,anddevelopinganargument,justificationorproofusingmathematicallanguage

cansolveproblemsbyapplyingtheirmathematicsto avarietyof routineandnon-routineproblemswithincreasingsophistication,includingbreakingdownproblemsintoaseriesofsimplersteps andperseveringinseekingsolutions.

Pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should apply their mathematical knowledge to science and other subjects.

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the securityof pupils’ understanding and their readiness to the next stage.

As a result Year groups are omitted from the policy to ensure the needs of the children are the priority at all times.

Recording

Recordingisdevelopedinarangeofways,includingthefollowing.Althoughinitiallythey willbedevelopedinthisorder, once awayofrecording,suchas ‘byshowing real objects’, isinplace, thatwillcontinuetobeusedthroughoutthePrimaryyears.InEYFSmostrecordingwillbebyshowingrealobjects,whereasinY6realobjectsmaybeusedtoshowanunderstandingofcalculationwithdecimals.

Developmentofrecording:

• byshowingreal objects
• byphotographingordrawingthecalculation activity
• countingonanumberline or 100 square
• apracticalcalculationactivityona numberline
• anumberbondonanumberline
• amentalcalculationona numberline
• apracticalactivityasanumber sentence
• anumberbondasa numbersentence
• amentalcalculationasanumbersentence
• awrittencalculation

Progressionincalculation

Addition

Childrenbegincalculationpurelywithpracticalactivities.Overtimetheylearntorecordtheseactivitiesinawaythatmakessenseto them. Thiswillbebyshowingortakingphotographsoftheequipment theyhaveused, leadingtodrawingsofwhattheydid.

Forinstance,withthepracticalactivity-Ihave3sweets, thenIget onemore. Thechilddrawsthesweets.They maydraw 3sweetsandthenanother.Theymayjustdraw4to startwith.

() () ()()

They won’tdraw3,then 1,then4,norshouldthey beexpectedtoatthisstage.

() () ()() = () ()() ()doesn’tmakemuch sense.You eitherhave3and1oryouhave

4.Youneverhaveboth.

Thismeansthatanyrecordingofthe format3+1=4isveryunhelpfulandisnot basedontheirexperiencebutonan abstractrecordingmethod.

Whenpupilsare readytorecordnumerals(possiblyat theend of the summer terminYearR,butprobably inYear1)theymaybeginto recordtheaboveexamplewithnumbersaswell:

() () ()()

31

or just as() () () ()4

butnotyetas3+1, andcertainlynotas3+1=4.

Aswellasusingobjects,pupilswillbegintousenumber tracks , 100 squares andthen numberlinesbothaspracticalequipment that makesthe calculationtransparent andaswaystorecordwhattheydid.

Forcalculations itis usefultohave‘lollipop’numbertracks,wherecounterscanbeplaced inthecircleswithoutcoveringoverthenumerals.

3plus1, forinstance:

Atfirst childrenwillrecordtheircountingonnumberlines, latermovingtorecordingofcalculationon anumberline.

Pupilswillusenumberednumber linestorecordjumps,forexample for3+2,beforerecordingonblanknumber lines.

Children should beconfidentusingnumberlines to‘play’with numbers.Theyhavecountedinstepsof100, 1000,50and500, andhaveseentheirteacherrecordthenumberstheyhave counted.Theybelieve thatmathsisaboutplayingwithnumbersandtryingthingsout ratherthanjust finding therightanswer.

Recordingnumbersentences

Beforepupilsmove torecording3+1they willneedlotsofexperienceofpracticaladdition,andanabilitytorespondtomathematicalvocabularypractically.Forinstance, if youaskachildtoshowyou5and2more, or3plus1, or1add4, they canusetheteddies,countersornumber tracksto show you.They will alsobedevelopingtheiruse ofmathematicalvocabularytoexplainwhat they havedone.

From thisitwillbepossibletodevelopanunderstandingofthe +sign, whichwillenablepupilstobegintorecordintheform5+2.

Pupilsthenneedtounderstandtheconcept ofequalitybeforeusingthe= sign.Thismeanstheycanseeanexamplesuchas7=6+1, or 5=5,aswellasthemore commonarrangement3+1=4,and knowthatitmakessense.

Pupilswillstillworkpracticallywithequipmentandrealobjects,butnowcanrecordtheirexplanationofwhat theyhave doneasaconventionalnumbersentence:

3+ 14= 1717= 14+317– 3=143+ 14= 14+ 3andsoon.

However,pupilswillstillrecordwithobjects, drawingsand numberlines onafrequentbasis,andwhenevertheyarelearningnewconceptsor startingtouse awider rangeofnumberstheywillneedto returntousingthese easilyunderstoodandexplainedmethodsofrecording.

Mentalmethods

Pupilsneed todeveloptheiruse of jottingsto support mentalcalculation.Thesejottingsmaybeasdrawings,numberlinesornumbersentences.

Oncechildrenhavean understandingofplacevaluein2-digitnumbers,inotherwordstheyare convincedthat23is20and3, or59is50and9,theycanbegintousepartitioningintheirmentalandrecordedcalculations. Children may find that using a 100 square leads onto the partitioning method well.

Partitioning

Partitioningmay berecordedin anumberofways, suchas:

The importantthingto considerwhenchildrenarerecordingpartitioningisthat they recordhowtheythoughtaboutthenumbers,anddon’talltry todoitthesameway. This isnotabout findinglotsofwaystorecord,butofrecordingwhat makessensetoa child.

Partitioningisalsoanappropriatestrategyforlargernumbers,eventuallyincludingdecimals.

Partitioningusingnumberlines

Keyunderstanding–Anumberlineisatool,nota rule.

Childrenpartitionnumbersto counton,mainlyinmultiplesof100, 10or1,onanumberline.Numberlineswillbeusedfor calculationsrightthroughKeyStage 2.

Initialattemptsmay be alittleslowaschildrenchooseeasynumbersto counton:

Whatmatters,however,isthat childrenmake theirownchoicesofwhichnumbersto useandthattheyusetheirunderstandingofnumberandplacevaluetofindaway that worksforthem.Thismaycontinueinto 3-digitnumbersforsome children.

Astheybecome moreconfident,childrenstartto jumpinmultiplesof100, 10and1.Theyuse theirownchoiceof numbers,doinganyjumpsonthenumberline,instepsof100, 10,1ormultiplesofthese, dependingontheirmentalstrategiesandability.

Childrenneedtodevelopunderstandingofcalculationinarangeofcontexts, forinstancemeasures,includingmoneyand time.

Time isparticularlydifficult, andat firstchildrenwillusenumberlinesto recordcountinginstepsofhoursorminutes.

Countingacross boundariesisparticularlyimportant.

Non-statutoryguidanceforY6suggests:‘Usingthenumberline,pupilsuse, addandsubtractpositive andnegativeintegersfor measuressuchastemperature.’Itwouldalsobesensible tocontinue touse thenumberlineforcalculationswith time.

Thetemperature is-80C. Itincreasesby150.Whatisthe temperaturenow?

Expandedverticalmethod

In key stage 2 or earlier if the child demonstrates advanced understanding,maybegintorecordadditioncalculationsvertically,atfirst recordingcalculationsbothasthepartitioningtheyhavebeenusingandasanexpandedverticalcalculation,addingnumbers incolumns,beginningwiththehundreds, then tensandthenaddingtheones.Thevocabularyusedwillalwaysbe wholenumberplacevaluevocabulary,so254 wouldbe200, 50and4,never2hundreds,5tensand4ones.Alwaysuse‘ones’,asthe term‘units’is onlyusedforunits of measurementand notfor placevalue.

Childrendiscusswhatisthesameandwhat isdifferent about eachof these waysofrecording.Theyrealisethatitdoesn’tmatterin whatorderyou addthetotals fortheones,tensorhundreds.Thefinal total isalwaysthe same.

Once pupilsareveryconfident withthismethodofrecordingtheymayextend ittonumberswith moredigits, providingtheirunderstandingofplacevalue issufficientto support this.

There isnoneedtoheadeachcolumnwithH, Tor O,aswritingthisdoesnothelppupilswhodonotunderstandplace value, and isunnecessary for thosewhodo.

Itispossibletorecordtheverticalmethodmorequicklybymakinganotewhenthe additionoftwoor morenumbersgoesabove1, 10or100 andsoon, ratherthanwritingitallout.

Thiscompact method isbest leftuntilchildrenhaveusedpartitioningwithdecimalsandareveryconfidentwiththis.

Whenchildren usethe‘shortcut’orcompactmethodtheyneedtoknowthatit works inasimilar waytopartitioning,butthatyouaddtheonesfirst.Ifchildrenreallyunderstandtheexpendedverticalmethodandhaveusediffor ayearormoreit shouldbe possibleto teachthe compact methodinoneortwolessons.

Keyunderstanding–It’s besttotalkabout‘makingone’from adding0.2 and0.9, andputtingthiswiththeother ones,and‘making100’from adding50+50andputtingthiswiththe otherhundreds, ratherthan ‘carrying’ one.

Subtraction

Aswithaddition, subtractionisinitially recordedasdrawingtheresultofapracticalactivity,movingonto recordthisusingnumbers,onnumber tracksor linesorasnumbersentences.

Initiallynumbertracksorlineswillbeusedtosubtract smallnumberssuchas5 – 2.

Whenpupilsmoveontousejottingsthenumber linewillbecomeespeciallyimportant.Jottingsasnumbersentencesarelessusefulforsubtraction aspartitioningcannotgenerallybeused.

In theexample73– 26it ispossibleto start with70–20,but3–6islessuseful!

Keyunderstanding–Pupils need torealisethatpartitioning is notappropriateforsubtraction.

Numberlines,however,make the calculationeasy.

Keyunderstanding–Puttingthezero ona numberlineforsubtractionandcrossingoutwhathasbeensubtractedmakesthesubtractionobvious.

You’llnoticethatthereisazero placedonthenumberline.This helps tostop childrenwritingthe73onthe lefthandsideofthenumberline,butmoreimportantlyenablesyou tocross outand‘takeaway’ the26. It makes iteasiertounderstandthatthis isasubtraction,

andyouarecountingonto findouthowmanyare left.Sothisuseofnumberlinesbuildsonthe understandingofsubtractionasdifference orascomplementaryaddition.

The jumpsonthisfirstnumberlineareintensandones. Thisisa goodstartingpoint asitbuildsonthedailycountingthat childrenwillbedoing,includingcountingonintensandonesfromanynumber.Italsomeansthatcalculatinghowmany youhavejumpedaltogetheriseasy.Ofcourse childrenmaydodifferentjumps

Whentheyareconfidentwiththisstage,pupilscanreducethe numberofsteps.

Theabovemethodcanbeextendedtolarger numbersbyusingcomplementsto100.

Childrenmaystillchoose to countinmultiplesof100,10and1.

Subtractionofdecimalsisjustassimpleusing thenumberline.

Don’t forgetthatchildrenwillstillencountercalculations whereit’s equallysensibletocountback.

3004–96= 2908

Continuetousenumber linesforsubtractioncalculationsina rangeof contexts, suchastime,money,mass, lengthandcapacity.

Verticalcalculationforsubtraction

Vertical calculationforsubtractioncancreaterealdifficultiesforbothchildrenandteachers.It’s easy tothinkthatteachingchildrento rememberaprocess,perhaps developedthroughthe useofplace valueresources, will work.Some childrenmaybeabletoremember this,but, eveniftheydo,learningwithoutunderstandingisnever abasisfor future development.

The followingmethod,whichcanbeusedifyoudecide youhavetoteacha verticalmethod,isstillmathematically‘transparent’.There arenotricks, thereisnoneedtoswap10 onesfor atenor 10tens forahundredandit’spossibletokeeptrack ofthenumberyouaresubtractingfromallthewaythrough.Having saidthat,don’tmove tothismethodunlesschildrenhaveathoroughgraspofsubtractionwithdecimalsonanumberlineandarealunderstandingof placevalue.For thefirsttimeit’spossible,andnecessary,tousepartitioningforsubtraction.

I’vesetthis calculationoutinanumberof stepstoclarifyhow itworks,thoughin practiceitwouldbe writtenasone step.

Step1

4subtract8ispossible,buttheanswerwouldbenegative.Partitionthe 50into 40and10.Nowyoucan putthe10withthe4, soyouhaveenoughtosubtract 8withoutgivinganegativeanswer.

Step2

You’llnoticethatit’s still10+4. Thereis noneedto changethis to14.Ifyouhaveboth10and4you cansubtract 8andthisleaves6.

Step3

The nextstepisto subtract 80from 40. Againthiswouldgiveanegativeanswer,so partitionthe300into200and100andput the100with the40.

Checkthat200+100and 40and10+4stillmake the354youstartedwith.

Step4

Nowyoucan subtract80from100+40,which gives60,andthen subtract 100from 200,leaving 100.

Thisfinalstepshowswhat thecompletecalculationwouldlook like.Allstepswouldberecordedinjustonecalculation.

So354-188=166

It’s possibletomovefromthis,oncechildren haveextensiveexperienceworkinginthisway,toa compactdecompositionmethod,thoughthismaynotbenecessaryandmay notbeanimprovement.

Multiplication

Children’sfirst recordinginmultiplicationwillbebyplacingobjectsinarraysandcountinginstepsonnumberlinesfromzero.

Conceptsofmultiplicationdevelopusingdoublingandcountinginsteps,andareextendedusingthearray. Objects, arrays,numberlinesandnumbersentenceswillcontinueto bethemainmethodsofrecording.

8x2meansIstart fromzero andcount on8twice.

Oncepupilsbegintomultiplyone-digitbytwo-digitnumbersthiswillbebyusingpartitioning.Pupilswillbeunlikely tohave usedbracketsat thisstage anditisbestto letthemrecordwithoutbrackets,butwithaclear understandingofwhattheyaredoing,basedonanunderstandingofarraysandadiagram toexplainthecalculation.

8x23

8x23 =8x10+8x10+8x3

= 80+80+ 24

=184

Thisleadstothegridmethodofmultiplication:

Oncechildrencanshowan understandingofa1-digitby2-digit multiplicationbothwith anarrayandagridmultiplicationtheycanexploremultiplying amultiple of 10bya1-digitnumber.Usingthisdecreasesthenumberofstepsneededto complete themultiplication.

The gridmethodcanthenbeusedfor2-digitby2-digitmultiplication. At first justusenumbersbetween11and19.Forinstance try16x 14:

Usingnumbers11to 19keepsthe mentalcalculationsrelativelysimple.

When addingtogetherthe fourtotals, thiscanbedoneeitherhorizontallyorvertically.

Later childrencanmoveontoother2-digit numbersanddecimals.

66x34

Additionoftheproductsmaybecomea separateadditioncalculation.

Movingchildrentoa verticalmultiplicationcalculationneeds to be donewithcare,ensuringthat theyunderstandwhattheyaredoingandwhy theyaredoingit.Tostartwithit’simportantthat all the stepsthatwouldoccur inthegridmethodarereplicatedintheverticalone.

WithinthePrimaryyears it’s besttostayatthis levelforallchildrenwhiledevelopingtheirabilityto calculatewithdecimalsandinarangeofcontexts.Moving ontomorecompactmethodsrequiresexcellentestimation, placevalueandmental calculationskillsandshouldnotbeattemptedunless childrenhavetheseskills and can learnthe‘shortcut’method inoneortwolessons.

Division

Aswith multiplication,divisionisrecordedwithobjects,arrays, numberlinesornumbersentences.

I startatzero andcountin8s untilI getto16. That’stwoeights.

Calculationswithremainders inthequotient arealsorecordedonanumberline.17÷8=2with1leftover

Istart at zeroandcountin8suntilIgetto16.Then thereis1moreto getto 17,so Ihave2jumpsof8and1leftover (remainder).It’s importantthattheremainderisneverrecordedasa jumpasthejumpsshowhow manyeightshavebeenmade.Usinga crossfor eachnumberleftovertendsto workwell.

33÷9=3with6leftover.

Whenchildrenaredividingnumberswhich aremorethan10timesthedivisoritbecomesusefultoworkwithmultiplesofthedivisor.

Inthisexamplechildrenwouldcountinstepsof 70, showing7ten timesequals70,thendecidinghow todothenext stepof56÷7. Itcouldbeonejump of7eighttimes,or couldbesmallerjumpsof7, 14, 21andsoonuntilthe 196isreached.

Formanypupils,the additionofan‘IKnow’boxmakesthecalculationeasy.For259÷6 thereareacoupleof waysofdoing the‘I know’box.

In thefirst examplethechildcountsinmultiplesof60, soneeds to beabletodothisupto6x90 if necessary. Inthesecondexamplethe‘I know’boxalwayshas x2,x5,x10,thenx20,x50, x200, x500and so onasnecessary.These areeasymental calculationsandchildrenarelesslikelyto makecalculationerrors.Thefinal calculationwillvarydependingonwhichfactsthe childuses.

Keyunderstanding–Children mustalways recordwhathappensin theirown mind,andnottry to guessandrecordwhat’sin yours.

Whendividingdecimalsitisusefultobeginbyadaptingacalculationthatcanalreadybeunderstood.

The‘I know’boxis particularlyimportantwhen dividingdecimals,and can leadtoaverticalmethodif youfeelthisisnecessary.Formanychildrenthenumberlinecontinuesto bethemethodusedthroughouttheprimaryyears.

1736÷14

Appendix

Examplesofwrittenmethodsforaddition,subtraction,multiplicationanddivision,suitablefortheendofKeyStage2 or when a child demonstrates advanced understanding.

Theseexamplescanbetaughtwithunderstandingratherthanasremembered processesonly.

Addition

Subtraction

Multiplication

Division