Subtraction structures
Partitioning and Reduction: Same number line form: different language and meaning.
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0
25
Partitioning: A quantity/group is split up (partitioned)
in some way.
-6
There are 25 sweets in a tin. Bob takes 6 sweets.
How many are left for Julie?
Reduction: The opposite of augmentation.
Reduce 25 by 6. Start at 25 and count back 6.
Comparison: Using subtraction (or complementary addition) to compare two groups of
objects or quantities.
Key language: What is the difference?
Tank 1
How many more fish are in Tank 1?
Tank 2
Subtraction of the smaller number from the larger number will find the difference.
However, when comparing (for example when using number lines) pupils may use
complementary addition to ‘count up’ to the higher number in order to find the difference.
Subtraction as the inverse of addition:
23
-4
19
23
+4
19
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Subtraction: Number lines
Subtraction/Taking away
Additional notes and guidance.
This policy covers both
subtraction/taking away and
complementary addition. See the
additional guidance in the appendix.
Counting back in ones
Link to a counting stick, multiples
of a given number etc.
Include arrows at either end
and zero whenever possible
to support conversations and
a deeper understanding of number.
Regular increments/spaces
on the number line.
More efficient jumps: Bridging through multiples of 5, 10, 100/partitioning etc.
Pupils should be encouraged to
make informal jottings during
number line work. Horizontal
jottings/simple recordings/written
methods will emerge from
number line work.
From example: 14 – 4 – 5 – 2 = 3
The move to blank number lines;
away from regular increments.
Link to partitioning.
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Complementary addition: KEY WORD = DIFFERENCE
What is the difference between 12 and 5 (or 5 and 12)?
Counting on in ones
Additional notes and guidance.
This policy covers both
subtraction/taking away and
complementary addition. See the
additional guidance in the appendix.
More efficient jumps: Bridging through multiples of 5, 10, 100/partitioning etc.
Finding the difference
The move to blank number lines;
away from regular increments.
Link to partitioning.
64 – 27 = 37
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Informal written methods.
Subtraction/Taking away
As pupils move beyond simple number line work – informal methods may emerge.
Additional notes and guidance.
Subtractive informal, vertical
written approaches are best
illustrated through vertical
number lines.
Link to partitioning.
(Informal written methods may emerge for complementary addition.
Care needs to be taken to emphasise the relationship between these additive approaches
and the concept of subtraction/taking away.)
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Partitioning and partial (subtraction) sums.
Additional notes and guidance.
Partitioning one number:
Progression from horizontal to vertical written methods
Moving partitioning and partial subtraction sums into a vertical form:
Care needs to be taken to make clear
the relationship between the additive
elements of these methods (e.g. the
use of the ‘+’ sign indicating
partitioning) and the concept of
subtraction/taking away.
Note:
No decomposition/exchanging is
needed in this example. This
partitioning approach can be
extended to HTU – TU or HTU –
HTU etc.
Working RIGHT to LEFT
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Partitioning with decomposition/exchanging.
Additional notes and guidance.
Notes: a) Establish that the subtraction can be carried out.
i.e. the smaller number can be subtracted from the larger number.
b) Ask: “How can [the larger number] be written differently?”
This discussion is supported by the language of place value.
Working RIGHT to LEFT
Note:
The word decompose means “to
separate into smaller and simpler
parts or elements”.
The word exchange can be used in
both directions.
“One ten has been exchanged for ten
ones”
“Ten units can be exchanged for one
ten”
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Progression to compact vertical subtraction: Formal decomposition
Notes: a) Establish that the subtraction can be carried out.
i.e. the smaller number can be subtracted from the larger number.
b) Ask: “How can [the larger number] be written differently?”
This discussion is supported by the language of place value.
Additional notes and guidance.
Illustrate partitioned and compact
methods side-by-side.
See demonstration PowerPoint.
Working RIGHT to LEFT
Note:
The word decompose means “to
separate into smaller and simpler
parts or elements”.
The word exchange can be used in
both directions.
“One ten has been exchanged for ten
ones”
“Ten units can be exchanged for one
ten”
Note: We are working RIGHT to LEFT.
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Similar principles to those above can be applied to money and decimals.
For example:
Subtracting/taking away:
Finding the difference using complementary addition:
Compact formal subtraction with decomposition:
The difference
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Appendix
‘Taking away’ vs. ‘Complementary addition’
Note 1: Pupils may not immediately associate “finding the difference” by using
complementary addition with “subtracting” or “taking away”.
Note 2: Pupils should experience both ‘subtractive’ ‘taking away’ and ‘additive’
‘complementary addition’ approaches.
It would be inefficient for a pupil to use ‘taking away’ on a number line to calculate
91 – 85 and likewise it would be inefficient for a pupil to use complementary addition
to calculate 91 – 7, say.
(Clearly mental approaches may be preferable in both cases)
The concept of ‘difference’ can be the link between the two methods.
Context and question format
The setting/context for a question and how a question is asked (format/structure) will
influence how pupils visualise the problem. This may then influence the informal or
formal methods they use. For example: consider the various formats below and how
they may influence a pupils’ visualisation and subsequent working.
What is 45 take away 6?
What is the difference between 45 and 6?
What is the difference between 6 and 45?
Reduce 45 by 6.
A pile of 45 sweet goes down by 6. How many are left?
Jane takes 6 sweets from a jar of 45. How many are left for Daniel?
Jane and Daniel have a jar of 45 sweets. How many are left for Daniel if Jane eats 6?
Jane needs 45 sweets. She has 6 in a packet. How many more sweets does she need?
What must be added to 6 to get 45?
How many bigger is 45 compared to 6?
How many less is 6 compared to 45?
All these questions could be written as ‘45 – 6’.
As with all calculations finding the numerical answer is only the first step.
Understanding and progression is supported when the numerical answer is interpreted
correctly within the given context/question.
Opportunities should always be taken to make links between the different operations:
Example: “The difference between 19 and 7 is 12.
What is 7 add 12?” etc.
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