1. Mathematics
Calculation Policy Document

2. Mental and written calculation progression
It is important that children’s mental methods of calculation are practised on a regular basis and secured alongside their learning and use of written methods of calculations.
The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads, they use a written method accurately and with confidence.
At each stage in their learning, children are taught and learn secure mental methods of calculation in addition to a written method, which they know they can rely on when mental methods are not appropriate.
This policy shows the stages of each written method of calculation which children progress through towards building the most efficient method for them to use.

3. Using and applying
Using and applying is a key theme and one of the aims of National Curriculum and before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts, these include:
using inverse
missing box questions
using units of measure including money and time
word problems
open ended investigations

4. Addition and Subtraction
There are some key basic skills that children need to help with addition and subtraction, which include:
Counting
Estimating
Recalling all addition pairs to 10, 20 and 100 along with their inverses (7 + 3 = 10, 10 – 3 = 7, = 20, 20 – 3 = 17, = 100, 100 – 30 = 70)
Knowing number facts to 10 and their inverses ( = 8, = 6)
Subtracting multiples of 10 ( ) using the related subtraction fact, , and their knowledge of place value
Partitioning two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways (432 into and also into )
Understanding and using subtraction and addition as inverse operations

5. Progression in Addition
Context based experiences at each level of development - money, measures, real-life
Developmental (FS & Yr1)
Expanded (Yr 2)
Compact (Yr 3/4)
Standard
Algorithm
(Yr 4/5/6)
MM – Larger number 1st
3 + 4 = 7
Use of objects, number tracks &
number lines + =
See progression – counting to calculating
= 27
Number lines – secure partitioning 2nd number & counting on T then U 14
+10 1 +3 =
30 + 3
20 + 3 50 6 56 =
40 + 6
20 + 8 60 74 =
40 + 3
169
1 2 6
+ 4 3 9
6 0
1 0 0
1 6 9
2 8
1 1
7 1
171.23 1
U+U TU+U TU+TU HTU+U HTU+TU HTU+HTU ThHTU U+0.t t+0.t U+0.t h
Mixed whole numbers
& decimals

6. + Misconceptions Models & Images Linked Vocabulary Add More Sum Total
Make
Greater
Plus
Addition
Increase
Not estimating first to see if their answer ‘makes sense’
Setting out when working in columns –
confusion over the place value
Confusion of ‘teen’ and ‘ty’
Using in number line – count start number so calculation is out by 1 +

7. Progression in Subtraction (removal from set, decomposition)
Context based experiences at each level of development - money, measures, real-life
Developmental (FS,KS1)
Expanded (Yr 3/4)
Standard
Algorithm (Yr 4/5/6)
7 – 4 = =
10 4 20 3 23 - =
40 6
20 8 10 8 18 - 30 1 =
Use of objects, number tracks & number lines
4 6
1 8 1 3
See progression – counting to calculating
= 13 =
40 3
80 3 83 -
120 =
4 2 9
1 3 2
2 9 7 3 1 -
Number lines – secure partitioning 2nd number & counting back T’s then U
= 57 57 60 64 84 -3 -4
-20
U-U TU-U TU-TU HTU-U HTU-TU HTU-HTU ThHTU U-0.t t-0.t U-0.t h
Mixed whole numbers
& decimals

8. Misconceptions Models & Images Linked Vocabulary Take Take-away Leave
Not estimating first to see if their answer ‘makes sense’.
Incorrect setting out when working in columns – confusion over the place value
Confusion of ‘teen’ and ‘ty’
Using number line incorrectly – count starting number so calculation is out by 1
Misunderstanding regarding place value and concept of exchanging T for ones, H for Tens etc
Not understanding that subtracting a number will result in a smaller number
Children switch the digits around to be able to ‘do’ the calculation (believe it is commutative as with +/x)
Take
Take-away
Leave
Left
Fewer
Less than
Decrease
Difference between
Minus
Subtract
Subtraction

9. Multiplication and Division
There are some key basic skills that children need to help with multiplication, which include:
counting
estimating
understanding multiplication as repeated addition
recalling all multiplication facts to 12 × 12 (by the end of Year 4)
partitioning numbers into multiples of one hundred, ten and one
working out products (70 × 5, 70 × 50, 700 × 5, 700 × 50) using the related fact 7 × 5 and their knowledge of place value
adding two or more single-digit numbers mentally
adding multiples of 10 ( ) or of 100 ( ) using the related addition fact, 6 + 7, and their knowledge of place value
adding combinations of whole numbers
understanding and using division and multiplication as inverse operations

10. Progression in Multiplication (Short multiplication)
Context based experiences at each level of development - money, measures, real-life
Developmental (FS, KS1)
Expanded (Yr 3)
Compact (Yr4)
Standard
Algorithm (Yr 5/6)
Grid Method
3 x 5 = 15
3 x 15 =
Use of objects, number tracks & number lines. Link initially to repeated +
1 5
x 3
1 5 (3x5)
3 0 (3x10)
4 5
1 5
x 3
4 5 1 3 10 5 30 15
= 45 + +5 3
100 20 6
300 60 18
= 378
3 x 126 =
Representation as an array using a variety of apparatus (Dienes, pegs, counters etc)
1 2 6 x
1 8 (3x6)
6 0 (3x20)
(3x100)
3 7 8
1 2 6 x
3 7 8 1
4 x 6 =
4 x 13 =
NB. As children become secure in this
method there may no longer be a need
to write the calculations in brackets
UxU UxTU UxHTU UxThHTU Ux0.t U+0.t h
Mixed whole numbers
& decimals

11. Progression in Multiplication (Long multiplication) Year 5 and 6 only
Context based experiences at each level of development - money, measures, real-life
Expanded
Standard
Algorithm
Compact
Developmental
A written method of long multiplication is not required until Year 5
Grid Method
1 4
x 1 3
1 8 2
1 2 (3x4)
3 0 (10x3)
4 0 (10x4)
(10x10)
14 x 13 = 10 3 4
100 40 30 12
= 182
1 4
x 1 3
4 2 1
1 4 0
1 8 2
14 x 123 = 10
100 3 4
1000 80 30 12
= 1722 20
200
400
NB. As children become secure in this
method there may no longer be a need
to write the calculations in brackets
1 2 3 x 1
1 2 (4x3)
8 0 (4x20)
(100x4)
3 0 (10x3)
(10x20)
(100x10)
1 2 4 x
7 4 4 1 2
TUxTU TUxHTU ThHTUxTU Decimals up to 2dp x whole numbers

12. Misconceptions Models & Images Linked Vocabulary Repeated addition
Groups of
Lots of
Multiply
Times
Multiplication
Product
Array
Understanding of multiplying by 10/100
and what happens to place value of the number
Rapid recall of multiplication tables is not secure and impacting of accuracy of calculation
Interpretation of digits in the T/H columns as single digits eg 4x3 instead of 4x30

13. Progression in Division (short division)
Context based experiences at each level of development - money, measures, real-life
Developmental (FS & KS1)
Expanded (Yr 3&4)
Compact (Yr3&4)
Standard
Algorithm (5/6)
School will need to decide to
‘chunk’ either forwards or backwards
15 ÷ 5 = 3
Children are encouraged to
chunk with larger multiples of the
divisor → 10x, 5x, 20x, 4x 5
7 5
5 0
2 5
1 5
‘Stand alone’ method
→ for efficiency
75 ÷ 5 = 15
(10 lots of 5) 75 50
10x 2x 60 70 1x
= 15x
3’s into 8 goes 2x with 2 remainder
Use of objects, number tracks & number lines. Link initially to repeated subtraction
(5 lots of 5) 3
8 5 2 8
r 1
63 ÷ 4 = 15 r3 63 40
10x 5x 60
= 15x r 3
15 ÷ 5 = 3 15 5 10 1 2 3
3 groups of 5
I need to get from 0 to 15 in jumps of 5 +5 -5
I need to get from 15 to 0 in jumps of 5
3’s into 25 go 8 remainder 1
575 ÷ 5 = 5
5 7 5
5 0 0
7 5
5 0
2 5
1 1 5 12 1 2 r4
12 into 1 won’t go
12s in 12 is 1 remainder 2
12 into 28 is 2 remainder 4 85 84
85 ÷ 3 = 28 r1 60
20x 8x
= 28x r1
Remainders should also be recorded as a decimal or fraction
(100 lots of 5)
(10 lots of 5)
If moving to ÷ 3digit number by a single digit,
then the compact method should be used.
(5 lots of 5)
TU ÷ U → HTU ÷ U → ThHTU ÷ U → U.t ÷ U → U.t h ÷ U → TU.u ÷ U → HTU ÷ TU

14. Therefore they MUST have an alternative method they can use.
Progression in Division (long division) Year 6 only
Context based experiences at each level of development - money, measures, real-life
Developmental
Expanded
Compact
Standard
Algorithm
Secure division strategies for short division, recall and application of known & derived facts and making sensible approximations to support accuracy when calculating 23
1 5 r1
(5 x 23)
(1 x 23) 33
2 9 r10
- 6 6
1 0
(X20)
(X8)
Remainders should also be recorded as a decimal or fraction 15 4 2 13 8
r12
15 into 4 won’t go
15 into 43 is 2 remainder 13
15’s in 132 is …
Children will need to be aware that the method for long division will not work with every calculation e.g.
Therefore they MUST have an alternative method they can use. 24
2 3 7
At each stage children may
need to make jottings to
support the use of known and
derived facts when calculating

15. Misconceptions Models & Images Linked Vocabulary Divisor Divisible
Divide
Group
Share
Chunk
Remainder
Sharing / shared
Equal groups
Lack of understanding of ‘remainders’ and their importance to the context of the problem
Insecure understanding of place value to know what each digit is representing
Unable to derive facts from known facts and ‘play’ with numbers
Approximations are wildly inaccurate so answers cannot be judged in the context of the problem/calculation
No method to ‘fall back’ on where use of a formal method won’t work

17. National Curriculum 2014, Progression through the PoS: +/-
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Pupils should be taught to:
 read, write and interpret mathematical statements involving addition (+), subtraction (–) and equals (=) signs
 represent and use number bonds and related subtraction facts within 20
 add and subtract one-digit and two-digit numbers to 20, including zero
 solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems
solve problems with addition and subtraction:
 using concrete objects and pictorial representations, including those involving numbers, quantities and measures  applying their increasing knowledge of mental and written methods
 recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100
add and subtract numbers using concrete objects, pictorial representations, and mentally, including:
 a two-digit number and ones
 a two-digit number and tens
 two two-digit numbers  adding three one-digit numbers
 show that addition of two numbers can be done in any order (commutative) and subtraction of one number from another cannot
recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems.
add and subtract numbers mentally, including:
 a three-digit number and ones
 a three-digit number and tens
 a three-digit number and hundreds
add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction
estimate the answer to a calculation and use inverse operations to check answers
solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction.
add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate
estimate and use inverse operations to check answers to a calculation
solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why.
add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)
add and subtract numbers mentally with increasingly large numbers
use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy
solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why.
use their knowledge of the order of operations to carry out calculations involving the four operations
solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

18. National Curriculum 2014, Progression through the PoS: x/÷
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Pupils should be taught to:
solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher
recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers
calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs
show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot
solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in context
recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables
write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods
solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects.
recall multiplication and division facts for multiplication tables up to 12 × 12
use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together three numbers
recognise and use factor pairs and commutativity in mental calculations
multiply two-digit and three-digit numbers by a one-digit number using formal written layout
solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects.
multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers
multiply and divide numbers mentally drawing upon known facts
divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context
multiply and divide whole numbers and those involving decimals by 10, 100 and 1000
solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes
solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign
solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates.
multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context
divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context
use their knowledge of the order of operations to carry out calculations involving the four operations
solve problems involving addition, subtraction, multiplication and division
use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.

19. Progression across the year groups - Addition
Typical calculations
Suitable methods Y1
U+U
TU + U (to 20 including zero)
Practical
Number line Y2
TU + U
TU + multiples of 10
TU + TU
U + U + U
Expanded columnar Y3
HTU + U
HTU + TU
HTU + HTU
Compact columnar Y4
THTU + HTU
THTU + THTU Y5
THTU.t + THTU.t
THTU.th + THTU.th
Standard algorithm Y6
THTU.tht + THTU.tht

20. Progression across the year groups - Subtraction
Typical calculations
Suitable methods Y1
U-U
TU - U (to 20 including zero)
Practical
Number line Y2
TU – U
TU - multiples of 10
TU - TU
U - U - U
Expanded columnar Y3
HTU – U
HTU - TU
HTU - HTU Y4
THTU – HTU
THTU - THTU
Standard Algorithm Y5
THTU.t - THTU.t
THTU.th - THTU.th Y6
THTU.tht - THTU.tht

21. Progression across the year groups - Multiplication
Typical calculations
Suitable methods Y1
U x U
Practical (repeated addition)
Practical and pictorial arrays Y2
Practical (repeated addition), pictorial arrays
Expanded grid Y3
TU x U
Compact Y4
HTU x U Y5
THTU x U
TU x TU
Standard Algorithm Y6
HTU x TU
THTU x TU
U.t x U
U.th x U
U.t x TU

22. Progression across the year groups - Division
Typical calculations
Suitable methods Y1
U ÷ U
TU ÷ U
Practical sharing
Number-line grouping Y2
Practical sharing, Number-line grouping
Expanded method Y3
Expanded method (Short)
Compact Y4
HTU ÷ U
Compact (remainders to be expressed as r) Y5
THTU ÷ U
Standard Algorithm - Short (remainders to be expressed as r, then as a fraction and as a decimal) Y6
 HTU ÷ TU
THTU ÷ TU
 U.th ÷ U
TU.th ÷ U
HTU.th ÷ U
THTU.th ÷ U
 Standard Algorithm - Long (remainders to be expressed as r, then as a fraction and as a decimal)
 Standard Algorithm Standard Algorithm - Short (remainders to be expressed as a decimal)