Addition and Subtraction
This sheet gives an overview of addition and subtraction strategies used in our school and how these progress to
‘formal written strategies’ as expected by the end of Key Stage Two. Often, the strategies of addition and subtraction
are taught alongside each other or, at a similar time as learning the opposite (inverse) operation helps us consolidate
our understanding.
Addition – using a number line and 100 square
Pupils begin by using a number line to add
numbers and are taught to start with the
largest number first. Such as 5 + 4 here:
Using a 100 square, pupils count on along the columns as shown
and also can add multiples of ten by jumping down the rows.
This helps when visualising and also when applying this in
reverse to
subtract.
Pupils move
to addition
where, when
adding, the numbers cross ‘tens boundaries’ which means that in order
to find the total on a number line, you cross over a multiple of ten.
Here pupils add the multiples of ten first by counting in tens and then
add the remaining jumps.
Addition – partitioning
Here, when adding 243 + 725, numbers are partitioned into their ‘place
value holders’ which means that we show these as their separate hundreds,
tens and units values. This layout begins
to look like the standard written method
that we would be familiar with.
We start with numbers which won’t need
‘carrying’ when adding. As this skill is
refined, pupils move to show ‘carried’ amounts which are placed in the next ‘place
value holder’ (see right). As this becomes fluent, we begin to use the same layout
when adding numbers involving decimals. In this strategy, it is made really clear exactly what is happening when we
‘carry’ and what the values of these digits are.
Addition – ‘formal written strategies’
When adding, the expectation is that by the end of Key Stage Two, the strategies used are ‘formal, written’ which means
that they look the way we would expect them to as adults – numbers stacked on top of each other and added underneath
starting with the units on the right.
Now, pupils will be familiar with this strategy and will be fluent when using it to
add amounts of money and other
measures involving decimals such as
kilometres or grams.
Now that we are familiar with exactly what is happening to the digits within these
calculations, they look more formal and the amounts we add grow – by the end of
Year 6, we are able to add numbers including millions and to three decimal places
(three digits after the decimal point).
Subtraction – using a number line and 100 square
When subtracting, children become familiar with counting
backwards to subtract (or take away) amounts such as 15 –
2 shown here. Note here that when counting back, this is
shown under the number line as opposed to on top when adding.
This strategy will help when counting back in larger jumps as they become more
confident.
Using the opposite as we would when adding, pupils use a 100 square to subtract such as
counting back along the columns when finding 29 – 6 or, when taking 20 away from 78 by
counting back up the rows. This helps children to visualise the jumps and also emphasise
how the numbers get smaller each time.
Now, using number lines without individual
numbers shown (right), pupils can apply their
‘counting back’ skills and their ‘partitioning’
skills to subtract numbers which ‘cross
tens boundaries’.
The number line is also used when ‘counting up’ to find the difference (shown
left). Here, when finding 70 – 56, we ‘find the difference’ by counting up (adding)
from the smallest number to the largest and then counting the jumps we make.
Here, you will see that we jump to the next ten and then jump on from here. This emphasises the inverse link between –
and +.
Subtraction – using a number line to find the difference and partitioning
Here, counting on to find the difference on a number line is developed further
and used when subtracting increasingly large numbers and those involving
decimals. Here, we continue to jump to the next ten (or whole number) and then
make larger, sensible jumps in tens or known tables facts.
To begin progressing towards a ‘formal written layout’, we use practical
resources (shown right) to show exactly what happens when we subtract
numbers and to reiterate the value of each digit in a
number. Here, when calculating 346 – 123, we would
show this using practical ‘Deanes’ materials alongside
a partitioned example of the calculation:
Showing these calculations using practical resources is vital when securing an understanding of
what is happening to the numbers when we subtract, particularly as we move towards a layout
which is increasingly familiar and formal such as that shown left.
Subtraction – ‘formal written strategies’
As strategies become increasingly formal, yet numbers become increasingly
more complex, we still refer back to early skills such as the use of the number
line when subtracting. This is because this supports visualising ‘counting on’
when calculating mentally. Here, when subtracting decimals, we count to the
next whole before continuing to jump on to find the difference.
Now, strategies are becoming increasingly formal and to
ensure that pupils are completely secure with the layout and
the process, we use place value headers (hundreds, tens and
units etc) so that exchange is emphasised and reinforced. We
then begin to show our subtractions in a compacted, efficient
and ‘formal’ way (shown right). Here, this shows a secure understanding as more than
one exchange is needed to complete this calculation.