Contents
Calculation and the Number system: An Introduction
2
Standardisation for the use of the Number Line
5
Addition
Addition: Progression within Written and Mental Methods
The Number Line: Holistic Approach to Calculation (Addition)
The Written Formal Method: Partitioned and Expanded Columns
8
9
10
Subtraction
Subtraction: Progression within Written and Mental Methods
The Number Line: Holistic Approach to Calculation (Subtraction)
Counting on Method
Formal Written Methods: Expanded and Short Column Method
12
12
14
15
Multiplication
Multiplication: Progression within Written and Mental Methods
Mathematical Understanding: Conceptual Developmental Framework
How do I do a sum like 48 x 27? (Child’s Version)
How do I break down a sum like 48 x 27? (Child’s Version)
The Number Line: Holistic Approach to Calculation (Multiplication)
Progression on to Formal Written Methods
17
19
20
21
22
24
Division
Division: Progression within Written and Mental Methods
Division: Whole School Approach
The Number Line: Holistic Approach to Calculation (Division)
Division: Progression of Written Methods
27
28
31
33
Calculation strategies: Summary
36
1
Calculation and the Number System
The Numeracy Framework (1999) has, in the school’s opinion, focused on Calculation at the
expense of the larger more global concept of Number. As a consequence children have
become very proficient at strategies that involve partitioning and use these with a fair degree
of competence but this has led to many children seeing Numeracy as a lesson where
numbers are manipulated in their constituent parts to produce the answer to a calculation.
These are powerful and essential building blocks in a child’s numerical understanding,
however its over emphasis has led to a situation where few children see numbers as a whole.
Hence 427 is seen as 400, 20 and 7 (or worse still 4,2 and 7) as this has become the
precursor to any calculation. However when it comes to a calculation such as 427x12, it may
be easier to solve the calculation as 427x10 and 427x2 as opposed to the partitioning method
(whether this be in the form of the grid or a written long multiplication method.) The truth is
that whilst the framework states that the first question children should ask is “Can I do this in
my head?” the children are too readily reverting to partitioning and related pen and paper
methods. Through this they are losing a feel for number and it is this true sense of the number
system that underpins mathematical understanding.
How do we correct this?
There probably needs to be greater focus on the teaching of number for the sake of number.
The 100 square in the Reception class may not assist the child’s ability to add two digit
numbers but it gives children the holistic feel of where a number like 76 fits into the number
system. Teachers should be aware that activities like this, whilst not necessarily enhancing
the children’s ability to calculate per se, are under-girding children’s understanding of number
and the number system as a whole.
More importantly there needs to be a change of emphasis on the presentation of written
methods to children. The children should meet the written methods through the following
progression:
1. The emphasis in any calculation should be mental in the wherever possible and
children should always be challenged to tackle the calculation without reverting, in the
first instance, to pen and paper.
2. The first written method children should meet in all areas of calculation is the number
line. This is because it is a powerful visual tool and gives provides a supportive
framework for children to develop their own thinking in “drawing” form.
3. Ultimately children will need to be introduced to the more formal written methods.
However these should not be introduced ahead of the number line and other informal
recording methods. The danger is that children get “stuck” on this perceived “adult”
form of calculation and through this lose the sense of number which they can develop
in the earlier strategies outlined above.
The number line acts as a good bridging point between the “Mental” and the “Formal”
methods of calculation. It is interesting to note that although in Year 2 it is introduced as an
introduction to “Pen and Paper methods” in the KS3 strategy it comes under the heading of
“Mental Methods”. It is, in a very real sense, a mental method that is enhanced through the
visual use of the line. It is incumbent upon us therefore to use it as a method that is
supportive of the mental approach and a precursor to the formal written methods.
2
It may well be that the teachers delay the teaching of the written formal methods to a time
later than that suggested in the Numeracy framework. This must be viewed as a positive
move forward rather than being seen in terms of holding children back. The rationale behind
the delay is that children should be kept away from the formal written strand of calculation as
it has a natural tendency to narrow mathematical thinking towards partitioning and restricts
children’s freedom to work with whole numbers. Also experience has shown that once
children learn the “trick” of reducing all number calculations to U+U through partitioning they
are reluctant to return to the key question which should be “Can I do this in my head?” When
children tackle sums such as 2465 + 999 using the column method, it should alert the teacher
to the fact that they may have moved onto the column method too quickly and that children
are not seeing the holistic nature of the numbers involved.
There is an argument for not teaching some of the written methods at all. Certainly the More
Able pupils in Year 6 have proved that in larger calculations involving multiplication and
division the number line has proved infinitely more reliable than the more traditional “column
method” or the partitioning of the grid. This is an area that the school will need to review as it
develops more informal methods. Bearing in mind that no school is an island one major factor
that may well impinge on this is the recognition that Secondary schools have an expectation
that children will be secure in the written methods and this may disadvantage our own
children. However where calculation is taught well, through informal and holistic methods then
the introduction of the written methods should not cause undue concern or misunderstanding.
An evaluation report commissioned by the government to look at the implementation of the
Numeracy strategy found that most children, notably the more able, tended to be overloaded
by the wealth of strategies that the old Framework offered. This led to them, at best, choosing
inefficient strategies for certain calculations and at worse, confusing strategies themselves.
“Schools are recognising, increasingly, the importance of adopting a common approach to
the recording and layout of pupils’ work, but much remains to be done to put policies into
practice.” (Teaching of Calculation in Primary Schools, A report by HMI, April 2002)
It is interesting to note in the light of this that the guidance document (DfES Renewed
Numeracy Framework: Calculation 2006) states in the introduction for each of the four rules
that:
“Children are entitled to be taught and to acquire secure mental methods of calculation
and one efficient written method of calculation which they know they can rely on when
mental methods are not appropriate.”
In one sense we made this decision independently some years ago. The written method
chosen was the number line and we should continue to use this as the “one efficient written
method of calculation”. This method will form the backbone of the curriculum throughout both
Key Stages. Its selection as a foundational strategy was based on the following basis:

It allows the children to remain on the “mental calculation” strand for longer. As stated
above the number line is a good staging post between mental calculation and the
more formal written methods.

The Number line is a powerful visual tool

The Number line puts children in charge of the calculations they undertake when
presented with any given problem. This is more in line with the school’s child centred
philosophy.
3

Related to this is the fact that the Number line allows children to explore calculation for
themselves and is therefore, by definition, more “emergent” in its teaching approach

It has been found (within school) to be the most reliable strategy in terms of accuracy
when used with the more able pupils in particular. This may be because it does not
have as many steps of calculation as its written counterparts. Something the guidance
document recognises when it states… “this expanded method is cumbersome, with six
multiplications and a lengthy addition of numbers with different numbers of digits to be
carried out.” (DfES Renewed Numeracy Framework: Calculation 2006)
4
Standardisation for the use of the Number Line
Basic Guidelines
The Number Line provides a visual map for children to build numerical understanding.
Therefore wherever possible the visual appearance should attempt to reflect the numbers
they represent e.g. the jump of 10 should be 10 times bigger than the jump of one. This
allows the child not only to see the Number Line as a calculation tool but simultaneously gives
them a “feel” for the size and relation of the number themselves.
The totals should be placed along the bottom line whilst the number to be added should be
placed above. This allows children to differentiate what it being added and the numbers they
are adding on. In multiplication this allows children to see x10 and x10 as x20 rather than
x100.
For Multiplication and Division they should also include the multiplying factors as this allows
them to “check” all aspects of the calculation.
23 x 18 = 414
This should be written out in full i.e. 23x10 not simply x10. They can then check whether they
have 18 lots of 23 by totalling the numbers in the jumps. They can check each individual sum
by the number above the jump e.g. is 23 x 10 =230 as the sum implies. The bottom numbers
are left for the running total in accordance with a standard number line. The children should
also be encouraged to include a zero on the number line to confirm the calculation’s start
point. Many will place a 23 there believing this to be part of the calculation.
Addition
The number line should have the numbers to be added across the top and the running totals
underneath.
48 + 36 = 84
Subtraction
5
In the first instance the subtraction number line should start on the right with the highest
number and decrease to the left. This allows children to visually create the concept that
“subtraction is the inverse of addition”.
74 – 27 = 47
However as children move through the school they will be introduced to the “Counting up”
method for calculations such as 99-94 as it is a far more appropriate method. This should be
linked to the concept of “difference” where the number line facilitates this imagery.
Again the numbers being subtracted should be placed on top and the total run along
underneath.
Multiplication
Whereas the Number line could be used for quite basic multiplication methods, it is probably
true to say that these should be limited, as the array provides a far more powerful and potent
image of multiplication. So whilst the number line might include the following method:
4 x 3 =12
Its use should emphasise recording and concept consolidation rather than a tool for
portraying multiplication in a learning context.
The use of the number line for the teaching of multiplication should be confined to the
“holistic” approach to calculation. Meaning that it uses the first number in each calculation as
a whole entity, it is not to be used in conjunction with partitioning, where the grid method and
the formal written methods are both more efficient and effective.
234 x 24
6
This method allows flexibility for the child as they are in control of the calculation. They can
multiply by 10 or any number they wish if they find it easier. The example above uses two
steps of 10 and two steps of two, but could have been done through two steps of 11 if
children found it easier. It has fewer steps than methods involving partitioning and
consequently children have been proved to gain more accurate answers using this approach.
Division
Like its counterpart, subtraction, division in the first instance should be calculated from right to
left on the line.
1344 ÷ 48
As children move towards the top end of the school they can be taught the principle of
“counting up” (as long as there is a secure understanding of the nature of inverse between
multiplication and division). Indeed at the top end of Year 6 the children can solve all
calculations using this method and in one sense can bypass the need for “division sums”
completely. Having said that, although using the number line to “count up” will deliver fast and
accurate results in a test situation, the children need a secure knowledge and understanding
of a range of division methods and these should be taught alongside the number line method.
7
Addition: Progression within Written and Mental Methods
Once the children move into an arena where they need a written method to support their
calculation they should be presented with the number line as a means of recording their
calculation in written form. (See below The Number Line: The Wyche School Holistic
approach to calculation)
The formal written column methods outlined later should be reserved until Year 5 when the
children should be introduced to the method of partitioned and expanded columns. The delay
in the more formal approach is so that true understanding can be secured and the number
line developed as the children’s “default” strategy. This should not hinder more able children
in lower year groups undertaking rigorous and challenging tasks in the area of addition, it
should simply lead to them developing a wealth of efficient mental strategies to solve a range
of calculations.
The Number Line: The Wyche School Holistic Approach to Calculation (Addition)
Stage 1:
Basic Introduction to Number Line
 The number line is a powerful visual
tool to aid the calculation process
Stage 1
The concept of the number line is introduced and the
two numbers added.
8 + 7 = 15
Stage 2: Partitioning (TU)
Stage 2

This calculation allows secure consolidation of the
use of the tens in addition
32+12=44


The main feature here is for children
to gain confidence in the partitioning
and breaking down of numbers into
their constituent parts (e.g. Tens and
Units)
The addition is undertaken using
these constituent parts without
further breaking down the numbers
This consolidates the basic pattern
of adding 10 i.e. 32 + 10 = 42
Stage 3: Partitioning (Bridging TU)

The empty number line helps to
record the steps on the way to
calculating the total.

The examples here show “bridging
through a ten” this builds on work
undertaken above where simple
partitioning of the additional number
is used.
Stage 3
Steps in addition can be recorded on a number line.
The steps often bridge through a multiple of 10.
8 + 7 = 15
48 + 36 = 84
or:
8
Stage 4: Half Partitioning (HTU)

Before moving onto larger numbers
the children should return to the
number line. There is a danger that
once the children understand that
partitioning can reduce any sum to a
series of U+U calculations then they
may well lose sight of the sense of
number.
Stage 4
256+226=482
9
10
The Written Formal Method: Partitioned and Expanded Columns
(These methods to be taught in Years 5 and 6 only)
Stage 1: Partitioning using number (TU)

The teacher may wish the children to
record their partitioning in number
form.
Stage 1
Record steps in addition using partitioning:
47 + 76 = 47 + 70 + 6 = 117 + 6 = 123
47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123
Stage 2: Partitioned Columns (TU)

Partitioning both numbers into tens and
ones mirrors the column method where
ones are placed under ones and tens
under tens. This also links to mental
methods.
Stage 2
Partitioned numbers are then written under one
another:
47  40  7
 76
70  6
110  13  123
Stage 3: Expanded Columns (TU)
Stage 3
Write the numbers in columns.
Adding the tens first:
47
 76
110
13
123



Move on to a layout showing the
addition of the tens to the tens and the
ones to the ones separately. To find
the partial sums either the tens or the
ones can be added first, and the total
of the partial sums can be found by
adding them in any order. As children
gain confidence, ask them to start by
adding the ones digits first always.
The addition of the tens in the
calculation 47 + 76 is described in the
words ‘forty plus seventy equals one
hundred and ten’, stressing the link to
the related fact ‘four plus seven equals
eleven’.
The expanded method leads children
to the more compact method so that
they understand its structure and
efficiency.
Stage 4: Column Method (TU)

In this method, recording is reduced
further. Carry digits are recorded below
the line, using the words ‘carry ten’ or
‘carry one hundred’, not ‘carry one’.
Adding the ones first:
47
 76
13
110
123
Discuss how adding the ones first gives the same
answer as adding the tens first. Refine over time to
adding the ones digits first consistently.
Stage 4
47
 76
123
11
Column addition remains efficient when used with
larger whole numbers and decimals. Once learned,
the method is quick and reliable.
11
Stage 5: Column Method (HTU)

The children should have a clear
understanding of the numbers behind
the calculation. They should therefore
know that the column method is not an
“efficient or effective method for sums
such as 234+99. Similarly they should
still be encouraged to ask the question
“Can I do this in my head?”
Stage 6: Column Method (TH H T U)

The children could move straight onto
the column method for larger numbers
when it is felt that they are fully
conversant with the rationale behind
the column method. That is:
1. They have a strong sense of place
value and are not seeing the sum in
the context of a series of U+U sums in
a variety of columns.
2. They are able to estimate accurately
the calculation they are about to
undertake because they have a “feel”
for the size of the numbers involved.
3. Also they should be dextrous in their
use of more appropriate strategies for
certain calculations. e.g. 2458+999
should be seen as a mental
calculation.
Stage 5
258
 87
345
366
 458
824
11
11
Stage 6
2345
+1467
3812
34,568
24,243
58,811
12
Subtraction: Progression within Written and Mental Methods
Like its addition counterpart the children should be introduced to the numberline as the
primary form of written recording. The numbers should be used holistically; namely that in
278-123, the 278 remains one number whilst the 100, 20 and 3 are partitioned to create the
calculation.
A further development in subtraction is the “Counting up method”. Children will come to see
that in a calculation such as 302-298, the most effective and efficient method is “Counting on”.
However this is a development and should be taught once the principles of subtraction on a
number line are secure. This is to allow children a true understanding of what “taking away”
actually means rather than simply presenting them with an easier option for a given
calculation.
The Number Line: The Wyche School Holistic Approach to Calculation (Subtraction)
Stage 1: Using the empty number line
Basic Introduction to Number Line

The number line is a powerful visual
tool to aid the calculation process
Stage 1
The concept of the number line is introduced and the
two numbers added.
15 - 7 = 8
Stage 2: Half Partitioning (TU)

The main feature here is for children
to gain confidence in the partitioning
and breaking down of numbers into
their constituent parts (e.g. Tens and
Units)

The subtraction is undertaken using
these constituent parts without further
breaking down the numbers
Stage 2
The use of the “whole 10” allows for consolidation of
the place value concept in subtraction
24 – 12 = 12

Subtraction can be recorded using partitioning:
74 – 27 = 74 – 20 – 7 = 54 – 7 = 47
74 – 27 = 70 + 4 – 20 – 7 = 60 + 14 – 20 – 7 = 40 + 7
This requires children to subtract a single-digit
number or a multiple of 10 from a two-digit number
mentally. The method of recording links to counting
back on the number line.
This consolidates the basic pattern of
subtraction of 10 i.e. 24 - 10 = 14
Stage 3: Half Partitioning (Bridging TU)

The empty number line helps to
record or explain the steps in
mental subtraction
Steps in subtraction can be recorded on a number
line. The steps often bridge through a multiple of 10.
15 – 7 = 8
13

A calculation like 74 – 27 can be
recorded by counting back 27 from 74
to reach 47.
74 – 27 = 47 worked by counting back:
The steps may be recorded in a different order:
or combined:
Stage 4: Half Partitioning (HTU)
 The same principle applies with larger
numbers.
 The primary number remains intact and
the digits of the second number are
subtracted from it. These should be
used in a manner that expresses their
value e.g. 300 is 300 not 2.
Stage 4
568 – 344 = 224
Stage 5: Half Partitioning (HTU)
An Alternative to Decomposition
Stage 5: Half Partitioning (HTU)
An Alternative to Decomposition
543 – 378 = 165
 There are issues with this method when
it comes to the secondary number being
larger than the primary, however the
solution (which involves bridging of
“real” numbers) has the ability to
develop greater mathematical
understanding than its paper based
decomposition counterpart, which
focuses on more abstract strategies.
 The solution for most children will lie on
the process of bridging as illustrated in
the second of the two sums opposite.
543 – 378 with bridging
14
Counting on Method
When the concept of subtraction is firmly embedded, the children may be taught the
“Counting On” method as a means of consolidating the fact that addition is the inverse of
subtraction.
The children should develop a good understanding of when the “Counting On” might be a
more appropriate strategy e.g. in a sum such as 304-298 where the numbers are close
together.
It will also consolidate the concept of “difference”.
It should not be taught in isolation as an alternative ‘easier’ method to subtraction.
The Counting on Method
Stage 1: Counting Up Method (TU)


This introduces the children to the
presentation of the “‘Counting On” method
within the context of a numberline
The teaching emphasis should be upon the
concept of “difference”
Stage 2: Partitioning (Bridging TU) with the
Counting On Method

Stage 2
74 – 47 = 27
The Counting on Method can be combined
with the bridging of tens
Stage 3: Partitioning but combining steps

Stage 1
44 – 12 = 32
The number of rows (or steps) can be
reduced by combining steps. With two-digit
numbers, this requires children to be able
to work out the answer to a calculation
such as 30 +  = 74 mentally.
Stage 3
74 – 47 = 27
Stage 4: Partitioning (HTU)
 With three-digit numbers the number of
steps can again be reduced, provided that
children are able to work out answers to
calculations such as 178 +  = 200 and
200 +  = 326 mentally.
 The most compact form of recording
remains reasonably efficient.
Stage 4
Stage 5: Partitioning Decimals
Stage 5

or:
The method can be used with decimals
where no more than three columns are
required. However, it becomes less efficient
when more than three columns are needed.
or:
15
Formal Written Methods: The Expanded and Short Column Method
There is a recognition in the Renewed framework paper on Calculation that:
Partitioning the numbers into tens and ones and writing one under the other mirrors the
column method, where ones are placed under ones and tens under tens. This does not
link directly to mental methods of counting back or up but parallels the partitioning method
for addition. It also relies on secure mental skills.”
If this is so then teachers should not be overly quick to introduce the column method to children as
they will, by the Framework’s own admission, be introducing a new conceptual form of recording
that the children have not yet undertaken.
The children should therefore remain using the number line and exploring the numerical concepts
thoroughly before being introduced to the column method. The school has made a policy decision
that the column method will be taught in Year 6. The rationale for this is that the column method
involves a greater abstraction of number and should therefore only be taught once the more
fundamental concepts are totally secure. This policy will allow children both the freedom and the
time to develop a range of mental strategies and a depth of understanding in the concept. The
column method should be seen as an extension activity rather than a strategy taught alongside its
mental counterpart.
There is an argument that some calculations (basically those that do not involve decomposition
e.g. 784-253) are more straightforward when presented through the column method than the
numberline. This is true, however the primary question related to all sums should be “Can I do this
in my head?” The answer to that question should be “Yes” and to that end the children should
therefore not be using any written method to solve such a calculation.
Progression for teaching the Column Method: The text is taken from the Framework Guidance
(These methods to be taught in Year 6 only)
Expanded layout, leading to column
method
• Partitioning the numbers into tens and ones
and writing one under the other mirrors the
column method, where ones are placed
under ones and tens under tens.
• This does not link directly to mental
methods of counting back or up but parallels
the partitioning method for addition. It also
relies on secure mental skills.
• The expanded method leads children to the
more compact method so that they
understand its structure and efficiency. The
amount of time that should be spent
teaching and practising the expanded
method will depend on how secure the
children are in their recall of number facts
and with partitioning.
Partitioned numbers are then written under one
another:
Example: 74 − 27
70  4
 20  7
60
6 14
14
70  4
 20  7
40  7
7 4
27
4 7
Example: 741 − 367
700  40  1
 300  60  7
600
130
11
700  40  1
 300  60  7
300  70  4
6 13 11
7 41
 3 67
3 74
16
The expanded method for three-digit numbers
Example: 563 − 241, no adjustment or decomposition needed
Expanded method
leading to
563
500  60  3
 241
 200  40  1
322
300  20  2
Start by subtracting the ones, then the tens, then the hundreds. Refer to subtracting the tens, for
example, by saying ‘sixty take away forty’, not ‘six take away four’.
Example: 563 − 271, adjustment from the hundreds to the tens, or partitioning the hundreds
500  60  3
 200  70  1
400  160  3
 200  70  1
200  90  2
400
160
500  60  3
 200  70  1
200  90  2
4 16
5 63
 2 71
2 92
Begin by reading aloud the number from which we are subtracting: ‘five hundred and sixty-three’. Then
discuss the hundreds, tens and ones components of the number, and how 500 + 60 can be partitioned
into 400 + 160. The subtraction of the tens becomes ‘160 minus 70’, an application of subtraction of
multiples of ten.
Example: 563 − 278, adjustment from the hundreds to the tens and the tens to the ones
500  60  3
400  150  13
 200  70  8  200  70  8
200  80  5
400
500
150
50
13
13
500  60  3
 200  70  8
200  80  5
4 15 13
5 6 3
 278
285
Here both the tens and the ones digits to be subtracted are bigger than both the tens and the ones digits
you are subtracting from. Discuss how 60 + 3 is partitioned into 50 + 13, and then how 500 + 50 can be
partitioned into 400 + 150, and how this helps when subtracting.
Example: 503 − 278, dealing with zeros when adjusting
500  0  3
 200  70  8
400  90  13
 200  70  8
200  20  5
400
400
90
100
13
3
500  0  3
 200  70  8
200  20  5
4
9 13
5 0 3
 278
2 25
Here 0 acts as a place holder for the tens. The adjustment has to be done in two stages. First the 500 + 0
is partitioned into 400 + 100 and then the 100 + 3 is partitioned into 90 + 13.
17
Multiplication: Progression within Written and Mental Methods
It is interesting to note that in both the old framework and in the guidance provided with the new
framework there is no requirement for the children to undertake any written calculations in
multiplication before Year 4. All calculations including those such as 32 x 3 = 96 are to be worked
out mentally. This allows the children to gain a good sense of calculation using the “whole number”
and prevents both the early use of partitioning and the potential problem of children seeing
numbers over 10 as a series of digits as opposed to a number in its own right.
The use of the number line is the school’s preferred strategy for calculation in multiplication. The
rationale for this is as follows:
1.
2.
3.
4.
5.
The numberline is used in conjunction with the school’s “half partitioning” strategy. This
allows children to calculate in real numbers, rather than reducing each sum to a series of
unit calculations. (See Appendix: The Partitioning Predicament) This gives them a “feel for
number” which is so crucial in terms of their overall numerical understanding.
Related to this the school has undertaken a brief study of the elements of calculation. (See
Appendix: Elements of Calculation) Whilst it may be true that in certain contexts and
certain scenarios other strategies may be more efficient and/or effective in the short term,
the numberline remains the one strategy that hits nearly all the calculation elements every
time.
The numberline is a very powerful visual tool and children have testified to the fact that it
greatly assists both their ability to calculate accurately and their conceptual understanding
of number calculation
It has proven to be the most effective and efficient strategy when working with children,
especially those in the upper reaches of KS2 where the calculations can become unwieldy
and cumbersome when the more traditional methods are used.
Finally and perhaps the most significantly it marries the school’s desire to have a core
strategy for each of the four rules of number and yet maintains a high level of creativity for
the children to operate within. The following four examples are work undertaken by children
at the top end of KS2. They are seeking to solve the calculation 48x28. Whilst they are all
using the central strategy of the number line they are using a variety of methods to
elucidate the final total.
Eric’s Method
Eric has multiplied by 10 and 10 to get 48 x 20. He has then multiplied 48x2 and used the answer
of 96 as a total to use throughout so that he does not have so many multiplication steps to
calculate. He will, however, have to add up the totals but he is hoping to double 96 and double the
total again as a quicker mental method.
Jane’s Method
Jane has multiplied by 10 again but she has realised that she can double 48 and double it again
easily in her head and has therefore gone for jumps of 48x4.
18
Dave’s Method
Dave has decided that if he knows that 48x10=480 then to calculate 48x5 is relatively
straightforward because it will simply be half of that total i.e. 240. This only leaves him 48x3 which
he decides to take in two jumps; one of 48x2 and the other of 48x1 because he feels he can solve
both of these mentally
Georgina’s Method
Georgina has noticed that 48x28 is only two 48’s away from 48x30. She can calculate the former
mentally and then knows that she only needs to take 96 away from the total. Again she has
appreciated that subtracting 96 is similar to taking away 100 and adding on four.
Concluding Comments on the children’s Calculation Methods
These calculations demonstrate that whilst there is a central core strategy being introduced to the
children by the teacher, there remains much scope for the children to use their own mathematical
understanding to solve the problem in hand. This keeps the teaching well within the “emergent”
continuum and allows children to creatively explore solutions.
One of the aspects of the number line that the children particularly commented on when they were
asked to evaluate its effectiveness is that they felt they were in control of the numbers. If they got
stuck calculating it one way there were always other options for them to turn to. They compared
this with the more traditional column method/grid method where the partitioned numbers are set for
you and allow for no flexibility.
The key to this teaching approach is to focus the children’s minds carefully when choosing the
jumps they wish to use. This is where the creativity is held and where the children can make the
calculation a swift and effective process or conversely a nightmarish scenario of awkward and
complex numbers.
19
Mathematical Understanding: Conceptual Developmental Framework
The Conceptual Developmental Framework was designed to enable easy tracking of children
through the learning process as well as an aid to the continuity and progression between classes
at the end of each academic year.
Mathematical Understanding: Conceptual Developmental Framework
Calculation Progression
Purpose of Strategy
2x3
CDF 1
3+3
Understanding
CDF 2
2x3=6
See it
CDF 3
2x3=6
Know it i.e. Rapid Recall
CDF 4
12 lots of 3
Probably drawn out in an array
Understanding
CDF 5
4 x 3; 4 x 3; 4 x 3 or 2 x 6; 2 x 6
Understanding
CDF 6
10 x 3 and 2 x 3
Effective and Efficient
CDF 7
Written
Number Line not column method
Written/Consolidation
With comprehension straight onto
number line
Written Method
Consolidation or Calculation
Straight onto Number Line
Written Method
Consolidation or Calculation
12 x 3
25 x 12
CDF 8
258 x 24
CDF 9
Notes for Teacher
1. The only place where we should need to model teaching is between CDF (Conceptual
Developmental Framework) 5 and 6 where the children can be introduced to a more
“efficient and effective” method of their own version if they don’t see it for themselves. All
other stages should be scaffolded by the teacher in such a manner that the children learn
through exploration.
2. CDF 1-5 and possibly 6 should have a heavy emphasis on the array as this is the most
visually correct method of “showing” the calculation
3. The number line is introduced as a written method but has the advantage that it can be
used as a calculation tool as well in later stages such as CDF 9.
4. The children should not be introduced to the number line until their concepts of
multiplication and the use of “half partitioning” are in place, especially the half partitioning
of tens and units.
20
How do I do a sum like 48 x 27?
(Child’s Version)
i. Can I do this in my head?
ii. Estimate the answer
iii. Take the sum and half partition the second number 4 8 x 2 7
iv. Draw the number line and decide what jumps you are going to take. Try to make jumps
which will allow you to do all the calculations in your head.
v. Next do the multiplication for all the parts. Remember wherever possible try to calculate
all the parts “in your head”
vi. Then calculate the final total by adding up all the parts you have multiplied. Again
wherever possible try to do these calculations “in your head” where this is not possible
then use pen and paper for some of them
vii. Finally check your answer with your estimate and check it more accurately using a
strategy such as “the inverse operation”
21
How do I break down a sum like 48 x 27?
(Child’s Version)
The children need to see that there are certain ways of breaking down the numbers into
multiples that are more effective than others. For instance, whilst it is possible in the sum
48x27 to take the 48 and multiply it by 3 and 4 to get the 7 required in the units column this is
by no means the most “efficient and effective” method. Children seem at a loss at times to
choose the appropriate means to break down such a sum.
The table below offers the most appropriate calculations to be used for any given number in
the unit’s column. For some there is only one reasonable solution for others there are viable
alternatives, but the children must come to see that those listed below are tried, tested and
the most efficient ones to use. They should fall back on these as a natural default option, this
is all part of the school’s overall strategy to streamline the methods used in calculation.
Whilst the table illustrates how the numbers in the unit’s column may be broken down, the
identical strategy should be adopted when using numbers in the tens, hundreds or thousands
column. So if breaking down 7 into x5 and x2 is deemed “efficient and effective” then the
same would apply to breaking down 70 when x50 and x20 would be used.
Unit Number
Effective and Efficient Operations
Multiply by 1
Multiply by 1
Multiply by 2
Multiply by 2
Multiply by 3
Multiply by 1 and Multiply by 2.
Total
Multiply by 4
Multiply by 2 and Multiply by 2.
Total
Multiply by 5
Multiply by 10 and divide answer
by 2
Multiply by 6
Multiply by 5 and Multiply by 1.
Total
Multiply by 7
Multiply by 5 and Multiply by 2.
Total
Multiply by 8
Multiply by 5, Multiply by 2,
Multiply by 1.Total
Multiply by 9
Multiply by 10 and multiply by 1.
Subtract
Multiply by 10
Multiply by 10
Double number, double and
double again
22
The Number Line: The Wyche School Holistic Approach to Calculation (Multiplication)
Stage 1: Multiplication on a Number Line
Those teachers teaching younger children
may wish to use the number line to illustrate
concepts of multiplication. Whilst we would all
recognise that the array is a far more
powerful visual tool, the use of the number
line in this way will give children a sense of
continuity rather than seeing a number line for
multiplication for the first time later on. It also
allows the school’s format and layout to be
rehearsed.
It is crucial that the children are able to
calculate these mentally and that the number
line is not seen as a calculation tool at this
stage.
Stage 1
4x2=8
12 x 4 = 48
Stage 2: Number Line (TU x U)
The number line can also be used to
calculate in the following manner where a TU
number is followed by a U. This is because
whilst the 12 is being partitioned the 4 is
being used holistically as a whole number.
Stage 2
12 x 4 = 48
Stage 3: Number Line (TUxTU)
The Number line should never be used
where both numbers to be multiplied are
partitioned. In this instance the grid method is
far more appropriate and if used in this way it
will only serve to confuse children when they
are asked to use the numbers holistically. In
this sense this is a single track method that is
only to be used when one number is
partitioned.
In the example illustrated the 27 is kept as a
whole number and multiplied by the 10 and
the 2 respectively.
Stage 3
27 x 12
Stage 4: Number Line (HTUxTU)
The children must be encouraged to see that
the power of this method is that they are in
control of the sum not the numbers. In the
grid method the calculations are determined
by the numbers in the sum. Here the children
have the freedom to choose the calculations
they feel are easiest. This is important to
stress. There are no right or wrong methods
only those that are more “efficient or
effective”.
If children end up using small jumps to make
the sum easier then one must conclude that
the calculation itself is probably at a level
beyond them and the teacher should redress
the level of differentiation offered to that child.
Stage 4
48 x 27
Given freedom to tackle this sum one child might break
down the calculations in to a series of 10’s and then
multiply by 5 and 2.
However another child may find the 5 calculation a little
complex and prefer to multiply by 10 and a series of 2’s
with a single number to finish.
23
Stage 5: Number Line (HTU x HTU)
The children need to see that using 100’s and
10’s in multiplying the numbers is a means of
quickly knocking the sum down to a
manageable size. The key factors remain that
they see themselves as controlling the
calculations and that there are no right or
wrong answers in terms of method
Stage 5
48 x 256 = 12288
24
Progression on to Formal Written Methods
The school has made a policy decision that the formal column method for multiplication will not be
taught until Year 6. The rationale for this is held in the fact that the school believes that to move
children on to the written method prematurely reduces the opportunity for true mathematical
understanding and creates pockets of misunderstanding that teachers will later have to backfill.
This is mainly due to the abstract nature of the calculation where a “full partitioning” of the numbers
reduces them to a series of unit measures.
Therefore the school will only present the formal written calculations to children for whom there is a
wealth of understanding with regard to the whole concept of multiplication; and for whom the
column method might present a more effective and efficient solution for specific calculations. In
this sense it will simply become another tool in the child’s armoury of calculation strategies.
Multiplication: Progression of Written Methods
(These methods to be taught in Year 6 only)
The following stages lean heavily on the material from the Renewed Framework document on
Calculation. The criteria for using these written methods are that the children are secure in their mental
strategies for the calculations they are set. The key question for children to ask remains the same “Can I
do this in my head?” Indeed many of the strategies rely on the use of mental calculation embedded within
them. For instance, the grid method (TUxTU) relies on children being able to mentally add the
calculations in the rows of the grid.
Stage 1: Mental multiplication using
partitioning

Mental methods for multiplying TU × U
can be based on the distributive law of
multiplication over addition. This allows
the tens and ones to be multiplied
separately to form partial products. These
are then added to find the total product.
Either the tens or the ones can be
multiplied first but it is more common to
start with the tens.
Stage 1
Informal recording in Year 4 might be:
Also record mental multiplication using partitioning:
14  3  (10  4)  3
 (10  3)  (4  3)  30  12  42
43  6  (40  3)  6
 (40  6)  (3  6)  240  18  258
Note: These methods are based on the distributive law.
Children should be introduced to the principle of this
law (not its name) in Years 2 and 3, for example
when they use their knowledge of the 2, 5 and 10
times-tables to work out multiples of 7:
7  3  (5  2)  3  (5  3)  (2  3)  15  6  21
Stage 2: The grid method

This is based on the distributive law and
links directly to the mental method. It is
an alternative way of recording the same
steps.

It is better to place the number with the
most digits in the left-hand column of the
grid so that it is easier to add the partial
products.
Stage 2
38 × 7 = (30 × 7) + (8 × 7) = 210 + 56 = 266
25
Stage 3: Grid Method in columns

The next step is to move the number
being multiplied (38 in the example
shown) to an extra row at the top.
Presenting the grid this way helps
children to set out the addition of the
partial products 210 and 56.
Stage 3
38 x 7
Stage 4: Expanded short multiplication
• The next step is to represent the method of
recording in a column format, but showing
the working. Draw attention to the links
with the grid method 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
38 × 7 is ‘thirty multiplied by seven’, not
‘three times seven’, although the
relationship 3 × 7 should be stressed.
Stage 4
38 x 7 = 266
Stage 5: Short multiplication
Stage 5
38
 7
266
• The recording is reduced further, with carry
digits recorded below the line.
• If, after practice, children cannot use the
compact method without making errors,
they should return to the expanded format
of stage 3.
30  8
 7
210
56
266

30  7  210
8  7  56
38
7
210
56
266
5
The step here involves adding 210 and 50 mentally with
only the 5 in the 50 recorded. This highlights the need
for children to be able to add a multiple of 10 to a twodigit or three-digit number mentally before they reach
this stage.
Stage 6: Two-digit by two-digit products
• Extend to TU × TU, asking children to
estimate first.
• Start with the grid method. The partial
products in each row are added, and then
the two sums at the end of each row are
added to find the total product.
• As in the grid method for TU × U in
stage 4, the first column can become an
extra top row as a stepping stone to the
method below.
Stage 6
56 × 27 is approximately 60 × 30 = 1800.
Stage 7: Introduction to the Columns (TU)
Stage 7
56 × 27 is approximately 60 × 30 = 1800.
56
 27
1000
50  20  1000
120
6  20  120
350
50  7  350
42
6  7  42
1512
• Reduce the recording, showing the links to
the grid method above.
1
26
Stage 8: Traditional Column Method
• Reduce the recording further.
• The carry digits in the partial products of
56 × 20 = 120 and 56 × 7 = 392 are usually
carried mentally.
Stage 8
56 × 27 is approximately 60 × 30 = 1800.
56
 27
1120
56  20
392
56  7
1512
1
Stage 9: Three-digit by two-digit products
• Extend to HTU × TU asking children to
estimate first. Start with the grid method.
• It is better to place the number with the
most digits in the left-hand column of the
grid so that it is easier to add the partial
products.
Stage 9
286 × 29 is approximately 300 × 30 = 9000.
Stage 10: Expanded Columns (HTU)
• Reduce the recording, showing the links to
the grid method above.
• This expanded method is cumbersome,
with six multiplications and a lengthy
addition of numbers with different numbers
of digits to be carried out. There is plenty
of incentive to move on to a more efficient
method.
Stage 10
286
 29
4000 200  20  4000
1600
80  20  1600
120
6  20  120
1800 200  9  1800
720
80  9  720
54
6  9  54
8294
1
Stage 11: Traditional Column Method
(HTU)
• Children who are already secure with
multiplication for TU × U and TU × TU
should have little difficulty in using the
same method for HTU × TU
• Teachers must discern whether this is
because the children are following the
“trick” of using numbers as digits (which in
essence there in no problem with) as long
as they have the corresponding
understanding of the calculation as whole
numbers.
• Again, the carry digits in the partial
products are usually carried mentally.
Stage 11
286 × 29 is approximately 300 × 30 = 9000.
286
 29
5720
286  20
2574
286  9
8294
1
27
Division: Progression within Written and Mental Methods
Division remains in children’s minds the most conceptually difficult of the four rules to comprehend.
This is a national as well as a local issue as “difficulties with fully understanding division persist
into secondary school” (Hart, 1981). It has become apparent from our early tracking of children in
KS2 that the issues with division far outweigh those in addition, multiplication and subtraction. As
one child said “It is because in division you have two things to concentrate on; taking away and
your times tables” (Olivia) There is probably much truth in this observation.
On top of that the division calculation involves a greater abstraction of numbers. 32x12 offers two
sums that can be readily seen as 32x10 and 32x2. However whilst 84÷7 can be calculated as
7x10 and 2x7 these calculations are “hidden” from the children as they are not explicit within
the numbers used in the original calculation. Therefore the children must “see” them and
“create” them for themselves. In this sense the calculation for division is potentially more
abstract than for multiplication.
Also whilst in multiplication it is possible for children to enter a transition phase of “Full
Partitioning” on their path to calculating with TU this is not possible in division. As we saw
above, the calculation 32x12 provides the opportunity for children to calculate as follows; 32x10
and 32x2, however solving 384÷12 using “Full Partitioning” is not possible. The child who
calculates 384÷10=38.4 and 384÷2=192 finds the resultant total is 230.4. Hopefully even a
limited understanding of calculation should allow the child to discern that the answer is far
from correct. This reveals a basic lack of understanding as to the process that occurs
physically in the operation of division. This then leads children to lean heavily on prior
knowledge of taught learning strategies from the other four rules, these are often found sadly
wanting when there is no understanding of what is happening within the calculation. This has
important ramifications for the teaching process as it means teachers will not be able to move
through “Full Partitioning” as a staging post, en-route to the more efficient and effective “Half
Partitioning” method.
Similarly the concept of the remainder also adds a further complication as it is hidden in the
calculation until the end, where an additional calculation step is required (often addition and/or
subtraction) before being able to attain the answer. There are no such additional steps within
multiplication, which is based around the much more straightforward idea of “What you see is
what you get.”
In this sense it is not surprising that division remains an Achilles heel for most children.
Therefore there should be a great emphasis placed upon the preparatory work of conceptual
understanding before the child moves on to any written calculating strategies.
28
Division: Whole School Approach
In Key stage 1, the children are introduced to the concept of division through the notion of
“sharing”. This is the foundation for all understanding and is less abstract compared with its
“grouping” counterpart. The reason being that in sharing 12÷4 the child can see how many objects
are to be shared and how many people they are to be shared with. The quotient being how many
each person has in the end.
However the approach the school has adopted at present involves the children having a firm
understanding of the concept of grouping. This is more abstract for whilst the initial number of
objects to be shared is known the number of people to be shared with remains an unknown. This
method relies on the principles of division being seen as a series of repeated subtractions based
on the divisor. So 12÷4 becomes 12 -4 -4 -4 the answer being 3 because it took 3 lots of 4 to get
back to zero from 12. For grouping the key question therefore is “How many 4’s are there in 12?”
rather than “I have 12 how many will 4 people get?”
It is important that children have a clear path set for them as they make the transition between one
concept of division to the other. In recent lessons the children have been taught the difference
between the two and shown clearly that there are two methods by which the quotient can be
calculated.
This is important because the number line method teaches division in relation to its multiplication
counterpart. The key strategy for the calculation 324÷14 will be to answer the question “How many
14’s are there in 324?” and children will be encouraged to calculate this mentally using the number
line as a support framework.
In one sense this is the identical strategy to multiplication, except that they will be starting the
number line on the right and subtracting the calculated totals; thereby seeking to underscore the
relationship between division and repeated subtraction. It is important that the children “see”
division as a pure mathematical concept, rather than using their knowledge of multiplication as a
means to bypass their true understanding of the process of division.
Having said that, the school recognises that this approach does not offer a full replication of the
division process for whilst the method readily replicates the concept of grouping it is not cognitively
secure with regard to sharing. “Division comes in two conceptual frameworks namely sharing and
grouping. Whilst the former is well understood by primary children the latter is not.” (Nunes and
Bryant,1996). In that sense it is a pragmatic approach which offers children a strategy of
calculation, however the school recognises that this approach does not provide a framework for
secure understanding of the concepts behind it. This must be done elsewhere.
The role of concrete materials in the teaching of division will be a key to successful learning. We
have tended to move away from cubes and Dienes material, especially in the upper reaches of
Key Stage 2 but the abstract nature of the subject matter means that there should be much
underpinning of the concepts with as great a variety of practical materials as is possible. The
child’s ability to “see” and visualise 10 groups of 24 will greatly enhance their ability to calculate
effectively when this is transferred to the more abstract number line.
29
I am rapidly coming to the conclusion that the best way to develop a secure cognitive
understanding is actually through the use of practical, real-life topic based maths (as advocated by
the renewed framework). It is only when children experience Maths in real contexts that they can
truly understand what 14 divided by 3 really means, with its groups of 3 and its 2 left over. These
are concepts we should not “teach” in a traditional lesson based environment, as they are too
abstract for children to appreciate the nuances of what is happening to the numbers. For instance,
if you were to ask two pre-school children to share 3 sweets between them despite the apparent
conceptual complexity you soon appreciate they have an intricate working of both division and
remainders! The school is in a good position to deliver this through its desire to reinvent a more
“real-life” based curriculum and the central strand of number-line calculation should therefore be
underpinned by work of a practical nature.
Related to this is the oft-quoted fact running throughout this document that, in the past we have
moved children onto calculation too quickly before understanding is fully in place. We should see
the acquisition of these skills in a whole school context. To this end the curriculum process of
calculation acquisition should be tackled as a long distance race not a sprint. Those who start too
fast often burn out before the end. Whilst we would want to challenge children in every year group
we should resist the pressure (especially in the early years of KS2) to rush into calculation
strategies when the children would probably benefit in the longer term from further work on
developing their understanding of division.
Whole School Approach: Standardisation of setting out a calculation
The school has adopted the above approach for setting out division on a number line. Using
1344 ÷ 48 as an illustration the children should undertake the following:
i. The jumps should outline the “groups of 48” that the children have chosen. The
calculation is written inside the jump i.e. 10 x 48 (10 lots of 48). It should be written as 10
x 48, not 48 x 10 to consolidate the fact that this is 10 groups of 48
ii. The number to be subtracted should be placed underneath the jump. This is so that the
child is quite clear on what they are subtracting.
iii. The number of groups is also placed above the jump. This is so that at the end of the
calculation the children can see clearly how many groups of 48 it has taken to get to zero.
In this instance the answer would be 10 + 10 + 4 + 4.
30
Progression in Upper Key Stage 2
Increasingly within the upper reaches of Key Stage 2 the children should be encouraged to
use a range of mental strategies to support their work on the number line. In the example
above it is hoped that most children working at this level will “know” that 48x2=96 and that
therefore the two jumps of 10 are superfluous and that one jump of 20 would suffice. Similarly
if they can calculate that 48x4 is 192 they should be able to double that total mentally through
a strategy such as 200+200-16.
Thereby creating a number line that looks more like that below:
Increasingly it is hoped that children will be calculating mentally and using the number line
purely as a visual support. In the latter stages of Year 6 it has been noted that the children
dispense with the number line yet still using the principles of half partitioning are calculating
answers mentally alongside informal jottings. So that the above calculation is reduced to
48x20=960 and 48x8= 48x2x2x2 = 96; 192; 384. Therefore the answer is 960+384 This too
they should be able to add mentally, or quickly resort to a column addition sum.
31
The Number Line: The Wyche School Holistic Approach to Calculation (Division)
Introduction: Division on a Number Line
Those teachers teaching younger children
may wish to use the number line to illustrate
concepts of division. Whilst we would all
recognise that there are more powerful tools,
the use of the number line in this way will give
children a sense of continuity rather than
seeing a number line for division for the first
time later in the school. It also allows the
school’s format and layout to be rehearsed.
It is crucial that the children are able to
calculate these mentally and that the number
line is not seen as a calculation tool at this
stage.
Introduction
8÷4=2
Stage 1: Division of TU÷ U using numbers
within the times table
Stage 1
The use of the number line for large multiples
needs to be used judiciously. Whilst it may
consolidate the concept of repeated
subtraction it is not the most powerful tool
either visually, kinaesthetically or
conceptually to illustrate division. It may well
be argued that children at this stage in their
development should be making greater use of
concrete materials and real life situation to
flesh out their basic understanding.
27 ÷ 3 = 9
Stage 2: Partitioning using the number
line TU ÷ U
Stage 2
Where the children have a clear
understanding that the original number is
being shared into groups of the divisor then
they can be encouraged to use multiples of
factors to allow for ease of calculation.
It should be borne in mind that this is a major
cognitive and abstract step and should not be
undertaken without a secure understanding of
the grouping principle.
48 ÷ 4 = 12
32
Stage 3: Division HTU ÷ U
At this stage in their development the children
should have a thorough understanding that
what they are seeking to achieve through the
sum 224 ÷ 8 is to find out how many 8’s there
in 224.
The process could involve “counting up” to
224 and although in essence this is probably
what the children will be doing cognitively, the
presentation on the number line is one of
“repeated subtraction” in the sense that the
numbers are taken away from 224 and
reduced to zero.
The key is to get the children to see that they
can “easily lose” groups of 8 by multiplying by
10. This should allow them to get quickly into
an arena where they can use their knowledge
of times tables to assist the final calculation
i.e. 64÷8.
Stage 3
Stage 4: Division HTU ÷ U with remainder
The number line provides a clear opportunity
for children to understand the concept of the
remainder, as the number that remains when
the total has been sorted into equal groups.
Stage 4
It will also enable a rich discussion when
children are dividing a number where they (for
ease of calculation) take the number into
negative numbers and then seek to calculate
the remainder. In the second example (see
right) the children will often assume that the
remainder is 6.
224 ÷ 8
228 ÷ 8
234 ÷ 8
Stage 5: Division HTU ÷ TU
The principle here is the same as previous;
namely that the children should find creative
ways of “grouping” the numbers so as to
reduce the total down to a place where they
can readily access the final answer through
their knowledge of times tables.
Stage 5
336 ÷ 14
Stage 6: Division ThHTU ÷ TU
As we move into the larger numbers the
opportunities for a wide range of strategies to
solve a given calculation come to the fore. As
with multiplication this allows the children to
develop a range of mental strategies that the
number line can support and thus marry the
need to establish a single default strategy
alongside an enriched creative maths
curriculum.
Stage 6
1344 ÷ 48
33
Division: Progression of Written Methods
(These methods to be taught in Year 6 only)
The following stages lean heavily on the material from the Renewed Framework document on
Calculation. The criteria for using these written methods are that the children are secure in their mental
strategies for the calculations they are set. There is a definite linear progression in the teaching of
division and security of knowledge and understanding of each stage must precede the teaching of the
next. The key question remains the same “Can I do this in my head?” Indeed many of the strategies rely
on the use of mental calculation embedded within them.
As stated elsewhere the children should not be moved on to the written method prematurely as it reduces
the opportunity for true mathematical understanding and creates pockets of misunderstanding that
teachers will later have to backfill.
Stage 1: Mental division using partitioning
Stage 1
• Mental methods for dividing TU ÷ U can be
based on partitioning and on the
distributive law of division over addition.
This allows a multiple of the divisor and the
remaining number to be divided
separately. The results are then added to
find the total quotient.
• Many children can partition and multiply
with confidence. But this is not the case for
division. One reason for this may be that
mental methods of division, stressing the
correspondence to mental methods of
multiplication, have not in the past been
given enough attention.
• Children should also be able to find a
remainder mentally, for example the
remainder when 34 is divided by 6.
One way to work out TU ÷ U mentally is to partition TU
into a multiple of the divisor plus the remaining ones,
then divide each part separately.
Stage 2: Mental Division using the grid
Stage 2
Using the grid method to make links with
multiplication as well as reinforcing the mental
approaches with questions such as ‘How
many sevens in seventy?’ and: ‘How many
sevens in fourteen?’
Another way to record is in a grid, with links to the grid
method of multiplication.
Informal recording in Year 4 for 84 ÷ 7 might be:
In this example, using knowledge of multiples, the 84 is
partitioned into 70 (the highest multiple of 7 that is
also a multiple of 10 and less than 84) plus 14 and
then each part is divided separately using the
distributive law.
As the mental method is recorded, ask: ‘How many
sevens in seventy?’ and: ‘How many sevens in
fourteen?’
Stage 3: Mental Division with expanded
recording (including remainders)
This approach simply allows for a numerical
recording of the stages the children go
through in the partitioning process.
Stage 3: Mental Division with expanded recording
Also record mental division using partitioning:
64 ÷ 4 = (40 + 24) ÷ 4
= (40 ÷ 4) + (24 ÷ 4)
= 10 + 6 = 16
87 ÷ 3 = (60 + 27) ÷ 3
= (60 ÷ 3) + (27 ÷ 3)
= 20 + 9 = 29
34
Stage 4: Mental Division with expanded
recording (including remainders)
This stage includes the use of remainders
Stage 4: Mental Division with expanded recording
(including remainders)
Remainders after division can be recorded similarly.
96 ÷ 7 = (70 + 26) ÷ 7
= (70 ÷ 7) + (26 ÷ 7)
= 10 + 3 R 5 = 13 R 5
Stage 5: Short division of TU ÷ U
Stage 5
• ‘Short’ division of TU ÷ U can be
introduced as a more compact recording of
the mental method of partitioning.
• Short division of a two-digit number can be
introduced to children who are confident
with multiplication and division facts and
with subtracting multiples of 10 mentally,
and whose understanding of partitioning
and place value is sound.
• The accompanying patter is ‘How many
threes divide into 80 so that the answer is
a multiple of 10?’ This gives 20 threes or
60, with 20 remaining. We now ask: ‘What
is 21divided by three?’ which gives the
answer 7.
For 81 ÷ 3, the dividend of 81 is split into 60, the highest
multiple of 3 that is also a multiple 10 and less than
81, to give 60 + 21. Each number is then divided by
3.
81 ÷ 3 = (60 + 21) ÷ 3
= (60 ÷ 3) + (21 ÷ 3)
= 20 + 7
= 27
The short division method is recorded like this:
20  7
3 60  21
This is then shortened to:
27
3 8 21
The carry digit ‘2’ represents the 2 tens that have been
exchanged for 20 ones. In the first recording above it
is written in front of the 1 to show that 21 is to be
divided by 3. In second it is written as a superscript.
The 27 written above the line represents the answer:
20 + 7, or 2 tens and 7 ones.
Stage 6: ‘Expanded’ method for HTU ÷ U
• This method is based on subtracting
multiples of the divisor from the number to
be divided, the dividend.
• For TU ÷ U there is a link to the mental
method.
• As you record the division, ask: ‘How many
nines in 90?’ or ‘What is 90 divided by 9?’
• Once they understand and can apply the
method, children should be able to move
on from TU ÷ U to HTU ÷ U quite quickly
as the principles are the same.
• This method, often referred to as
‘chunking’, is based on subtracting
multiples of the divisor, or ‘chunks’. Initially
children subtract several chunks, but with
practice they should look for the biggest
multiples that they can find to subtract.
• Chunking is useful for reminding children
of the link between division and repeated
subtraction.
Stage 6
97 ÷ 9
9 97
 90 9  10
7
Answer:
10 R 7
6 196
 60 6  10
136
 60 6  10
76
 60 6  10
16
 12 6  2
4
32
Answer:
32 R 4
35
• However, children need to recognise that
chunking is inefficient if too many
subtractions have to be carried out.
Encourage them to reduce the number of
steps and move them on quickly to finding
the largest possible multiples.
Stage 7: Shortened Chunking Method and
Estimation
Stage 7: Shortened Chunking Method and
Estimation
• The key to the efficiency of chunking lies in
the estimate that is made before the
chunking starts. Estimating for HTU ÷ U
involves multiplying the divisor by multiples
of 10 to find the two multiples that ‘trap’ the
HTU dividend.
• Estimating has two purposes when doing a
division:
– to help to choose a starting point for the
division;
– to check the answer after the
calculation.
• Children who have a secure knowledge of
multiplication facts and place value should
be able to move on quickly to the more
efficient recording on the right.
To find 196 ÷ 6, we start by multiplying 6 by 10, 20,
30, … to find that 6 × 30 = 180 and 6 × 40 = 240. The
multiples of 180 and 240 trap the number 196. This
tells us that the answer to 196 ÷ 6 is between 30 and
40.
Start the division by first subtracting 180, leaving 16, and
then subtracting the largest possible multiple of 6,
which is 12, leaving 4.
Stage 8: Short division of HTU ÷ U
Stage 8
• ‘Short’ division of HTU ÷ U can be
introduced as an alternative, more
compact recording. No chunking is
involved since the links are to partitioning,
not repeated subtraction.
• The accompanying patter is ‘How many
threes in 290?’ (the answer must be a
multiple of 10). This gives 90 threes or
270, with 20 remaining. We now ask: ’How
many threes in 21?’ which has the answer
7.
• Short division of a three-digit number can
be introduced to children who are confident
with multiplication and division facts and
with subtracting multiples of 10 mentally,
and whose understanding of partitioning
and place value is sound.
6 196
 180 6  30
16
 12 6  2
4
32
Answer:
32 R 4
The quotient 32 (with a remainder of 4) lies between 30
and 40, as predicted.
For 291 ÷ 3, because 3 × 90 = 270 and 3 × 100 = 300,
we use 270 and split the dividend of 291 into
270 + 21. Each part is then divided by 3.
291 ÷ 3 = (270 + 21) ÷ 3
= (270 ÷ 3) + (21 ÷ 3)
= 90 + 7
= 97
The short division method is recorded like this:
90  7
3 290  1  3 270  21
This is then shortened to:
97
3 2 9 21
The carry digit ‘2’ represents the 2 tens that have been
exchanged for 20 ones. In the first recording above it
is written in front of the 1 to show that a total of 21
ones are to be divided by 3.
The 97 written above the line represents the answer:
90 + 7, or 9 tens and 7 ones.
36
Stage 9: Long division
The next step is to tackle HTU ÷ TU.
Stage 9
How many packs of 24 can we make from 560 biscuits?
The layout on the right, which links to
chunking, is in essence the ‘long division’
method. Recording the build-up to the
quotient on the left of the calculation keeps
the links with ‘chunking’ and reduces the
errors that tend to occur with the positioning
of the first digit of the quotient.
Start by multiplying 24 by multiples of 10 to get an
estimate. As 24 × 20 = 480 and 24 × 30 = 720, we
know the answer lies between 20 and 30 packs. We
start by subtracting 480 from 560.
Conventionally the 20, or 2 tens, and the 3
ones forming the answer are recorded above
the line, as in the second recording.
In effect, the recording above is the long division
method, though conventionally the digits of the
answer are recorded above the line as shown below.
23
24 560
480
80
72
8
Answer: 23 R 8
24 560
20  480
24  20
80
3
72
24  3
8
Answer: 23 R 8
37
The following are an illustrative summary of the sequencing approaches to be used for the four
rules. For brevity they are illustrated by strategies that might be taught to children in the middle
of Key Stage 2. The full progression is outlined in the document “Calculation: School Version”
Addition
The first number in the addition sentence is
kept as a 3 digit figure and only the second
numeral is partitioned into its constituent parts
of 200, 20 and 6. The focus is on the place
value of each digit remaining intact so 200 is
seen as such not a 2 in the hundreds column.
Calculation Exemplar
Subtraction
As with addition counterpart the first numeral
maintains its sense of totality as a single
figure whilst the second number is partitioned
into its place valued digits. The children will
choose a simple counting back or a bridging
strategy where the answer needs bridging
through a multiple of 10
Calculation Exemplar
Multiplication
The first number remains intact whilst the
child creates a series of “sequenced jumps”
to attain the answer. These jumps should be
as logical as possible e.g. For x26 two jumps
of 10, a jump of 5 and a jump of 1 provides a
judicious and efficient strategy
Calculation Exemplar
Division
The key strategy will be to answer the
question “how many 14’s are there in 324?” in
seeking to solve the calculation 324÷14.
Children will be encouraged to calculate this
mentally using the number line as a support
framework. In one sense this is the identical
strategy to multiplication, except that they will
be starting the number line on the right and
subtracting the calculated totals; thereby
seeking to underscore the relationship
between division and repeated subtraction.
Calculation Exemplar
256 + 226
74 - 27
27 x 12
48 ÷ 4
38