Written Calculation Methods Hierarchy
The written calculation methods at BIS have been selected and arranged to support the learning of
Mathematics. In demonstrating greater initial value for the process than the outcome, we aim to ensure
that students develop a deep understanding of the four core mathematical operations – to find a
solution, but perhaps more importantly to understand the process involved in doing so.
Not only do the methods themselves become increasingly sophisticated, the numbers students may be
asked to explore should also move hierarchically. Not only progressing from smallest to largest, but also
through decimal fractions and particular numbers used to highlight potential challenges faced when
using certain methods.
It is not unusual for students to be working at differing levels of sophistication in each of the four
operations. In many cases skills honed through addition can be used to assist with more complex
calculations. They are therefore best explored in the order shown below, although each operation may
also be taught in conjunction with its inverse.
Use the links below to skip directly to relevant content.
> Written Calculation Methods Hierarchy: Addition
> Written Calculation Methods Hierarchy: Subtraction
> Written Calculation Methods Hierarchy: Multiplication
> Written Calculation Methods Hierarchy: Division
Requisite Skills
Many of the methods outlined below rely heavily on one or more of the following three key skills:
Counting up and back
Students need to develop the ability to count confidently up and back in units, tens and hundreds from a
variety of starting points and have a clear understanding of the impact of doing so.
Combining and removing
Students need to develop the ability to combine several groups of objects or numbers together and
understand the impact of removing one or more.
Partitioning
Students need to develop the ability to break numbers into equivalent parts. In doing so they should
understand the true meaning of the equals sign.
> Number Facts
Number facts are taught explicitly within the classroom to all students. Non-negotiable outcomes have
been agreed to ensure underlying knowledge of number facts supports mathematical learning. Use the
link above to find out more.
Written Calculation Methods Hierarchy: Addition
Use the links below to skip directly to relevant content:
> Pictures or Marks
Combining two sets
> Using a Number Line
Starting with the largest number, jumping on in appropriate amounts
> Column by Column Addition
Vertical addition beginning with the right hand column, to form totals which are then themselves
added together
> Pure Column Addition
Vertical addition beginning with the right hand column, carrying up to higher value columns
when required
> Back to top
> Subtraction
> Multiplication
> Division
1. Pictures or Marks
Addition: Combining two sets
Pictures and marks can be useful tools in the initial stages of developing a written method for addition.
They underpin the knowledge that addition involves a combination of two or more sets of objects. It
also marks a significant move from literal to figurative understanding.
Steps to success
Combined objects can be depicted by representative images, before moving onto dots or marks which
can then be counted as a single whole group.



Represent the two sets discretely
Combine or ring them
Count up the remaining group
The limitations of this method are clear, particularly when working with numbers of objects larger than
ten, but it remains an important first step in recorded addition.
> Back to Addition
> Phase 2
> Phase 3
> Phase 4
2. Using a Number Line
Addition: Starting with the largest number, jumping on in appropriate amounts
Number lines can offer support to students learning about the process of addition. They encourage
manipulation of information and require an ability to partition numbers into two or more parts.

5+ 8
It can help if you use marked number lines to
begin with

Place the largest number on the left and jump on
above the line in single units
13
8

24 + 38
+10
though - partitioning or breaking up larger
numbers can help you to combine them
+10
+4
38
48
You don’t want to jump on in single units forever
58

The separate parts can then be added in turn
using relatively sized jumps starting with the
62
largest one

When you are feeling confident, you can increase
326 + 485
the size of your jumps, working with numbers
773 + 268
could even try working with vertical number lines
39.7 + 83.6
partitioned into a greater number of parts - you

Before moving onto a more sophisticated method,
try starting your addition by adding the smallest
partitioned part first – does it make a difference?
Steps to success
Number lines should be drawn neatly with a ruler – any marks placed for support should be evenly and
proportionately placed.



Place the larger of the two numbers at the beginning (the left hand side) of the number line
If required, partition the number to be added
Count on each partitioned group in turn marking above the line
Number lines can also be used to model near doubles (such as 8+7) and adjustment (adding 9 being the
same as adding 10 and subtracting 1). These processes however are primarily aimed at developing
mental calculation strategies.
> Back to Addition
> Phase 1
> Phase 3
> Phase 4
3. Column by Column Addition
Addition: Vertical addition beginning with the right hand column, to form totals which are then
added together
Column by column addition is a precursor to pure column addition, providing as it does some clarity as
to the true value of each digit. It also helps to develop a preceding understanding as to why we may
‘carry’ digits when working with more sophisticated methods.
26 + 37
37
+ 26
37
+ 26

Place the larger number above the smaller one –
be careful to line up the digits carefully!

Always begin by combining the smallest digits,
recording your full total underneath the initial
calculation

Other pairs of digits can then be added in turn

Once each column has been accounted for the
totals can be combined for a final solution
13
+ 50
63

94 + 368
77.6 + 247
85.6 + 38.9
When you are feeling confident, you can begin to
work with more unusual pairs of numbers – just
be sure to line their digits with great care!

You might want to move on from this phase quite
quickly as you develop your own shorthand
Steps to success
Columns should be organised carefully, particularly when dealing with two numbers containing differing
numbers of digits.



Place the larger number above the smaller number, making sure to align the units digit
Starting with the smallest digits, sum each pair marking their total clearly below the initial
calculation – columns can be ticked off once completed
Make a final total by combining all individual column totals
> Back to Addition
> Phase 1
> Phase 2
> Phase 4
4. Pure Column Addition
Addition: Vertical addition beginning with the right hand column, ‘carrying’ up to higher value
columns when required
Pure column addition is the most commonly used written method for addition and for many of our
parents, the most familiar. It is both quick and effective, but does not help to clarify the meaning of
addition, nor the process by which two numbers are combined.
24 + 68

Line your digits up carefully according to their size

Always begin by combining the smallest digits,
recording your full total underneath the initial
68
+ 24
calculation
12
1
68
+ 24
92
94 + 67
236 + 187

Any total larger than
9 will require you ‘carry up’
to the next column – when you do, make sure you
record very clearly!

Once each column has been accounted for the
total should be recorded beneath the initial
calculation

When you are feeling confident, you can try to
combine a more complex collection of numbers
which require you to carry more frequently
82.6 + 47.8
Steps to success
Columns should be organised carefully, particularly when dealing with two numbers containing differing
numbers of digits.




Place the larger number above the smaller number, making sure to align the units digit
Starting with the smallest digits, sum each pair
When the total found is larger than ten, the smallest digit is recorded below, while the additional
larger digit is carried into the preceding higher value column
The final solution should be displayed clearly underneath the initial calculation
> Back to Addition
> Phase 1
> Phase 2
> Phase 3
Written Calculation Methods Hierarchy: Subtraction
Use the links below to skip directly to relevant content:
> Pictures or Marks
Removing from a given set
> Using a Number Line
Starting with the largest number, jumping back in appropriate amounts
> Column by Column Subtraction
Partitioned vertical subtraction beginning with the right hand column
> Column Subtraction
Vertical subtraction beginning with the right hand column, partitioning when required
> Back to top
> Addition
> Multiplication
> Division
1. Pictures or Marks
Subtraction: Removing from a given set
Pictures and marks can be useful tools in the initial stages of developing a written method for
subtraction. They underpin the knowledge that subtraction involves the removal of a certain number of
items from a given set. It also marks a significant move from literal to figurative understanding.
Steps to success
Accurate representation of the initial group is key, before moving onto dots or marks which can then be
circled, crossed or rubbed out as they are counted away.



Represent the larger set
Removal or cross marks one by one
Count up the remaining group or count back from the initial set
The limitations of this method are clear, particularly when working with numbers of objects larger than
ten, but it remains an important first step in recorded subtraction.
> Back to Subtraction
> Phase 2
> Phase 3
> Phase 4
2. Using a Number Line
Subtraction: Starting with the largest number, jumping back in appropriate amounts
Number lines can offer support to students learning about the process of subtraction. They encourage
manipulation of information and require an ability to partition numbers into two or more parts.

15 - 7
It can help if you use marked number lines to
begin with

15
8
back below the line in single units

51 - 24
Place the largest number on the right and jump
You don’t want to jump back in single units
forever though - partitioning or breaking up
larger numbers can help you to make larger jumps
27
31
51
41

using relatively sized jumps starting with the
-4
-10
The separate parts can then be subtracted in turn
largest one
-10

When you are feeling confident, you can increase
426 + 218
the size of your jumps, working with numbers
327 + 179
could even try working with vertical number lines
23.4 + 16.7
partitioned into a greater number of parts- you

Before moving onto a more sophisticated method,
try starting your subtraction by taking the
smallest partitioned part away first
Steps to success
Number lines should be drawn neatly with a ruler – any marks placed for support should be evenly and
proportionately placed.



Place the larger of the two numbers at the end (the right hand side) of the number line
If required, partition the number to be subtracted
Count back each partitioned group in turn marking below the line
Number lines can also be used for finding a difference between two numbers. This involves counting up
from the smallest to the largest number. This strategy is most frequently used when two numbers are
close in value, but it a significantly more complex concept that subtracting one group from another and
primarily aimed at developing mental calculation strategies.
> Back to Subtraction
> Phase 1
> Phase 3
> Phase 4
3. Column by Column Subtraction
Subtraction: Partitioned vertical subtraction beginning with the right hand column
Column by column subtraction is a precursor to pure column subtraction, providing as it does some
clarity as to the true value of each digit. It also helps to develop a preceding understanding as to why
we may ‘borrow’ digits when working with more sophisticated methods.
85 - 53

be careful to line up the digits carefully!

85
- 53
Place the larger number above the smaller one You will then need to partition your numbers
carefully, separating out the values of each column
80 + 5
50 + 3

80
- 50
5
-3
30
2
Once placed above one another, partitioned pieces
can then be subtracted, bottom from top
85
- 53

The remaining numbers can then be combined to
form your solution!
32
345 - 172
3 00
- 100
1 40
- 70
5
-2
100
70
3
2

When you are feeling confident, you can work
with more complex numbers, including those that
may require you to move values across columns

If you do need to ‘borrow’ make sure that you are
careful to record the changes in value accurately
always starting with the column of smallest value
Steps to success
Columns should be organised carefully, particularly when dealing with two numbers containing differing
numbers of digits.




Place the larger number above the smaller number, making sure to align the digits
Partition each number in turn, placing relatively sized pieces above one another
Subtract one piece from another, moving from smallest to largest group, ‘borrowing’ where
required
Make a final total by combining all individual column totals
> Back to Subtraction
> Phase 1
> Phase 2
> Phase 4
4. Pure Column Subtraction
Subtraction: Vertical subtraction beginning with the right hand column, ‘borrowing’ from higher value
columns when required
Pure column subtraction is the most commonly used written method for addition and for many of our
parents, the most familiar. It is both quick and effective, but does not help to clarify the meaning of
subtraction, nor the process by which one number may be removed from another.
97 - 46
97
- 46

have to be placed carefully with digits aligned and
larger number above smaller one before you begin

514
- 132
4
the top recording the result underneath the initial
calculation


Once each pair of digits has been accounted for,
the final result should be recorded beneath the
initial calculation

204 - 129
If you need to ‘borrow’ from the previous column,
be sure to adjust both affected columns clearly
382
324 - 136
Starting with the smallest pair of digits over on
the right hand side take the bottom number from
51
1
Similar to column by column subtraction, numbers
When you are feeling confident, you might want
to explore calculations involving place holding
zeros – these require sequential ‘borrowing’!
50.6 - 34.8
Steps to success
Columns should be organised carefully, particularly when dealing with two numbers containing differing
numbers of digits.




Place the larger number above the smaller number, making sure to align the digits
Starting with the smallest digits, subtract the number below from the number above
If this is not possible, you may borrow from the preceding higher value column, making a correction
to the remaining value as you do so
The final solution should be displayed clearly underneath the initial calculation
> Back to Subtraction
> Phase 1
> Phase 2
> Phase 3
Written Calculation Methods Hierarchy: Multiplication
Use the links below to skip directly to relevant content:
> Pictures or Marks
Constructing a given number of sets, moving towards an array
> Using a Number Line
Repeated addition – jumping on from zero
> The Grid Method
Partitioning of numbers before multiplying each on a grid and summing the given totals
> Vertical Multiplication
Partitioning of numbers before multiplying each vertically and summing the given totals
> Back to top
> Addition
> Subtraction
> Division
1. Pictures or Marks
Multiplication: Constructing a given number of sets, moving towards an array
Pictures and marks can be useful tools in the initial stages of developing a written method for
multiplication. It is important to recognise that early multiplication skills build upon the foundations of
core addition knowledge which need to be in place before exploring what is a far more multifaceted
concept. Multiplication will therefore initially be explored in terms of repeated addition.
Steps to success
Multiple sets need to be depicted as distinct groups, before they are ready to be combined. Greater
sophistication can be developed through the way in which marks are gathered. Forming a grid like array
represents a significantly more advanced method of recording and allows for extended exploration of
supporting multiplication as well as associated division facts.



Represent a correct number of same sized sets
Combine or ring them
Count up the remaining group
The limitations of this method are clear, particularly when working with larger arrays, but it remains an
important first step in recorded multiplication.
> Back to Multiplication
> Phase 2
> Phase 3
> Phase 4
2. Using a Number Line
Multiplication: Repeated addition – jumping on from zero
Number lines can offer support to students learning about the process of multiplication. They
encourage manipulation of information and provide a very visual aid for those developing conceptual
understanding while building upon a secure knowledge of addition.

6 x 8
It can help if you use marked number lines to
begin with

Mark a zero on the left hand side before counting
up your first set marking above the line
0

48
correct total number of sets required

35 x 4
5x 4
120
0
You can even turn the calculation around if it
helps you to make fewer jumps!

30 x 4
Continue the process until you have jumped up the
140
Larger numbers can be partitioned before
multiplying to save time

Don’t forget that 30 x 4 = 3 x 4 x 10

When you are feeling confident, you can increase
2
224 x 6
the size of your jumps, working with numbers
467 x 9
working with vertical number lines
partitioned into more parts - you could even try
62.7 x 8
Steps to success
Number lines should be drawn neatly with a ruler – any marks placed for support should be evenly and
proportionately placed.



Place a zero at the beginning (the left hand side) of the number line
If required, partition the number to be multiplied
Count up in sets of the initial number presented - 3 x 4 would mean counting four sets of three
unless the calculation had been manipulated
> Back to Multiplication
> Phase 1
> Phase 3
> Phase 4
3. The Grid Method
Multiplication: Partitioning of numbers before multiplying each on a grid and summing the given totals
The grid method of multiplication is a precursor to vertical multiplication, providing as it does some
clarity as to the process involved in undertaking calculations involving larger numbers.
35 x 6
30

them around an appropriately shaped grid
5

6
x
x
5
180
30
The more pieces a number partitions into, the
larger numbers the grid will need to be!

30
Partition both numbers involved before placing
You then need to fill your grid, multiplying the
two corresponding numbers together

6
The numbers inside the grid then need to be
combined to deliver a final solution
35 x 6 = 210

When you are feeling confident, you can work
with a more complex combination of numbers,
including those with decimal places

Just remember to make sure that your grid is the
right shape!
Steps to success
Columns should be organised carefully, particularly when dealing with two numbers containing differing
numbers of digits.



Partition both numbers and place them around an appropriately shaped grid
Multiply the partitioned pairs, recording the results inside the relevant sections of the grid
Make a final total by combining all individual grid totals
> Back to Multiplication
> Phase 1
> Phase 2
> Phase 4
4. Vertical Multiplication
Multiplication: Partitioning of numbers before multiplying each vertically and summing the given totals
Vertical multiplication is the most commonly used written method for multiplication and for many of
our parents, the most familiar. It is both quick and effective, but does not help to clarify the process by
which one number may be multiplied by another.
46 x 7
46
7
x
6
40
x
x
x
42
280
7 =
7=
356
4
2

multiplication is very similar to the grid method

638 x 9
27.6 x 8
Begin by placing the larger number above the
smaller one with the digits aligned

The small number can then be multiplied with
each of the parts of the number above it, before
the results are combined

You can use notes to avoid the majority of the
addition required above

This time the results are recorded with the
smallest digit placed and the larger digit ‘carried’
2
1424
Although quite different in appearance, vertical

As higher value columns are multiplied out, you
must add any ‘carried’ digits to the result!

When you are feeling confident, you can work
with a more sophisticated combination of
numbers, including those with decimal places
46 x 52
Steps to success
Columns should be organised carefully, particularly when dealing with two numbers containing differing
numbers of digits.



Place the larger number on top of the smaller one, making sure to align the digits
Starting with the digit of lowest value, multiply the smaller number by each of the digits above
Record the smallest digit of each result in the relevant column and any remaining digits above to be
added to any following results
> Back to Multiplication
> Phase 1
> Phase 2
> Phase 3
Written Calculation Methods Hierarchy: Division
Use the links below to skip directly to relevant content:
> Pictures or Marks
A large set split into small groups
> Using a Number Line
Repeated subtraction – jumping back from the largest number
> Chunking: Division though Multiplication and Subtraction
Repeated subtraction – repeated vertical subtraction
> Long and Short Division
Digit by digit division sequence beginning with the largest digit
> Back to top
> Addition
> Subtraction
>Multiplication
1. Pictures or Marks
Division: A large set split into small groups
Pictures and marks can be useful tools in the initial stages of developing a written method for division.
It is important to recognise that early multiplication skills build upon the foundations of core addition
and subtraction knowledge which need to be in place before exploring what is a far more multifaceted
concept. Division will therefore initially be explored in terms of repeated subtraction.
4
4 4
Steps to success
Multiple groups need to be highlighted or circled one at a time from an accurately represented initial set
until an incomplete group or remainder is found. Greater sophistication can be developed through the
way in which marks are initially gathered. Forming a grid like array from the initial set represents a
significantly more advanced method of recording and allows for far more efficient calculation.




Represent the initial set
Remove or cross same sized groups one by one
Count up the number of groups removed
Identify any remainder
The limitations of this method are clear, particularly when working with larger initial sets, but it remains
an important first step in recorded division.
> Back to Division
> Phase 2
> Phase 3
> Phase 4
2. Using a Number Line
Division: Repeated subtraction – jumping back from the largest number
Number lines can offer support to students learning about the process of division. They encourage
manipulation of information and provide a very visual aid for those developing conceptual
understanding while building upon a secure knowledge of subtraction.

24 ÷ 3
begin with - place the number you would like to
divide (the largest one) on the right
24
0
3
3
3
3
3
It can help if you use marked number lines to
3
3

You then need to subtract your smaller number as
many times as you can, recording your jumps
3
under the line – your solution can be found by
8 jumps
counting the number of jumps you have made!

427 ÷ 4
r
are dividing – this means less work for you to do!
-10
3

Any small amount left is called the remainder

When you are feeling confident, you can increase
427
27
4x 6
You can jump back in multiples of the number you
4 x 100
106 jumps
327 ÷ 3
473 ÷ 4
the size of your jumps and work with a more
complex array of numbers - you could even try
working with vertical number lines
623 ÷ 9
Steps to success
Number lines should be drawn neatly with a ruler – any marks placed for support should be evenly and
proportionately placed.



Place the number to be divided at the end (the right hand side) of the number line
Subtract the number you are dividing by and continue to do so as many times as possible
Count up the number of times you have been able to make the subtraction
> Back to Division
> Phase 1
> Phase 3
> Phase 4
3. Chunking: Division though Multiplication and Subtraction
Division: Repeated subtraction – repeated vertical subtraction
Chunking represents an opportunity to simplify and extend preceding number line work, while
developing an effective mental method for more complex calculation.

12 ÷ 3
the number line work you will be familiar with

12
- 3

second subtraction

12
3
-
6
3
-
9
3
-
3
3
324 ÷ 6
-
As the result of the subtraction is uncovered, place
the smaller number underneath and carry out a
-
24
24
Place the larger number above the smaller one with
the digits carefully aligned ready for subtraction
9
324
- 300
Although it looks different, chunking is similar to
Keep repeating the subtraction until you have
nothing left, or a remaining number that is too
= 54
small to subtract from

Counting up the number of times a subtraction
has been made provides you with a final solution

As you become more confident, you can use your
multiplication knowledge to subtract increasingly
large multiples (or chunks) of the number you are
dividing by
50 x 6

4x 6
Try to solve the division calculation using as few
steps as possible!
Steps to success
This method requires good column subtraction skills – numbers should be recorded carefully along with
any annotations to support a final solution.




Place the larger number on top of the smaller one, making sure to align the digits
Find a suitable multiple of the smaller number to subtract, clearly marking the value of the
subtraction in terms of the number you are dividing by
Continue to subtract multiples or ‘chunks’ of the number you are dividing by until you are unable to
remove any more
Count up the number of times you have been able to make the subtraction
> Back to Division
> Phase 1
> Phase 2
> Phase 4
4. Long and Short Division
Division: Digit by digit division sequence beginning with the largest digit
Short division is the most commonly used written method for multiplication and for many of our
parents, the most familiar. It is both quick and effective, but does not help to clarify the process by
which one number may be divided by another.
726 ÷ 3
2
3 726

the smaller one beside it - tackle each digit in turn,
beginning with the one holding greatest value

3 726

If there is a remaining amount, it is recorded below

The second digit from the larger number is then
carried down before you repeat the process


6 313
3
1
Your final solution should be placed neatly above
the calculation
1 2
0 6
5 2 r1
Record the multiple of the smaller number that could
be subtracted from the first digit of the larger one
1
2 4 2
Place the larger number beneath an inverted ‘L’ and
As you become more confident, you can develop a
shortened recording system

Just be careful to keep your annotations clear!
Steps to success
Digits should be appropriately spaced to ensure that columns are easily recorded and any annotations
required noted accordingly



Place the larger number beneath the inverted ‘L’ with the smaller one beside it on the left hand
side
Subtract the largest possible multiple of the smaller number from the first digit of the larger one
recording any remainder beneath
Continue to subtract multiples ‘carrying’ any remaining amounts forward until the final column
where any amount that cannot be subtracted is referred to as a remainder
> Back to Division
Back to top
> Phase 1
> Phase 2
> Phase 3>