Manley Park Primary School Calculation Policy Handbook Division.

Manley Park Primary School Calculation Policy Handbook

Division

Division Strategies

• 15 ÷ 5 - Pictures/Tallies (Sharing)• 20 ÷ 4 - Jottings (Sharing)• 54 ÷ 2 - Halving• 54 ÷ 4 - Halve twice• 380 ÷ 10 - Moving digits (Dividing by 10, 100 and 1000)• 380 ÷ 5 - Moving digits and doubling (Dividing by 5)• 42 ÷ 6 - Arrays (Grouping)• 84 ÷ 6 - Number line (Grouping)• 344 ÷ 6 - Chunking (Grouping)

• Children should always seek to apply known multiplication facts when possible to solve division calculations.

Basic Principles

• Basic principles are GROUPING and SHARING.• Good multiplication skills are vital to children being

successful with division.• It is essential that children have a strong knowledge of

Multiplication Facts.• There are few mental strategies to support division. The

clearest is to use Multiplication Facts.

Basic Principles• Sharing is the natural, early experience

that young children are exposed to.• Sharing becomes inefficient as the size of

calculations increase.• Grouping is linked to knowledge of

multiplication.• Grouping allows for mental strategies to

be applied when solving division calculations.

• KS2 children should be familiar with Grouping.

Division Vocabulary• DIVIDEND – The number you are dividing.

• DIVISOR – The number you are dividing by.

• QUOTIENT – How many full groups of the divisor you can make.

• REMAINDER – How many you have left over after you have made the maximum number of full groups possible.

• 38 ÷ 6 = 6 r2

Sharing• This is children’s earliest

experience of division.• Much practical early division is

based on sharing.• Sharing will take place in

EYFS as an introduction to division.

• Sharing is a correct division strategy. It does become inefficient though as numbers get larger.

• Most children should not be using sharing as their primary strategy in KS2.

• Dienes cubes, Counters and Coins can all be used to support Sharing.

Sharing through pictures or tallies•For example, 18 ÷ 6 = ?

•If attempting to solve this question using sharing, the calculation would be asking ‘If 18 Counters were shared into 6 equal groups, how many Counters would be in each group?’

•Begin by getting 18 Counters.

•Then start placing the Counters into six groups, putting one counter into each group at a time.

•When all the counters have been placed into equal groups, count up the number of Counters in each group. That number will be your answer.

•So, when 18 Counters are shared into 6 equal groups, there are 3 Counters in each group.

•So 18 ÷ 6 = 3

Grouping• Grouping builds on knowledge

from other operations, in particular addition and multiplication.

• The link to multiplication allows knowledge of multiplication facts to be used effectively.

• Grouping becomes much more efficient than sharing as the numbers in calculations get larger.

• Children should initially be introduced to grouping as a division strategy in Year 2, then this should be consolidated throughout KS2.

• Dienes cubes, Counters, Coins and Beadstrings can all be used to support Grouping.

Grouping through jottings•For example, 18 ÷ 6 = ?•This calculation is asking ‘How many groups of 6 are there in 18?’•First identify what you are finding groups of. In this case it is groups of 6.•Next, place 6 dots/Counters/Cubes together to create a group of 6.•Repeat this until there are no more counters left to place into groups.•The number of groups you have created is the answer to the question ‘How many groups of 6 are there in 18?’ or 18 ÷ 6 = ?•So, 18 ÷ 6 = 3•If there had been any dots/Counters/Cubes left over so that a whole group of 6 could not have been created, these would be a ‘remainder’.

Link to Multiplication Facts• Although Division can be

represented as repeated subtraction, it is logical that linking it to multiplication (repeated addition) will make it more accessible for children.

• When solving Division calculations, children should be encouraged to use the multiplication facts that they know.

• Using Multiplication Facts is the most efficient mental strategy to solve division calculations.

• Arrays and Counting On can support children’s understanding of multiplication facts.

• Children need to be taught to derive division facts from known multiplication facts.

Link to Multiplication Facts• For example, 24 ÷ 3 = ?• First identify what the question is

asking. ‘How many groups of 3 are there in 24?’

• Then think about which multiplication fact this links to: 3 x ? = 24

• Using Counting On, Repeated Addition, Arrays or knowledge of the 3 Times Table, calculate how many 3s are in 24. (3 x 8 = 24)

• The missing number gives you the number of groups of 3 that are in 24.

• So, 24 ÷ 3 = 8• This can also be used for

calculations where the answer will have a remainder.

Arrays• Arrays are an important tool to

enhance children’s understanding of division.

• Arrays develop children’s understanding of division as ‘grouping’ which can be solved using ‘repeated addition’.

• Arrays provide children with a concrete representation or an image of the process of division through grouping.

• Arrays provide a link between division and multiplication.

• The use of arrays can be modelled using counters, Dienes cubes and Multilink.

• Having strong knowledge of grouping using an array should provide successful progression to a number line.

Arrays• For example, 24 ÷ 6 = ?• This calculation is asking ‘How

many groups of 6 are there in 24?’

• First identify what you are finding groups of. In this case it is groups of 6.

• Next, place 6 dots/Counters/Cubes together in a line to create a row of 6. (The first row of your array)

• Repeat this until you have placed dots/Counters/Cubes into rows of 6 on the array.

• The number of whole groups (rows) of 6 that you have created is the answer to the question ‘How many groups of 6 are there in 24?’ or 24 ÷ 6 = ?

• So, 24 ÷ 6 = 4• If there had been any

dots/Counters/Cubes left over so that a whole group (row) of 6 could not have been created, these would be a ‘remainder’.

Alternatives to grouping or sharing.

• Halving – 48 ÷ 2

• Halve twice – 64 ÷ 4

• Moving digits – 370 ÷ 10

• Divide by 10 and double – 370 ÷ 5

Halving• Halving is used for dividing any

number by 2.• Halving is one of the earliest

experiences children have of sharing.

• Halving builds on children’s knowledge of early halves.

• Halving builds on children’s knowledge of number facts and number bonds.

• Halving relies on children’s understanding of the relative position of numbers, place value and the ability to partition and recombine.

• Children need to be able to halve all numbers up to 20 to be able to access further halving.

Halving • For example, 48 ÷ 2 = ?• First partition the digits. (48

becomes 40 and 8)• Next find half of 40. Half of

40 = 20.• Then find half of 8. Half of 8

= 4.• Finally add together 20 and

4. 20 + 4 = 24.• So, 48 ÷ 2 = 24 • Children need to recognise

that when halving an odd number there answer will not be a whole number.

• Halving can be supported using Coins, Counters, Multilink and Dienes cubes.

Halving – Progression and Examples

• 24 ÷ 2• 224 ÷ 2• 1044 ÷ 2• 60 ÷ 20• 240 ÷ 20• 600 ÷ 200• 4.6 ÷ 2• 4.36 ÷ 2

Moving Digits• When dividing a number by 10,

100 and 1000, the calculation can be completed efficiently using knowledge of place value.

• Moving digits builds on children’s understanding of counting in steps of 10, 100 and 1000.

• When dividing a positive whole number by 10, all of the digits move one place to the right.

• The new number will be 10 times smaller in size than the old number.

• To divide a positive whole number by 100, each digit is moved two places to the right.

• To divide a positive whole number by 1000, each digit is moved three places to the right.

Moving Digits• For example, 380 ÷ 100 = ?• First draw a place value chart,

placing the digit 3 in the Hundreds column, the digit 8 in the Tens column and the digit 0 in the Units column.

• Next identify how many places to the right each digit has to move. (As 380 is being divided by 100 each digit will need to move two places to the right.

• Then move each digit two places to the right. The 3 will move to the Units column, the 8 will move to the Tenths column and the 0 will move to the Hundredths column.

• So, 380 ÷ 100 = 3.8• Place value charts, Dienes

cubes, Coins (£1, 10p and 1p) 100 square with cubes and Place value cards can be used to support Moving Digits.

Moving Digits – Progression and Examples

• 40 ÷ 10• 400 ÷ 10/100• 4 ÷ 10/100• 42 ÷ 10/100• 1420 ÷ 10/100• 60 ÷ 30• 600 ÷ 300• 4000 ÷ 1000• 400 ÷ 100• 4 ÷ 1000• 4.2 ÷ 10/100/1000

Dividing by 5 (Divide by 10 and Double)

• Divide by 10 and Double is a strategy that can be used to divide by 5, 50 and 500.

• When knowledge of moving digits to divide by 10, 100 and 1000 is secure, this knowledge can be used to divide by 5 (÷ 5).

• Divide by 10 and Double builds on children’s understanding of doubling any number.

• Divide by 10 and Double builds on children’s understanding of place value and the relative position of numbers.

Dividing by 5 (Divide by 10 and Double)

• For example, 42 ÷ 5 = ?• First divide 42 by 10. (Double the 5

to get 10 to make the calculation easier.) 42 ÷ 10 = 4.2

• Next double the answer (4.2) Effectively, we have found what the quantity would be in 10 groups, yet as we want to know what the quantity would be in 5 groups we have to double our answer as these groups would be double the size. Double 4.2 = 8.4

• So 42 ÷ 5 = 8.4• It is vital that children have a secure

strategy for doubling numbers before using Divide by 10 and Double.

• Arrays, Counters, Coins and Dienes cubes can all be used to support Divide by 10 and Double.

Dividing by 5 (Divide by 10 and Double) – Progression and

Examples• 40 ÷ 5• 400 ÷ 5• 1420 ÷ 5• 700 ÷ 50• 7000 ÷ 500• 4 ÷ 5• 42 ÷ 5• 730 ÷ 50• 8200 ÷ 500

Number Line• Number lines are a formal written

method used to support children when completing division calculations.

• Number lines build on the concept of repeated addition, grouping, counting on and place value.

• Number lines can be supported with Dienes cubes, Beadstrings and Counters.

• Pre-numbered and part-numbered lines can also be used to support children using Number lines.

• A number line can be linked to an array by showing each group on the line as a group from the array.

Number line•When using a Number line to solve division calculations, use grouping to support.•For example, 18 ÷ 6 = ?•This question is asking ‘How many groups of 6 are there in 18?’•When dividing or multiplying on a number line, always begin at 0.•Count on in groups of the divisor until you have reached the dividend e.g. count on in groups of 6 until you reach 18.•When you have reached the dividend count how many whole groups there are and identify any remainders (left over numbers).•In this example there are 3 whole groups of 6, so 18 ÷ 6 = 3

Number line – Progression and Examples

•Individual groups of the divisor 26 ÷ 4

•Chunks of 10 of the divisor – the tens digit of the dividend the same 78 ÷ 7

•Chunks of 100 of the divisor – the hundreds and tens digits of the dividend the same 779 ÷ 7

•Chunks of 10 of the divisor, the tens digit of the dividend being different 74 ÷ 6

•Chunks of 10 of the divisor, the hundreds and tens digits of the dividend being different 342 ÷ 6

•Division of decimal numbers 5.4 ÷ 0.6

Chunking Method• This strategy should

only be used when children have a completely secure knowledge of the number system and multiplication facts.

• It is not necessary for children to have to use this method, ‘chunking on a number line’ is a perfectly acceptable strategy.

• This strategy focuses on multiplication, addition and ‘Counting On’.

Chunking Method• Begin by finding ‘chunks’

of the divisor. • Show the multiplications

that you are doing to find these chunks and show what your total is.

• Repeat this and add the total of this chunk to the total of the previous chunk.

• Continue until all of the possible whole groups of the divisor have been found in the dividend.

• Add all of the groups together using column addition.

• Identify any remainders by counting on from your current number to the dividend.