Stages in Multiplication
Multiplication – Stage 1
Children will engage in a wide variety of songs and rhymes, games and activities.
In practical activities and through discussion they will begin to solve problems involving doubling.
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●
Count in twos from 0 to 10, will we say 9?
Count round the circle of children in twos, who do you think will say 60?
‘Three apples for you and three apples for me. How many apples altogether?’
autumn
Count in 10s from 0 to 100. Count
in 10s to 50, holding up a finger for
each 10 we count. How many 10s
did we count?
How many 10p coins are here?
How much money is that?
● What is double 2?
● How many socks in 2 pairs?
● Share 6 biscuits between two people
so each person has the same number
spring
Count in 2s to 30. Count in 2s to
10, holding up a finger for each
2 we count.How many 2s have
we counted?
● Count in 5s to 15.How many 5s
have we counted?
● How many toes are there on
two feet?
● How many gloves in 3 pairs?
● This domino is a double 4.How
many spots does it have?
summer
Count in 3s from 0 to 30.How
many 3s did we count?
● How many sides are there on 4
triangles? Count them in 3s.
● What is double 5?
● What is half of 10?
● I roll double 3.What’s my score?
● There are 8 raisins.Take half, how
many have you taken?
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Multiplication – Stage 2
Solve one-step problems involving multiplication by calculating the answer using
concrete objects, pictorial representations and arrays with the support of the
teacher
Count in multiples of twos, fives and tens (to the 10th multiple)
Children will count repeated groups of the same size in practical contexts.
They will use the vocabulary associated with multiplication in practical contexts.
They will solve practical problems that involve combining groups of 2, 5 or 10. E.g. socks, fingers
and cubes
‘Six pairs of gloves. How many gloves altogether? 2, 4, 6, 8, 10, 12’
‘Three boxes of ten pencils. How many pencils altogether? 10, 20, 30’
Use arrays to support early multiplication
‘Five groups of two bees. How many bees altogether? 2, 4, 6, 8,10’
Two groups of five bees. How many bees altogether? 5, 10’
Continue to solve problems in practical contexts and develop the language of early multiplication, with
appropriate resources, throughout stage 2.
Autumn
Spring
Summer
What is 5 + 5 + 5?
How can we write this as a
multiplication sentence?
● What numbers go in the boxes?
6 x 2 = ■; ■ x 10 = 40;
■ x ▲ = 12
● How many 2s make 8?
● What numbers go in the boxes?
6 x 2 = ■;
20 x ■ = 2;
■ x 10 = 3
● There are 12 cubes. Make 3
towers of the same height. How tall
is each tower?
●
What multiplication sentences
could we write about this array?
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● ● ● ●●
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What are two 5s? Double 6?
6 times 2? 6 multiplied by 2?
● What is half of 12?
● If 6 grapes are shared equally
between two people, how many will
they get each?
● If Sarah counts in 2s, and Nigel
counts in 5s, when will they reach the
same number?
● How many 10s can you subtract from
60?
If double 11 is 22, what is half of 22?
If 12 x 2 = 24, what is 24 x 2?
● A giant is twice as tall as a 10m high
house. How tall is the giant?
● One snake is half the length of
another snake which is 20 cm long.
How long is the shorter snake?
● I doubled 3, then doubled the
answer. What number did I get? I
halved this number and then halved it
again. What was my answer
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Multiplication – Stage 3
Recall and use multiplication facts for the 2, 5 and 10 multiplication tables
Calculate mathematical statements for multiplication within the multiplication tables
and write them using the multiplication (x) and equals (=) signs
solve problems involving multiplication, using materials, arrays, repeated addition,
mental methods, and multiplication facts, including problems in contexts
show that multiplication of two numbers can be done in any order (commutative)
REMEMBER: if at any time, children are making significant errors, return to the previous stage.
Children will use a range of vocabulary to describe multiplication and use practical resources, pictures,
diagrams and the x sign to record.
‘3 groups of 10 pencils’
‘How many altogether?’
’10 + 10 + 10 = 30’
‘3 groups of 10’ is the same as‘ 3 times 10’
‘3 x 10 = 30’ is the same as ’10 x 3 = 30’
‘5 groups of 3’ ‘5 lots of 3’ ‘3 + 3 + 3 + 3 +3 = 15’
‘5 times 3’
‘3 multiplied by 5’ ‘5 x 3 = 15’ ‘3 x 5 = 15’
Use arrays to support multiplication
6 x 5 = 30
‘5 + 5 + 5 + 5 + 5 + 5 = 30’
‘6 rows of 5’
‘6 groups of 5’
‘5 groups of 6’
‘5 x 6 = 30’
‘6 x 5 = 30’
Use an empty number line:
0
5
10
6 x 5 = 30 Make the link to repeated addition.
15
20
25
30
Autumn
Spring
Summer
What is 10 added together 4 times?
How can we write this as an
addition? And as a multiplication?
Describe this array.
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If 7 x 5 = 35, what is 35 x 5?
● What four multiplication and
division sentences can you write to
using 2, 5 and 10?
● Share 18 between 2. Divide 25 by 5.
How many fives make 45? How
many 5p coins do you get for 35p?
● How many 5s make 35?
● Share 18 between 2. Divide 25 by 5.
● How many lengths of 10m can you
cut from 80m of rope?
● A baker bakes 24 buns. She puts 6
buns in every packet. How many
packets can she fill?
What numbers go in the boxes?
16 x 2 = ■;
30 x ■ = 6;
■x5=7
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I have 4 stickers. If you had 3 times
as many stickers how many would you
have?
● What is the product of 6 and 2? Of 2
and 6? What is 5 multiplied by 3?
● Is 35 a multiple of 5?
● What numbers go in the boxes?
4 x 4 = ■;
10 x ■ = 80;
■ x 5 = 30;
6 x 20 = ■; ■ x ▲ = 60
●
How many lengths of 10 cm can you
cut from 81 cm of tape? How many
will be left?
● If you put 25 eggs in boxes of 10,
how many boxes would you fill?
How many eggs would be left over?
● Count a handful of pasta pieces by
grouping them in 5s.How many 5s
were there? How many left over?
How many pasta pieces altogether?
What division sentence could
you write?
What numbers go in the boxes?
17 = 5 x 3 + ■;
■ = 2 x 8 + 1;
29 = 5 x ■ + 4;
REMEMBER: if at any time, children are making significant errors, return to the previous stage.
Multiplication – Stage 4
Recall and use multiplication facts for the 3, 4 and 8 multiplication tables (continue to practise the 2,
5 and 10 multiplication tables)
Write and calculate mathematical statements for multiplication using the
multiplication tables that they know, including for two-digit numbers times one-digit
numbers, using mental and progressing to a formal written method
Continue to use number lines and arrays to support multiplication, as appropriate (see stage 3guidance).
4 x 3 = 12
0
3
6
9
4 x 3 = 12
12
Demonstrate the partitioning method using a number line:
13 x 5 = 65
Partitioning method for multiplication of a ‘teen’ number by a one-digit number:
13 x 5 = 65 (Partition 13 into 10 + 3)
10 x 5 = 50
3 x 5 = 15
50 + 15 = 65
10 x 5 = 50
0
3 x 5 = 15
50
6
Grid Method (teen number multiplied by a one- digit number):
13 x 8 = 104
X
10
3
8
80
24
‘Partition 13 into 10 + 3 then multiply each number by 8. Add the partial products (80
and 24) together.’
This will lead into expanded short multiplication:
13 x 8 = 104
10 3
X
8
24
+80
1 04
(3 x 8)
(10 x 8)
Refine the recording in preparation for formal short multiplication:
13 X 8 = 104
13
x 8
24 (3 x 8)
+80 (10 x 8)
104
Use the language of place value to ensure understanding
Include an addition symbol when adding partial products
Model the same calculation using a number line, if necessary, to ensure understanding.
Formal short multiplication:
13
x 8
104
Ensure that the digit ‘carried over’ is written under the
line in the correct column.
Use the language of place value to ensure understanding
Continue to develop the formal written method of multiplication throughout year three using ‘teen’
numbers multiplied by a one-digit number.
If children are confident progress to multiplying other two-digit numbers by a one-digit number (see
stage 5).
REMEMBER: if at any time, children are making significant errors, return to the previous stage.
What is the product of 15 and 6?
If you add 10 x 5 and 8 x 5, what multiple of 5 do you get?
● Is 40 a multiple of 5? How do you know?
● What numbers go in the boxes?
4 x ■ + 8 = 24;
33 = 4 x ■ + ▲
spring
autumn
Understand and use the principles (but
Understand that multiplication is the
not the names) of the commutative,
inverse of division (multiplication
associative and distributive laws as
reverses division and vice-versa) and
they apply to multiplication.
use this to check results.
8 x 15 = 15 x 8;
● Demonstrate understanding of
6 x 15 = 6 x (5 x 3)
multiplying or dividing a whole
= (6 x 5) x 3 = 30 x 3 = 90;
number by 10.
18 x 5 = (10 + 8) x 5 = (10 x 5) + (8 x
● Respond rapidly to oral or written
5)
questions, explaining the strategy used.
= 50 + 40 = 90
Two elevens; double 16;
● Understand that:
9 multiplied by 3; divide 69 by 3.
86 + 86 + 86 is equivalent to 86 x 3 or
Is 40 a multiple of 5? How do you
3 x 86; multiplication by one leaves a
know?
number unchanged and that
Is 72 divisible by 3? How do you
multiplication by 0 results in zero.
know?
● Understand the operation of division
What is the product of 15 and 6? Tell
either as sharing equally or as grouping me two numbers with a quotient of 5.
and that division by 1 leaves a number
Use known facts to answer:
unchanged
7x2=■;
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●
summer
Use mental or mental with jottings to
answer:
90 x 6 = ■ ;
8 x ■ = 560;
■ x 90 = 270;
320 ÅÄ 4 = ■ ;
240 x ■ = 60;
■ x 30 = 8
Give a remainder as a whole number.
41 x 4 = 10 remainder 1
● Make sensible decisions about
rounding up or down after division.
I have £62. Tickets cost £8 each.
62 x 8 = 7 remainder 6. I can only buy
7 tickets.
●
10 x ■ = 80;
■ x 5 = 35;
36 x 4 = ■ ;
60 x ■ = 6;
■ x3= 7
Multiplication- Stage 5
Recall multiplication facts for multiplication tables up to 12 x 12
Multiply two-digit and three-digit numbers by a one-digit number using formal written layout
NB Ensure that children are confident with the methods outlined in the previous
stage’s guidance before moving on.
Continue to use empty number lines, as appropriate (see stage 4 guidance).
Further develop the grid method for two-digit numbers multiplied by a one- digit number.
36 x 4 = 144
X
4
30
120
6
24
120 + 24 = 144 (add the partial products)
Expanded short multiplication (two-digit number by a one-digit number):
36 x 4 = 144
30 6
x
4
24
+ 120
144
(4 x 6 = 24)
(4 x 30 = 120)
Include an addition symbol when adding partial products.
This leads to short multiplication (formal method) of a two–digit number multiplied by a one digit
number:
36 x 4 = 144
36
X 4
144
Use the language of place value to ensure understanding
Ensure that the digit ‘carried over’ is written under the line
in the correct column.
Continue to practise the formal method of short multiplication of a two-digit number by a one –digit
number throughout stage 5
If children are confident, continue to develop short multiplication with three- digit numbers multiplied
by a one-digit number. If necessary, return to the grid method and/or expanded method first:
4
X
100
30
24
6
600
180
600 + 180 + 24
This leads to expanded short multiplication:
127 x 6 = 762
2
127
x 6
42 (6 x 7)
+ 120 (6 x 20)
+ 600 (6 x 100)
762
This will lead into short multiplication (formal method):
127
x 6
762
Use the language of place value to ensure understanding
Ensure that the digits ‘carried over’ are written under
14
the line in the correct column.
REMEMBER: if at any time, children are making significant errors, return to the previous stage.
autumn
spring
summer
Understand that, with positive whole
numbers,multiplying makes a number
larger, and division makes a
number smaller.
● Demonstrate understanding of
multiplying or dividing a whole
number by 10 or 100.
● Understand that division is non
commutative.
72 divided by 9 is not the same as 9
divided by 72
● Understand that a number cannot be
divided by zero.
● Start to use brackets: know that they
determine the order of operations, and
that their contents are worked out first.
3 + (6 x 5) = 33, where as
(3 + 6) x 5 = 45
Respond rapidly to oral or written
questions, explaining the strategy used.
Double 32;multiply 31 by 8; divide 56
by 7; what is the remainder when 74 is
divided by 8?
What is the product of 25 and 4?
What are the factors of 36? Is 81 a
multiple of 3? Is 156 divisible by 6?
How do you know?
Use mental or mental with jottings to
answer:
80 x 9 = ■ ;
■ x 9 = 0.36;
172 divided by 4 = ■ ;
54/? = 18
Use written methods or a calculator to
answer:
134 x 46 = ■ ;
Begin to give a quotient as a fraction or
decimal fraction.
43 divided by 9 = 4 7/9;
61divided by 4 = 15.25
Relate division and fractions.
1/3 of 24 is equivalent to 24 divided by
3
Solve simple problems using ideas of
ratio and proportion.
Zara uses 3 tomatoes for every litre of
sauce. How much sauce can she make
from 15 tomatoes?
2.7 x ■ = ■ ;
(14 x 60) + ■ ;
900 divided by 36 = ■ ;
(125 divided by ■) + 2 = 27
Multiplication – Stage 6
Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including
long multiplication for two-digit numbers
NB Ensure that children are confident with the methods outlined in the previous
stage’s guidance before moving on.
Build on the work covered in stage 5 with the formal method of short multiplication (two-digit number
multiplied by a one-digit number).
When children are confident introduce multiplication by a two-digit number.
If necessary, return to the grid method and/or expanded method first.
Grid method (two-digit number multiplied by a ‘teen’ number):
35 x 13 = (30 + 5) x (10 + 3) = 455
X
30
5
10
300
50
3
90
15
350 + 105 = 455
This leads into … Compact long multiplication (formal method):
23 x 13 = 299
23 Use the language of place value to ensure understanding
X 13
+ 69 (3 x 23)
230 (10 x 23) Add the partial products
299
Extend to larger two-digit numbers (returning to the grid method and then expanded
method if necessary)
When children are confident with long multiplication extend with three-digit numbers multiplied by a
two-digit number, returning to the grid method first, if necessary:
124 x 26 = 3224
124
X 26
Use the language of place value to ensure understanding
12
+ 744 (6 x 124)
2480 (20 x 124) Add the partial products
3224
11
The prompts (in brackets) can be omitted if children no longer need them.
Extend with short and long multiplication of decimal numbers (initially in the context of money and
measures), returning to an expanded method first, if necessary (see stage 7guidance).
REMEMBER: if at any time, children are making significant errors, return to the previous stage.
Demonstrate understanding of
multiplying or dividing a whole
number by 10, 100 or 1000.
Respond rapidly to oral or written
questions, explaining the strategy used.
Double 75; multiply 25 by 8; divide 15
into 225; what is the remainder when
104 is divided by 12?
What is the product of 125 and 4?
What are the factors of 98?
Use mental or mental with jottings to
answer:
0.7 x 20 = ■;
0.3 x ■ = 2.4;
17.2 divided by 4 = ■;
?/25 = 39
Use written methods or a calculator to
answer:
738 x 639 =13;
(41 x 76) + ■ = 4000;
4123 divided by 365;
(100 divided by ■) + 5 = 7.5
Give a quotient as a decimal fraction,
rounding where appropriate to one
decimal place.
676 divided by 8 = 84.5;
85 divided by 7 =12.1 to one
decimal place.
Solve simple problems involving ratio
and proportion.
At the gym club there are 2 boys for
every 3 girls. There are 30 children at
the club. How many boys are there?
Multiplication – Stage 7
Multiply multi-digit numbers (including decimals) up to 4 digits by a two-digit whole
numbers using the formal written method of multiplication
NB Ensure that children are confident with the methods outlined in the previous
stage’s guidance before moving on.
Continue to practise and develop the formal short multiplication method and formal long multiplication
method with larger numbers and decimals. Return to an expanded forms of calculation initially, if
necessary.
The grid method (decimal number multiplied by a two-digit number):
X
20
3
50
1000
150
3
60
9
0.3
6
0.9
1066
159.9
1225.9
The formal written method of long multiplication:
53.2
X 24.0
It is an option to include .0 in this example, but not essential.
1
212.8
1064.0
1276.8
(53.2x4)
(53.2x20)
The prompts (in brackets) can be omitted if children no longer need them
REMEMBER: if at any time, children are making significant errors, return to the previous stage.
Our aim is that by the end of stage 7 children use mental methods (with jottings) when appropriate, but
for calculations that they cannot do in their heads, they confidently and accurately use an efficient
formal written method with speed.
Potential difficulties Understanding multiplication and division
Children may:
● be able to double four by pairing two groups of four and counting the group of eight but do not
associate this addition with the equivalent multiplication four multiplied by two;
● halve by sharing or forming pairs and counting but may not associate it with division by two or
division between two;
● halve and double independently, without recognising they are inverse operations, for example,
knowing that half of eight is four means that double four is eight;
● interpret 12 ÷ 3 as 12 shared between 3 and use objects or pictures to share out the 12,but lose
track of their recording as the numbers increase as they have no other strategy available such as
counting in steps or groups;
● interpret 6 × 2 as six lots of 2 and apply repeated addition of two, rather than doubling the six;
● not be proficient in counting forwards and backwards in equal steps, and so make mistakes
when carrying out repeated addition or repeated subtraction;
● understand multiplication as repeated addition and resort to this method without using known
facts, for example writing 6 × 7 as 7 + 7 + 7 + 7 + 7 + 7 and counting in 7s despite knowing that 5 ×
7 = 35; ● understand division as repeated subtraction and use a method of counting back but lose
track of the number of steps to the answer;
● carry out division by sharing or grouping but cannot cope with a remainder and do not
recognise that a remainder must always be less than the divisor;
● associate × with multiplication and ÷ with division and do calculations 8 × 2 and 16 ÷ 2 but are
not able to find missing numbers in statements such as 6 × ■ = 12 and ■ ÷ 5 = 3;
● recognise what calculation to do when word problems include the words times or share, but
are less confident when other language is used such as product, divided by, remainder, and
mistakenly associate ‘how many?’ and ‘how much?’ with addition or subtraction;
● understand multiplication as repeated addition, and division as repeated subtraction, but not as
scaling up and down to prepare the way for later work in measures and on ratio.