Download Calculation - Progression in Multiplication 2014

Progression in Multiplication
STEP ONE
Children will need to keep using arrays, number lines and pictures to help them.
They must learn their multiplication tables in order to have instant recall of them in any order.
Multiplication Tables
e.g. What is 5 x 6?
During their time at Velmead children should learn their times
What is 3 x 6?
tables facts. Instant recall means knowing the table facts out
What is 6 x 4?
of order and being able to answer a question within five
seconds! It also helps to link these facts to division too and to
What is 24 ÷ 6?
insert missing numbers too.
4 x ? = 24
Children will start to work through the Velmead times table
award scheme.
STEP TWO: Number Lines and Arrays
Using the Number Line to Show Multiplication as Repeated
Addition
By doing this, the children learn the patterns of the times
tables and can visualise the equal steps. They also use
apparatus, such as number beads and number lines so that it
helps them learn kinaesthetically.
Arrays
Arrays are a helpful model/image for developing the idea of
commutativity (see below) and an understanding of
multiplication and division.
They will be taught that multiplication is the inverse of
division. They will be reminded that multiplication can be done
in any order.
Commutativity means that 4 x 3 = 3 x 4
e.g. 6 x 3 = 18
+3
0
+3
3
e.g.
3 x 4 = 12
12 ÷ 4 = 3
+3
6
+3
9
+3
12
+3
15
4 x 3 = 12
12 ÷ 3 = 4
This array shows
4 x 3 = 12 and 3 x 4 = 12
1
18
STEP THREE: Partitioning - Multiplying Two Digit Numbers by One Digit Numbers
Partitioning in Preparation for Multiplying Two Digit
numbers by a One Digit number
Children will be confident in partitioning numbers and will be
introduced to the associative property of multiplication.
e.g. Double 6 is the same as double 5 add
double 1
12 x 3 is the same as (10 x 3) added to (2 x 3)
e.g. 32 x 3 = 96
is the same as 30 x 3 added to 2 x 3
Larger arrays allow demonstration of how a number can be
partitioned into tens and ones. This enables children to
visualise the image as an aid to mental calculation and is a
helpful introduction to the grid method of multiplication.
x
30
3
90
90 + 6 = 96
2
6
STEP FOUR: The Grid Method of Multiplication
This method builds on a firm understanding of partitioning
and arrays. It requires that children are confident with their
tables facts to 10 x 10, and also that they can add numbers
accurately.
NB: This is why it is important to teach children to estimate
their answer first, and then carefully check their calculations
at the end.
Also children must be given numbers that will use the tables
facts that they are confident with to start with.
Children also need to be confident with recognising factors of
a number in order to increase their ability to multiply
different numbers.
e.g. 20 x 6 is the same as 10 x 2 x 6 or 5 x 2 x 2 x 6 or 5 x 4
x6
e.g. 23 x 7
Estimation: I know that 20 x 7 is 140, and 30
x 7 is 210 so my answer will be between 140
and 210, and closer to 140.
e.g. 23 x 65 =
Estimation: 20 x 60 is 1200
In teaching the grid method, it is essential to
ensure that children have a sound
understanding of place value and of factors.
X
60
5
20
6 x 2 x 10 x 10
1200
5 x 2 x 10
100
3
180
15
1200 + 180 + 100 + 15 = 1495 (or use the
column method)
Children may also be partitioning with some calculations. It
depends how confident they are with mental calculation.
e.g. 47 x 6 = (40 x 6) + (7 x 6)
= (240) + (42)
= 282
2
Hundreds, Tens and Ones x Tens and Ones
e.g. 127 x 54
Estimate: 130 x 50 is half of 130 x 100
= 6500
x
50
100
I know that
100 x 5 = 500,
so 100 x 50 is
5000
400
4
20
2 x 5 is 10, so
20 x 5 is 100
and 20 x 50
is 1000
4 x 2 is 8
4 x 20 is 80
7
7 x 5 is
35, and
7 x 50
is 350
28
Now add the columns using column addition
5000
1000
350
400
80
28
6858
1
e.g. 127 x 54
X
50
4
100
5000
400
20
1000
80
7
350
28
They may also add the columns across as they work
5000 + 1000 + 350 = 6 3 5 0
400 + 80 + 28 =
508
= 6858
The Grid Method with Decimals
Once children have a better understanding of decimals and
place value they can then be taught to use the grid method
with decimals.
54 x 5.6
X
5
0.6
NB Children must be taught to use zeros as place holders and
to line up the decimal points carefully when carrying out
column addition.
e.g
Children will progress onto using the grid method without
needing the explanation in the boxes
50
250
5 x 10
x 6 ÷ 10
4
20
4 x 6 ÷ 10
50
250
30
4
20
2.4
250.0
20.0
30.0
2.4
302.4
1
X
5
0.6
Children may add across the columns
270.0
32.4
302.4
1
3
STEP FIVE: Vertical Multiplication
Partitioning
Children will be reminded of partitioning with some
multiplication sums.
e.g. 97 x 8 = (90 x 8) + (7 x 8)
= (720) + (56)
= 776
Vertical Multiplication
They will be taught to set out multiplication sums in a formal
way.
e.g. 97 x 8
X
For some calculations, children may find it easier to continue
using the grid method.
97
8
56
7 20
7 76
7x8
90 x 8
e.g. 97 x 8
Children will then be taught to use vertical multiplication for
TO and TO
Estimate 100 x 8 = 800
97
X
8
56
720
7x8
90 x 8
776
e.g. 87 x 26
Estimate 90 x 30 = 2700
87
X
26
42
480
140
1600
2262
6x7
6 x 80
20 x 7
20 x 80
11
STEP SIX : Short Multiplication
Children will now be taught to carry starting with HTO by O
436 x 8
Estimate 400 x 8 = 3200
436
X
8
3488
3 2 4
4
STEP SEVEN : Long Multiplication
Children will then be taught to use Long multiplication
5