Download Maths - Progression in Multiplication

Progression in Multiplication
Previous Learning
Multiplication Tables
Children will be familiar with counting in 2s, 3s, 5s, 10s,
using counters, numbers lines or a counting stick.
They will be aware that they need to learn these
patterns and may have already begun to gain instant recall
of some of the multiplication tables. (e.g. Two times table,
Five times table and Ten times table). If not, we do lots
of work at Velmead on this!

Progression in Understanding Multiplication
Children will have been introduced to multiplication as
repeated addition. Drawing a picture can help visualise
the problem.
e.g. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
3, 6, 9, 12, 15, 18, 21, 24, 27, 30,
5, 10, 15, 20, 25, 30, 35, 40, 45, 50,
10, 20, 30, 40, 50, 60, 70, 80, 90, 100
e.g. Can you tell me what 4 x 5 = (five seconds
to answer)
What about 8 x 2 = (five seconds to
answer)
e.g. Each child has two eyes. How many eyes
do five children have?
2




+

+
2
+
2
+
2
Children will also understand multiplication as scaling.
(a number of times as wide, tall …)
Children will be able to draw an array (showing the
sum in rows and columns). This develops understanding
that 4 x 3 is the same as 3 x 4
They may also have used the number line to carry out
multiplication calculations. This is explained in step
one of our progression in multiplication.
e.g. There are four cakes in a pack. How many
in five packs?
e.g. The blue ribbon is 5cm long. Find the
ribbon that is four times as long.
e.g. One sweet costs 4p. How much do three
sweets cost?
e.g. 5 lots of 4
0

2
Dots or tally marks can also be used.
Facts about Multiplication
They will know that multiplication can be carried out in
any order. (commutative)
They will understand that doubling a number is the
same as multiplying it by 2, and that halving a number
reverses doubling
4
8
12
16
20
e.g. 3 x 4 is the same as 4 x 3
e.g. Double 11 is 22 means 2 x 11. Half of 22 is
11
1
Year Three
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 Year Three children should learn their 2,3,4,5,6 and 10
What is 3 x 6?
times tables facts. Instant recall means knowing the table
What is 6 x 4?
facts out of order and being able to answer a question within
five seconds! It also helps to link these facts to division too
What is 24 ÷ 6?
and to insert missing numbers too.
4 x ? = 24
Year Three
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
0
e.g.
3 x 4 = 12
12 ÷ 4 = 3
3
6
9
12
15
4 x 3 = 12
12 ÷ 3 = 4
This array shows
4 x 3 = 12 and 3 x 4 = 12
2
18
Year Three
STEP THREE: 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.
Real Life Problems
Children will be given word problems to apply their methods
x
30
3
90
90 + 6 = 96
2
6
e.g. Alex has four stickers. Jo has 3 times as
many stickers as Alex. How many stickers
does Jo have?
e.g. A box of biscuits contains twenty four
biscuits. How many biscuits in four boxes?
Year Four
Multiplication Tables
Multiplication Tables
Children will need to constantly reinforce and revise their
instant recall of the 2,3,4,5,6 and 10 times tables to keep
their knowledge sharp. They should also be applying their
knowledge to deduce other facts.
By the end of Year 4 they should also know 7,8,9 times tables
too.
Constant Practice and Derived Facts
e.g.
7 x 8 = ? so I can work out 70 x 8 = ?
56 ÷ 7 = ? so I can work out 560 ÷ 7 = ?
6 x ? = 42 so I can work out 6 x ? = 4200
If 8x 4 = 32, I know that 8 x 8 is double 32
If 8 x 10 is 80, I know that 8 x 9 is 8 less
than 80
Year Four
STEP FOUR: Multiplication of Two Digit numbers by One Digit numbers
Using Arrays
Children will continue to use arrays to visualise how numbers
can be partitioned in order to carry out multiplication sums.
e.g. 18 x 4 can be shown on an array and
partitioned into
10 x 4
8x4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
.
.
.
.
.
.
.
.
.
.
.
.
Year Four
STEP FIVE: 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
4
Year Four
STEP SIX: Working With Decimals
Decimals in a Context
Children will be working with decimals in the context of
money.
As using the grid method with decimals requires a huge
amount of background work, children will be taught to start
with using repeated addition to add decimals.
e.g. £2.75 x 3
£2.75 + £2.75 + £2.75 (Use the column
method)
2.75
+2.75
+2.75
0. 1 5
2.10
6.00
£8 . 2 5
During Year Four as well, children will be extending their
knowledge of place value to include decimals, and it is
important that they have a very firm understanding of how
multiplying by 10, 100, 1000 affects the position of the
number on a place value chart, by moving it places to the left.
(Using a place value chart)
Make 4 ten times bigger = 40
Make 4 ten times smaller makes 0.4
Making 90 one hundred times smaller makes
0.9
NB: IT IS IMPORTANT that children see this as a move,
rather than adding or taking off a 0, as this brings problems
when working with decimals!
Real Life Problems
e.g. A biscuit box holds 74 biscuits. How many
biscuits in 6 boxes?
Children will apply these methods into real life problems
e.g. I buy 5 toys costing £1.40 each. How
much change do I get from £10.00?
Year Five
Multiplication Tables
Multiplication Tables
Children should now have instant recall of all multiplication
facts to 10 x 10 and should learn the 11 and 12 times tables.
They should also be confident with their division facts too,
and need plenty of practise at deriving facts.
e.g. 12 x 11 =?
If 7 x 7 is 49, what is 7 x 700 ?
9 x ? = 108 , and 9 x ? = 10800
5
Year Five
STEP SEVEN: The Grid Method TO x TO and HTO x TO
Children will continue to reinforce their knowledge and
understanding of the grid method and they will use larger
numbers in their calculations.
They will be taught techniques to check their multiplication of
multiples of 10.
NB: They must be taught to estimate their answers each time
and must check their calculations thoroughly
e.g. 127 x 54
Estimate: 130 x 50 is half of 130 x 100
= 6500
x
50
4
They will also need to be proficient in column addition.
100
I know that
100 x 5 =
500, so 100
x 50 is
5000
400
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
7x5
is 35,
and 7
x 50 is
350
28
Now add the columns using column addition
5000
1000
350
400
80
28
6858
1
Children will progress to using the grid method without the
explanations
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
6
Year Five
STEP EIGHT: Working With Decimals
Children will need a great deal of reinforcement in deriving
facts about decimals.
This must be linked to place value work on the effects of
multiplying by 10, 100, 1000 and dividing by 10, 100 and 1000.
Some children may need to see this visually on a place value
chart.
e.g. 7 x 6 = 42 ….. so what?
0.7 x 6 = 4.2
(7 ÷ 10) x 6
7 x 0.6 = 4.2
7 x (6 ÷ 10)
0.7 x 0.6 = 0.42
(7 ÷ 10) x (6 ÷ 10)
Year Five
STEP NINE: 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.
e.g. 54 x 5.6
X
5
0.6
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
NB Children must be taught to use zeros as place holders and
to line up the decimal points carefully when carrying out
column addition.
1
e.g
Children will progress onto using the grid method without
needing the explanation in the boxes
X
5
0.6
Children may add across the columns
270.0
32.4
302.4
1
7
Year Five
STEP TEN: 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.
Real Life Problems
Children will apply these methods into real life problems
97
8
56
7 20
7 76
7x8
90 x 8
e.g. There are 32 children in each class at
Velmead. How many children in the school?
e.g. Petrol costs £1.23 a litre. How much
would it cost to fill a 3 litre can?
e.g. I buy 8 tins of soup that costs 96p a tin.
How much change would I get from £20?
8
Year Six
Multiplication Tables
Multiplication Tables and Derived Facts
Children will need to know their multiplication tables to 12 x
12.
(Even though they only really need tables to 10 x 10 in our
metric system, we have decided that it is still good practice
for children to know their 11 and 12 times table too.)
They will also know division facts to 12 x 12
They will use these facts to derive facts about decimals and
multiples of 10, 100 and 1000
e.g. 40 x 6 = 240
240 ÷ 6 = 40
I know that 4 x 8 = 32
0.4 x 8 = 3.2
0.4 x 0.8 = 0.32
Year Six
STEP ELEVEN: The Grid Method
Children will be confidently using the grid method with whole
numbers and will revise the method.
NB: It is important that children are taught to estimate and
then check their answers carefully.
It may help to explain the individual sums to the children to
ensure that their knowledge of place value is secure.
They will then carry out the sums confidently and use the
column addition method to find the answer.
e.g. 372 x 24 =
372 x 24 is approximately 400 x 20 = 8000
(4 x 2 is 8, 400 x 2 is 800, 400 x 20 is 8000)
Teaching Grid
e.g.
x
300
20
2 x 10 x
3 x 100
4
4x3x
100
70
2 x 10 x
7 x 10
4x7x
10
2
2 x 10 x
2
2x4
e.g.
x
20
4
70
1400
280
2
40
8
300
6000
1200
6000+1400+40=7440
1200+ 280 + 8 =1488
Answer
=8928
1
9
Year Six
STEP TWELVE : The Grid Method with Decimals to Two Decimal Places
Children will need to have carried out a great deal of
reinforcement work ensuring that their knowledge of
multiplying decimals is sound.
They will then be taught to use the grid method to multiply
decimals to 2 decimal places
This example is one of the hardest examples that they may
face, but the principle of the method remains the same.
Obviously, most calculations of this nature would be carried
out on the calculator, but it is useful for children to know how
they could attempt it, without a calculator too.
e.g. 5 x 5 = 25, so what …
I know that …
0.5 x 5 = 2.5
0.05 x 5 = 0.25
500 x 50 = 5 x 100 x 5 x 10 = 25000
e.g. 34.37 x 29.6
Estimate 30 x 30 = 900
(with teaching explanation)
X
30
4
0.3
20
2 x 10 2 x 10 2 x 10
x3x
x4
x3÷
10
10
9
9x3
4x9
9x3÷
x 10
10
0.6
6 ÷ 10 6 ÷ 10 6 ÷ 10
x3x
x4
x3÷
10
10
e.g. (without explanation)
X
30
4
0.3
0.07
20
600
80
6
1.4
9
270
36
2.7
0.63
0.6
18
2.4
0.18
0.042
They will use column addition to work out the
answer
600 + 80 + 0.3 + 0.07 = 6 8 7. 4 0
270 + 36 + 2.7 + 0.63 = 3 0 9. 3 3
18 + 2.4 + 0.18 + 0.042 = 2 0 .6 2 2
Answer
1 0 1 7. 3 5 2
1 1 1
YEAR SIX
Step THIRTEEN :Vertical Method with Two Digit Whole Numbers
Children will have been introduced to the vertical method for
TU x U and will need reminding of it.
0.07
2 x 10 x
7 ÷ 10 ÷
10
9x7÷
10 ÷ 10
6 ÷ 10
x7÷
10÷ 10
e.g. 97 x 8
Estimate 100 x 8 = 800
97
X
8
56
720
776
7x8
90 x 8
10
They will then be taught to use it for TO and TO
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
Year Six
STEP FOURTEEN : 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
Real Life Problems
Children will apply these methods into real life problems
e.g. Find the cost of 121 bottle of lemonade at
21p each. What change would you get from
£50?
e.g. There is space in the multi-storey car
park for 17 rows of 30 cars on each of 4
floors. How many cars can park?
11
EXTENSION YEAR 6 PLUS:
Short Multiplication with Decimals to Two Places
Once children have a solid knowledge of place value and of
short and long multiplication, they will be introduced to using
long and short multiplication with decimals.
e.g. 4.92 x 3
Estimate 5 x 3 = 15
4.92 x 3
4.00 x 3 = 1 2 . 0 0
0.90 x 3 = 2 . 7 0
0.02 x 3 = 0 . 0 6
14.76
Children need to use zeros as place holders and must be
taught to estimate before they calculate and then check their
answers carefully.
Long Multiplication with Decimals to Two Places
Children will also extend their long multiplication methods to
include decimals too.
e.g. 34.9 x 23
Estimate 35 x 20 = 700
34.9
23.0
104.7
34.9 x 3
6 9 8 . 0 34.9 x 20
802.7
X
1 1
Children will carry out investigative work which will use some
of these methods.
e.g.
This four digit number is a square number
9__9
Find the missing digits.
e.g.
A plank of wood weighed 1.4kg. 25cm was cut
off its length. The plank then weighed 0.8kg.
What was the length of the original plank?
12