Download Multiplication and Division

Auckley School
Mathematics Policy
Calculation: multiplication and division
Foundation, Key Stage 1 and 2 mathematics
Updated: November 2016
This policy contains the key pencil and paper procedures that are to be taught throughout the school. It has
been written to ensure consistency and progression throughout the school. This policy is a suggested guide
for what method to teach in each year group and also to give each teacher an idea of the skills they can
expect children to come with at the start of each school year. The policy allows teachers to use an appropriate
method for each child in their class, depending on their ability and mathematical understanding, not their age.
It also aids parents in understanding their child’s stage of learning.
 Use Numicon and other practical resources to support calculations and help
to secure children’s understanding of place value, where appropriate.
 Ensure the children are confident with previous calculation methods before
moving on.
 Children should be encouraged to approximate their answers before
calculating.
 Children should be encouraged to check their answers after calculation using
an appropriate strategy.
 Children should be encouraged to consider if a mental calculation would be
appropriate before using written methods.
Note regarding place value terminology – throughout this policy we refer to “units”; however the National
Curriculum refers to “ones”. It is our opinion that both terminologies should be used with the pupils.
Stages in Multiplication
Method
Counting
songs and
rhymes
Pictorial,
practical
multiplication
Example
Double Rap
A fisherman catches 3 fish on Monday, 3 on Tuesday and 3 on
Wednesday. How many fish did he catch altogether?
Group and
count all
(Counting to
follow stages
in addition)
1, 2, 3
and
1, 2, 3
make 1, 2, 3, 4, 5, 6, 7, 8, 9
nine fish
and
1, 2, 3
Pictorial,
practical
multiplication
A fisherman catches 3 fish on Monday, 3 on Tuesday and 3 on
Wednesday. How many fish did he catch altogether?
Group and
count
multiples
3
and
3
and
3
make 3 and 6 and 9 = nine fish
Use of
numbered
lines
5 x 2 = 10
+2
+2
+2
+2
+2
Count on in
multiples
0
1
2
3
4
5
6
7
8
9
10
Use of blank
number line
5 x 2 = 10
+2
Count on in
multiples
0
+2
2
+2
4
+2
6
+2
8
10
Introduce
arrays
Three lots of two make 6
two lots of three make six
Leading to
3x2=6
2x3=6
Informal
recording:
partition the
number into
tens and
units/ones
multiply the
parts then
recombine
13 x 4
Possible eg.
Partition 13 into 10 and 3
13 x 4
10 x 4 = 40
3 x 4 = 12
10 x 4
3x 4
Recombine
40
12
Or
13 x 4
10 x 4 = 40
+
3 x 4 = 12
52
13 x 4 = 52
40 + 12 = 52
52
13 x 4 = 52
Progress to 13 x 4
using
previous step Verbally explain e.g. To calculate thirteen times four, partition thirteen into ten and three. Multiply
as a mental these parts by four. Ten times four is forty, three times four is twelve. Add the answers together:
method
forty add twelve is fifty two. So thirteen times four equals fifty two.
Formalise
TU x U
recording in
to the
expanded
vertical
method
Vertical
recording
compact
method
TU x U
38
X 7
56 (8 x 7)
210 (30 x 7)
266
Extend to higher numbers
38
X 7
266
5
Extend to higher numbers
Formalise
TU x TU
recording in
to the
expanded
vertical
method
38
X 17
266
5
380
646
1
Vertical
recording
compact
method TU x
TU
(38 x 7)
38
X 17
266
380
646
1
(38 x 10)
Extend to larger numbers
Extend to larger numbers
For decimals, the method used depends on the individual’s confidence with decimals. All partitioning methods will
work with a decimal partition – the informal partitioning is the most straightforward to do. An alternative would be to
deal with the decimal as a whole number and then replace the decimal in the final answer.
1.3 x 4
1.3 x 4
1.0 x 4 = 4
+
0.3 x 4 = 1.2
5.2
13 x 4 =
13
X 4
52
1
1.3 is ten times smaller than 13 so 1.3 x 4 is
ten times smaller than 13 x 4.
1.3 x 4 = 5.2
13 x 4 = 52
1.3 x 4 = 52 ÷ 10 = 5.2
Multiplication
Key Stage 1
The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency
with whole numbers, counting and place value. This should involve working with numerals, words and the four operations,
including with practical resources [for example, concrete objects and measuring tools].
By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value.
An emphasis on practice at this early stage will aid fluency.
Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling
knowledge at key stage 1.
Reception and Year 1
Objectives:
Pupils should be taught to:
 Solve one-step problems involving multiplication, by calculating the answer using concrete objects, pictorial representations and arrays
with the support of the teacher.
Notes and guidance (non-statutory):


Through grouping small quantities, pupils begin to understand: multiplication; doubling numbers and quantities; and finding simple
fractions of objects, numbers and quantities.
They make connections between arrays, number patterns, and counting in twos, fives and tens.
Resources and methods:





Children will experience equal groups of objects and will count in 2s, 5s and 10s and begin to count in 5s. They will work on practical
problem solving activities involving equal sets or groups, e.g. Count five hops of 2 along this number track. What number will you
reach? (Children will begin to move from using number tracks to number lines as appropriate through year 1 and 2).
Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of
recording calculations using pictures, etc.
They use number lines and practical resources to support calculation and teachers demonstrate the use of the number line.
Children then begin to use numbered lines to support their own calculations using a numbered line to count on in equal steps.
Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes,
number tracks, numbered lines
Year 2
Objectives:




Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers
Calculate mathematical statements for multiplication within the multiplication tables and write them using the multiplication (×) and
equals (=) signs
Show that multiplication of two numbers can be done in any order (commutative)
Solve problems involving multiplication, using materials, arrays, repeated addition, mental methods, and multiplication facts, including
problems in contexts.
Notes and guidance (non-statutory):


Pupils use a variety of language to describe multiplication.
Pupils are introduced to the multiplication tables. They practise to become fluent in the 2, 5 and 10 multiplication tables and
connect them to each other. They connect the 10 multiplication table to place value, and the 5 multiplication table to the divisions
on the clock face. They begin to use other multiplication tables and recall multiplication facts, including using related division facts
to perform written and mental calculations.
Pupils work with a range of materials and contexts in which multiplication and division relate to grouping and sharing discrete and
continuous quantities, to arrays and to repeated addition. They begin to relate these to fractions and measures (for example, 40 ÷
2 = 20, 20 is a half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 =
20 and 20 ÷ 5 = 4).

Resources and methods:


Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes,
number tracks, numbered lines.
Children will develop their understanding of multiplication and use jottings to support calculation.
Lower Key Stage 2
The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent
with whole numbers and the four operations, including number facts and the concept of place value. This should
ensure that pupils develop efficient written and mental methods and perform calculations accurately with
increasingly large whole numbers.
At this stage, pupils should develop their ability to solve a range of problems, including with simple fractions and
decimal place value. Teaching should ensure that pupils can make connections between measure and number.
By the end of year 4, pupils should have memorised their multiplication tables up to and including the 12
multiplication table and show precision and fluency in their work.
Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading
knowledge and their knowledge of spelling.
Year 3
Objectives:
Pupils should be taught to:
 Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables
 Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for
two-digit numbers times one-digit numbers, using mental and progressing to formal written methods
 Solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems
and correspondence problems in which n objects are connected to m objects.
Notes and guidance (non-statutory):




Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order
to improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables.
Pupils develop efficient mental methods, for example, using commutativity and associativity (for example, 4 × 12 × 5 = 4 × 5 × 12
= 20 × 12 = 240) and multiplication and division facts (for example, using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts
(for example, 30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).
Pupils develop reliable written methods for multiplication, starting with calculations of two-digit numbers by one-digit numbers and
progressing to the formal written methods of short multiplication.
Pupils solve simple problems in contexts, deciding which of the four operations to use and why. These include measuring and
scaling contexts (for example, four times as high, eight times as long etc.) and correspondence problems in which m objects are
connected to n objects (for example, 3 hats and 4 coats, how many different outfits?; 12 sweets shared equally between 4
children; 4 cakes shared equally between 8 children).
Resources and methods:


Children understand the relationship between multiplication and division. For example, they state two multiplication
sentences and two division sentences that relate to a particular array.
Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes,
number tracks, numbered lines.
Year 4
Objectives:
Pupils should be taught to:
 Recall multiplication and division facts for multiplication tables up to 12 × 12
 Use place value, known and derived facts to multiply mentally, including: multiplying by 0 and 1; multiplying together three
numbers
 Recognise and use factor pairs and commutativity in mental calculations
 Multiply two-digit and three-digit numbers by a one-digit number using formal written layout
 Solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit,
integer scaling problems and harder correspondence problems such as n objects are connected to m objects.
Notes and guidance (non-statutory):





Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.
Pupils practise mental methods and extend this to three-digit numbers to derive facts, (for example 600 ÷ 3 = 200 can be derived
from 2 x 3 = 6).
Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers.
Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and
associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental
and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60.
Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This
should include correspondence questions such as the numbers of choices of a meal on a menu.
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes,
number tracks, numbered lines.
Upper Key Stage 2
The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding
of the number system and place value to include larger integers. This should develop the connections that pupils
make between multiplication and division with fractions, decimals, percentages and ratio.
At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex
properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation.
With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of
problems. Teaching in geometry and measures should consolidate and extend knowledge developed in number.
By the end of year 6, pupils should be fluent in written methods for all four operations, including long multiplication
and division, and in working with fractions, decimals and percentages.
Pupils should read, spell and pronounce mathematical vocabulary correctly.
Year 5
Objectives:
Pupils should be taught to:










Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
Establish whether a number up to 100 is prime and recall prime numbers up to 19
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
Multiply numbers mentally drawing upon known facts
Multiply whole numbers and those involving decimals by 10, 100 and 1000
Recognise and use square numbers and cube numbers, and the notation for squared 2 and cubed 3
Solve problems involving multiplication including using their knowledge of factors and multiples, squares and cubes
Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of
the equals sign
Solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates.
Notes and guidance (non-statutory):





Pupils practise and extend their use of the formal written methods of short multiplication.
They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger
calculations.
Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by
powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres.
Distributivity can be expressed as a(b + c) = ab + ac.
They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for
2
example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9 x 10).

Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13 + 24 = 12
+ 25; 33 = 5 x 6 + 3).
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number
tracks, numbered lines.
Year 6
Objectives:
Should be taught to:
 Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication.
 Perform mental calculations, including with mixed operations and large numbers
 Identify common factors, common multiples and prime numbers
 Use their knowledge of the order of operations to carry out calculations involving the four operations
 Solve problems involving addition, subtraction, multiplication and division
 Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.
Notes and guidance (non-statutory):






Pupils practise multiplication for larger numbers, using the formal written methods of short and long multiplication.
They undertake mental calculations with increasingly large numbers and more complex calculations.
Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.
Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50 etc., but not to a specified number
of significant figures.
Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.
Common factors can be related to finding equivalent fractions.
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes,
number tracks, numbered lines.
Stages in Division
Method
Pictorial, practical
division – sharing
based on stories /
simple word
problems.
Note: there is no
mathematical way of
showing this form of
division other than by
practical and pictorial
means. It is a
necessary step in
understanding
division though and
so should be taught.
Example
6 fish shared between 3 bears:
Giving:
Each bear has two fish.
Pictorial, practical
division – grouping as
repeated subtraction.
6 fish are put into baskets and each basket has 3 fish in it. How many
baskets are needed?
6 divided by 3
Six take away one group of 3 then another group of three.
6 divided by 3 = 2
2 baskets are needed.
Number lines
Children will be taught
to use both repeated
addition and
subtraction on a
number line using the
term ‘groupings’ –
How many groups of 3
can we get from 15?
How many groups of 3
can we subtract from
15?
15 ÷ 3 = 5
1x3
0
3
- 1x3
0
It would be helpful to
refer to “lots of” e.g. 1
lot of 3, two lots of 3
etc.
1x3
1x3
6
- 1x3
3
1x3
9
- 1x3
6
1x3
12
- 1x3
9
15
- 1x3
12
15
Introduce problems
with remainders
pictorially.
8 fish shared between 3 sealions:
remainder
Giving:
Each sealion has two fish with 2 left over. This is the remainder.
Number lines and
remainders
Children will be taught
to use both repeated
addition and
subtraction on a
number line using the
term ‘groupings’ –
27 ÷ 6 = 4 remainder 3
1x6
0
1x6
6
1x6
12
1x6
18
1
24
1
1
27
How many groups of 6
can we get from 27?
How many groups of
six can we subtract
from 27?
Informal vertical
division - Chunking
Children will be taught
to use the ‘chunking’
method initially by
single multiples of the
divisor. Then
progressing to larger
multiples, ideally 10, 5
and 2, but also other
known multiples.
- 1x6
0
3
24 ÷ 5 = 4 remainder 4
24
- 5
19
- 5
14
- 5
9
- 5
4
- 1x6
9
- 1x6
15
- 1x6
21
Progressing to:
24 ÷ 5 = 4 remainder 4
(1 x 5)
(1 x 5)
(1 x 5)
(1 x 5)
Add the
multiples
to find
the answer
1+1+1+1 = 4
24
- 10
14
- 10
4
(2 x 5) Add the
multiples to
(2 x 5) find the
answer 2+2=4
27
Condensed Chunking
Once children are
confident in
‘chunking’, they may
condense the number
of steps by carrying
out the minimum
number of
subtractions –
subtract multiples of
powers of 10 in
descending size order.
Eg 3000, 500, 20, 7
7427 ÷ 3 = 2475 rem 2
7427
-6000 (2000 x 3)
1427
-1200 (400 x 3)
227
- 210 (70 x 3)
17
- 15 (5 x 3)
2
The children should
be discouraged from
finding the answer in
“one go” as this is an
inefficient use of time.
Informal recording:
partition the number
into useful multiples
of the divisor, divide
the parts then
recombine
27÷6 = 4 remainder 3
1 + 2 + 3 + 4 remainder 3
6 )6 + 12 + 18 + 24 r 3
Progressing to:
2473 rem 2
3)7427
-6000
1427
-1200
227
- 210
17
- 15
2
Formal recording –
Short division
Model with base 10
how to group and
exchange.
24÷6 = 4
TU
4
6 )224
Narration – can I split 2 tens into groups of 6? No: then we move the 2 tens into the next column to
make 24 units. Can 24 units be split into groups of 6? Yes: 4 groups. Record 4 in the units column.
Answer 4.
68 ÷ 4 =
T U
1 7
4 )6 28
Narration – can I split 6 tens into groups of 4? Yes, there is 1 group of 4 tens and 2 tens remaning.
Record the 1 group in the tens column and then move the 2 remaing tens into the units column to
make 28 units. Can 28 units be split into groups of 4? Yes 7 groups with no remainder. Record 7 in
the units column of the answer line. Answer 17.
435 ÷ 3 =145
HTU
1 4 5
3 )4 131 5
Narration – can I split 4 hundreds into groups of 3? Yes - there is 1 group of 3 hundreds and 1
hundred remining. Record the 1 group in the hundreds column and then move the the remaining 1
hundred into the tens column to make 13 tens. Can 13 tens be split into groups of 3? Yes - 4 groups
with 1 ten remaining. Record the 4 groups in the tens column and then move the reminaing 1 ten
into the units column to make 15 units. Can 15 units be split into groups of 3? Yes - 5 groups with
no remainder. Record the 5 groups in the units column. Answer 145.
Condensed Chunking
Chunking can also be
introduced for
divisors greater than
10
124 ÷ 15 = 8 r 4
124
- 90
34
- 30
4
(6 x 15) Add the
multiples to
(2 x 15) find the
answer 6+2=8
Formal recording –
Long division
Link to chunking.
494 ÷ 15 = 33 r 4
33
15 ) 4 9 9
-4 50
49
- 45
4
Help box 15x table
15,30,45,60,75,90,105,120,135,150
Narration – can I subtract a hundreds multiple of
15? (use help box to work out 100 x 15 = 1500).
No. What is the largest tens multiple of 15 I can
subtract? (Use the help box to work out 30 x 15 =
450). Record 3 in the Tens column then carry out
the subtraction. Now ask - What is the largest
units multiple of 15 I can take away from what is
left? (Use the help box to work out 3 x 15 = 45).
Record the 3 in the units column and then carry
out the subtraction.
Answer is 33 remainder 4.
For decimals, the most straightforward method would be to deal with the decimal as a
whole number and then replace the decimal in the final answer.
e.g. 12.4 ÷ 4 = 3.1
124 ÷ 4 = 31
124
- 120 (30 x 4)
4
- 4 (1 x 4)
0
12.4 is ten times smaller than 124 so
12.4 ÷ 4 is ten times smaller than 124÷ 4
so 12.4 ÷ 4 = 31 ÷10 = 3.1
Progression in
Remainders
Model using number
line, chunking and
short/long division
Initially remainders should be expressed as e.g. “remainder 3”
This should progress to expressing them as a fraction of the divisor.
5÷3
1x3
0
3
4
5
6
5÷3=1 2
(5 is two steps towards the next multiple of 3)
3
Knowledge of fraction and decimal equivalences also allows these
remainders to be converted to decimals e.g. 1 2/3 = 1.66
Extending the dividend beyond the decimal point:
e.g. 24 ÷ 5 = 4.8
24.0
- 20.0
4.0
- 4.0
0
(4 x 5)
(0.8 x 5) - from knowing 8 x 5=40
43.5 ÷ 3 =14.5
T U. te
1 4. 5
3 )4 13.1 5
Division
Key Stage 1
The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency
with whole numbers, counting and place value. This should involve working with numerals, words and the four operations,
including with practical resources [for example, concrete objects and measuring tools].
By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value.
An emphasis on practice at this early stage will aid fluency.
Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling
knowledge at key stage 1.
Reception and Year 1
Objectives:
Pupils should be taught to:
 Solve one-step problems involving division, by calculating the answer using concrete objects, pictorial representations and arrays with the
support of the teacher.
Notes and guidance (non-statutory):


Through grouping and sharing small quantities, pupils begin to understand: multiplication and division; doubling numbers and quantities;
and finding simple fractions of objects, numbers and quantities.
They make connections between arrays, number patterns, and counting in twos, fives and tens.
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number
tracks, numbered lines.
Year 2
Objectives:
Pupils should be taught to:
 Recall and use division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers
 Calculate mathematical statements for division within the multiplication tables and write them using the multiplication (×), division (÷) and
equals (=) signs
 Show that division of one number by another cannot be done in any order
 Solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and
division facts, including problems in contexts.
Notes and guidance (non-statutory):



Pupils use a variety of language to describe division.
Pupils are introduced to the multiplication tables. They practise to become fluent in the 2, 5 and 10 multiplication tables and connect them
to each other. They connect the 10 multiplication table to place value, and the 5 multiplication table to the divisions on the clock face. They
begin to use other multiplication tables and recall multiplication facts, including using related division facts to perform written and mental
calculations.
Pupils work with a range of materials and contexts in which division relates to grouping and sharing discrete and continuous quantities, to
arrays and to repeated addition. They begin to relate these to fractions and measures (for example, 40 ÷ 2 = 20, 20 is a half of 40). They
use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 = 20 and 20 ÷ 5 = 4).
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number
tracks, numbered lines.
Lower Key Stage 2
The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent with
whole numbers and the four operations, including number facts and the concept of place value. This should ensure that
pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole
numbers.
At this stage, pupils should develop their ability to solve a range of problems, including with simple fractions and decimal
place value. It should ensure that they can use measuring instruments with accuracy and make connections between
measure and number.
By the end of year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table
and show precision and fluency in their work.
Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading knowledge
and their knowledge of spelling.
Year 3
Objectives:
Pupils should be taught to:
 Recall and use division facts for the 3, 4 and 8 multiplication tables
 Write and calculate mathematical statements division using the multiplication tables that they know,
 Solve problems, including missing number problems, involving division, including positive integer scaling problems and correspondence
problems in which n objects are connected to m objects.
Notes and guidance (non-statutory):




Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to
improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables.
Pupils develop efficient mental methods, for example, using commutativity and associativity (for example, 4 × 12 × 5 = 4 × 5 × 12 = 20 ×
12 = 240) and multiplication and division facts (for example, using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (for example,
30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).
Pupils develop reliable written methods for division, starting with calculations of two-digit numbers by one-digit numbers and progressing
to the formal written methods of short multiplication and division.
Pupils solve simple problems in contexts, deciding which of the four operations to use and why. These include measuring and scaling
contexts, (for example, four times as high, eight times as long etc.) and correspondence problems in which m objects are connected to n
objects (for example, 3 hats and 4 coats, how many different outfits?; 12 sweets shared equally between 4 children; 4 cakes shared
equally between 8 children).
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number
tracks, numbered lines.
Year 4
Objectives:
Pupils should be taught to:
 Recall division facts for multiplication tables up to 12 × 12;
 Use place value, known and derived facts to divide mentally, including: dividing by 1;
 Recognise and use factor pairs and commutativity in mental calculations.
Notes and guidance (non-statutory):





Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.
Pupils practise mental methods and extend this to three-digit numbers to derive facts (for example 600 ÷ 3 = 200 can be derived from 2 x
3 = 6).
Pupils practise to become fluent in the formal written method of short division with exact answers.
Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law
(2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations
for example, 2 x 6 x 5 = 10 x 6 = 60.
Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should
include correspondence questions such as three cakes shared equally between 10 children.
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number
tracks, numbered lines.
Upper Key Stage 2
The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the
number system and place value to include larger integers. This should develop the connections that pupils make between
multiplication and division with fractions, decimals, percentages and ratio.
At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex
properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this
foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems.
Teaching in geometry and measures should consolidate and extend knowledge developed in number.
By the end of year 6, pupils should be fluent in written methods for all four operations, including long multiplication and
division, and in working with fractions, decimals and percentages.
Pupils should read, spell and pronounce mathematical vocabulary correctly.
Year 5
Objectives:
Pupils should be taught to:
 Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
 Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
 Establish whether a number up to 100 is prime and recall prime numbers up to 19
 Divide numbers mentally drawing upon known facts
 Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the
context
 Divide whole numbers and those involving decimals by 10, 100 and 1000




2
3
Recognise and use square numbers and cube numbers, and the notation for squared ( ) and cubed ( )
Solve problems involving division including using their knowledge of factors and multiples, squares and cubes
Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the
equals sign
Solve problems involving division, including scaling by simple fractions and problems involving simple rates.
Notes and guidance (non-statutory):




Pupils practise and extend their use of the formal written methods of short division.
They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger
calculations.
Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions,
as decimals or by rounding (for example, 98 ÷ 4 =98/4 = 24 r 2 = 24 ½ = 24.5 ≈ 25).
Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10
in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres.
Distributivity can be expressed as a(b + c) = ab + ac.
They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x

35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9 x 10).
Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13+24 = 12 + 25; 33 = 5 x).


2
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number
tracks, numbered lines.
Year 6
Objectives:
Pupils should be taught to:
 Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as
whole number remainders, fractions, or by rounding, as appropriate for the context
 Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting
remainders according to the context
 Perform mental calculations, including with mixed operations and large numbers
 Identify common factors, common multiples and prime numbers
 Use their knowledge of the order of operations to carry out calculations involving the four operations
 Solve problems involving addition, subtraction, multiplication and division
 Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.
Notes and guidance (non-statutory)






Pupils practise division for larger numbers, using the formal written methods of short and long division.
They undertake mental calculations with increasingly large numbers and more complex calculations.
Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.
Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50 etc., but not to a specified number of
significant figures.
Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.
Common factors can be related to finding equivalent fractions.
Resources and methods:

Practical resources could include: bead bars, threading beads, Numicon, counters, base 10, digit cards, arrow cards, dominoes, number
tracks, numbered lines.