PROGRESSION THROUGH CALCULATIONS
INFORMAL/WRITTEN METHODS FOR ADDITION
Children are encouraged to develop a mental picture of the number system in their heads to use
for calculation. They develop ways of recording calculations using pictures, etc.
They use numberlines and practical resources to support calculations.
3+2=5
+1
+1
___________________________________________
0
1
2
3
4
5
6
7
8
9
Children then begin to use numbered lines to support their own calculations using a numbered
line to count on in ones.
8 + 5 = 13
0
1
2
3
+1
4
5
6
7
8
9
+1
+1
+1
+1
10 11 12 13 14 15
Bead strings or bead bars can be used to illustrate addition including bridging through ten by
counting on 2 then counting on 3.
WRITTEN CALCULATIONS
Children will recognise that addition can be done in any order and that it is often best to start
with the biggest number.
Children will begin to use ‘empty number lines’ themselves starting with the larger number and
counting on.
First counting on in tens and ones.
34 + 23 = 57
+10
34
+10
+1
44
54
+1
55
+1
56
57
Then helping children to become more efficient by adding the units in one jump (by using the
known fact 4 + 3 = 7).
34 + 23 = 57
+10
+10
+3
34
44
54
57
Followed by adding the tens in one jump and the units in one jump.
34 + 23 = 57
+3
+20
34
54
57
Bridging through ten can help children become more efficient.
37 + 15 = 52
+10
37
+3
47
+2
50
52
Partitioning Method

This method involves partitioning the numbers into tens and ones and writing one under
the other.
60
7
+ 20
4
______
80 + 11 = 91
______
200 60
7
+
80
5
___________
200 + 140 + 12 = 352
___________
Compact Method
Children will then move onto a compact method and carry below the line.
625
+ 48
673
783
+ 42
825
1
367
+ 85
452
1
11
INFORMAL/WRITTEN METHODS FOR SUBTRACTION –
Children are encouraged to develop a mental picture of the number system in their heads to use
for calculation. They develop ways of recording calculations using pictures etc.
They use numberlines and practical resources to support calculation. Teachers demonstrate
the use of the numberline.
6–3=3
0
1
2
3
4
-1
5
-1
6
-1
7
8
9
10
WRITTEN CALCULATIONS
Children then begin to use numbered lines to support their own calculations - using a numbered
line to count back in ones.
13 – 5 = 8
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
-1 -1
-1 -1
-1
Children will begin to use empty number lines to support calculations.
Counting back
Counting back in tens and ones.
47 – 23 = 24
24
-1
25
26
27
-1
-1
37
47
- 10
- 10
Then helping children to become more efficient by subtracting the units in one jump (by using
the known fact 7 – 3 = 4).
47 – 23 = 24
24
27
37
47
-10
-3
-10
Subtracting the tens in one jump and the units in one jump.
47 – 23 = 24
24
27
-3
47
-20
Bridging through ten can help children become more efficient.
42 – 25 = 17
17
20
-3
22
42
-2
-20
Partitioning Method
Numbers are partitioned into tens and ones and recorded underneath one another, before
subtracting.
72
- 21
Step 1
Step 2
70
- 20
2
1
70
- 20
50
2
1
1
=
51
This method is used to explain decomposition, when numbers are broken down to exchange tens
and hundreds .
71
- 46
Step 1
70
- 40
1
6
Step 2
60
- 40
20
11
6
5
=
25
754
- 86
Step 1
700
50
80
700
40
14
80
6
Step 2
Step 3
600
140
600
80
60
-
4
6
(adjust from T to O)
14
(adjust from H to O)
We exchange 54 for 40
and 14
We exchange 754 for 600
and 140 and 14
6
8 = 668
Compact method
Children will then move onto the compact method, using decomposition to exchange tens and
hundreds when appropriate.
Exchanging 754 for 600 and 140 and 14
614 1
//
754
- 86
668
MULTIPLICATION
Children will experience equal groups of objects and will count in 2s and 10s and begin to count
in 5s. They will work on practical problem solving activities involving equal sets or groups.
Children will develop their understanding of multiplication and use jottings to support
calculation:
Repeated addition
3 times 5
is
5 + 5 + 5 = 15
Repeated addition can be shown easily on a number line:
+5
0
1
2
+5
3
4
5
6
7
or 3 lots of 5 or 5 x 3
5x3=5+5+5
+5
8
9
10
11
12
13 14
15
Commutivity
Children should know that 3 x 5 has the same answer as 5 x 3. This can also be shown on the
number line.
+5
0
1
2
+5
3
+3
4
5
6
7
+5
8
9
+3
+3
10
+3
11
12
13
14
15
+3
Arrays
5 x 3 = 15
3 x 5 = 15
Partitioning
Children will then partition to multiply using grid method.
38 x 5 = (30 x 5) + (8 x 5)
= 150 + 40
= 190
Grid method
Tens and Ones x Ones
23 x 8
Children will approximate first
23 x 8 is approximately 25 x 8 = 200
x
20
3
8
160
24
160
+ 24
184
HundredsTensOnes x Ones
346 x 9
Children will approximate first
346 x 9 is approximately 350 x 10 = 3500
x
9
300 2700
40
360
6
54
2700
+ 360
+ 54
31 1 4
1 1
Short multiplication will then be used as a more efficient method of multiplying numbers.
1 3
X 7
____
9 1
____
2
DIVISION
Children will understand equal groups and share items out in play and problem solving. They will
count in 2s and 10s and later in 5s.
Children will develop their understanding of division and use jottings to support calculation

Sharing equally
6 sweets shared between 2 people, how many do they each get?
Grouping or repeated subtraction
There are 6 sweets, how many people can have 2 sweets each?
Repeated subtraction using a number line or bead bar
12 ÷ 3 = 4
0
1
2
3
4
-3
5
6
7
-3
8
-3
9
10 11
12
-3
Children should also move onto calculations involving remainders.
13 ÷ 4 = 3 r 1
0
1
r1
5
-4
9
-4
13
-4
Then onto the vertical method using ‘chunking’ and known facts:
Short division Tens Ones ÷ Ones
72 ÷ 3
Method
Known facts
3 ) 72
- 30
42
- 30
12
- 6
6
6
0
10 x 3 = 30
20 x 3 = 60
2x3=6
Answer :
10x3
10x3
2x3
2x3
24
.
96 ÷ 6
16
6 ) 96
- 60
36
- 36
0
16
10x6
6x6
Any remainders should be shown as whole numbers, i.e. 14 remainder 2 or 14 r 2.
This method will then be extended to the short division method as shown below.
6
12
1
72
Or with a remainder as below.
12
6
1
7
r3