ProgressioninCalculationsJune2016

AyresomePrimarySchool

Elements of Number Early Learning Goal

(ELG)

Progression Notes

1. Children count reliably with numbers from 1 to 20.

Uses some number names accurately in play. Recites numbers in order to 10. Counts up to 3 or 4 objects saying one number name for each item.

Counts out up to six objects from a larger group. Counts objects or actions which cannot be moved. Counts an irregular arrangement of up to 10 objects. Children count reliably with numbers from 1 to 20.

Children need to see and hear numbers being used in their learning environment. As their knowledge of number increases they will begin to recognise that numbers identify how many objects are in a set.

One, two, three, four… there are four teddies.

EarlyLearningGoalChildrencountreliablywithnumbersfromoneto20,placetheminorderandsaywhichnumberisonemoreoronelessthanagivennumber.Usingquantitiesandobjects,theyaddandsubtracttwosingle-digitnumbersandcountonorbacktofindtheanswer.Theysolveproblems,includingdoubling,halvingandsharing.

Using number names beyond 20.

2. They place them in order.

Shows an interest in numerals in the environment.

Shows an interest in representing numbers. Recognise some numerals of personal significance. Recognise and order numbers to 5. Recognise and order numbers to 10. Recognise and order numbers to 20.

Note: Numicon supports number recognition, ordering and counting.

3. They say which number is one more or one less than a given number.

One more Saying one more using songs and rhymes. Continue to count on in 1’s by saying the next number.

Note: Find 1 more/less up to 5, then 10, then 20 and

11p

I am three!

5, 6, 7, 8…

Using equipment (including objects and fingers) ask the children to say one more. Using groups of objects, ask the children one more.

One less Saying one less using songs and rhymes. Continue to count back in 1’s by saying the next number. Using groups of objects, ask the child to find one more.

beyond. Initially the children may need a prompt to know that “one less” is

9, 8, 7, 6…

If I give you one more, how many will

you have?

If I give you one less, how many will

you have?

Using equipment (including objects and fingers) ask the children to say one less.

“take one away.”

4. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer.

Addition Addition songs and rhymes. e.g 1, 2, 3, 4, 5 Once I caught a fish alive and makes 5 At a party, I eat 2 cakes and my friends eat 3. How many do we eat altogether?

Count out 3 apples. Count out 2 apples. How many apples all together? Once confident children will be encouraged to draw simple pictured to help them combine sets without objects.

Addition vocabulary is introduced: add addition plus and count on more sum total altogether increase Early addition is about combining 2 sets of objects physically.

Children will then be introduced to the formal method of recording addition. 2+3=5 Count along a number line to add numbers together.

Subtraction Subtraction songs and rhymes. e.g. 5 little speckled frogs Begin to relate subtraction to “taking away”.

Children are encourages to “one touch one count.” Children are encouraged to use a large number line counting on in 1’s, often using a finger to mark each jump. Initially this method would be used alongside previous methods until children are confident in using a number line. Subtraction

3 presents take away 2 presents leaves 1 present.

Children will then be introduced to the formal method of recording subtraction. 3-2=1 Using a number line to subtract. Start on 3 count back 2 jumps.

vocabulary is introduced: subtract take away minus count back less fewer difference between left/leaves Early subtraction is about physically removing objects.

5. They solve problems, including doubling, halving and sharing.

Doubling (Multiplication) Counting songs and rhymes. Counting on in 2s, 5s and 10s

2 4 6 etc…

Multiplication vocabulary is introduced: lots of groups of times multiply multiplication multiple

5 10 15 20

10 20 30 Counting in 2’s along a number line or track. Adding in groups.

once, twice, three times double At this early stage children are counting objects or pictures. They are also being introduced to the language of multiplication. It is important children have the opportunity to count repeatedly in groups of the same size. Children physically make sets or groups and then add them by counting up from 1 until all

Thereare2cakesinapack.Howmaycakesin3

packs?

Halving and Sharing (Division) Number songs and rhymes. Sharing games and activities, including: Sharing fruit Teddy Bear’s Picnic Grouping activities, including: Asking the children to get into groups (pairs etc…)

the objects have been used. Division vocabulary is introduced: lots of groups of share group halve half divide division divided by remainder Visual images for division are important. Sorting objects and people into groups is the main skill to be taught.

Correspondingcriteriatoindicatereadinesstolearnwrittenmethodsformultiplicationanddivisionare:

Ø Dothechildrenunderstandmultiplicationasrepeatedadditionanddivisionasrepeatedsubtraction?Ø Dotheyunderstandthecorrectvocabularyformultiplicationanddivision?Ø Dothechildrenknowthe2,3,4,5and10timestablesandcorrespondingdivisionfacts?Ø Dotheyknowtheresultofmultiplyingby0or1?Ø Dotheyunderstandplacevalue?Ø Dotheyunderstand0asaplaceholder?Ø Cantheymultiply2and3digitsmentallyby10and100?Ø Cantheyusetheirknowledgeofallthemultiplicationtablestoapproximate?Ø Cantheyfindproductsusingmultiplesof10?Ø Dotheyusethecommutativeandassociativelawsofmultiplication?Ø Cantheyhalveanddouble2digitnumbersmentally?Ø Cantheyusemultiplicationfactstoderivementally,othermultiplicationfactstheydon’tknow?Ø Cantheyexplaintheirmentalstrategiesorallyandrecordthemusinginformaljottings?

Addition

+

Ashortlistofcriteriainreadinessforwrittenmethodsofadditionandsubtractionwouldinclude:

Ø Canchildrencountforwardsandbackwardsinsequence?Ø Dotheyunderstandthecorrectvocabularyforadditionandsubtraction?Ø Dochildrenknowadditionandsubtractionfactsto20?Ø Dotheyunderstandplacevalueandcantheypartitionnumbersintohundreds,tensandunits?Ø Dotheyuseandapplythecommutativeandassociativelawsofaddition?Ø Cantheyaddatleastthree1-digitnumbersmentally?Ø Cantheyaddandsubtractanypairof2-digitnumbersmentally?Ø Cantheyexplaintheirmentalstrategiesorallyandrecordthemusinginformaljottings?

ProgressioninmentalCalculationsforAddition

MentalCalculationsTheseareaselectionofmentalcalculationstrategies:Mentalrecallofnumberbonds6+4=10 +=1025+75+100 19+ =20 +=15Usingneardoubles6+7=double6+1=13Additionusingpartitioningandrecombining34+45=(30+40)+(4+5)=79Countingonorbackinrepeatedstepsof1,10,100,100086+57=143(bycountingonintensandtheninones)460–300=160(bycountingbackinhundreds)Addthenearestmultipleof10,100and1000andadjust24+19=24+20–1=43458+71=458+70+1=529Usetherelationshipbetweenadditionandsubtraction36+19=55 19+36=55

55–19=36 55–36=19Countingonorbackinfractionalstepsappropriatetostage⅕,⅖,⅗,⅘,1,1⅕,1⅖etc.Alsopractisewithpercentagesanddecimalsrelatedtofractions.

Mentalcalculationstrategieswillcontinuetobeused.Theyarenotreplacedbywrittenmethods.

Year1Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line Key skills for addition at Y1: · Read and write numbers to 100 in numerals, incl. 1—20 in words · Recall bonds to 10 and 20, and addition facts within 20 · Count to and across 100 · Count in multiples of 1 2, 5 and 10 · Solve simple 1-step problems involving addition, using objects, number lines and pictorial representations. Mental

* Addandsubtractone-digitandtwo-digitnumbersto20,includingzero

Children are encouraged to develop amental picture of the number system in their heads to use for calculation. They developways ofrecordingcalculationsusingpictures,etc.

Storyof6(Allstorynumbersupto10)

Theyusenumberedlinesandpracticalresourcestosupportcalculationandteachersdemonstratetheuseofthenumberline.3+2=5Childrenthenbegintousenumberedlinestosupporttheirowncalculationsusinganumberedlinetocountoninones.Increasethenumbers–lessthan100.

2and4 3and3 4and2

0and61and5 5and1

0 1 2 3 4 5 6 7 8 9

+1 +1

8+5=13

0123456789101112131415

+1+1+1+1+1

Objective and Strategies

Concrete Pictorial Abstract

Combining two parts to make a whole: part- whole model

4 + 3 = 7 10= 6 + 4

Starting at the bigger number and counting on

Start with the larger number on the bead string and then count on to the smaller number 1 by 1 to find the answer.

12 + 5 = 17

Start at the larger number on the number line and count on in ones or in one jump to find the answer.

5 + 12 = 17 Place the larger number in your head and count on the smaller number to find your answer.

8 1

5

3

Usecubestoaddtwonumberstogetherasagrouporinabar.

Usepicturestoaddtwonumberstogetherasagrouporinabar. Usethepart-part

wholediagramasshownabovetomoveintotheabstract.

Regrouping to make 10.

6 + 5 = 11 Start with the bigger number and use the smaller number to make 10.

Use pictures or a number line. Regroup or partition the smaller number to make 10.

7 + 4= 11 If I am at seven, how many more do I need to make 10. How many more do I add on now?

Year2

Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, addition, column, tens boundary Key skills for addition at Y2:·Add a 2-digit number and ones (e.g. 27 + 6) · Add a 2-digit number and tens (e.g. 23 + 40) · Add pairs of 2-digit numbers (e.g. 35 + 47) · Add three single-digit numbers (e.g. 5 + 9 + 7) · Show that adding can be done in any order (the commutative law). · Recall bonds to 20 and bonds of tens to 100 (30 + 70 etc.) · Count in steps of 2, 3 and 5 and count in tens from any number. · Understand the place value of 2-digit numbers (tens and ones) · Compare and order numbers to 100 using < > and = signs. · Read and write numbers to at least 100 in numerals and words. · Solve problems with addition, using concrete objects, pictorial representations, involving numbers, quantities and measures, and applying mental and written methods. Mental

Addandsubtractnumbersusingconcreteobjects,pictorialrepresentations,andmentally,including:* atwo-digitnumberandones* atwo-digitnumberandtens* twotwo-digitnumbers

* addingthreeone-digitnumbersWrittenNonewwrittenmethodstobetaughtinYear2

Concrete Pictorial Abstract

Adding three single digits

4 + 7 + 6= 17 Put 4 and 6 together to make 10. Add on 7.

Following on from making 10, make 10 with 2 of the digits (if possible) then add on the third digit.

Combine the two numbers that make 10 and then add on the remainder.

Adding 2 digits and single digit

27+8= 35 Image!!!!

Image!!!

Addtogetherthreegroupsofobjects.Drawapicturetorecombinethegroupstomake10.

+ +

+

Adding 2 digits and 2

24 + 15= Add together the ones first then add the tens. Use the Base 10 blocks first before moving onto place value counters.

After practically using the base 10 blocks and place value counters, children can draw the counters to help them to solve additions. T O

Usingnumberlines

Ø Firstcountingonintensandones.34+23=57Thenhelpingchildrentobecomemoreefficientbyaddingtheunitsinonejump(byusingtheknownfact4+3=734+23=57

+10+10 +1+1+1

34 44 54555657

+10 +3+10

Followedbyaddingthetensinonejumpandtheunitsinonejump.34+23=57

34 54 57

34 44 54 57

+20 +3

Year3

Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, vertical, ‗carry‘, expanded, compact Key skills for addition at Y3:·Read and write numbers to 1000 in numerals and words. · Add 2-digit numbers mentally, incl. those exceeding 100. · Add a three-digit number and ones mentally (175 + 8) · Add a three-digit number and tens mentally (249 + 50) · Add a three-digit number and hundreds mentally (381 + 400) · Estimate answers to calculations, using inverse to check answers. · Solve problems, including missing number problems, using · number facts, place value, and more complex addition. · Recognise place value of each digit in 3-digit numbers (hundreds, tens, ones.) · Continue to practise a wide range of mental addition strategies, ie. number bonds, adding the nearest multiple of 10, 100, 1000 and adjusting, using near doubles, partitioning and recombining. Mental

Addandsubtractnumbersmentally,including:* athree-digitnumberandones* athree-digitnumberandtens* athree-digitnumberandhundredsWritten

• Addandsubtractnumberswithuptothreedigits,usingformalwrittenmethodsofcolumnaradditionandsubtraction.

Year3Childrenwillcontinuetouseemptynumberlineswithincreasinglylargenumbers,includingcompensationwhereappropriate.

Ø Countonfromthelargestnumberirrespectiveoftheorderofthecalculation.38+86=124

Ø Compensation49+73=122

73 124123

86 116 120 124

+30+4 +4

+50 -1

Adding 2 digits and 2

24 + 15= Add together the ones first then add the tens. Use the Base 10 blocks first before moving onto place value counters.

After practically using the base 10 blocks and place value counters, children can draw the counters to help them to solve additions. T O

Column method- regrouping

Make both numbers on a place value grid.

Add up the units and exchange 10 ones for one 10.

Add up the rest of the columns,

Children can draw a pictoral representation of the columns and place value counters to further support their learning and understanding.

As the children move on, introduce decimals with the same number of decimal places and different. Money can be used here.

exchanging the 10 counters from one column for the next place value column until every column has been added. This can also be done with Base 10 to help children clearly see that 10 ones equal 1 ten and 10 tens equal 100 and Numicon pegs. As children move on to decimals, money and decimal place value counters can be used to support learning.

Examplesof3digitplus2digitaddition 625 783 367+48 +42 +85673 825 4521111Usingsimilarmethods,childrenwill:

Ø Addseveralnumberswithdifferentnumbersofdigits(uptothreedigits)Ø Begintoaddtwoormorethree-digitsumsofmoney,withorwithoutadjustmentfromthepencetothepoundsØ Knowthatthedecimalpointsshouldlineupundereachother,particularlywhenaddingorsubtractingmixedamounts,eg£3.59+78p.

Year 4

Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, vertical, „carry‟, expanded, compact, thousands, hundreds, digits, inverse Key skills for addition at Y4:·Select most appropriate method: mental, jottings or written and explain why. · Recognise the place value of each digit in a four-digit number. · Round any number to the nearest 10, 100 or 1000. · Estimate and use inverse operations to check answers. · Solve 2-step problems in context, deciding which operations and methods to use and why. · Find 1000 more or less than a given number. · Continue to practise a wide range of mental addition strategies, ie. number bonds, add the nearest multiple of 10, 100, 1000 and adjust, use near doubles, partitioning and recombining. · Add numbers with up to 4 digits using the formal written method of column addition · Solve 2-step problems in contexts, deciding which operations and methods to use and why. · Estimate and use inverse operations to check answers to a calculation. MentalContinuetodevelopskillsfromyears1,2and3Written

Addandsubtractnumberswithupto4digitsusingtheformalwrittenmethodsofcolumnaradditionandsubtractionwhereappropriate

PleasemakereferencetoYear3concretemodelsandpictorialrepresentationstosupportunderstanding.

Childrenshouldextendtheformalwrittenmethod(whereappropriate)tonumberswithatleastfourdigits587 3587+475 +6751062 426211 111Usingsimilarmethods,childrenwill:

Ø AddseveralnumberswithdifferentnumbersofdigitsØ BegintoaddtwoormoredecimalfractionswithuptothreedigitsandthesamenumberofdecimalplacesØ Knowthatdecimalpointsshouldlineupundereachother,particularlywhenaddingorsubtractingmixedamountseg3.2m–280cm.

Year5

Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, „carry‟, expanded, compact, vertical, thousands, hundreds, digits, inverse & decimal places, decimal point, tenths, hundredths, thousandths Key skills for addition at year 5:·Add numbers mentally with increasingly large numbers, using and practising a range of mental strategies ie. add the nearest multiple of 10, 100, 1000 and adjust; use near doubles, inverse, partitioning and re-combining; using number bonds. · Use rounding to check answers and accuracy. · Solve multi-step problems in contexts, deciding which operations and methods to use and why. · Read, write, order and compare numbers to at least 1 million and determine the value of each digit. · Round any number up to 1, 000, 000 to the nearest 10, 100, 1000, 10,000 and 100,000. · Add numbers with more than 4 digits using formal written method of columnar addition. Mental

addandsubtractnumbersmentallywithincreasinglylargenumbersWrittenaddandsubtractwholenumberswithmorethan4digits,includingusingformalwrittenmethods(columnaradditionandsubtraction)

PleasemakereferencetoYear3concretemodelsandpictorialrepresentationstosupportunderstanding.

Childrenshouldextendthecarryingmethodtonumberwithanynumberofdigits.7648 6584 42+1141816 +15181418 64329134 12432 786 3 +4162811 119447648 6584 42+1486 +5848 64329134 12432 786111 1111 3 +4681 11944 121

Usingsimilarmethods,childrenwill

Ø AddseveralnumberswithdifferentnumbersofdigitsØ BegintoaddtwoormoredecimalfractionswithuptofourdigitsandeitheroneortwodecimalplacesØ Knowthatdecimalpointsshouldlineupundereachother,particularlywhenaddingorsubtractingmixedamountseg401.2+26.85+

0.71

Year6

Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, „carry‟, expanded, compact, vertical, thousands, hundreds, digits, inverse, decimal places, decimal point, tenths, hundredths, thousandths Key skills for addition at Y6:·Perform mental calculations, including with mixed operations and large numbers, using and practising a range of mental strategies. · Solve multi-step problems in context, deciding which operations and methods to use and why. · Use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy. · Read, write, order and compare numbers up to 10 million and determine the value of each digit. · Round any whole number to a required degree of accuracy. · Pupils understand how to add mentally with larger numbers and calculations of increasing complexity. Adding several numbers with different numbers of decimal places (including money and measures): · Tenths, hundredths and thousandths should be correctly aligned, with the decimal point lined up vertically including in the answer row. Adding several numbers with more than 4 digits. Empty decimal places should be filled with zero to show the place value in each column Mental Performmentalcalculations,includingwithmixedoperationsandlargenumbers

PleasemakereferencetoYear3concretemodelsandpictorialrepresentationstosupportunderstanding.

InsertexemplificationfromncadditionandsubtractionSnapshotadditionmethods

Subtraction

-

ProgressioninCalculationsforSubtraction

MentalCalculations(ongoing)Theseareaselectionofmentalcalculationstrategies:Mentalrecallofadditionandsubtractionfacts10–6=4 17-=1120-17=3 10- =2Findasmalldifferencebycountingup82–79=3Countingonorbackinrepeatedstepsof1,10,100,100086-52=34(bycountingbackintensandtheninones)460–300=160(bycountingbackinhundreds)Subtractthenearestmultipleof10,100and1000andadjust24-19=24-20+1=5458-71=458-70-1=387Usetherelationshipbetweenadditionandsubtraction36+19=55 19+36=5555–19=36 55–36=19

Mentalcalculationstrategieswillcontinuetobeused.Theyarenotreplacedbywrittenmethods.

Year1VocabKeyskillsMentalWrittenYear1

Children are encouraged to develop amental picture of the number system in their heads to use for calculation. They developways ofrecordingcalculationsusingpicturesetc.Childrenunderstand‘subtraction’as‘takingaway’andfindingouthowmanyareleft.Theyusenumberlinesandpracticalresources(egcubesandNumicon)tosupportcalculation.Teachersdemonstratetheuseofthenumberlineandchildrenusethem.6-3=3Thenumberlineshouldalsobeusedtoshowthat6–3meansthedifferencebetween6and3orthedifferencebetween3and6andhowmanyjumpstheyareapart.Countonandbackfrom3to6andbackfrom6to3+1+1+1

0 1 2 3 4 5 6 7 8 9 10

-1-1 -1

0 1 2 3 4 5 6 7 8 9 10

Childrenthenbegintousenumberedlinestosupporttheirowncalculations–usinganumberedlinetocountbackinones.13–5=8

Taking away ones

Use physical objects, counters, cubes etc to show how objects can be taken away. 6 – 2 = 4

Cross out drawn objects to show what has been taken away.

18 -3= 15 8 – 2 = 6

Counting back Make the larger number in your subtraction. Move the beads along your bead string as you count backwards in ones. 13 – 4

Count back on a number line or number track

Start at the bigger number and count back the smaller

Start at 13 then count back 4. What number are you at? .

0123456789101112131415

-1-1-1-1-1

Use counters and move them away from the group as you take them away counting backwards as you go.

number showing the jumps on the number line.

This can progress all the way to counting back using two 2 digit numbers.

Part Part Whole Model

Link to addition- use the part whole model to help explain the inverse between addition and subtraction.

If 10 is the whole and 6 is one of the parts. What is the other part?

10 - 6 =

Use a pictorial representation of objects to show the part part whole model.

Move to using numbers within the part whole model.

Make 10

14 – 9 =

16 – 8=

105

Make 14 on the ten frame. Take away the four first to make 10 and then takeaway one more so you have taken away 5. You are left with the answer of 9.

Start at 13. Take away 3 to reach 10. Then take away the remaining 4 so you have taken away 7 altogether. You have reached your answer.

How many do we take off to reach the next 10?

How many do we have left to take off?

Year2KeyvocabSkillsMentalwritten

Year2Childrenwilluseanumbersquare,Dienes,numberednumberlines,thenastheyprogress,emptynumberlinestosupportcalculations.Theseshouldbedifferentiatedasnecessary.Countingback

Ø Firstcountingbackintensandones.47–23=24

Ø Thenhelpingchildrentobecomemoreefficientbysubtractingtheunitsinonejump(byusingtheknownfact7–3=4).47–23=24

-1-1-1

24252627 37 47

-10 -10

24 27 37 47

-3 -10 -10

Ø Subtractingthetensinonejumpandtheunitsinonejump.47–23=24

Ø Bridgingthroughtencanhelpchildrenbecomemoreefficient(higherlevel2/3).42–25=17CountingonIfthenumbersinvolvedinthecalculationareclosetogetherorneartomultiplesof10,100,etc,itcanbemoreefficienttocounton.

24 27 47

-3 -20

17 20 22 42

-20

-3 -2

Countupfrom47to82injumpsof10andjumpsof1.Thenumberlineshouldstillshow0sochildrencancrossoutthesectionfrom0tothesmallestnumber.Theythenassociatethismethodwith‘takingaway.’82–47Helpchildrentobecomemoreefficientwithcountingonby:

Ø SubtractingtheunitsinonejumpØ SubtractingthetensinonejumpandtheunitsinonejumpØ Bridgingthroughten.

Part Part Whole Model

Link to addition- use the part whole model to help explain the inverse between addition and subtraction.

If 10 is the whole and 6 is one of the parts. What is the other part?

10 - 6 =

Use a pictorial representation of objects to show the part part whole model.

Move to using numbers within the part whole model.

0 47484950 60 70 808182

+1 +1 +1 +1+1+10 +10+10

105

Find the difference

Compare amounts and objects to find the difference.

Use cubes to build towers or make bars to find the difference

Use basic bar models with items to find the difference

Count on to find the difference.

Draw bars to find the difference between 2 numbers.

Hannah has 23 sandwiches, Helen has 15 sandwiches. Find the difference between the number of sandwiches.

Year3VocabKeyskillsMental

addandsubtractnumbersmentally,including:* athree-digitnumberandones* athree-digitnumberandtens* athree-digitnumberandhundredsWritten

Year3

Column method without regrouping

Use Base 10 to make the bigger number then take the smaller number away.

Show how you partition numbers to subtract. Again make the larger number first.

Draw the Base 10 or place value counters alongside the written calculation to help to show working.

This will lead to a clear written column subtraction.

Column method with regrouping

Use Base 10 to start with before moving on to place value counters. Start with one exchange before moving onto subtractions with 2 exchanges. Make the larger number with the place value counters

Start with the ones, can I take away 8

Draw the counters onto a place value grid and show what you have taken away by crossing the counters out as well as clearly showing the exchanges you make.

When confident, children can find their own way to record the exchange/regrouping.

Children can start their formal written method by

from 4 easily? I need to exchange one of my tens for ten ones.

Now I can subtract my ones.

Now look at the tens, can I take away 8 tens easily? I need to exchange one hundred for ten tens.

Now I can take away eight tens and complete my subtraction

Just writing the numbers as shown here shows that the child understands the method and knows when to exchange/regroup.

partitioning the number into clear place value columns. Moving forward the children use a more compact method. This will lead to an understanding of subtracting any number including decimals.

Show children how the concrete method links to the written method alongside your working. Cross out the numbers when exchanging and show where we write our new amount.

Year4VocabularyKeyskillsMental

Continue to addandsubtractnumbersmentally,including:* athree-digitnumberandones* athree-digitnumberandtens* athree-digitnumberandhundredsWritten

addandsubtractnumberswithupto4digitsusingtheformalwrittenmethodsofcolumnaradditionandsubtractionwhereappropriate

Year4

Column method without regrouping

Use Base 10 to make the bigger number then take the smaller number away.

Show how you partition numbers to subtract. Again make the larger number first.

Draw the Base 10 or place value counters alongside the written calculation to help to show working.

This will lead to a clear written column subtraction.

Column method with regrouping

Use Base 10 to start with before moving on to place value counters. Start with one exchange before moving onto subtractions with 2 exchanges. Make the larger number with the place value counters

Start with the ones, can I take away 8 from 4 easily? I need to exchange one of my tens for ten ones.

Draw the counters onto a place value grid and show what you have taken away by crossing the counters out as well as clearly showing the exchanges you make.

When confident, children can find their own way to record the exchange/regrouping.

Children can start their formal written method by

partitioning the number into

Now I can subtract my ones.

Now look at the tens, can I take away 8 tens easily? I need to exchange one hundred for ten tens.

Now I can take away eight tens and complete my subtraction

Just writing the numbers as shown here shows that the child understands the method and knows when to exchange/regroup.

clear place value columns. Moving forward the children use a more compact method. This will lead to an understanding of subtracting any number including decimals.

Show children how the concrete method links to the written method alongside your working. Cross out the numbers when exchanging and show where we write our new amount.

Year5/6VocabularyKeyskillsMentalWrittenContuihetoiusemanipulativetodevelopundrersrtanding

Multiplication

x

ProgressioninCalculationsforMultiplication

MentalCalculations(ongoing)Theseareaselectionofmentalcalculationstrategies:DoublingandhalvingApplyingtheknowledgeofdoublesandhalvestoknowfacts.Eg8x4isdouble4x4UsingmultiplicationfactsTablesshouldbetaughteverydayfromY2onwards,eitheraspartofthementaloralstarterorothertimesasappropriatewithintheday.UsingandapplyingdivisionfactsChildrenshouldbeabletoutilisetheirtimestableknowledgetoderiveotherfacts.EgIfIknow3x7=21,whatelsedoIknow?30x7=210,300x7=2100,3000x7=21000,0.3x7=2.1etc.Usecloselyrelatedfactsalreadyknown13x11=(13x10)+(13x1) =130+13 =143Multiplyingby10or100Knowingthattheeffectofmultiplyingby10isashiftinthedigitsoneplacetotheleft.Knowingthattheeffectofmultiplyingby100isashiftinthedigitstwoplacestotheleft.

Partitioning23x4=(20x4)+(3x4)=80+12=92Useoffactors8x12=8x4x3

Mentalcalculationstrategieswillcontinuetobeused.Theyarenotreplacedbywrittenmethods.

Year1

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count Key skills for multiplication at Y1: Count in multiples of 2, 5 and 10. Solve one-step problems involving multiplication, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher. Make connections between arrays, number patterns, and counting in twos, fives and tens. Begintounderstanddoublingusingconcreteobjectsandpictorialrepresentations.Mental

countinmultiplesoftwos,fivesandtensWritten

Year1Childrenwillexperienceequalgroupsofobjectsandwillcount in2sand10sand5s. Theywillworkonpracticalproblemsolvingactivitiesinvolvingequalsetsorgroups.

2 4 6etc…

5 10 15 20

10 20 30

Counting in multiples

Count in multiples supported by concrete objects in equal groups.

Use a number line or pictures to continue support in counting in multiples.

Count in multiples of a number aloud. Write sequences with multiples of numbers. 2, 4, 6, 8, 10 5, 10, 15, 20, 25 , 30 10s

Repeated addition

Write addition sentences to describe objects and pictures.

Usedifferentobjectstoaddequalgroups.

Year2

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times... Key skills for multiplication at Y2: · Count in steps of 2, 3 and 5 from zero, and in 10s from any number. · Recall and use multiplication facts from the 2, 5 and 10 multiplication tables, including recognising odds and evens. · Write and calculate number statements using the x and = signs. · Show that multiplication can be done in any order (commutative). · Solve a range of problems involving multiplication, using concrete objects, arrays, repeated addition, mental methods, and multiplication facts. · Pupils use a variety of language to discuss and describe multiplication. MentalWritten

Childrenwilldeveloptheirunderstandingofmultiplicationandusejottingstosupportcalculation:

Ø Repeatedaddition3times5 is 5+5+5=15 or 3lotsof5 or 5x3Repeatedadditioncanbeshowneasilyonanumberline:3x5=5+5+5

Ø CommutativityChildrenshouldknowthat3x5hasthesameansweras5x3.Thiscanalsobeshownonthenumberline.

0123456789101112131415

1

1

1x5

1x3

1

1

1

1

0123456789101112131415

1

1

1x5

Ø ArraysChildrenshouldbeabletomodelamultiplicationcalculationusinganarray. Thisknowledgewillsupportwiththedevelopmentofthegridmethod.Childrenshoulduseemptynumberlinestosupporttheirunderstanding.4x6=

5x3=15

3x5=15

0 6 12 18 24

1

1

1

1

Arrays- showing commutative multiplication

Create arrays using counters/ cubes to show multiplication sentences.

Draw arrays in different rotations to find commutative multiplication sentences.

Link arrays to area of rectangles.

Use an array to write multiplication sentences and reinforce repeated addition.

Year3

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated ad-dition, column, row, commutative, sets of, equal groups, times, _times as big as, once, twice, three times..., partition, grid method, multiple, product, tens, units, value Key skills for multiplication: · Recall and use multiplication facts for the 2, 3, 4, 5, 8 and 10 multiplication tables, and multiply multiples of 10. · Write and calculate number statements using the multiplication tables they know, including 2-digit x single digit, drawing upon mental methods, and progressing to reliable written methods. · Solve multiplication problems, including missing number problems. · Develop mental strategies using commutativity (e.g. 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240) · Solve simple problems in contexts, deciding which operations and methods to use. · Develop efficient mental methods to solve a range of problems e.g using commutativity (4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and for missing number problems ? x5=20,3x?=18,?x?=32MentalWritten

Year3Gridmethodasaninterimsteptotheformalwrittenmethod.

Grid Method Show the link with arrays to first introduce the grid method.

4 rows of 10 4 rows of 3

Move on to using Base 10 to move towards a more compact method.

4 rows of 13

Move on to place value counters to show how we are finding groups of a number.We are multiplying by 4 so we need 4 rows.

Children can represent the work they have done with place value counters in a way that they understand. They can draw the counters, using colours to show different amounts or just use circles in the different columns to show their thinking as shown below.

Start with multiplying by one digit numbers and showing the clear addition alongside the grid.

Move 210 +25 =245 to vertical addition Moving forward, multiply by a 2 digit number showing the different rows within the grid method.

Fill each row with 126.

Add up each column, starting with the ones making any exchanges needed.

Then you have your answer.

Year4

Vocab Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, groups of, sets of, lots of, equal groups, times, multiply, times as big as, once, twice, three times... partition, grid method, total, multiple, product, sets of, inverse Key skills for multiplication at Y4: · Count in multiples of 6, 7, 9, 25 and 1000 · Recall multiplication facts for all multiplication tables up to 12 x 12. · Recognise place value of digits in up to 4-digit numbers · Use place value, known facts and derived facts to multiply mentally, e.g. multiply by 1, 10, 100, by 0, or to multiply 3 numbers. · Use commutativity and other strategies mentally 3 x 6 = 6 x 3 , 2 x 6 x 5 = 10 x 6 , 39x7 = 30 x 7 + 9 x 7.

· Solve problems with increasingly complex multiplication in a range of contexts. · Count in multiples of 6, 7, 9, 25 and 1000 · Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and units) multiplytwo-digitandthree-digitnumbersbyaone-digitnumberusingformalwrittenlayout

Progressingfromgridmethodtoformalwrittenmethod

Grid Method Show the link with arrays to first introduce the grid method.

4 rows of 10 4 rows of 3

Move on to using Base 10 to move towards a more compact method.

4 rows of 13

Move on to place value counters to show how we are finding groups of a number.We are multiplying by 4 so we need 4 rows.

Fill each row with 126.

Add up each column, starting with the

Children can represent the work they have done with place value counters in a way that they understand. They can draw the counters, using colours to show different amounts or just use circles in the different columns to show their thinking as shown below.

Start with multiplying by one digit numbers and showing the clear addition alongside the grid.

ones making any exchanges needed.

Then you have your answer.

Column multiplication

2by1digit

Dotscountersplacevalue

Up to 3 by 1 digitexamplewritten

Graemeexample…..

written progression 123

Year5

Key vocabulary groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, _times as big as, once, twice, three times..., partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short/long multi-plication, ‗carry‘ Key skills for multiplication at Y5: Identify multiples and factors, using knowledge of multiplication tables to 12x12. Solve problems where larger numbers are decomposed into their factors Multiply and divide integers and decimals by 10, 100 and 1000 Recognise and use square and cube numbers and their notation Solveproblemsinvolvingcombinationsofoperations,choosingandusingcalculationsandmethodsappropriately.

Copyfromyear4Asyear4upto4digitsby2digits

Bytheendofyear6,childrenwillhavearangeofcalculationmethods,mentalandwritten.Selectionwilldependuponthenumbersinvolved.Childrenshouldnotbemadetogoontothenextstageif:

1. Theyarenotready.2. Theyarenotconfident.

Childrenshouldbeencouragedtoapproximatetheiranswersbeforecalculating.Childrenshouldbeencouragedtochecktheiranswersaftercalculationusinganappropriatestrategy.Childrenshouldbeencouragedtoconsiderifamentalcalculationwouldbeappropriatebeforeusingwrittenmethods.

Year6

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times... partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short / long multiplication, „carry‟, tenths, hundredths, decimal Key skills for multiplication at Y6: · Recall multiplication facts for all times tables up to 12 x 12 (as Y4 and Y5). · Multiply multi-digit numbers, up to 4-digit x 2-digit using long multiplication. · Perform mental calculations with mixed operations and large numbers. · Solve multi-step problems in a range of contexts, choosing appropriate combinations of operations and methods. · Estimate answers using round and approximation and determine

Division

÷

ProgressioninCalculationsforDivision

MentalCalculations(ongoing)Theseareaselectionofmentalcalculationstrategies:DoublingandhalvingKnowingthathalvingisdividingby2.DerivingandrecallingdivisionfactsTablesshouldbetaughteverydayfromY2onwards,eitheraspartofthementaloralstarterorothertimesasappropriatewithintheday.UsingandapplyingdivisionfactsChildrenshouldbeabletoutilisetheirtimestableknowledgetoderiveotherfacts.EgIfIknow3x7=21,whatelsedoIknow?30x7=210,300x7=2100,3000x7=21000,0.3x7=2.1etc.Dividingby10or100Knowingthattheeffectofdividingby10isashiftinthedigitsoneplacetotheright.Knowingthattheeffectofdividingby100isashiftinthedigitstwoplacestotheright.Useoffactors378÷21 378÷3=126 378÷21=18 126÷7=18UserelatedfactsGiventhat1.4x1.1=1.54

Whatis1.54÷1.4or1.54÷1.1?

Mentalcalculationstrategieswillcontinuetobeused.Theyarenotreplacedbywrittenmethods.

Year1Childrenwillunderstandequalgroupsandshareitemsoutinplayandproblemsolving.Theywillcountin2sand10sandlaterin5s.

Year2Childrenwilldeveloptheirunderstandingofdivisionandusejottingstosupportcalculation.

Ø Sharingequally6chocolatessharedbetween2people,howmanydotheyeachget?

Ø GroupingorrepeatedsubtractionThereare6chocolates,howmanypeoplecanhave2chocolateseach?

Ø Repeatedsubtractionusinganumberedline12÷3=4

Ø Repeatedsubtractionusinganemptynumberline24÷4=6

1

0123456789101112

1x3

1

1(lotof3)

0 4 8 12 16 20 24

1x4

1

1

1(lotof4)

1

1

Year3Childrenshouldalsomoveontocalculationsinvolvingremainders.13÷4=3r1Childrenwilldeveloptheiruseofrepeatedsubtractiontobeabletosubtractmultiplesofthedivisor. Initially,theseshouldbemultiplesof10s,5s,2sand1s–numberswithwhichthechildrenaremorefamiliar.72÷5=14r2Movingonto:

0 1 5 9 13

1

1

1(lotof4)

02712172227323742475257626772

1

1

1(lotof5)

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1

1

1

1

1

1

1

1

1

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r2

10(lotsof5)

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r2

Thenontotheverticalmethod:ShortdivisionTU÷U72÷3 24____ 3)712Answer:24Leadingtosubtractionofothermultiples96÷6 16___ 6)936Answer:16Anyremaindersshouldbeshownasintegerseg14remainder2or14r2.Childrenneedtobeabletodecidewhattodoafterdivisionandroundupordownaccordingly.Theyshouldmakesensibledecisionsaboutroundingupordownafterdivision.Forexample62÷8is7remainder6,butwhethertheanswershouldberoundedupto8orroundeddownto7dependsonthecontext.

027121722

EgIhave62p.Sweetsare8peach.HowmanycanIbuy?Answer:7(theremaining6pisnotenoughtobuyanothersweet).Applesarepackedintoboxesof8.Thereare62apples.Howmanyboxesareneeded?Answer:8(theremaining6applesstillneedtobeplacedintoabox).

Level4ChildrenwillcontinuetousewrittenmethodstosolveshortdivisionTU÷U.Childrencanstarttosubtractlargermultiplesofthedivisor,eg30x.ShortdivisionHTU÷U196÷6 032r4_ 6)1916Anyremaindersshouldbeshownaasintegers,eg14remainder2or14r2.Childrenneedtobeabletodecidewhattodoafterdivisionandroundupordownaccordingly.Theyshouldmakesensibledecisionsaboutroundingupordownafterdivision.Forexample240÷52is4remainder32,butwhethertheanswershouldberoundedupto5orroundeddownto4dependsonthecontext.

Year5ChildrenwillcontinuetousewrittenmethodstosolveshortdivisionTU÷UandHTU÷U.LongdivisionHTU÷TU972÷36 27_ 36)972-360 (36x10) 56112-360 (36x10) 2152-180 (36x5) 72 -72 (36x2) 0Answer:10+10+5+2=27Anyremaindersshouldbeshownasfractionsegifthechildrenweredividing32by10theanswershouldbeshownas32/10whichcouldthenbewrittenas31/5initslowestterms.Extendtodecimalswithuptotwodecimalplaces.Childrenshouldknowthatdecimalpointslineupundereachother.87.5÷7

12.5 7)87.5-70.0 10x 17.5-14.0 2x 3.5-3.5 0.5x 0Answer:12.5or87.5÷7 12.5 7)817.35Answer:12.5

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Bytheendofyear6,childrenwillhavearangeofcalculationmethods,mentalandwritten.Selectionwilldependuponthenumbersinvolved.Childrenshouldnotbemadetogoontothenextstageif:

1. Theyarenotready.2. Theyarenotconfident.

Childrenshouldbeencouragedtoapproximatetheiranswersbeforecalculating.Childrenshouldbeencouragedtochecktheiranswersaftercalculationusinganappropriatestrategy.Childrenshouldbeencouragedtoconsiderifamentalcalculationwouldbeappropriatebeforeusingwrittenmethods.