Division Methods

Please note that all guidance below is the minimum expectations for the teaching within each year group. It is expectated that many children will move through these steps faster (although understanding of all prerequisit steps is essential before moving to abstract calculations.

Previous steps (Pre-Prep)

CONCRETE

Children have been taught to halve by sharing an identical amount between two groups. They share cubes one by one, alternating groups

ABSTRACT

They have also been taught to halve by partitioning numbers into 10s and 1s and halving each part before recombining (only simple, whole number answers).

ABSTRACT

The children used a counting stick to subtract repeatedly for their 2x, 5x and 10x tables. They will have recited the sequences forwards and backwards.

CONCRETE

By creating 'arrays' The children discussed division facts: if they have 10 cubes in two groups, there are 5 cubes in each group. This array shows 10÷5 = 2 and 10÷2 = 5.

CONCRETE

Previously, the children will have looked at finding multiplication and division facts for a number by creating multiply arrays. Here are 3 examples for the number 12 (12÷1 = 12, 12÷2=6, 12÷3 = 4)

CONCRETE

Children used manipulatives to create EQUAL groups when dividing

CONCRETE

Children used different objects to subtract equal groups

PICTORIAL

Children used number lines and repeated subtraction to complete division calculations.

ABSTRACT

Children began to use multiplication fact triangles. Here, you can derrive 4 x 2 = 8, 2 x 4 = 8, 8 ÷ 4 = 2 and 8 ÷ 2 = 4. These are known as multiplication fact families

Year 3 and 4

Division with remainders

CONCRETE

Children use cubes or counters to share equally between two groups. Any that cannot be shared equally are the 'remainder'. In Year 3, remainders are presented as 'r2', for example.

PICTORIAL

When using a number line, pupils can jump forwards or backwards in equal jumps then see how many more you need to jump to find a remainder.

ABSTRACT

Children learn the language of 'dividend', 'divisor', 'quotient', 'remainder'

Year 4

Division without a remainder

PICTORIAL

(Example to the right)

Use place value counters to divide using the bus stop method alongside.

a) Start with the biggest place value, we are sharing 40 into three groups. We can put 1 ten in each group and we have 1 ten left over.

b) We exchange this ten for ten ones and then share the ones equally among the groups.

We look how much in 1 group so the answer is 14.

ABSTRACT

Children move to short division where, although remainders may happen internally, the answer will not have a remainder.

Division with remainders

PICTORIAL

(Example to the right)

Use place value counters to divide using the bus stop method alongside.

a) Start with the biggest place value, we are sharing 40 into three groups. We can put 1 ten in each group and we have 1 ten left over.

b) We exchange this ten for ten ones and then share the ones equally among the groups.

We look how much in 1 group so the answer is 14 but there is 1 left over. The answer is 14 remainder 1. This can also be presented as 14 and 1 third (remainder over the divisor).

ABSTRACT

Children move to short division where the answer can result in a remainder. Initially, this remainder is presented as a fraction.

Year 5 and 6

Remainders presented as decimals

ABSTRACT

As before, the 364 is divided by 5. Ultimately a remainder of 4 is found. A decimal point is placed to the right of the quotient (364.0) and directly above in the dividend (72.). The remainder is placed in front of the zero. 40 is divided by 5 with an answer of 8. As there is no further remainder, the answer is complete, 72.8. If, however, there is a remainder still, another zero is placed in the quotient and the calculation continues.

Long division

ABSTRACT

Long division is used to divide by 2- or more digits. DMSB is used (divide, multiply, subtract, bring down)

It is often helpful, as you can see in the example, for children to write down the first few multiples of the divisor.

D: 37 divided by 28 is 1 (ignore remainders)

M: 28 multiplied by 1 is 28

S: 37 subtract 28 is 9

C: Carry down the '2' in the next column

D: 92 divided by 28 is 3 (ignore remainders)

M: 28 multipled by 3 is 84

S: 92 subtract 84 is 8

C: As there are no more digits to carry down, 8 is the remainder.

Dividing large numbers

ABSTRACT

There are often ways of avoiding the use of long division altogether by reducing the size of both the divisor and quotient by the same rate.

In the example, as both 372 and 28 are even, we can half them to reduce the scale of the calculation. As both dividends are even, we can repeat the process until we are left with a short division calculation.

Essential to success in this 'short-cut' is the ability to apply division facts to large numbers.

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