Mathematics is essential to everyday life: It is critical to science, technology and
engineering, necessary for financial literacy and most forms of employment. At St Peter’s,
our aim is to provide each child with a high quality mathematics education; a foundation for
understanding the world, the ability to reason mathematically, appreciate the beauty and
power of mathematics and develop a sense of enjoyment and curiosity about the subject.
Children are encouraged to make connections across mathematical ideas to develop fluency,
mathematical reasoning and competency in solving increasingly sophisticated problems.
This information will provide you with some guidance about how written strategies for
calculation are taught at our school in line with the National Curriculum expectations. The
strategies that are taught build upon the mental skills used by the children.
Within Mathematics, your child will be learning through play, using writing, pictures, diagrams
or jottings to support their thinking before they begin to use a formal written method.
Mental and written calculation methods are taught alongside each other. When children are
learning to calculate, emphasis is placed on choosing and using the most efficient method.
The aim of written methods is to help children to solve problems they are unable to do in
their heads – not to replace mental calculation strategies.
It is important that children continue to practise their instant recall facts to enable them to
carry out calculations efficiently and accurately. A sound understanding of place value and
the number system is central to all of the methods.
A correct answer to a problem is important: however a child being able to understand
and explain what they are doing and why they are doing it is of greater importance.
Developing Mental Methods - Learning Through Play (Developed throughout Early Years)
Before children begin to use written methods for addition, they are encouraged to develop a
mental picture of the number system in their heads and will experience practical calculation
opportunities using a wide range of equipment. This example shows how they will develop
ways of recording calculations using pictures.
One bear and two more bears
makes
one,
two,
three bears altogether.
Children will experience a range of representations such as number tracks and number lines,
as well as using visual resources such as hundred squares and bead strings.
Number track
1
Bead string
100 square
2 3 4 5 6 7 8 9 10
Number line
0
1
2
3
4
5
6
7
8
9
10
Mental strategies with jottings (Introduced in KS1 and developed throughout KS2)
A labelled number line
Children begin to use numbered lines to support their calculations counting on in ones.
They select the biggest number first and count on in ones.
Example:
8 + 5 = 13
There is progression to using number lines where only the 5’s and 10’s are labelled which then
further develops where intervals are marked but not labelled.
An empty number line
Blank number lines are visual representations which are used as informal jottings to support
mental calculations. They are an important tool when solving calculations. Bead strings are
also used as a tactile resource to support the use of the number line. In this example
children are required to bridge through a multiple of 10. Therefore, seven is partitioned into
2 and 5, so that 2 creates a number bond to 10 with 8 and then the 5 is added on.
Example:
8 + 7 = 15
Children may continue to use this method in future year groups, learning how to count
forward and backwards in 1s or 10s, selecting ‘jumps’ which they are comfortable with.
Example:
34 + 23 = 57
Children are encouraged to count on from the largest number. The jumps that the child uses
become more efficient as their understanding of the number system develops: knowing key
number bonds (e.g. numbers that add together to make 10 or 100) will support this process
and is also vital in more advanced methods.
Mental Methods that lead to Formal Written Methods - Partitioning
Children learn how to partition numbers using their knowledge of place value. They will also
use a range of practical apparatus, such as arrow cards and dienes, to support their
calculations. As children become more confident, they may record fewer jottings: this is an
important step in understanding the compact column method.
Example:
(Arrow cards)
48 + 36 = 84
(Dienes)
When talking your child through
this method, it is important to
reinforce the relative value of
the digits.
i.e. The 4 in 48 is worth 40 etc.
Standard Written Method (Introduced and Developed throughout KS2)
Once children have a firm understanding of the previous methods, the standard written
method for addition, adding the least significant digits first, is introduced. To ensure that
this method is grounded in understanding, a range of practical equipment may still be used.
The vocabulary of exchanging is used when crossing the tens barrier with ones. In this
method, recording is reduced and the digits that are grouped and exchanged are recorded.
Example:
348
+ 275
623
1 1
“8 units and 5 units together
make 13 units, so I will group 10
of my units and exchange them
for 1 ten. I now have 1 ten and 3
units.”
Remember to continue to reinforce the
place value of the digits involved.
Vocabulary – counting on, add on,
partitioning, exchanging and grouping
Column addition remains efficient when used with larger numbers and with decimals and once
learned, is quick and reliable. However it is not always the most efficient. Later, the method
is extended when adding more complex combinations such as several numbers of different
sizes. It is extremely important that your child understands and can explain what they are
doing.
Developing Mental Methods – Learning through Play (Developed throughout Early Years)
Before children begin to use written methods for subtraction, they are encouraged to
develop a mental picture of the calculation in their heads and will experience practical
activities to model these using a variety of equipment. They will develop ways of recording
calculations using models and pictures.
The ‘difference between’ is introduced through practical situations and images.
Mental Strategies with Jottings
Number Lines (Introduced in KS1 and developed in KS2)
Finding out how many items are left after some have been ‘taken away’ is initially supported
with a number track followed by labelled, unlabelled and finally empty number lines, as with
addition. Subtraction is the inverse to addition.
Number track
1 2 3 4 5 6 7 8 9 10
Number line, all numbers labelled
0
1
2
3
4
5
6
7
8
9
10
A labelled number line
Linked with previous experiences, this is used to support calculations where the answer is
less by counting back (ie taking away).
The number line is also used to make comparisons
between numbers. It can be used to show the
‘difference in value between 6 and 3’ or the
‘difference between 6 and 3’.
An empty number line
Steps in subtraction can be recorded on a number line and often bridge through a multiple of
10. In the example below, the seven is partitioned into five and two. This enables children to
count back to 10, a “friendly number”, before taking away the two.
Example:
15 – 7 = 8
COUNTING BACK
In the example below, children are counting back from one number to arrive at the answer.
The steps may be recorded in a different order or combined as in the example below.
Example
74 – 27 = ?
These examples show how children are taught to use jumps of different sizes and in an order
that is most helpful depending on the numbers they are calculating with.
COUNTING ON
The steps can also be recorded by counting on from the smaller number to the larger number
to find the difference. The example below counts up from 27 to 74 in steps totalling 47.
This is useful for comparisons;
For example: Jill has knitted 27cm of her scarf, Alex has knitted 74cm. How much longer is
Alex’s scarf?
Example:
74 – 27 = 47
The ‘jumps’ should be added, either mentally or with jottings according to confidence,
beginning with the largest number.
Mental Methods that lead to Formal Written Methods
(Introduced and Developed in KS2)
It is important when beginning to subtract in columns that the children have a secure
understanding of place value and partitioning. The expanded method leads the children to
the more compact column method so that they understand its structure and efficiency.
Example:
54 - 37 = 17
40
50 +
- 30 +
10 +
10+
4
7
7
Developing the concept of exchanging: in this example the 50 is shown as
four tens and ten ones which is important to understanding the
decomposition in vertical subtraction. For this example, this is now “fortyfourteen”.
Throughout the use of this method, practical resources such as Dienes and
place value counters are essential to model exchanging tens and units.
Standard Written Method (Introduced and Developed in KS2)
Finally, children complete the compact columnar subtraction method as the most efficient
form.
Example:
4
1
5 4
- 3 7
1 7
Column subtraction remains efficient and once confident, children
are extended to use solve calculations with larger numbers.
Helping your child to learn their times tables is very important. Children need to be able to
recall multiplication and related division facts with confidence and speed.
They are shown how multiplication is repeated addition.
For example:
Using multiplication as repeated addition
4 + 4 + 4 + 4 + 4 = 20
5 lots of 4 = 20
5 x 4 = 20
5 + 5 + 5 + 5 = 20
4 lots of 5 = 20
4 x 5 = 20
Also if you know 4 x 5 = 20, then you
also know that....
5 x 4 = 20
20 ÷ 4 = 5
20 ÷ 5 = 4
This knowledge, along with a secure understanding of place value,
helps children to find a whole variety of number facts from each
multiplication fact they can recall.
The more confident your child is with tables, the easier they will
find multiplying and dividing bigger numbers and those involving
decimals.
Using place value
5 x 40 = 200
4 x 50 = 200
200 ÷ 5 = 40
200 ÷ 40 = 5
40 x 50 = 2000
0.5 x 4 = 2
and so on.
Multiplying and Dividing by 10, 100 and 1000.
To support their ability to multiply and divide larger numbers, children need to be able to
multiply and divide by 10, 100 and 1000.
It is important that they understand the effect of the multiplication on the digits (rather
than learning to just “add a zero” which will not work when introducing decimals).
When multiplying by 10, each digit gets 10 x bigger, with the effect of all the digits moving
one place to the left (the decimal point stays still). Likewise, when dividing by 10, each digit
gets 10 x smaller – moving one place to the right.
Examples:
37 x 10 = 370
25 ÷ 10 = 2.5
H T U
3 7
3 7 0
T U.t
2 5
2 .5
Children learn to see how division and multiplication are inverse operations.
Developing Mental Methods – Learning through Play
(Introduced in Early Years and developed in Year 1)
Before children begin to use written methods for multiplication, they will experience equal
groups of objects and see everyday versions of arrays such as egg boxes, ice cube trays etc.
5 + 5 = 10
10 – 5 = 5
2 x 5 = 10
10 ÷ 5 = 2
5 x 2 = 10
10 ÷ 2 = 5
An array is a set of objects or
numbers arranged in order, often in
rows or columns. They make counting
and calculating easier.
Children will experience practical calculation opportunities involving equal sets or groups using
a wide variety of equipment including small world play, counters and cubes. Through practical
activities, children will develop ways of recording calculations using models and pictures.
Mental Methods with Jottings
Repeated addition (Introduced in KS1 and developed throughout KS2)
Children will use repeated addition to carry out multiplication supported by the use of
objects such as counters or cubes. Children will use pictorial representations to support
their calculations.
Examples:
5 +
2+2+2=6
or
3+3=6
Arrays
5
+
5
= 15
10p + 10p + 10p + 10p + 10p = 50p
10p x 5 = 50p
It is important to be able to visualise multiplication as a rectangular array.
In this way, children learn about the commutative law of multiplication
ie. 3 x 4 = 4 x 3. The relationship between the array and the number line
showing both repeated additions should be demonstrated alongside each
other.
4 lots of 3 = 12
4 x 3 = 12
3 lots of 4 = 12
3 x 4 = 12
Mental Methods with Jottings
Grid Method (Introduced and Developed in KS2)
Introduction to the Grid Method
Using an array, children look for ways to split them up
using number facts that they are familiar with. Over
time this leads to children partitioning two digit
numbers into tens and ones, making the link to grid
multiplication which is a pre cursor to short and long
multiplication.
After lots of practical experience with a focus on mental strategies using jottings, arrays
and repeated addition, children will be introduced to the ‘grid method’.
Children move to the grid method without arrays, once they can confidently explain the
relationship between an array and its corresponding grid, even when the array is no longer
visible.
Example:
18 x 13 = 234
x
10
3
10
100
30
8_
80_ =
24 =
180
54
234
1
Having calculated the sections of the grid,
children will then add the sub totals
together to arrive at the answer. Children
will choose the most efficient method to do
this, which could include jottings, informal
or formal written methods.
Formal Written Method - (Introduced and Developed in KS2)
Short Multiplication
Example: 23 x 7
20 + 3
x
7
140 (20 x 7)
21 (3 x 7)
161
Leading to:
23
x
7
161
Expanded Short Multiplication
The first step is to represent the method of recording in a column
format but showing the working. This highlights the links with the
grid method as described above. Children should describe what they
do by referring to the actual values of the digits in the columns. For
example, the first step in 23 x 7 is ‘twenty multiplied by seven’, not
‘two times seven’, although the relationship 2 × 7 should be stressed.
Short Mutiplication
The recording is reduced further. Children learn to start from the
right hand side, with the carried digits recorded.
2
Long Multiplication
Example: 53 x 16
53
x 16
18 (6 x 3)
300 (6 x 50)
30 (10 x 3)
500 (10 x 50)
848
Expanded Long Multiplication
To ensure understanding, links are made to the grid method and
teachers initially model both methods side by side to highlight the
relationship. There is an emphasis on making sure that each part of
each number is multiplied by each part of the other number.
Children are moved towards working from right to left along the
bottom number and are taught to maintain this systematic approach
to multiplication to ensure that mistakes are not made by ‘missing’
parts.
Long Multiplication
Each digit continues to be multiplied by each digit, but the totals are recorded in a more
compact form, using ‘carrying’. Children should be able to explain each step of the process,
initially relating it back to previous methods and experiences. They should be able to
articulate the different stages of this calculation with the true values of the digits they are
dealing with.
Example:
124 x 26=
1
2
124
x 26
744
2480
3224
1 1
6 x 4 = 24 so record the 4 in the units and carry the 20 (2) into the tens.
6 x 20 = 120 + (the carried) 20 = 140.
Record the 40 in the tens and carry the 100 (1) into the hundreds column.
6 x 100 = 600 + (the carried) 100 = 700. Record the 7 in the hundreds.
The children will then understand that 6 x 124 = 744
20 x 4 = 80, so record this on a new row in the correct columns.
20 x 20 = 400. Record the 4 in the hundreds column.
20 x 100 = 2000. Record the 2 in the thousands column.
The children will then understand that 20 x 124 = 2480
Using column addition, children add the two totals together. Final answer is 3224.
Developing Mental Methods – Learning Through Play (Introduced in
Early Years and developed in KS1)
Before children begin to use written methods for division, they are
encouraged, through practical experiences, to develop physical and mental
images. They begin to make recordings of their work (with adult support)
using pictures. The vocabulary of sharing and grouping is used.
In this example, 12 strawberries are shared into 4 equal groups so there are 4 groups of 3.
Mental Methods That Lead To Formal Written Methods – Use Of Arrays (Introduced in
KS1 and developed throughout KS2)
After lots of practical experience of sharing items equally and organising items into equal
groups, children will explore division as the inverse of multiplication. Division is taught as the
inverse of multiplication and a secure knowledge of multiplication facts is essential in
developing more advanced and efficient written methods for division.
Arrays are used to help reinforce the link between
multiplication and division.
The vocabulary used for division is dividend, divisor and
quotient so for example, 56 ÷ 7 = 8.
Dividend ÷ Divisor = Quotient
Children then begin to construct the arrays using place
value equipment to represent the dividend. With adult
support, this leads the children towards the method of short division.
Example: 363 ÷ 3
In this example, to represent 363, children have selected
3 place value counters with a value of 100, 6 place value
counters with a value of 10 and 3 place value
counters with a value of 1.
Children will then organise the counters equally into 3
groups. Children are encouraged to use the arrangement
of an array as this will support their understanding when
moving into formal written methods.
In this example, children can see that when 363 is divided
into 3 equal groups, there is 1 place value counter worth
100, 2 place value counters worth 10 and 1 place value
counter worth 1. Therefore, there are 3 groups of 121.
Alongside this visual image, children will record the division as;
This then becomes more complex when an exchange is needed.
Example: 345 ÷ 3
In this example, the four, place value counters with a value of 10, cannot be grouped equally
into 3. There is one counter too many. Therefore, that counter is exchanged into 10 place
value counters with a value of 1.
Becomes ..
Then the counters are organised into the arrangement of an array.
Alongside this visual image, children will record the division as;
Short Division (Introduced and developed in KS2)
Once children have developed a sound understanding of division, using practical resources,
they will use the formal written method of short division.
Example:
432 ÷ 5 becomes
With short division, children are expected to
Answer:
‘internalise’ the working described in the previous
86 remainder 2
methods.
Long Division (Introduced in Year 6)
This method is used for calculations involving
larger numbers. (ie when dividing by a 2 digit
number). Children may choose to record the
‘chunks’ alongside to help them calculate the
final answer.
For example
432 ÷15 =
In this example:
We ask ourselves ‘How many groups of 15
hundreds can we make?’ None, so we exchange
the 4 hundreds for 40 tens.
‘How many groups of 15 tens can we make?’ The
answer is 2 which is equivalent to 300. We
record the 2 and subtract the 300 from 432.
This leaves us with 132 ‘ones’. ‘How many groups
of 15 can we make with these?’ The answer is 8,
which is equivalent to 120. We record the 8 and
subtract 120 from 132. This leaves us with 12
‘ones’. ‘How many groups of 15 can we make with
these?’ None, so we have 12 left over.
Therefore 432 divided into equal groups of 15
is 28 with a remainder of 12.
Children will start to interpret the ‘remainder’ in the most appropriate way to the context of
the question.
Remainders
As children become more confident with division, they will be expected to record their
quotient (answer) in different ways.
Example:
21 ÷ 4 = 5 remainder 1
However, the remainder also needs to be divided by 4
Therefore, the quotient could be written as …
21 ÷ 4
=5r1
=5¼
= 5.25
Word Problems
With division, as with all calculations, it is important to think about what the actual problem
is asking, when you come to record an answer. It is especially important when considering a
remainder - does it need to be rounded up or down, or given as a fraction or decimal?
Some example word problems involving division may be:


Alan has 45 beans. If he plants 4 beans in each pot, how many pots would he need in total?
The answer to the calculation of 45 divided by 4 equals 11 remainder 1, but the answer to
the problem needs to be rounded up. Alan would need 12 pots to plant all of his beans.
Alan has 13 eggs. He puts them into egg boxes that can hold 6 eggs. How many full boxes
will he have?
The answer to the calculation of 13 divided by 6 equals 2 remainder 1, but the answer to
the problem needs to be rounded down. Alan would only be able to fill 2 full boxes with
eggs.
Imagery to support solving word problems using bar modelling
To support children’s conceptual understanding of word problems, bar model images have been
introduced. The mantra ‘if in doubt, draw it out’, is encouraged.
Example 1,
Alan has 44 beans. If he plants 4 beans in each pot, how many pots would he need in total?
44 beans
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
4 in
each
pot
There are 11 pots.
Example 2,
Alan has 44 beans and he plants them into 4 pots, how many beans are in each pot?
44 beans
? in each pot
Answer of 11
? in each pot
Answer of 11
? in each pot
Answer of 11
? in each pot
Answer of 11
Example 3,
Alan spends two fifths of his pocket money on a CD. The CD cost him £10, how much pocket
money does he get?
1/5
1/5
Money spent on a CD = £10
£5
£5
Pocket money total = ?
1/5
1/5
1/5
£5
£5
£5
Therefore Alan gets £25 pocket money.