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PROGRESSION THROUGH CALCULATIONS FOR DIVISION

Knowing and using number facts

(ongoing)

Using multiplication facts

Year 1 Count on or back in ones, twos, fives and tens and use this knowledge to

derive the multiples of 2, 5 and 10 to the tenth multiple;

Recall the doubles of all numbers to at least 10

Year 2 Derive and recall multiplication facts for the 2, 5 and 10 times-tables and the

related division facts; recognise multiples of 2, 5 and 10;

Derive and recall doubles of all numbers to 20, and the corresponding halves

Year 3 Derive and recall multiplication facts for the 2, 3, 4, 5, 6 and 10 times-tables

and the corresponding division facts;

Recognise multiples of 2, 5 or 10 up to 1000

Year 4 Derive and recall multiplication facts up to 10 10, the corresponding division

facts and multiples of numbers to 10 up to the tenth multiple

Calculate doubles of multiples of 10 and 100 and derive the corresponding

halves

Year 5 Recall quickly multiplication facts up to 10 10 and use them to multiply pairs

of multiples of 10 and 100;

Derive quickly corresponding division facts

Year 6 Use knowledge of place value and multiplication facts to 10 10 to derive

related multiplication and division facts involving decimals (e.g. 0.8 7, 4.8 6)

Using and applying: To support planning, refer to the renewed framework and

curricular targets on using and applying multiplication and division strategies.

Target link

YR Related objectives: Count repeated groups of the same size; Share objects into equal

groups and count how many in each group, e.g.

Add trays with small compartments for sorting to the making area. Add collections of

things: bottle tops, sequins, threads, tiny pieces of fabric, etc. Model sharing out the

objects equally. For example: do you all want sequins? I'll put 5 each on your trays. Can you

give everybody the same number of these? Have you got the same?

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Hang up 3 bags outside for making collections. Put a number 2 on each bag. Encourage the

children to collect 2 of any treasured object in each bag, for example fir cones or smooth

pebbles. The collections could be used inside and outside in the learning environment for

different purposes, for example as a gallery of natural objects or for adding to the making

area.

Y1 Related objectives: Solve practical problems that involve combining groups of 2, 5 or 10, or

sharing into equal groups

Children will experience equal groups of objects and will count in 2s, 5s and 10s. They will

work on practical problem solving activities involving equal sets or groups,

e.g. If the frog hops in 2s, how many hops will there be

before he lands on 10?

Here are 20 counters. Arrange them in equal rows. Is there

a different way to arrange them in equal rows?

15 children sit at 3 tables. There is the same number of children at each table. How many children sit at each table?

Millie had 6 toffees, she gave half to her friend. How many toffees do they each get?

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Y2

Related objectives: Represent repeated addition and arrays as multiplication, and sharing

and repeated subtraction (grouping) as division; use practical and informal written methods

and related vocabulary to support multiplication and division, including calculations with

remainders

Use the symbols , -, , and to record and interpret number sentences involving all four

operations; calculate the value of an unknown in a number sentence (e.g. 2 6, 30 - 24)

Children will develop their understanding of multiplication and use jottings to support

calculation:

Sharing equally

Use sharing to answer division questions; Suppose 15

pencils were to be shared out between three children.

How many pencils would each child get? Explain to me

how you could work it out.

Experience divisions that give rise to remainders, such as:

Three friends share 16 marbles equally. How many marbles does each friend get? How many marbles are left over?

Grouping or repeated subtraction

Use practical equipment or objects to answer questions such as: How many 2s make 12?

Relate this to the division 12 2.

Use objects or a number line to support, record or explain this.

For example, starting from 12, jump back in steps of 2, or starting with 12 counters, keep

on taking away 2 counters.

Record this as repeated subtraction and as division:

12 - 2 - 2 - 2 - 2 - 2 - 2 0

12 2 6

12 divided by 2 equals 6

Children explain how they use equipment, objects or a number line to carry out division.

Or count forward, e.g. How many tens make 80?

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Repeated subtraction using a number line or bead bar

Show me on a number line how you could do: 12 ÷ 3 = 4

3 3 3 3

The bead bar will help children with interpreting division calculations such as 12 ÷ 3 as ‘how

many 3s make 12?’

Using symbols to stand for unknown numbers to complete equations using inverse

operations

÷ 2 = 4 20 ÷ = 4 ÷ = 4

A number of marbles

divided between 2 groups

gives each group 4 each

20p is divided between some

children. Each child gets 4p.

How many children are

there?

On a number line, I do four

equal jumps. What numbers

could I land on?

Y3

Related objectives: Multiply one-digit and two-digit numbers by 10 or 100, and describe

the effect;

Use practical and informal written methods to multiply and divide two-digit numbers (e.g.

13 3, 50 4); round remainders up or down, depending on the context;

0 1 2 3 4 5 6 7 8 9 10 11 12

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Understand that division is the inverse of multiplication and vice versa; use this to derive

and record related multiplication and division number sentences;

Sharing

Children understand division as sharing. They solve problems such as:

42 crayons are divided equally between six pots. How many crayons are there in each pot?

Three children want to buy their grandmother a present costing 1.50. They each give the same amount. How much does each child give?

An 80 cm length of ribbon is cut into four equal pieces. How long is each piece?

Repeated subtraction

Children review multiplication as repeated addition and division as repeated subtraction

by counting hops on a number line. For example, they find how many fours make 24, either

by counting on or back 6 hops of 4.

Children divide a number of objects by using grouping . They understand that one way

to find 30 6 is to find how many sixes there are in 30 . Through practical experience,

they understand that some division calculations have a remainder, for example 13 4 3 R 1:

Grouping ITP may be a useful resource

Children understand the relationship between multiplication and division . For example,

they state two multiplication sentences and two division sentences that relate to a

particular array, for example: Multiarray ITP

5 2 10, 2 5 10

10 2 5, 10 5 2

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They use the image of an array to explain why, for example, 2 5 gives

the same answer as 5 2. They also use the image to show how many fives

make 10 and how many twos make 10.

Children should use number lines or bead bars to support their understanding.

0 6 12 18 24

Remainders

Children work out calculations that divide exactly and those that give rise to remainders.

They discuss the images in the ITP 'Remainders'.

Children decide whether to round up or down to answer word problems such as:

We have 21 building block wheels. How many four-wheeled cars can we make?

Peaches come in packs of six. I want 20 peaches. How many packs do I need to buy?

How many 30 cm lengths of ribbon can I cut from a ribbon measuring 2 metres?

Children model such problems with objects or draw a sketch to help them. They discuss

their answers and give reasons why they decided to round up or down.

Using symbols to stand for unknown numbers to complete equations using inverse

operations

6 6 6

6 6 6 6

6

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x 5 = 20 3 x = 18 x = 32

20 ÷ 5 = 18 ÷ 3 = 32 ÷ =

complete this division

in as may ways as you can.

Y4

Related objectives: Multiply and divide numbers to 1000 by 10 and then 100 (whole-

number answers), understanding the effect; relate to scaling up or down

Develop and use written methods to record, support and explain multiplication and division

of two-digit numbers by a one-digit number, including division with remainders (e.g. 15 9,

98 6)

Children will develop their use of repeated subtraction to be able to subtract multiples of

the divisor. Initially, these should be multiples of 10s, 5s, 2s and 1s – numbers with which

the children are more familiar.

72 ÷ 5

-2 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5

0 2 7 12 17 22 27 32 37 42 47 52 57 62 67 72

Moving onto:

-50

_______________________________________________________

0 2 7 12 17 22 72

or counting forward(linked to multiplication)

Solve: 72 5 using jumps on a number line

-2 -5 -5 -5 -5

10 1 1 1 1

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or

Leading to mental division using partitioning

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.

Another way to record is in a grid, with links to the grid method of multiplication.

As the mental method is recorded, ask: ‘How many sevens in seventy?’ and: ‘How many

sevens in fourteen?’

Could also be recorded vertically, for example:

When dividing 64 by 4 children approximate first. They recognise that the answer must lie

between 40 4 10 and 80 4 20, and use this approximation to do a calculation such as:

64 ÷ 4 = 16

40 (4 x 10)

24 (4 x 6)

And similarly for remainders

72 ÷5

50 (5x10)

20 (5x4)

2

14r2

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Y5

Related objectives:

Recall quickly multiplication facts up to 10 10 and use them to multiply pairs of multiples

of 10 and 100; derive quickly corresponding division facts

Extend mental-methods for whole-number calculations, for example to multiply a two-digit

by a one-digit number (e.g. 12 9), to multiply by 25 (e.g. 16 25);

Use understanding of place value to multiply and divide whole numbers and decimals by 10,

100 or 1000

Refine and use efficient written methods to multiply and divide HTU U, TU TU, U.t U

and HTU U

Use understanding of place value to multiply and divide whole numbers and

decimals by 10, 100 or 1000

Use calculators (possibly by setting a constant function) or the ITP 'Moving digits' to

explore the effect of repeatedly multiplying/dividing numbers by 10.

Answer questions such as:

32 500 325

How many 10 notes would you need to make 12 000? Find the missing number in 0.42 42. Play 'Stepping stones': Work out what operation to enter into a calculator to

turn the number in one stepping stone into the number in the next stepping stone.

use and discuss mental strategies for special cases of harder types of

calculations, for example to work out

- using factors

e.g. 90 ÷6 360 ÷5 £2.48 ÷4

90 ÷ 3 ÷ 2 360 ÷10 x2 £2.48 ÷2 ÷2

(90 ÷ 3) ÷ 2 (360 ÷10) x2 (£2.48 ÷2) ÷2

30 ÷ 2 36 x 2 £1.24 ÷2

Ans: 15 Ans: 72 Ans: 62p

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Refine and use efficient written methods to multiply and divide HTU U, TU

TU, U.t U and HTU U Short division of TU ÷ U

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 (3x20)

21 (3x7)

ans=27

The short division method is recorded like this:

20 7

3 60 21

This is then shortened to:

2

2 7

3 8 1

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.

6 196

60 6 10

136

60 6 10

76

60 6 10

16

12 6 2

4 32

Answer: 32R 4

To find 196 ÷ 6, 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.

6 196

180 6 30

16

12 6 2

4 32

Answer: 32R 4

The quotient 32 (with a remainder of 4) lies between 30 and 40, as predicted.

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Y6

Related objectives:

Calculate mentally with integers and decimals: U.t U.t, TU U, TU U, U.t U, U.t U;

Use efficient written methods to add and subtract integers and decimals, to multiply and

divide integers and decimals by a one-digit integer, and to multiply two-digit and three-

digit integers by a two-digit integer

Calculate mentally with integers and decimals: U.t U.t, TU U, TU U, U.t

U, U.t U;

Use the fact that 96 6 = 16

to solve 9.6 6 = ∆

or 9.6 0.6 = ∆

Calculate 3600 25

(3600 100) x2 x2

(36 x2) x2

72 x2

Ans: 144

84 6

60 6x10

24 6x4

Ans: 14

9.5 5

95 5 =19

so 9.5 5=1.9

Use efficient written methods to add and subtract integers and decimals, to

multiply and divide integers and decimals by a one-digit integer, and to multiply

two-digit and three-digit integers by a two-digit integer

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 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:

2

9 7

3 2 9 1

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.

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A ribbon is 87.6m long. It is cut into 6

eqaul pieces. How long is each piece?

Start by multiplying 6 by multiples of 10

to get an estimate. As 6 × 10 = 60 and

6 × 20 = 120, we know the answer lies

between 10 and 20 packs. We start by

subtracting 60 from 87.6

6 87.6

60.0 6 x10

27.6

24.0 6 x4

3.6

3.6 6 x0.6

Ans: 14.6m

How many packs of 24 can we make from

560 biscuits? 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.

24 560

20 480 24 20

80

3 72 24 3

8

Answer: 23 R 8

so 23 packs of 24 biscuits can be made.

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

Solve'missing-number' problems, using their knowledge of inverse operations:

Link to earlier trio work, e.g.

50 ÷∆= 2.5 = 371X

By the end of year 6, children will have a range of calculation methods, mental and

written. Selection will depend upon the numbers involved.

Children should not be made to go onto the next stage if:

1) they are not ready.

2) they are not confident.

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Children should always be encouraged to approximate their answers before calculating.

Children should always be encouraged to consider if a mental calculation would be

appropriate before using written methods.