Supporting Numeracy – Calculation methods and strategies
1) Mental Addition in KS1 (Year 1 and Year 2)



Build on the fact children can count on in 1s
Begin to count on in 10s using a beaded number line or 100 square using
‘spider’
E.g. 22 = 10, 20 and 2 more

Number bonds are taught – facts to 10 e.g. 8 = 4+4, 5+3, 6+2 … - covered
also in 5 minute starters in lessons to embed. Also use physical resources, e.g.
cubes to reinforce.

Pupils will use Place Value/100 square to add two 2-digit numbers using ‘fly’
-

E.g. 36 + 23
Count on in 10s from 36, then two more in 1s = 59
In Year 2 there is a greater development of mental addition strategies
E.g. 2-digit number + 1 digit number
E.g. 35 + 3
- use our number facts/bonds
- 5+3 = 8, so 35+3 = 38, 25+3=28, 65+3 = 68
They are also learning to add several 1-digit numbers, using number bonds
E.g. 8+7+8
Double 8 =16
16+7=23
E.g. 3+4+7+5
Use number bonds to 10 = 7+3
Use number bond = 5+4 = 9
= 19
Adding of two 2-digit numbers begins which bridges 100
E.g. 76+38
Count on in 10s and then 1s from 76
2) Mental addition in Key Stage 2 (Year 3-6)

Year 3 moves away from counting on in 1s to add
E.g. 77+5 would be tackled using number facts
- 3 more to 77 = 80, 2 more from 80 = 82

Concept of PARTITIONING is introduced in which numbers are split into their values
77+34
= 70 + 30 = 100
= 7 + 4 = 11
=> 111

Development of adding ‘nearly numbers’ by adding multiples of 10 first and adjusting
77 + 49 (49 is nearly 50)
Do 77+50 = 127 first, but you have been too greedy by adding 1 too many so
you need to give to give one back = 126

Into Y5-6 these strategies are extended but number facts and place value are key.
E.g. 7.5 + 0.7
.5 + .7 (we now know that 5+7=12, so this is 1.2) = 8.2
Or
7.5 + 0.5 = 8, add 0.2 (0.5+0.2 is the same as 0.7) = 8.2
3) Written addition in Key Stage 2
Year 1 and 2 is focused on mental addition as this is vitally important to understand
first before introducing written methods. Children need to have a good ability to bond
numbers and understand the value of digits in numbers first.
A) Partitioning method – the Foundation of the written arithmetic
67 +38
= 60 + 30
=7+8
= 90
= 15
= 90+15 = 105
In Y3
267 + 338
= 200+300
60+30
7+8
= 500
= 90
= 15
= 605
B) Expanded method – this makes the place value explicit to children
+
451
487
__________
__________
is 400 50 1
is 400 80 7
___________
800 130 8
___________
= 938
Stress to the children that it is 400 or 50 not 4 or 5
C) Hybrid method
e.g. 267 + 346
200
300
60
40
7
6
<- space needed to move digits across
100____10________
600
10
3
= 613
(7+6=13 so we move across 10; 60+40+10 is 110
so we move across 100)
Space and movement of numbers should be ABOVE the line to make it clear to the children
where the numbers belong – link to place value.
Place value should be embedded before method is used
D) Compact method
e.g. 357 + 426
357
426
+
__1___
<- space still given
783
NB – we still call the carried number a 10, not 1
Reinforced with Base ten Equipment:
Extends to
+
3682
2753
+
£37.56
£26.82
Extends to
importance of columns is made to stress the place value
Space is still left
1 _1________
£64.38
4) Mental Subtraction Methods in Key Stage 1 (Year 1-2)
The starting point for children is that they can count back in 1s from 100, as well as count
back in 10s using the 100 square and ‘spider’.
In parallel children are taught that addition facts are also subtraction facts, known as
inverse operations.
It is important that children understand firstly that:
5+
□= 8, is the same as 8 - 5=□ , or □ = 5 + 3; it’s just the box that changes place.
In Year 2
Subtraction is developed along two lines, and this will continue into Key Stage 2. That is
that subtraction is:
a) Taking away or counting back; or
b) Counting up or finding the difference
- they must develop both concepts
A) Counting back/taking away
-
Children will count back in 10s and 1s
E.g. 77 – 23 = from 77 count back in 10s (67, 57), then in 1s (56,
55, 54)
-
Children will count back just in 1s
E.g. 73-5 = 72, 71, 70, 69, 68
B) Counting up (using ‘Frog)
-
E.g. ‘You have 20p and spend 14p, how much money do you have
left?’
+ 6p
[_________________________________________________________________________]
14p
20p
5) Mental subtraction at Key Stage 2 (Year 3 – 6)
In Year 3 we do not want our children to be counting back in 1s any more with problems
such as 72 – 5. Instead we would encourage them to count back 2 to the next multiple of
10 (i.e. 70) and then 3 more to get to 67. We call this a ‘No work’ sum.
Again, following Key Stage 1, we will use counting back using ‘fly’ and ‘spider’ on the 100
square.
We introduce ‘nearly numbers’ on the number grid:
- E.g. 73 – 29
29 is nearly 30, so we take 30 instead = 43, we’ve
been too greedy and taken too much so we give that
back, therefore the answer is 44.
We count back in 100s
- 535 – 300, start on 535 to 435, 335, 235
- 535 – 302, do the same as above and then back two in 1s = 233
- 534 – 199, go back 200, give one back = 335
Counting up method
-
E.g. 121 – 87
Start on the ‘baby’ number which is 87, ‘frog’ jumps to 121
3
10
21
____________________________________________________________________
87
90
100
121
= 3 + 10 + 21 = 34
By the end of Y3 pupils can use this method to subtract two 3-digit numbers:
2
E.g. 500 – 378
20
100
_________________________________________________________________________
378
380
400
500
= 2 + 20 + 100 = 122
This extends to finding the difference between times.
For example, ‘It is 12:35pm and I need to be home for 1pm, how long do I have?’
5
20
= 25mins
12:35
12:40
1pm
15 mins
1:45pm
1 hour
15 mins
2:00pm
3pm
3:15pm
= 1h 30 minutes
Children should be able to identify the most appropriate strategy to use. So, for example,
they would not use the counting on method for 2003 – 5, we would say this is cruel to frog!
We would count back instead; but for 2003 – 1878, frog/counting on is a good method. In
fact, these are preferred strategies than any written method.
6) Written subtraction in Key Stage 2 (Year 3-6)
In Year 4 this is developed with a clear link to expanded addition (see section 3). We begin
by showing that partitioning is useful in subtraction.
E.g.
873
800
–
3
332
300
70
2
30
Using mental subtraction they should know: 800 – 300 = 500
70 – 30 = 40, 3 – 2 = 1
=> 541
Written methods involve decomposition, but children are taught that these are not always the
most effective methods. In an international test, a high number of Y6 pupils got the question
2003 – 5 wrong as they tried to answer it vertically, and got confused when the moved
across digits which involved the zeros. This is an example where counting back is more
effective.
a) Expanded Subtraction
653 – 478
-
500
140
40
13
600
400
50
70
3
8
__________________________
100
70
5
Begin with
-
the 1s column
3 – 8, can’t do this so we need to borrow a ten
Therefore, 50 is ten less (i.e. 40) and the ten borrowed makes 13
40 – 70, can’t do this, so we need to borrow a hundred
Therefore, 600 becomes one hundred less (500), and we carry the
hundred to make 140 in the tens column.
The calculation can now be completed.
b) Compact subtraction
Moves away from expanded place value, as children are now secure.
4
12
14
5 3 4
2 7 8
__________
2 5 6
__________
This is reinforced with Base 10 equipment.
As noted above, if a child faces a question such as £100 - £56.49, we would encourage them
to use ‘frog’ and count on the number line from £56.48 to £100 rather than use
decomposition which involves tackling zeros and is more confusing.
7) Multiplication in Key Stage 1 (Year 1-2)
Here we are focusing on children knowing ARRAYS or REPEATED ADDITION.
Children will learn that 3 x 5, or 3 lots of 5 are:
Children will use ‘Clever Counting’, so will count in 2s/5s/10s
-
E.g. 3 lots of 10 = 10,20,30
E.g. 4 lots of 5 are 5,10,15,20
Simultaneously, children will be doubling and halving. They will use vocabulary such as twice
a big or twice as long.
They will learn the 2x, 5x, and 10x table. Some children will be learning the 3x table.
These are often covered in our quick maths starters, and our scheme allows for systematic
practice of these key skills.
8) Mental multiplication in Key Stage 2 (Year 3-6)
For children in this age group, mental agility is dependent on partitioning.
A) Grid method
3 x 14
x
10
4
3
30
12
(split up the 14 into it’s values)
= 30 + 12 = 42
B) ‘Nearly Numbers’
3 x 29 is ‘nearly’ 3 x 30 – which is 90. As we have been too greedy we must
give back 3. So 90 – 3 = 87
Year 5 uses the grid method with decimals:
3 x £3.17
X
£3
10p
3
£9
30p
7p
21p = £10.51
C) Using doubling and halving for multiplication – linked to place value understanding
E.g. tackling 48 x 5
E.g. tackling 82 x 4
as
as
48 x 10 = 480
82 x 2 = 166
then ½ your answer = 240
then double your answer = 332
9) Written multiplication in Key Stage 2 (Year 3-6)
We build on what the times tables are all about. For example 7 x 8 is 7 lots of 8, or 8 lots of
7, which is also twice 4 lots of 7.
A) Grid Method
38 x 34
X
30
30
900
8
320
4
120
= 1020
32
= 352
______________
Would know this is ten times bigger than 3 x 8
B) Expanded method – again makes the place value explicit
x
637
8
______
4800
8 x 600
240
8 x 30
56
8x7
5096
C) Compact method – is very reliant on place value being secure
637
8
_2_ 5____
5096
29 tens
x
is 50 not 5
1372
NB with money e.g. £6.47 x 7 – we would say the most efficient strategy is the grid method:
X
£6
7
£42
40p
£2.80
7p
49p
= £42.00
£ 2.80
£ 0.49
£ 45.29
10)
Mental Division in Key Stage 1 (Year 1 and 2)
This can be a problem area for children. We introduce division as knowing their times
table.
□
x 5 = 15
Asking the children how many ‘clever counts’ to 15 = 3
So
15 ÷ 3 =
□
Children need to know that division is the inverse of multiplication. We would ask
them questions such as: ‘How many lots of 2 is 22?’ = Answer is 11
At Key Stage 1 children are introduced to basic fractions: ½ and ¼, and that this is
SHARING
½ of 8 = 4
or
¼ of 12 = 3
and this is the inverse of doubling.
Some children may know that 1/3 of 3 = 1 ½
As we are sharing, this is very practical and uses objects to demonstrate the process.
Children will also count in 1/2s or 1/4s e.g. ½ , 1 , 1 ½ , 2, 2 ½ ….
11)
Mental division in Key Stage 2 (Year 3-6)
Division facts and multiplication are covered in starters and quick maths activities.
Children continue to learn how division is the inverse of multiplication and they will
learn division facts alongside multiplication facts.
E.g. 56 ÷ 7 = 8, because 8 x 7 = 56 and 7 x 8 =56
A) Mental Chunking/subtraction method
Children also solve problems such as:
87 ÷ 5 =
Ask: ‘Is it going to be more than 10 steps in our 5x table?’ – Yes
So we will take 10 steps away from 87 (10 x 5 = 50), so we take away
50 from 87. This leaves 37 remaining. We know, using our 5x table
facts that 7x5=35, so we can take 35 away. We have 2 remaining and
can’t take away any more using the 5x table. Therefore, we have taken
away 10 + 7 which is 17, with 2 remaining.
87
10 x 5 =
50
37
7x5=
35
=17
r2
= 17 r 2
or even
17 2/5 or 17.4 (as pupils move into Y5/6)
Children would use this information to answer a different range of questions, which may
include rounding their answer or modifying it. For example, how many full boxes of toys
would be made if complete boxes had 5 toys in them, and there were 87 toys. The answer is
17.
Or: How many cars are needed to transport 87 people if each car holds 5 people. The answer
is 18, as we need that extra car even if there are only 2 people in it.
Children continue to relate fractions to decimals:
1/3 of 81 is the same as 81 ÷ 3
It can be hard for pupils to make these connections.
We also use a ‘Function Machine’ to demonstrate further doubling and halving with larger
numbers.
12)
Written division in Key Stage 2 (Year 3-6)
Children learn that some problems are too hard for our heads!
For example: 135 ÷ 3 =
Standard method is used:
45
1
3 135
Firstly, we cover the 35 in the box.
How many 3s go into 1?= none
Now show the 1 and the 3, how many 3s go into 13 = 4, and 1 little
step needed which moves to the 5 to make a 15. We know 3 goes into 15 perfectly =
5.
This method is applied to 375 ÷ 3 or even 1452 ÷ 12
121
21
12
1452
Again cover up each digit
12 doesn’t go into 1, but goes into 14 once
with two little steps forward…