Various Division Methods

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NEO Mon, Jan 24, 2022 07:49 AM UTC

Learn how to divide a number using different methods.

1. Written methods for division of whole numbers
The aim is that children use mental methods when appropriate, but for calculations that
they cannot do in their heads they use an efficient written method accurately and with
confidence. Children are entitled to be taught and to acquire secure mental methods of
calculation and one efficient written method of calculation for division which they know
they can rely on when mental methods are not appropriate.
These notes show the stages in building up to long division through Years 4 to 6 - first
long division TU ÷ U, extending to HTU ÷ U, then HTU ÷ TU, and then short division
HTU ÷ U.
To divide successfully in their heads, children need to be able to:
understand and use the vocabulary of division - for example in
18 ÷ 3 = 6,the 18 is the dividend, the 3 is the divisor and the 6 is the quotient
partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in
different ways
recall multiplication and division facts to 10 × 10, recognise multiples of one-
digit numbers and divide multiples of 10 or 100 by a single-digit number
using their knowledge of division facts and place value
know how to find a remainder working mentally - for example, find the
remainder when 48 is divided by 5
understand and use multiplication and division as inverse operations.
Note: It is important that children's mental methods of calculation are practised and
secured alongside their learning and use of an efficient written method for division.
To carry out written methods of division successful, children also need to be able to:
understand division as repeated subtraction
estimate how many times one number divides into another - for example,
how many sixes there are in 47, or how many 23s there are in 92
multiply a two-digit number by a single-digit number mentally
subtract numbers using the column method.
Method Example
Stage 1: Mental division using partitioning
Mental methods for dividing TU ÷ U
can be based on partitioning and on
the distributive law of division over One way to work out TU ÷ U mentally is to partition TU into a
addition. This allows a multiple of multiple of the divisor plus the remaining ones, then divide each
the divisor and the remaining part separately.
number to be divided separately.
The results are then added to find

2. Method Example
the total quotient. Informal recording in Year 4 for 84 ÷ 7 might be:
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 In this example, using knowledge of multiples, the 84 is
been given enough attention. 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
Children should also be able to find separately using the distributive law.
a remainder mentally, for example
the remainder when 34 is divided by Another way to record is in a grid, with links to the grid method of
6. multiplication.
As the mental method is recorded, ask: 'How many sevens in
seventy?' and: 'How many sevens in fourteen?'
Also record mental division using partitioning:
Remainders after division can be recorded similarly.
Stage 2: Short division of TU ÷ U
'Short' division of TU ÷ U can be
introduced as a more compact
recording of the mental method of For 81 ÷ 3, the dividend of 81 is split into 60, the highest multiple
partitioning. of 3 that is also a multiple 10 and less than 81, to give 60 + 21.
Short division of two-digit number Each number is then divided by 3.
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 short division method is recorded like this:
For most children this will be at the
end of Year 4 or the beginning of
Year 5.
The accompanying patter is 'How
many threes divide into 80 so that This is then shortened to:

3. Method Example
the answer is a multiple of 10?' This
gives 20 threes or 60, with 20
remaining. We now ask: 'What is 21
divided by three?' which gives the
answer 7. 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 3: 'Expanded' method for HTU ÷ U
This method is based on subtracting 97 ÷ 9
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
of the divisor that they can find to
subtract.
Chunking is useful for reminding
children of the link between division
and repeated subtraction.
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.
The key to the efficiency of chunking To find 196 ÷ 6, we start by multiplying 6 by 10, 20, 30, ... to find
lies in the estimate that is made that 6 × 30 = 180 and 6 × 40 = 240. The multiples of 180 and 240
before the chunking starts. trap the number 196. This tells us that the answer to 196 ÷ 6 is
Estimating for HTU ÷ U involves between 30 and 40.
multiplying the divisor by multiples
of 10 to find the two multiples that Start the division by first subtracting 180, leaving 16, and then

4. Method Example
'trap' the HTU dividend. subtracting the largest possible multiple of 6, which is 12, leaving
Estimating has two purposes when 4.
doing a division:
o to help to choose a starting point
for the division;
o to check the answer after the
calculation.
Children who have a secure
knowledge of multiplication facts
and place value should be able to The quotient 32 (with a remainder of 4) lies between 30 and 40,
move on quickly to the more as predicted.
efficient recording on the right.
Stage 4: Short division of HTU ÷ U
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
'Short' division of HTU ÷ U can be divided by 3.
introduced as an alternative, more
compact recording. No chunking is
involved since the links are to
partitioning, not repeated
subtraction.
The accompanying pattern is 'How The short division method is recorded like this:
many threes in 290?' (the answer
must be a multiple of 10). This gives
90threes or 270, with 20 remaining.
We now ask: 'How many threes in
21?' which has the answer 7. This is then shortened to:
Short division of a three-digit
number can be introduced to
children who are confident with
multiplication and division facts and The carry digit '2' represents the 2 tens that have been
with subtracting multiples of 10 exchanged for 20 ones. In the first recording above it is written in
mentally, and whose understanding front of the 1 to show that a total of 21 ones are to be divided by
of partitioning and place value is 3.
sound.
For most children this will be at the The 97 written above the line represents the answer: 90 + 7,or 9
end of Year 5 or the beginning of tens and 7 ones.
Year 6.
Stage 5: Long division
The next step is to tackle HTU ÷ TU, How many packs of 24 can we make from 560 biscuits? Start by
which for most children will be in multiplying 24 by multiples of 10 to get an estimate. As
Year 6. 24 × 20 = 480 and 24 × 30 = 720, we know the answer lies
between 20 and 30 packs. We start by subtracting 480 from 560.
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

5. Method Example
'chunking' and reduces the errors that
tend to occur with the positioning of
the first digit of the quotient.
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.

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