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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

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:

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

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

'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.

'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.