Mathematics
at
Longparish CE Primary School
A guide to calculation strategies
This booklet is designed to explain how some of the different methods of calculating are taught in school. Many of these may look very different to the traditional methods you may be familiar with, but are taught to facilitate a full understanding for each child.
ADDITION
When children first encounter addition, they will use physical objects. They will begin by counting sets of objects and then count on to combine the number of objects in two sets:
1 2 3 4 5
‘I have 3 apples and 2 apples, I have 5 apples altogether.’
When the child is ready, they will be introduced to recording this as: 3 + 2 = 5 , however it is important that this move is not made too soon. It must also be remembered that the symbols ‘+’ and ‘ = ‘ will be new.
The next step is to represent calculations on a number line:
+1+11 2 3 4 5 6 7 8 9 10
‘I start at 3 and I make 2 more jumps so I reach 5’
Initially, the number line will have digits on so children can visualise where they start at and where they are ‘jumping’ to. The children should always start with the larger number and count on.
When children are secure in this method, they will move onto blank number lines to enable them to work with larger numbers, for example when adding two 2-digit numbers: 28 + 16 =
+10+628 38 44
16 is added on in two steps, first by adding 10, then by adding 6, so reaching 44.
This method greatly assists the more visual learners and is used throughout the school:
+100+30+1
254 354 384385
By adding the hundreds, then the tens, then the units (ones) along the number line, the answer 385 is reached. 254 + 131 = 385
Partitioning
Children will be taught how to partition numbers into parts, e.g. hundreds, tens and units, and when adding, to add the parts and then recombine them to find the total.
45 + 13 =
Partition the numbers into tens and ones (units):
40 + 5 + 10 + 3
Add the tens together:
40 + 10 = 50
Add the units together:
5 + 3 = 8
Recombine the numbers to give the total:
50 + 8 = 58
It is important when adding 45 to 13 to emphasise that it is 4 tens add1 ten which equals 5 tens, the children can then translate this into 50. This will emphasise the place value and aid their understanding when moving to more formal methods.
This knowledge of partitioning can then be used in a vertical calculation, as shown in the example below with 67 + 24. As above, initially the largest part of the numbers will be added first, moving through to the smallest parts. The individual answers are then added to give the final total.
60 7 200 30 6
+20 4 + 100 40 7
80 + 11 = 91 300 + 70 + 13 = 373
In order to eventually lead to a compact method, the children need to move to adding the units first.
Compact Method
The compact method is more of the ‘traditional method’. It requires the child to have a very secure understanding of place value.
587+475
002
1 / Add the units:
seven add five is twelve.
One ten under the tens column and 2 in the units column.
587
+475
00062
1 1 / Add the tens: eighty add seventy is one hundred and fifty plus 1 ten underneath, is one hundred and sixty. One hundred under the hundreds column and sixty in the tens column.
587
+475
671062
1 1 / Add the hundreds: five hundreds add four hundreds is nine hundreds, plus one hundred underneath is ten hundreds which equals one thousand. One thousand in the thousand column and zero hundreds in the hundreds column.
This compact method can also be used with larger / decimal numbers:
58.7+47.5
.2
1 / Add the tenths:
sevententhsadd five tenthsis twelve tenths, which equals 1.2
One unit under the units column and 2 in the tenths column.
58.7
+47.5
6.2
1 1 / Add the units: eight add seven is fifteen plus 1 unit underneath, is sixteen.
One ten under the tens column and six in the units column.
58.7
+47.5
1 0 6.2
1 1 / Add the tens: five tens add four tens is nine tens, plus one ten underneath is one hundred. One hundred in the hundred column and zero tens in the tens column.
SUBTRACTION
As with addition, early years subtraction begins with physical objects.
‘If I have 5 apples, but I eat 2, how many apples will I have left?’They will physically take away two of the objects and count how many are left.
When the child is ready, they will be introduced to recording this as: 5 – 2 = 3
however it is again important that this move is not made too soon. It must also be remembered that the symbol ‘-’ and will be new.
The next step will be to use number lines to work out subtractions, jumping back from
the starting number to find the answer: ‘I start at 5, jump back 2 and the answer is 3’
-1 -11 2 3 4 5 6 7 8 9 10
Pupils will be taught different vocabulary for subtraction, for example ‘take-away’ and ‘find the difference’. With subtraction sentences involving larger numbers which are close together, e.g. 22 – 18, children will use a number line to count on and thereby ‘find the difference’. ‘I start at 18 and jump on until I reach 22’
+1 +1 +1 +118 19 20 21 22
‘I made four jumps, so 22 is 4 more than 18: 22 – 18 = 4’
This method can also be used with larger numbers such as 970 – 436, when one of the main methods that children are taught again involves finding the difference between the two numbers by counting up from 436 to 970 to see how much bigger 970 is than 436. In order to assist with this method and make it more visual, an ‘empty’ number line may be used. It is easier for children to work around the multiples of 10 and 100 when calculating.
Counting Up
+4 +60 +400 +70
436440 500 900970
First we look at how much we need to add to the starting number (436) to get to the next multiple of 10 (440), then from 440 to the next multiple of 100 (500). We can then add on in 100s to 900(+400) and finally count up to 970 (+70). The amounts needed for each stage are then added together in order to find out how much bigger 970 is than 436: 4 + 60 + 400 + 70 (this can be done in any order, so 6 tens and 7 tens = 13 tens (130), 130 + 400 = 530, 530 + 4 = 534).
As the children progress with this method, they will move onto adding in bigger ‘chunks’.
+64 +470
436500970
This method is favoured by many children and progression will be in the complexity of numbers used:
1234 1300 2000 5000 5678
2.44.0 6.0 6.1
£13.68 £14£20
8.714.24
Another method uses partitioning to subtract, for example, in the number sentence
970 – 436 first the 4 hundreds would be taken away, then the 3 tens and then the 6 units:
970570540534
(-400)(-30)(-6)
A third strategy involves compensation, whereby numbers which are close to a multiple of 10 or 100 or 1000 etc. are rounded up or down for the subtraction and then the amount involved in the rounding is either added or taken away from that answer as appropriate. For example:
84 – 56 = 84 – 60 + 4 = 24 + 4 = 28
The 56 is rounded up to 60 and then taken away, therefore 4 too much has been taken away, so this is then added to the answer.
Children are not required to use the traditional, vertical subtraction method (decomposition); the emphasis is on using an efficient accurate method with understanding e.g. 972 – 143 :
Partition both numbers and create a subtraction sum for each part.
Starting with the units (2 – 3) there are not enough units to subtract 3, therefore exchange one of the tens for 10 units, which leaves 6 tens (60) and makes 12 units. Note that this does not change the overall amount, there is still 972 from which to subtract 143.
6010
900702
-100403
800 + 20 +9 = 829
Then subtract 3 from 12 in the units column. Next subtract 40 from the 60 remaining in the tens column. After this, subtract 100 from 900. Finally add the individual answers together to reach the total.
Children will be introduced to the formal, compact decomposition method in the upper juniors, where it is taught with understanding as opposed to a trick, therefore language such as ‘borrowing’ and ‘paying back’ is not used. There is no pressure for children to adopt this method if they are not clear how it works or are happier with a different effective method.
MULTIPLICATION
Early multiplication skills begin in Reception with counting in steps of two.
Children in Year 1 are encouraged to count in twos, fives and tens and learning and recalling multiplication tables begins in Year 2.
Repeated addition
Early multiplication is repeated addition with the use of physical objects.
‘If I have six teddy bears, how many legs will there be?’
2 / + / 2 / + / 2 / + / 2 / + / 2 / + / 2=12 legs
A strategy to help children learn multiplication table facts from counting is to say or show the child a multiplication fact such as: 6 x 2 = .
Ask the child to put up six fingers and count across the six fingers in twos.
Six lots of 2 is 12.
Also with 7 x 10 = . Ask the child to put up seven fingers and count across the fingers in tens. Seven lots of 10 is 70.
It is important for children to know that 10 x 7 will give the same answer as 7 x 10, let them show this with their fingers.
Arrays
To support this learning and thinking, children are taught arrays. It is important that at this point there is still a link to repeated addition. There should also be an encouragement to learn the multiplication facts.
First group the circles
Then count up as repeated addition
2 + 2 + 2 + 2 =8
2
4
Or group them in the other way. It is important for children to learn that 4x2 =8 is the same calculation as
2x4=8.
Partitioning
Children use partitioning when multiplying large numbers e.g. 38 x 7 =
(30 x 7) + (8 x 7)
First multiply the tens by 7:
30 x 7 = 210 (3 x 7 = 21 and 21 x 10 = 210)
Then multiply the units by 7:
8 x 7 = 56
Then add the two totals together.
38 x 7 = (30x 7) + (8 x 7) = 210 + 56 = 266
Grid Method
When using a written method they will move on to using a grid layout using partitioning as above, as this helps them to ensure that they haven’t missed any parts out.
Thirty-eight is partitioned into tens and units and put into the grid:
30 / 8The multiplying number is then put into the grid:
x / 30 / 87
Multiply each part of the partitioned number by 7 and write the answer in the box underneath each part:
x / 30 / 87 / 210 / 56
Total the numbers and write the answer at the side:
x / 30 / 87 / 210 / 56 / = 266
This can then be extended to larger numbers e.g.356 x 27 =
Partition both numbers into hundreds, tens and units as appropriate, then multiply each part. Then total the rows to find the answers to 20 x 356 and 7 x 356.
Total the final column to get the answer.
x / 300 / 50 / 620 / 6000 / 1000 / 120 / = 7120
7 / 2100 / 350 / 42 / = 2492
9612
Progression will include decimals, for example £3.42 x 6.
The grid method is encouraged at school; the children are shown how this develops into thecompact methods when they are ready.
DIVISION
Early division begins with sharing in practical activities.
Children need to recognise that: 15 ÷ 3 =
can mean 15 shared between 3
or
how many lots of 3 are there in 15.
We can use a number line to find out how many threes there are in fifteen, by counting forwards or backwards in threes.
1 2 3 4 50 3 6 9 12 15
15 ÷ 3 = 5
By Key Stage 2, when appropriate, children will move onto giving a whole-number remainder when one number is divided by another, e.g. 16 ÷ 3 is 5 remainder 1.
As children move through Key Stage 2 they start to use written methods to support their calculations, but will always approximate first. For example, 72 lies between 50 and 100, therefore72÷ 5 lies between (50 ÷ 5 = 10) and (100 ÷ 5 = 20).
72 ÷ 5 or ‘How many lots of 5 are there in 72?’
What do I know from my 5 times Table?
I know 10 lots of 5 are 50, so I can split 72 into 50 + 22 and take off 10 lots of 5.
72-50 / 10 x 5
22
How many lots of 5 are there in 22?
I know 4 lots of 5 are 20, so I can take off 4 lots of 5.
72-50
22 / 10 x 5
-20 / 4 x 5
2
How many are left?
There is a remainder of 2, therefore the answer is 14 lots of 5 remainder 2
72 ÷ 5 = 14 r 2
Progression is again with the numbers involved in the division and by Year 6, children are expected to understand that remainders can be represented in different ways.
For example, in the number sentence: 72 ÷ 5 = 14 remainder 2 , the remainder can be divided by 5 as well and be represented by
Fraction: 2/5
Decimal: 0.4, so 72 ÷ 5 = 14.4
As always, careful thought needs to be given to the answer to ensure it is a ‘sensible’ solution. For example, look at this problem:
‘432 children and adults are going on a school trip. If each bus takes 30 people, how many are needed?‘
As above, children can use their multiplication knowledge to help:
432people going
-30010 x 30 (10 buses would mean 30 people transported)
132people still left
-1204 x 30 (another 4 buses)
12people left
So we see that the calculation would result in:
432 ÷ 30 = 14 remainder 12 or 432 ÷ 30 = 14.4
Neither of these are good answers for this question: either the 12 people left over can’t go or would need to travel in 0.4 of a bus! So we see that 15 buses are needed …. or 14 buses and some cars.
When the children really understand these expanded methods they can be shown how they are developed into a compact method. But as always, the informal written methods are very good ways of working out an answer and the emphasis is on the children using a method they understand, which will then usually be quicker for them and more accurate.
TIMES TABLE SQUARE
If there is any difficulty, please come in to school and ask a teacher for advice on how to help your child with their homework.