Orchard Way Primary School
Policy for Teaching Written Calculations
1
The following policy lays out the process for teaching children written calculation. When
beginning any calculation, children should be taught that the first question that they ask
themselves is whether the calculation should be done mentally, mentally with jottings or by using
a written method.
The stages below show the order in which children should be taught written calculation in order
to build up a firm understanding of the process. Age related expectations are shown next to
each stage. Teachers need to bear in mind that children may be a different stages in the
different operations. It is important to note that children should not be moved on to a new
stage without having a thorough understanding of the preceding one. Children should be
encouraged to approximate their answers before calculating and to check their answers after
calculation using an appropriate strategy. By the end of Year 6, children will have a range of
calculation methods, mental and written to choose from, depending on the numbers involved.
PROGRESSION THROUGH WRITTEN CALCULATIONS FOR ADDITION
THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO
ACHIEVE
Stage 1: Bead string and Numbered Number line
ARE: YR AND Y1
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 number lines and practical resources to support calculation and teachers demonstrate
the use of the number line. It is important that children have a secure understanding of what a
number line represents. They should be encouraged to use bead strings to form a concrete
concept of a number line.
2
3+2=5
+1
0
1
2
3
+1
4
5
6
7
8
9
Children then begin to use numbered number lines to support their own calculations using a
numbered line to count on in ones.
8 + 5 = 13
+2
0
1
2
3
4
5
6
7
8
+3
9 10
11
12
13
14
15
They will need to be encouraged to progress to ‘jumps’ of different sizes. Bridging 10 is an
important part of this process.
Stage 2: Empty Number Line
ARE: Y2+
Before moving on to this stage, children need plenty of experience with:
 Use of numbered number line
 Partitioning of two digit numbers
Children will begin to use ‘empty number lines’ themselves starting with the larger number and
counting on.
They may start by first counting on in tens and ones, and then should be helped to become more
efficient in their calculations by adding the units in one jump (using the known fact 4 + 3 = 7)
34 + 23 = 57
+ 10
+ 10
+3
34
44
54 55 56 57
Progressing to adding the tens in one jump.
Bridging through 10 can help children to become more efficient.
+ 10
+3
37
47
3
+2
50
52
Compensation
49 + 73 = 122
+ 50
-1
73
122
123
Stage 3: Partitioning
ARE: Y2/3
Children will begin to use informal pencil and paper methods.
84 + 33 =
=
=
84 + 30 + 3
114 + 3
117
47 + 76 = 40 + 70
= 110
= 123
(count on tens first to form partial sums)
+
7+6
+
13
(Partition to add most significant, i.e. tens, first)
Stage 4: Expanded Column Method
ARE: Y3+
Option 1 – Adding the most significant digits first, and then moving to adding least significant
digits, as this mirrors mental methods.
Option 2 – Adding the least significant digits first.
Option 1: (most significant)
Option 2: (least significant)
+
67
24
80 (60 + 20)
1 1 ( 7 + 4)
91
+
267
+ 85
200
140 (60 + 80)
12 ( 7 + 5)
352
67
24
11 (7+4)
80 (60 + 20)
91
4
267
+ 85
12 (7 + 5)
140 (60 + 80)
200
352
Stage 5: Compact Method
ARE: Y4+
From this, children will begin to carry below the line.
625
+ 48
673
783
+ 42
825
1
1
367
+ 85
452
11
When modelling this method, use the correct place value language e.g. carry the ten not carry
the one.
Stage 6: Compact Method – Decimals up to 2dp
ARE: Y5+
Children should extend the carrying method to number with any number of digits. (Only once
they have a secure understanding of decimals using number lines and place value cards and
charts). Using a real context like money will help the children to make sense of this.
£2.27 + £5.64
2.27
+ 5.64
7.91
Decimal points must be aligned.
PROGRESSION THROUGH WRITTEN CALCULATIONS FOR SUBTRACTION
5
THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO
ACHIEVE
The stages below show the order in which children should be taught written calculation in order
to build up a firm understanding of the process. Age related expectations are shown next to
each stage. It is important to note that children should not be moved on to a new stage without
having a thorough understanding of the preceding one. Teachers need to bear in mind that when
children encounter larger numbers or decimals, they may need to go back to a more expanded,
informal method of calculation.
Stage 1: Bead string and Numbered Number Line
ARE: YR and Y1
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 number lines and practical resources to support calculation. Teachers demonstrate use
of the number line. They should be encouraged to use bead strings to form a concrete concept of
a number line, relating ‘taking away’ to ‘counting back’.
6–3=3
-1
0
1
2
3
-1
4
-1
5
6
7
8
9
10
The number line should also be used to show that 6 – 3 means the ‘difference between 6 and 3’ or
‘the difference between 3 and 6’ and how many jumps they are apart, which can
also be calculated by ‘counting on’.
6
0
1
2
3
4
5
6
7
8
9
10
Stage 2: Empty Number Line and Partitioning
ARE: Y2+
Before moving on to this stage, children need plenty of experience with:
 Use of numbered number line
 Partitioning of two digit numbers
Children will begin to use empty number lines to support calculations.
Counting back
First counting back in tens and ones.
47 – 23 = 24
-3
-1 -1
24
25
-10
-10
-1
26
27
37
47
Then helping children to become more efficient by subtracting the units in one jump
(by using the known fact (7 – 3 = 4).
Bridging through ten can help children become more efficient.
42 – 25 = 17
-20
-3
17
-2
20
22
Stage 3: Finding the Difference
ARE Y3+
42
7
Counting on
Both counting on and counting back should be taught at this stage, so that children know that
both are used for subtraction.
i)
ii)
Subtracting T, then U, bridging through 10
Counting on, bridging through 10
Counting on is particularly effective with numbers close in size, and again builds on mental
calculation methods. This method should continue to be taught through to Y6 as an important
alternative to column subtraction.
2006 - 1997
+3
1997
+6
2000
2006
3002 – 1997 = 1005
+ 1000
+3
0
1997
+2
2000
3000
3002
When moving onto decimal calculations, children would benefit from revisiting number line
methods, which give a true sense of the value of the whole number.
15.3
- 12.9 = 2.4
+ 1.
+1
+ 0.1
12.8 12.9
1.0
+ 1.0
2.0
13.0
+
+ 0.3
13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8.13.9
0.1
+ 0.3
0.4
14.0
= 2.4
Stage 4: Expanded column subtraction
15.0
15.1 15.2 15.3
8
ARE: Y3+
Partitioning and decomposition
This process should be demonstrated using arrow cards to show the partitioning and base 10
materials to show the decomposition of the number.
89
- 57
=
=
80 and 9
50 and 7
30 and 2
= 32
Initially, teach the children with examples that do not need them to exchange.
From this, the children will begin to exchange. (NOT borrow or carry)
71
- 46
=
Step 1
70 and 1
- 40 and 6
Step 2
60 and 11
- 40 and 6
20 and 5
=
25
This could be recorded by the children as
60
70 + 11
- 40 + 6
20 + 5 = 25
Children should know that units line up under units, tens under tens, and so on.
Expanding helps prepare children for later column subtraction.
Stage 5 Compact method - Tens and Units
9
ARE: Y4+
This should be taught alongside Partitioning to demonstrate the similarities between the two
methods.
i) No exchanging
63
- 21
42
6
ii) Exchanging
-
714
27
47
Partitioning and decomposition
When moving onto larger numbers – HTU, ThHTU, children may need to revisit expanded
methods.
-
754 =
86
Step 1
700
-
+
50
80
+
+
4
6
Step 2
700
-
+
40
80
+
+
Step 3
600
-
+
140
80
+
+
600
+
60
+
14 (adjust from T to U)
6
14 (adjust from H to T)
6
8
= 668
This could be recorded by the children as
600
-
700
+
600
+
140
50
80
+
+
60
+
Stage 6 Compact method - HTU+
1
4
6
8
=
668
10
ARE: Y5+
Decomposition
When modelling this method, use the correct place value language e.g. exchange the ten for ten
ones/units (not borrow or carry)
-
754
86
668
=
6
714514
86
6 6 8
PROGRESSION THROUGH WRITTEN CALCULATIONS FOR MULTIPLICATION
11
THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO
ACHIEVE
Stage 1: Making Sets and Groups
ARE: YR and Y1
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.
Stage 2: Number Lines and Arrays
ARE: Y2 and Year 3
Children will develop their understanding of multiplication and use jottings to support
calculation:
Repeated addition
3 times 5
is
5 + 5 + 5 = 15 or 3 lots of 5 or 5 x 3
Repeated addition can be shown easily on a number line:
5x3=5+5+5
5
5
5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
and on a bead bar:
5x3=5+5+5
5
Commutativity
5
5
12
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
5
5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3
3
3
3
3
Arrays
Children should be able to model a multiplication calculation using an array. This knowledge will
support the development of the grid method and models the commutative nature of
multiplication:
5x3=3x5
5 x 3 = 15
3 x 5 = 15
Partitioning
As numbers beyond 10 are used partitioning is useful:
13 x 3 = (10 x 3) and (3 x 3)
= 30
+
9
= 39
Stage 3: The Grid Method: Multiplying by a single digit
ARE: Y3+
13
Children will continue to use arrays and partitioning, where appropriate, leading into the grid
method of multiplication.
X
10
4
(6 x 10) + (6 x 4)
60 + 24
84
6
60
24
Grid method
TU x U
(Short multiplication – multiplication by a single digit)
23 x 8
Children should approximate first
23 x 8 is approximately 25 x 8
X
8
20
3
160
24
24
160
+ 24
184
Grid method
HTU x U
(Short multiplication – multiplication by a single digit)
346 x 9
Children should approximate first
346 x 9 is approximately 350 x 10 = 3500
X
9
300
40
6
2700
360
54
Use related facts to complete the grid e.g.
If you know that: 9 x 3 = 27
Then: 9 x 30 = 270
And: 9 x 300 = 2700
14
2700
+ 360
+
54
3114
For children not ready for column addition, use partitioning:
2700
11
+
300
+
+
110
4
3114
Stage 4: The Grid Method: Multiplying by more than a single digit
ARE: Y5+
TU x TU
72 x 38
Children should approximate first
72 x 38 is approximately 70 x 40 = 2800
X
70
2
30
8
2100
21
60
560
16
Use related facts to complete the grid e.g.
If we know that 8 x7 = 56
Then we know that 8 x 70 = 560
2100
+ 560
+ 60
+ 16
2736
1
Or using partitioning for children
not ready for column addition
2100
+ 500
+
120
+
16
2736
Using similar methods, children will be able to multiply decimals with one decimal place by a single
digit, approximating first. They should know that the decimal points line up under each other.
e.g. 4.9 x 3
15
Approximating first:
4.9 x 3 is approximately 5 x 3 = 15
X
3
4
12
0.9
2.7
+
12
2.7
14.7
Use related facts and place value to complete the grid e.g.
If we know that 3 x 9 = 27
Then 3 x 0.9 = 2.7
Progress to ThHTU x U and HTU x TU using the same method
Stage 6: The Grid Method: Multiplying Decimals
ARE: Y5/6
Using similar methods, children will be able to multiply decimals with up to two decimal places by a
single digit number and then two digit numbers, approximating first, as it is a way of
organising/formalising partitioning. They should know that the decimal points line up under each
other.
For example:
4.92 x 3
Children should approximate first.
4.92 x 3 is approximately 5 x 3 = 15
X
3
4
12
0.9
2.7
0.02
0.06
+
+
12
2.7
.06
14.76
Stage 6: Column Multiplication (for children with secure understanding of the Grid Method
ONLY)
ARE: Y6
16
Initially, write as conventional number sentence then model working out alongside the grid
method to show how the two methods relate to one other.
72 x 38 = (70 x 30)
+ (70 x 8)
+ (2 x 30)
+ (2 x 8)
________
5
X
2
4
35
1 2
1 0 0
1 5 1
6
7
2
0
0
0
2
1
72
X 38
2100
+ 560
+ 60
+ 16
2736
(70 x 30)
(70 x 8)
(2 x 30)
(2 x 8)
Show the working at first.
1
Compact Method
ARE: Y6
Reduces working still further. Model working out starting with the units.
38
X 7
266
5
72
x 38
576
+ 21 60
2736
PROGRESSION THROUGH WRITTEN CALCULATIONS FOR DIVISION
17
THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO
ACHIEVE
The stages below show the order in which children should be taught written calculation in order
to build up a firm understanding of the process. Age related expectations are shown next to
each stage. It is important to note that children should not be moved on to a new stage without
having a thorough understanding of the preceding one.
Stage 1: Equal Groups
ARE: YR and Y1
Children will understand equal groups and share items out in problem solving. They will count in 2s
and 10s and later in 5s.
Stage 2: Sharing Equally and Grouping Equally
ARE: Y2
Children use practical resources to develop the concepts of both sharing and grouping.
They will develop their understanding of division and use jotting to support calculation.
Sharing equally
6 divide
2
=
3
6 sweets shared between 2 people, how many do they each get?
What is 6 shared between 2?
Stage 3: Repeated subtraction and repeated addition on a number line
ARE: Y2+
18
Build on concept of ‘sharing equally’ e.g.
12 sweets shared equally between 3 groups means each group has how many sweets? Answer 4
by moving onto ‘lots of’
12 ÷ 3 = 4
How many lots of 3 in 12? How many 3s in 12?
The number line can be empty.
0
3
6
9
12
“ 3…..6……9……12……that’s 4 lots of 3.”
3
3
3
3
The bead bar will help children with interpreting division calculations. Remainders can be
introduced at this stage.
e.g. “ How many 4s in 13? 3 fours with 1 left over.”
Stage 4: Greater Efficiency
ARE: Y2 and Y3
Children should be expected to learn multiplication tables. Once numbers used are
beyond those in the tables, children need to learn ‘chunking’. Children will develop their use of
repeated subtraction to be able to subtract larger ‘chunks’ which is more efficient. Initially,
these should be multiples of 10.
This could be shown on a number line.
How many lots of 5 are there in 72?
How many equal groups of 5 are there in 72?
72 ÷ 5
0
5
5
5
5
5
5
5
5
5
5
5
5
5
70
5
72
19
Steps can be combined to make this more efficient.
10 x 5 = 50
0
4 x 5 = 20
50
2
(10 + 4 = 14)
70
Stage 5: ‘Bus stop’ and ‘Chunking’
ARE: Y3 term 3+
When using the ‘bus stop’ method for division, it is important to remember to use the correct
place value language with the children.
So with the division sentence 72 ÷ 3,
model the bus stop method clearly showing the TU column headings above and remind the children
that when they say ‘3s into 7’, it is really ‘3s into 7 tens’. The ‘1’ left over is really 1 x 10 and
should be written in beside the unit 2, to show the new number 12 ( the left over 10 + the unit2).
The next question can then be ‘how many 3s in 12?’ and the answer written above.
TU
24
3 712
It may be useful to use the chunking method to prove there are twenty 3s in 72, or by
modelling using a number line, so children understand clearly what the chunks are.
chunking the 10s
24
3 72
-30 (10 x 3)
42
-30 (10 x 3)
12
- 12 ( 4 x 3)
adding the chunks together
3
24
72
60 (20 x 3)
+ 12 ( 4 x 3)
72 ( 24 x 3)
the completed calculation
3
TU
24
712
Steps can be combined to make this more efficient.
3) 72
- 60
12
- 12
0
Answer:
20
(20 x 3)
(4 x 3)
24
Any remainders should be shown as whole number remainder i.e. 14 remainder 2 or 14 r2.
Children need to be able to decide what to do after division and round up or down accordingly.
They should make sensible decisions about rounding up or down after division.
For example 62 ÷ 8 is 7 remainder 6, but whether the answer should be rounded up to 8 or
rounded down to 7 depends on the context. Children should be provided with a variety of ‘real
life’ context problems, so that they practice deciding what to do after division.
e.g. I have 62p. Sweets are 8p each. How many could I buy?
Answer: 7 (the remaining 6p is not enough to buy another sweet)
Apples are packed into boxes of 8. There are 62 apples. How many boxes are needed?
Answer: 8 (the remaining 6 apples still need to be placed in a box)
Stage 6: Greater Efficiency of ‘Chunking’
ARE: Y5+
HTU ÷ U
196 ÷ 6
32 r4
6) 196
- 180 (30 x 6)
16
- 12 (2 x 6)
4
Answer: 32 remainder 4 or 32 r4
Long Division HTU ÷ TU
ARE: Y6
21
972 ÷ 36
36) 972
720
1 1
2 52
180
072
0
Answer:
(20 x 36)
(5 x 36)
(2 x 36)
Use related facts. If we know that 10 x 36 = 360
Then we know 5 x 36 = 180
27
Depending on the context of the problem, remainders may be shown as whole numbers (e.g. people
left standing at a bus stop) or as fractions (e.g. apples, pizzas).
Extend to decimals with up to 2 decimal places. Children should know that the decimal places line
up under each other.
87.5 ÷ 7
12.5
7) 87.5
- 70.0
17.5
- 14.0
3.5
0
Answer:
(10 x 7)
(2 x 7)
(0.5 x 7 – can rephrase as half of 7)
12.5
Compact method
ARE: Y6
Some children may be ready for a compact or short method of division. It is important to refer
to the place value of the digits correctly.
e.g.
H T U
1 3 7 r5
7) 9 26 54
22

Begin by stating the whole calculation.
“We are finding out how many 7s there are in 964”

Decide on an appropriate answer –
“More than 100, less than 150”

“How many 7s in 900?” (NOT how many 7s in nine) This relates back to the ‘chunking’
method.