Multiplication and division: mental methods.
• Use the relationship between multiplication and division.
This includes having instant recall of the multiplication facts
to 10 x 10 and knowing these as division facts.
All of this is underpinned by a secure knowledge of place value. Understanding
what each digit is worth and that we are making a number so many times bigger –
this is why multiplication facts are known as ‘times tables’. However, ‘timesing’ is
not a verb or a maths word; the maths we are doing is ‘multiplying’!
Applying these facts to new contexts
e.g. ‘I know that 7 x 5 = 35
so 0.7 x 5 will be 3.5’
or ‘I know that 4 x 7 = 28 so 40 x 7 will be 280’
‘If 42 ÷ 6 = 7 then 4200 ÷ 6 = 700’
Multiplication and division: mental methods.

Understand the term ‘square number’ and know the
square numbers to 100.

1² = 1 x 1 = 1

2² = 2 x 2 = 4

3² = 3 x 3 = 9

4² = 4 x 4 = 16

5² = 5 x 5 = 25
etc.
Multiplication and division: mental methods.

Mental methods for multiplying larger numbers.
e.g. multiplying by 50 is the same as multiplying by 100 and
halving. Try it both ways:
18 x 50
= (18 x 100) ÷ 2
= 1800 ÷ 2
= 900
18 x 50
= (18 ÷ 2) x 100
= 9 x 100
= 900
By Year 6, children should be developing sufficient number
confidence and skills to be manipulating numbers mentally,
choosing the most appropriate strategy for the calculation.
Multiplication and division: mental methods.

How about multiplying by 25? That would be the same as
multiplying by 100 and dividing by 4 (which is also the
same as halving and halving again)

Try it:
14 x 25
(well that seems pretty hard…)
14 x 100 = 1400
1400 ÷ 2 = 700
700 ÷ 2 = 350 ( dividing by 2 again – so that we have divided by 4)
(Well, that wasn’t so hard. Was it?)
Multiplication and division: mental methods.
For our capable mathematicians we can show them more
unusual methods that can be helpful:

If your multiplication involves a multiple of 5 and an even
number, e.g. 35 x 18, then you can manipulate this
mentally into a manageable format too.

By doubling one number and halving the other, the
resulting answer will be correct.
Double 35 = 70. Half of 18 = 9.
70 x 9 = 630.
So 35 x 18 = 630. (Check it on a calculator if you don’t believe it!)
Multiplication and division: mental methods.

Mentally multiplying a 2 digit number by a single digit
number e.g. 14 x 9.
There are a number of possible strategies here, all of which
use known multiplication facts in combination.
A favourite method is to partition and recombine e.g.
Partition 14 = 10 + 4
Multiply each part by 9
10 x 9 = 90
4 x 9 = 36
Recombine the parts
90 + 36 = 126
Multiplication and division: mental methods

Multiply or divide a simple decimal number by an integer
e.g. 4.2 x 8 or 3.6 ÷ 6.

These calculations can be solved using multiplication and
division facts. If the decimal point is confusing us, we can
solve this by multiplying the number by 10 to begin with.
We must remember to divide by 10 again at the end in
order to ‘undo’ our process.
e.g.
3.6 ÷ 6
3.6 x 10 = 36.
36 ÷ 6 = 6
6 ÷ 10 = 0.6
so,
3.6 ÷ 6 = 0.6
Multiplication and division: written methods.
Once calculations become too cumbersome, we must learn
to use a reliable written method.
For multiplication there are two:
The Grid method and the Expanded method.
Multiplication and division: written methods.

Grid method multiplication.

The numbers in the calculation are partitioned and placed
on the borders of a grid.

Each component part is multiplied by each of the others
from the other edge of the grid.

The totals are added together.
37 x 46 Partition these: 30 and 7, 40 and 6. Place them on the grid.
Write the totals for each multiplication in the spaces.
x
30
7
40
(30 x 40)
(40 x 7)
6
(30 x 6)
(6 x 7)
180
42
1200
280
Finally, add the totals:
1200
+ 280
+ 180
+ 42
--------1702
Multiplication and division: written methods.
Common errors with the Grid method:

Some children find the construction of a grid fiddly or
they do not make the grid large enough to clearly write in
the numbers.

Errors in transcription can occur when transferring the
numbers from the grid ready for addition.
Multiplication and division: written methods.

The Expanded method, still partitions, multiplies and adds
the numbers but the layout is different.
37
x 46
------1200 (30 x 40)
280 (40 x 7)
180 (30 x 6)
42 (6 x 7)
----------1702
Common errors with the Expanded method:

Children forget to carry out one of the
multiplication sums.

The answers can be fiddly to line up
causing errors in addition.
Multiplication and division: written methods.

Short division involves using known division facts and
works from left to right, carrying forward any remainder.
Children are expected to interpret the remainder in 3
possible ways: as a ‘remainder’, as a fraction and, on
occasions, as a decimal.
4
21r3
8 7
The thought process for this calculation would be
something like this: How many 4s in 8 tens? That’s 2 tens so
I’ll write that in the tens column on top of the ‘bus
shelter’. It goes exactly so there’s no remainder to carry
forward. How many 4s in 7? Well, that’s 1 with 3 left over
so I’ll write that in the units column on top of the ‘bus
shelter’. If there’s 3 out of 4 left over, I could call that 21
and three-quarters or 21.75.
Multiplication and division: written methods.

‘Chunking’ (or long division) is used for longer division
sums – Warning: it can use up a lot of paper if you work
with chunks which are too small!
We are going
to subtract
‘chunks’
from our
number.
These
chunks are
multiples of
the divisor.
452 ÷ 13
452
- 130 (13 x 10)
--------322
- 130 (13 x 10)
--------192
- 130 (13 x 10)
--------62
52 (13 x 4)
--------10
In total we have
subtracted 34 lots of 13
from 452 with 10 left
over.
Therefore, 452 ÷ 13 = 34
remainder 10.
This could also be
recorded as 34 and ten
thirteenths.