Year1Group and sharesmall quantities

Using objects, diagramsand pictorialrepresentationstosolve problemsinvolvingboth groupingandsharing.

Howmany groupsof4can bemadewith 12stars?= 3

Grouping:

Sharing:

Pupilsshould:

Exampledivisionprobleminafamiliarcontext:

Thereare6pupilsonthistableandthereare18piecesoffruittosharebetweenus.Ifwesharethemequally,howmanywillweeachget?

Cantheyworkitoutandgiveadivisionstatement…?

“18sharedbetween6peoplegivesyou3each.”

  • Use lots ofpracticalapparatus, arraysandpicturerepresentations.
  • Betaughttounderstandthedifferencebetween‘grouping’objects(howmanygroupsof2canyoumake?)and ‘sharing’(sharethesesweetsbetween2people).
  • Be ableto countinmultiplesof2s, 5sand 10s.
  • Findhalfofa groupof objectsbysharinginto2 equal groups.

KeyVocabulary:share,shareequally,oneeach,twoeach…,group,groupsof,lotsof,array

Key numberskillsneeded for divisionatY1:

  • Solve one-step problemsinvolving multiplication anddivision,by calculatingtheanswer using concrete objects, pictorialrepresentationsarrayswith thesupport of the teacher.
  • Through grouping and sharingsmall quantities,pupilsbegintounderstanddivisionandfinding

simple fractionsof objects, numbersandquantities.

  • Theymake connectionsbetweenarrays,number patternsand countingintwos,fivesandtens.

Year2Group and share,using the÷ and=sign

Use objects, arrays,diagramsandpictorialrepresentationsand grouping on anumber line.

Arrays:

Thisrepresents 12 ÷3,posedas

howmanygroups of3 arein12?

Pupilsshould alsoshowthat the samearray canrepresent12÷4= 3ifgrouped horizontally.

Knowandunderstand sharing andgrouping:

Grouping

Sharing

Childrenshouldbetaughttorecognisewhetherproblemsrequiresharing orgrouping.

Grouping using anumber line:

Groupfromzeroinequaljumpsofthedivisortofindout ‘howmanygroupsof_in_?’Pupilscouldandusingabeadstringorpracticalapparatusto workoutproblemslike “ACDcosts£3, howmanyCDscanIbuywith£12?”Thisisanimportantmethodtodevelopunderstandingofdivisionasgrouping.

+3+3+3+3

Pose12÷3as‘Howmanygroupsof3arein12?’

KeyVocabulary:share,shareequally,oneeach,twoeach…,group,equalgroupsof,lotsof,array,divide,dividedby,dividedinto,division,grouping,numberline,left,leftover

KeynumberskillsneededfordivisionatY2:

  • Countinstepsof2,3and5from0
  • Recallandusemultiplicationanddivisionfactsforthe2,5and10multiplicationtables,includingrecognisingoddandevennumbers.
  • Calculatemathematicalstatementsformultiplicationanddivisionwithinthemultiplicationtablesandwritethemusingthex,÷and=signs.
  • Showthatmultiplicationoftwonumberscanbedoneinanyorder(commutative)anddivisionofone number byanothercannot.
  • Solveproblemsinvolvingmultiplicationanddivision,usingmaterials,arrays,repeatedaddition,mentalmethods,andmultiplicationanddivisionfacts,includingproblemsincontexts

.

Year3Divide2-digitnumbersbya singledigit (wherethereis noremainder inthefinal answer)

Groupingonanumberline:

4r 1

+3+3+3+3r 1

STEP1:Childrencontinuetoworkoutunknowndivisionfactsbygroupingonanumberlinefromzero.Theyarealsonowtaughttheconceptofremainders,asintheexample.Thisshouldbeintroducedpracticallyandwitharrays,aswellasbeingtranslatedtoanumberline.Childrenshouldworktowardscalculatingsomebasicdivisionfactswithremaindersmentallyforthe 2s, 3s, 4s,5s, 8s and 10s, ready for ‘carrying’remaindersacrosswithintheshortdivisionmethod.

Reallife contexts need tobe used routinelyto helppupils gainafull understand- in, andthe abilityto recognise the place of division and how toapply it to problems.

Shortdivision:Limitnumbersto NOremaindersinthe answerORcarried (each digitmustbe amultipleofthe divisor).

Shortdivision:Limitnumbersto NOremaindersinthe finalanswer,but with remaindersoccurringwithinthe

STEP2:Oncechildrenaresecurewithdivision asgroupingand demonstratethisusingnumberlines,arraysetc.,shortdivision forlarger2-digitnumbersshouldbeintroduced,initiallywithcarefullyselectedexamplesrequiringnocalculatingof remainders atall.Startbyintroducingthelayoutofshort divisionbycomparingitto anarray.

Remindchildren ofcorrectplacevalue, that96 isequal to90and6,butinshortdivision,pose:

  • Howmany3sin9?=3,andrecorditabovethe9tens.
  • Howmany3sin6?=2,andrecorditabovethe6units.

STEP3:Oncechildrendemonstrateafullunderstandingof remainders,andalsotheshortdivisionmethodtaught, theycanbetaughthowtousethemethodwhenremaindersoccur withinthecalculation(e.g.96÷4),andbetaughtto ‘carry’theremainder onto the nextdigit.Ifneeded, childrenshoulduse the number line toworkoutindividualdivisionfactsthat occurwhichtheyare notyetabletorecallmentally.

Step3Onlytaughtwhenpupilscancalculate’remainders‘

KeyVocabulary:share,shareequally,oneeach,twoeach…,group,equalgroupsof,lotsof,array,divide,dividedby,dividedinto,division,grouping,numberline,left,leftover,inverse,shortdivision,‘carry’,remainder,multiple

KeynumberskillsneededfordivisionatY3:
  • Recallandusemultiplicationanddivisionfactsforthe2,3,4,5,8and10multiplicationtables(throughdoubling, connectthe2,4and8s).
  • Writeandcalculatemathematicalstatementsformultiplicationanddivisionusingthemultiplicationtablesthattheyknow,includingfortwo-digitnumberstimesone-digitnumbers,usingmentalandprogressingtoformal written methods.
  • Solveproblems,incontexts,andincludingmissingnumberproblems,involvingmultiplicationanddivision.
  • Pupilsdevelopefficientmentalmethods,e.g.usingmultiplicationanddivisionfacts(e.g.using3×2= 6,

6÷3=2and2=6÷3)toderiverelatedfacts(30 ×2=60, so60÷3=20and20=60÷3).

  • Pupilsdevelopreliablewrittenmethodsfordivision,startingwithcalculationsof2-digitnumbersby1-digitnumbersandprogressingtotheformalwrittenmethodofshortdivision.

Year4Divide upto 3-digitnumbersbyasingledigit

(without remaindersinitially)

Continuetodevelopshortdivision:

Short division shouldonly betaughtonce children have securedtheskill of calculating ‘remainders’.

STEP1:Pupilsmustbesecurewiththeprocessofshortdivisionfordividing2-digitnumbersbyasingledigit(thosethatdonotresultinafinalremainder

—seestepsinY3),butmustunderstandhowtocalculateremainders,usingthisto ‘carry’remainderswithinthecalculationprocess(seeexample).

Reallife contexts need tobe used routinelyto helppupils gainafull understand- ing, andthe abilityto

STEP2:Pupilsmoveontodividingnumberswithupto3-digitsbyasingledigit;howeverproblemsandcalculationsprovidedshould notresultinafinalanswerwithremainderatthisstage.ChildrenwhoexceedthisexpectationmayprogresstoY5level.

Whentheanswerforthefirstcolumniszero

recognise the place of division and how toapply it to problems.

(1÷5,asinexample),childrencouldinitiallywriteazeroabovetoacknowledgeitsplace,andmustalways‘carry’thenumber(1) overtothenextdigitasaremainder.

Includemoneyandmeasure contextswhen confident.

KeyVocabulary:share,shareequally,oneeach,twoeach…,group,equalgroupsof,lotsof,array,divide,dividedby,dividedinto,division,grouping,numberline,left,leftover,inverse,shortdivision, ‘carry’,remainder,multiple,divisibleby,factor

Key numberskillsneeded for divisionat Y4:

  • Recallmultiplicationanddivisionfactsforallnumbersupto12x12.
  • Useplacevalue,knownandderivedfactstomultiplyanddividementally,including:multiplyinganddividingby10and100and1.
  • Pupilspractisetobecomefluentintheformalwrittenmethodofshortdivisionwithexactanswerswhen dividing by a one-digitnumber.
  • Pupilspractisementalmethodsandextendthistothree-digitnumberstoderivefacts,forexample

200x 3 = 600 so, 600÷3=200.

  • Pupilssolvetwo-stepproblemsincontexts,choosingtheappropriateoperation,workingwithincreasinglyhardernumbers.Thisshouldincludecorrespondencequestionssuchasthreecakessharedequally between 10 children.

Year5Divide upto 4digitsby asingledigit, including

those with remainders.

Shortdivision,including remainder answers:

Theanswerto5309÷8couldbeexpressedas663andfiveeighths,663r5, asadecimal,orroundedasappropriatetotheprobleminvolved.

Includemoneyandmeasure contexts.

Shortdivisionwithremainders:Nowthatpupilsareintroducedtoexamplesthatgiverisetoremainderanswers,divisionneedstohaveareallifeproblemsolvingcontext,wherepupilsconsiderthemeaningoftheremainderandhowtoexpressit,i.e.asafraction,adecimal,orasaroundednumberorvalue,dependinguponthecontextoftheproblem.

SeeY6for howtocontinuetheshort divisiontogiveadecimal answerfor childrenwhoareconfident.

Estimate, Calculate,

If childrenare confidentandaccurate:

Check!

  • Introducelong divisionforpupilswhoareready to divide any number bya2-digitnumber (e.g.2678÷ 19).ThisisaYear6expectation.

KeyVocabulary:share,shareequally,oneeach,twoeach…,group,equalgroupsof,lotsof,array,divide,dividedby,dividedinto,division,grouping,numberline,left,leftover,inverse,shortdivision,

‘carry’,remainder,multiple,divisibleby,factor,inverse,quotient,primenumber,primefactors,compositenumber(non-prime)

Key numberskillsneeded for divisionatY5:

  • Recallmultiplicationanddivisionfactsfor allnumbersupto12x12(asinY4).
  • Multiplyanddividenumbersmentally,drawingupon knownfacts.
  • Identifymultiplesandfactors,includingfindingallfactorpairsofanumber,andcommonfactors oftwonumbers.
  • Solveproblemsinvolvingmultiplicationanddivisionwherelargernumbersaredecomposedintotheirfactors.
  • Multiplyanddividewholenumbersandthoseinvolvingdecimalsby10,100and1000.
  • Usethevocabulary ofprimenumbers,primefactorsand composite(non-prime)numbers.
  • Work outwhetheranumberupto100isprime number,andrecallprimenumbersto19.
  • Dividenumbersupto4digitsbya one-digitnumberusing theformalwrittenmethod ofshortdivisionandinterpretremaindersappropriatelyforthecontext.
  • Usemultiplicationanddivisionasinverses.
  • Interpretnon-integeranswerstodivisionbyexpressingresultsindifferentwaysaccordingtothecontext,includingwithremainders,as fractions,as decimals orbyrounding (e.g.98 ÷4 =24 r2=241/2=24.5≈25).
  • Solveproblemsinvolvingcombinationsofallfour operations,includingunderstandingoftheequalssignandincluding division for scalingbydifferentfractionsandproblemsinvolvingsimplerates.

Year6Divideatleast4digitsbybothsingle-digitand 2-digit numbers(includingdecimal numbersandquantities)

Shortdivision,fordividingbyasingledigit:e.g.6497 ÷ 8

Shortdivisionwithremainders:Pupilsshouldcontinuetousethismethod,butwithnumberstoatleast4 digits, and understandhowtoexpressremainders asfractions,decimals,wholenumberremainders, orroundednumbers.Reallifeproblemsolvingcontexts needtobethestartingpoint,wherepupilshavetoconsiderthemostappropriatewaytoexpresstheremainder.

Calculatingadecimalremainder:Inthisexample,ratherthanexpressingtheremainderas r1,adecimalpointisadded aftertheunitsbecausethereisstillaremainder,and theoneremainderis carriedontozeroesafterthedecimalpoint(toshowthere wasnodecimal valueintheoriginalnumber). Keepdividingtoanappropriatedegree ofaccuracyfor theproblembeingsolved.

Mustbe

Introduce long divisionby chunking for dividingby 2digits.

  • Findout ‘Howmany36sarein972?’by

alignedin place value for subtracting.

subtracting‘chunks’of36,untilzeroisreached(oruntilthereisaremainder).

  • Teachpupilstowritea‘usefullist‘firstat

thesidethatwillhelpthemdecidewhat chunkstouse,e.g.:

‘Useful‘list: 1x=36

10x=360

100x=3600

  • Introducethemethodinasimplewayby

limitingthechoiceofchunksto ‘Canweuse10lots?’ ‘Canwe use100lots?’Aschildren becomeconfidentwiththeprocess,encouragemoreefficientchunkstogettotheanswermorequickly(e.g.20x,5x),and expandontheir‘useful’lists.

Whereremainders

occur, pupilsshould expressthem as fractions,decimalsor use rounding,depend- ingupon theproblem.

Estimate, Calculate, Check!

KeyVocabulary:Aspreviously,commonfactor

KeynumberskillsneededfordivisionatY6:

  • Recallandusemultiplicationanddivisionfactsforallnumbersto12x12formorecomplexcalculations.
  • Dividenumbersupto4digitsbyatwo-digitwholenumberusingtheformalwrittenmethodoflongdivision,andinterpretremaindersaswholenumberremainders,fractions,orbyrounding,asappropriatefor thecontext.Useshortdivisionwhereappropriate.
  • Performmentalcalculations,includingwithmixedoperationsandlargenumbers.
  • Identifycommonfactors,commonmultiplesandprimenumbers.
  • Solveproblemsinvolvingall4operations.
  • Useestimationtocheckanswerstocalculationsanddetermineaccuracy,inthecontextofaproblem.
  • Usewrittendivisionmethodsincaseswheretheanswerhasuptotwodecimalplaces.
  • Solveproblemswhichrequireanswerstoberoundedtospecifieddegreesofaccuracy.