Multiplication 1
This presentation looks at the background to and development of
multiplication in the Leicestershire Recommendation for Pencil & Paper
Procedures.
It emphasises the importance of children understanding repeated
addition, scaling and arrays.
Multiplication 2
The first part of the session looks at the language used for
multiplication and the impact this has on the images we show children.
Ask your staff to quickly illustrate this calculation – no detailed
drawings required!
Compare the diagrams. Have some people drawn 4 lots of 5 objects
and other 5 lots of 4?
Which image would you use for the statement 4 multiplied by 5?
Which image would you use for the statement 4 lots of 5?
Multiplication 3
Teachers may have drawn a diagram in an ‘array’.
For 5 x 4 would you loop 5 objects 4 times or 4 objects 5 times?
The answer depends on the language you are using as illustrated on
the following 2 slides.
Multiplication 4
This illustration shows four objects circled 5 times.
You will notice that when the words ‘multiplied’ and ‘times’ are used
the numbers in the calculation 4 x 5 stay in order. However when the
words ‘lots of’ and ‘groups of’ are used the order reverses.
Multiplication 5
This illustration shows five objects circled 4 times.
Again you can see how the words ‘lots of’ and ‘groups of’ reverses the
order of the numbers in the calculation.
Multiplication 6
This further exemplifies how different language implies a slightly
different calculation.
Since multiplication is commutative all interpretations are valid.
However care does not to be taken when you are modelling a
calculation – you need to ensure the words you are using match the
model/image you are drawing or demonstrating.
Multiplication 7
Ask staff to consider which of these words they use most frequently?
Which of these words would children have most difficulty with?
Is this because of the language we use as teachers? Do we need to
place more emphasis, than we currently do, on any of these words?
Multiplication 8
The first ‘multiplication’ objective is in Year 2 but preparation for the
concept of multiplication begins in Reception.
Multiplication 9
The first objectives for multiplication appear in Year 2. Although much
background work has taken place (counting in different steps and
sizes) it is in Year 2 that the sign is introduced.
Multiplication 10
Arrays are an important image to use in Year 2 and beyond.
Multiplication 11
As this slide demonstrates they are not just suitable for Key Stage 1
but can be applied to multiplication of fractions.
Multiplication 12
Number lines are another good image. They help reinforce counting in
equal steps/the link to repeated addition.
Note how this slide shows 2 multiplied by 5 and the next slide…
Multiplication 13
… shows 5 multiplied by 2. Multiplying by 2 should also be linked to
doubling.
Multiplication 14
And finally another useful image that can be matched to the number
line.
Multiplication 15
To review the use of language consider the two number lines.
Multiplication 16
As in Year 2 children need to appreciate multiplication is repeated
addition and the description of an array. They should also begin to
experience the idea of scaling (this will be developed later in Key
Stage 3 in measures and ratio).
Multiplication 17
This is one way that scaling could be modelled.
Multiplication 18
This is another model for scaling – this is how scaling links to
multiplication tables (in this instance the 5 times table).
Multiplication 19
The laws associated with multiplication should be understood in years
4 to 6.
The commutative law will have been experienced before Year 4. The
distributive law is used within the grid method of multiplication.
Multiplication 20
The distributive law uses the concept of partitioning and is used in grid
multiplication. It can also be modelled on an empty number line as
shown in the slide.
Multiplication 21
(animated slide)
This animation goes back to the idea of arrays. It shows how the
array can be partitioned (using the distributive law) and how the grid
method of multiplication is based on this.
Multiplication 22
Once the array is removed you are left with grid multiplication.
Multiplication 23
(animated slide)
For anyone unfamiliar with grid multiplication work through this two by
two-digit example.
Multiplication 24
This shows the power of the grid method at KS3 with algebra – a
brilliant way to demonstrate expanding brackets.
Multiplication 25
In order to carry out grid multiplication (such as the one previously
demonstrated) these are the key skills required.
Some of these are exemplified on the next slides.
Multiplication 26
This outlines the importance of teaching x10, x100 etc correctly.
Children often refer to adding zeros when multiplying by 10 etc. It is
worth checking that they understand what is really happening and
ensure that when they describe calculations to others they use the
correct explanation.
Multiplication 27
There will be some multiplication facts/tables children struggle to
recall. It is important to teach strategies for these, e.g. using a known
facts such as 6 x 6 to establish that 7 x 6 = 42.
It is also important to make links between the tables (e.g. the four
times-table is double the two times-table, the five times-table is half of
the 10 times-table) and ensure that children realise that knowing one
multiplication facts means they know three other facts (e.g. if you
know 7 x 6 = 42 you also know 6 x 7 = 46, 46  7 = 6 and 46 6 =
7).
Multiplication 28
Emphasise the importance of estimation in all calculations.
To do this children need to be very familiar and confident with the
numbers they are calculating with. It is important that approximation
is modelled frequently. Children also need to consider ‘the
reasonableness’ of answers reached.