1. Progression in
Written Calculation - x ÷ +

2. Introduction
Written methods of calculations are based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly. These skills lead on to more formal written methods of calculation. Strategies for calculation need to be supported by familiar models and images to reinforce understanding. When teaching a new strategy it is important to start with numbers that the child can easily manipulate so that they can understand the concept. The transition between stages should not be hurried as not all children will be ready to move on to the next stage at the same time, therefore the progression in this document is outlined in stages. Previous stages may need to be revisited to consolidate understanding when introducing a new strategy. A sound understanding of the number system is essential for children to carry out calculations efficiently and accurately.

3. Partitioning and recombining+
Progression in
written methods
for Addition +
Number Track
Number Line
Expanded method
Partitioning and recombining
Formal Compact Method

5. + + + = + = 7 3 + 2 = 3 1 1 + 2 = 3 and and Stage 1 – Number Track
add 3 1 2 3 4 5 8 7 6 10 9 7 + 3
and +
and
understand addition is combining
groups of objects
count on using a number track
use a puppet to accentuate jumps + = + = + 2 = 3 1 1 + 2 = 3
understand addition can be done in
any order

6. Stage 2a – Introducing the number line
+ 3 1 2 3 4 5 8 7 6 10 9 1 2 3 4 5 6 7 8 9 10 7 + 3
and
and +
link number track and number line
count on using a fully numbered
number line (start counting on in ones
and then move on to larger jumps)

7. Stage 2b – Using a number line Partition the smallest number.
Add the unit(s) first
+ 1
+ 10 13 14 24
+ 7
+ 1
+ 10 21 13 20 31
always encourage ESTIMATION first
teach and encourage children to
partition numbers in different ways in
order to bridge to the nearest multiple of 10
start from the largest number and then count on
progress from not bridging 10 to bridging through 10
progress from fully numbered line to partially
numbered line then blank

8. Stage 3 – Partitioning & Recombining
The Expanded Method 20 8 40 3 + = T U
Encourage ESTIMATION
reinforce place value by using place value cards to
partition alongside place value apparatus (Dienes)
reinforce ‘carrying’ use equipment alongside
expanded written method to bridge from concrete
to abstract (see appendix a for recording – transition
between equipment – pictorial recording and then
abstract)

9. Stage 4 – Expanded Method leading
to Formal Compact Method 20 8 40 3 +
= 7 1
Add the unit(s) column then the ten(s) column to calculate the final answer
4 3
7 1
Expanded method leading to compact method 1 9

10. Stage 4 – Expanded Method leading to Formal Compact Method (decimal)
0.20
0.08
0.40
0.03 + =
continue to encourage ESTIMATION
link to money (add more than two amounts) & measurement
link to using a calculator/interpreting calculator display C M √ AC % ÷ 7 4 1 8 5 2 . 9 6 3 = x - +
0.43
0.71
Remember to line up decimal points especially when number of digits differs 1 10

11. (Finding the difference Partitioning and recombining
Progression in
written methods
for Subtraction - -
Number Track
Number Line
(Finding the difference
and counting back)
Expanded method
Partitioning and recombining
Formal Compact Method

13. 7 3 - Stage 1 – Number Track take away 32 3 4 5 8 7 6 10 9 7 3 -
seven
three
understand subtraction is taking
away objects
jump/count back along a number track
use a puppet to accentuate jumps

14. Stage 2a – Introducing Number Line1 2 3 4 5 8 7 6 10 9
- 3 1 2 3 4 5 6 7 8 9 10 7 3 -
seven
three
link number track and number line
understand subtraction is taking away
objects
jump/count back along a fully numbered
number line (start counting back in ones
and then move on to larger jumps)

15. Stage 2bi – Using a number line (not bridging through 10)
finding the difference
23 – 18 5 18 23
counting back
37 – 13
Partition the smallest number.
Count back the units, then tens.
- 3
- 10 34 37 24
ESTIMATE first
understand ‘finding the difference’
AND ‘counting back’ has the same result
promote finding the difference when the
numbers involved are close together
progress from counting back in ones to
larger steps. 15

16. Stage 2bii – Using a number line
(bridging through 10)
43 – 27
- 10
- 10
- 4
- 3 16 26 36 40 43
ESTIMATE first
encourage children to partition numbers in
different ways
bridge through multiples of 10
ensure children have the opportunity to
solve subtraction problems in a range of
different contexts
encourage use of vocabulary and explanation

17. take away the units and then take away the tens
Stage 3a – Expanded Method
(no exchanging)
= 33
take away the units and then take away the tens
30 and 3
use place value apparatus (Dienes) to
re-inforce concept of exchanging
move from concrete apparatus to
expanded written method (see appendix a for
recording – transition between equipment – pictorial
recording and then abstract)
continue to encourage ESTIMATION 17

18. to subtract 7 units we need to exchange a ten for ten units
Stage 3b – Expanded Method (with exchanging)
= 16
to subtract 7 units we need to exchange a ten for ten units
and 6 30
10 +
use place value apparatus (Diennes) to
re-inforce concept of exchanging
move from concrete apparatus to
expanded written method (see appendix b for
recording – transition between equipment – pictorial
recording and then abstract)
continue to encourage ESTIMATION 18

19. - 20 7 10 and 6 - 2 7 1 6 Stage 4a – Formal Compact Method 4 3
10 and 6
10 + 30
4 3
1 6 3 1
move from expanded written method
to compact method
continue to encourage ESTIMATION

20. - 2 . 7 1 . 6 Stage 4 – Formal Compact Method (decimal) 4 . 3
1 . 6 1 3
Remember to line up decimal points especially when number of digits differs
continue to encourage ESTIMATION
link to money (giving change) and
measurement
link to using a calculator/interpreting
calculator display C M √ AC % ÷ 7 4 1 8 5 2 . 9 6 3 = x - +

21. Repeated addition, arrays
Progression in
written methods
for multiplication x x
Repeated addition, arrays
Grid method
(with imagery)
Long multiplication

23. 2 2 2 2 + 2 4 x 4 2 x Stage 1 – Repeated addition, arrays +
= 8
4 x 2 = 8
2 multiplied by 4
4 lots of 2 x
4 2
2 4 x
understand that multiplication is a
shortened form of repeated addition
understand multiplication as arrays
and jumps on a number line

24. 4 x 13 ‘four lots of thirteen’
Stage 2 – Modelling grid method with
place value equipment
4 x ‘four lots of thirteen’ 4 10 3 4 10 3 40 12
= 52
use place value apparatus to illustrate
grid method, encourage jottings
use digits of 5 and below to avoid
‘difficult’ tables and ensure method is secure

25. 4 x 23 ‘four lots of twenty three’
Stage 2a – Modelling grid method with place
value equipment (multiples of 10)
4 x ‘four lots of twenty three’ 20 3 4
20 ( 2 x 10 ) 3 80 12 4
(4 x 2 x 10)
(4 x 3)
= 92
use place value equipment to illustrate
grid method with multiples of 10
reinforce using known facts to
multiply e.g. 4 x 20 = 4 x 2 x 10

26. Stage 3 – Grid method (no apparatus)
45 x 6
40 ( 4 x 10 ) 6
240 36 6
(6 x 4 x 10)
(6 x 6)
= 276
47 x 52 50 2
2000 80
2000 40
350
(4 x 5 x 10 x 10)
(4 x 2 x 10) 80 80 14
350 7
+ 14 14
(7 x 5 x 10)
(7 x 2 )
2444
continue to reinforce using known
facts to multiply e.g. 40 x 50 = 4 x 5 x 10 x 10
progress to using the grid method
efficiently to multiply decimals

27. Stage 4 – Long multiplication
× (56 × 20) (56 × 7) 4 1 1
ONLY move on to this method if
understanding of grid method is
secure

28. Stage 4a – Long multiplication (decimal)
× (5.6 × 2.0) (5.6 × 0.07) 4 1 1
continue to encourage ESTIMATION
(re-inforce place value)
link to money and measurement
link to using a calculator and
interpreting display C M √ AC % ÷ 7 4 1 8 5 2 . 9 6 3 = x - +

29. ÷ ÷ Division as sharing and grouping Grouping on a number line
Progression in
written methods
for division ÷
Division as sharing and grouping
Grouping on a number line
Link division and multiplication
Vertical recording
Chunking (fact box)
Short/long division

31. ÷ ÷ 15 3 15 5 Stage 1 – Division as sharing and grouping sharing
one at
a time ÷
15 divided into
3 equal groups ÷
15 divided into
5 equal groups
understand division as sharing, understand
division as grouping
understand remainders

32. Stage 2 – Grouping on a number line - 
= 6 groups r1
understand that division is
repeated subtraction
show division as equal groups on a
number line
then begin to understand remainder

33. Vertical recording (teacher model only)3 6 9 12 15 18 18 15 12 9 6 3
- 3
18 ÷3 = 6 18
( 1 x 3 )
1 5
1 2 9
( 1 x 3 ) 6 3
( 1 x 3 )
turn horizontal number line vertical so children can see link to vertical calculation and model recording, use to illustrate need to take ‘chunks’ for efficiency

34. Stage 3 – Linking division & multiplication leading to chunking
– introducing fact box
What facts
do I know about the
5 times-table?
96  5
96 ÷ 5 = 19 r 1 96
( 10 x 5 ) 46
( 5 x 5 ) 21
- 20 1
Fact Box
1 x 5 = 5
2 x 5 = 10
5 x 5 = 25
10 x 5 = 50
children need to see that when numbers are larger
it is more efficient to subtract larger ‘chunks’
building a fact box will help children with the size
of the ‘chunks’
children need to work with and without remainders
considering if answer needs rounding up or rounding
down

35. Stage 4 – Chunking with a fact box
What facts
do I know about the
7 times-table?
100 ÷ 7 = 14 r 2
100
( 10 x 7 ) 30
( 4 x 7 ) 2
518 ÷ 7 = 74
518
( 50 x 7 )
168
( 20 x 7 ) 28
( 4 x 7 )
Fact Box
1 x 7 = 7
2 x 7 = 14
5 x 7 = 35
10 x 7 = 70
20 x 7 = 140
50 x 7 = 350
100 x 7 = 700
children need to see that when numbers are larger it is
more efficient to subtract larger ‘chunks’
building a fact box will help children with the size of the
‘chunks’
children need to work with and without remainders
considering if answer needs rounding up or rounding down

36. ONLY move on to this method if understanding is secure
Stage 5 – Long division
560 ÷ 24
2 3 r 8 8
Jottings – Fact Box 20 3 4
400 80 60 12 x
= 552
ONLY move on to this method if
understanding is secure
move on to show remainders as a
fraction and decimal

37. (showing remainder as a fraction)
Stage 5a – Long division
(showing remainder as a fraction)
560 ÷ 24
2 3 r 8/24 () 8
Jottings – Fact Box 20 3 4
400 80 60 12 x
= 552
ONLY move on to this method if
understanding is secure
move on to show remainders as a
fraction and decimal

38. (showing remainder as a decimal)
Stage 5b – Long division
(showing remainder as a decimal)
560 ÷ 24
8 0
7 2 8
Jottings – Fact Box 20 3 4
400 80 60 12 x
= 552
ONLY move on to this method if
understanding is secure
move on to show remainders as a
fraction and decimal

39. Start with apparatus then show children how to record pictorially
T - tens
U - units
concrete
appendix a
Start with apparatus then show children how to record pictorially
visual

40. Start with apparatus then show children how to record pictorially
appendix b
concrete
T - tens
U - units
Start with apparatus then show children how to record pictorially
visual

41. Link division and multiplication
12 divided into groups of 3 gives 4 groups
12  3 = 4
12 divided into groups of 4 gives 3 groups
12  4 = 3
3 x 4 = 12 or 4 x 3 = 12
12  4 = 3 or 12  3 = 4
understand that division is the
inverse of multiplication
reinforce division as grouping
emphasise link between times table facts
and division facts

42. Understanding the inverse/finding unknowns (empty boxes)
I can work out missing numbers in a number sentence (year 1 & Year 2)
Introducing the Inverse – Play Mrs/Mr Opposite. Every instruction the teacher gives the children have to do the opposite e.g. teacher says take one step forward, children take one step backwards or teacher says turn to the right, children turn to the left etc. Explain that in maths we call the opposite the inverse and that we are going to be looking at the inverse (opposite) of adding.
Addition and Subtraction – Numicon Families (Using the inverse) Ten is the same as/equals nine add/plus one.
Explain that the children are going to be using Numicon. Show them what it is and explain how it is structured. Model how this can be used to demonstrate the inverse
Once imagery is secure replace one piece of Numicon with an empty box. Remember to move the = sign!
Ten is the same as/equals nine add/plus one. + =
10 = 9 + 1

43. Understanding the inverse/finding unknowns (empty boxes)
I can work out missing numbers in a number sentence including where = sign is used to balance an equation e.g = 3 + ? or 6 x 4 = 3 x ? (Year 3)
Use Numicon and balance to model and for the children to practise in order to reinforce understanding of equality.
Once imagery is secure replace one piece of Numicon with an empty box. Remember to move the = sign AND begin to explore balancing different operations.
I can work out missing numbers in a number sentence including those where = balances an equation e.g. 10 – 3 = 3 + ? or 2 x 3 = 60  ? (Year 4)
Again use Numicon. Emphasis on exploring balancing equations with different operations.
I can work out missing numbers in more complex calculations e.g. 3?67 – 192? = 1539 or  ? = 325 (Year 5)
I can find the unknown in a calculation such as ? = or 5.6  ? = 0.7, drawing on knowledge of number facts and place value, including using a calculator and inverse operations to solve more complex problems such as  ? = 24.7 and explain my reasoning (Year 6) + = +