Progression in calculation Year N to Year 2: MULTIPLICATION AND DIVISION
 National Curriculum statements 2014 (statutory) in blue and bold
Nursery
Exemplification and activities taken from NCETM National Curriculum toolkit
Reception
Year 1
Year 2
EYFS
Programme of Study statement: Pupils should be taught to:
Programme of Study statement: Pupils should be taught to:
Mathematics development involves providing children with opportunities to practise and improve
their skills in counting numbers, calculating simple addition and subtraction problems, and to
describe shapes, spaces, and measures.
 solve one-step problems involving multiplication and division, by calculating the answer
using concrete objects, pictorial representations and arrays with the support of a teacher
 calculate mathematical statements for multiplication and division within the multiplication tables and
write them using the multiplication (x), division (÷) and equals signs
Children should be able to:
Children should be able to: Find missing numbers or symbols in a calculation: 4 x __ = 20, __ ÷ 10 = 3
Anna has 3 boxes of cakes. Each box contains 5 cakes. How many cakes does she have altogether? Show how
you worked this out.
Early Learning Goal 11: Numbers
[Expected outcomes] Children count reliably with numbers from 1 to 20, place them in order and
say which number is one more or one less than a given number. Using quantities and objects, they
add and subtract two single-digit numbers and count on or back to find the answer. They solve
problems, including doubling, halving and sharing.
EYFS Profile exemplification for the level of learning and development expected at the end of the
EYFS Mathematics
Examples of doubling, halving and repeated addition for ‘expected’ achievement related to ELG:11
taken from exemplification
No examples of these aspects in the ‘expected’ exemplification for ELG 11
Use practical apparatus, arrays and images to help solve multiplication and division problems
such as:
 Ben had 5 football stickers. His friend Tom gave him 5 more, how many does he have
altogether?
 Share 12 sweets between two children. How many do they each have?
 Find half of and double a number or quantity:
 16 children went to the park at the weekend. Half that number went swimming. How many
children went swimming?
 I think of a number and halve it. I end up with 9, what was my original number?
NCETM Actvity A - Noah’s Ark
Give the children the opportunity to count in twos, finding the total number of animals on board
the Ark. As the children gain fluency counting in twos, start at different numbers and perhaps
changing from using concrete objects, to jumping in twos along a number line. Further uses could
be to find the number of groups of two on the ark, again initially using tangible objects, then
moving on to using a number line and demonstrating repeated subtraction.
Activity B - Arrays powerpoint
This resource is based in ‘PowerPoint’. The teacher can set simple multiplication word problems
for the children to solve. It is also useful for modelling arrays and how to write a multiplication
sentence.
Activity C - W hole class counting sessions
For this activity the children themselves are the objects to count. You can count in twos to find the
number of shoes in a group, count fingers on hands in fives and number of toes in tens. To
extend the children you can ask them to model how to write down this calculation or alter it to
practise their division facts from the 2, 5 and 10 times tables.
Activity D - NRICH Share Bears:
A lovely investigation involving the children in division by sharing, and early introduction to the
concept of remainders.
NCETM Activity B

BBC KS1 Starship Number Jumbler game, a simple animation where children choose the missing operation
sign in the calculation.

Write the calculation: give the children different pictures of groups of items or arrays. They then have to write
the multiplication sentence to match the picture. 2p, 5p and 10p coins could be used for this activity. The
children can write down how many of each coin they have and the total amount. Using coins, the children
could also write division calculations to match the images.
 show that multiplication of two numbers can be done in any order (commutative) and division of one
number by another cannot
Children should be able to: Use their knowledge of triangles of numbers to show related number facts. e.g. If 6 x 2
= 12 then 2 x 6 = 12 and 12 ÷ 6 = 2.
NCETM Activity C

Triangle of numbers: a good activity to use as a starter or plenary, demonstrating the commutativity of
multiplication. It can be used to demonstrate one number fact and the children can suggest the others.

Function box: reinforcing the relationship of division being the inverse of multiplication. To build up to this
online activity, a function box could be made and used as a visual resource in class. For example, model
how a number goes in and doubles, but put the number back through the machine it will halve. There are
lots of different ways to use the function box in class to deepen the understanding of the relationship
between multiplication and division.

Class number sentence: using digit cards and x and ÷ and = cards. Get the children to show how we can
take one known fact and find others using those numbers. As the children move the digits around, they will
demonstrate their understanding of using and applying their tables knowledge. What numbers can we move
around in a division sentence? Can they spot the relationship between multiplication and division? Ensure
they really understand the concept behind this activity, by encouraging them to show you with practical
apparatus
 solve problems involving multiplication and division, using materials, arrays, repeated addition, mental
methods, and multiplication and division facts, including problems in contexts
Children should be able to: Use various methods and apparatus to help them solve word problems such as:
There are 10 lollies in a bag. Charlie needs 30 lollies for his party. How many bags does he need to buy? Show
how you worked this out.
NCETM Activity D

TES word problems with differentiated questions: resources submitted by a Year 2 teacher, focusing on
problem solving using multiplication.

BBC Class Clips: problem solving - how many chairs? Use multiplication facts to help the Chuckle Brothers
organise their tables ready for their guests’ arrival.

Give the children a multiplication or division fact. Can they write a word problem to match it? Now swap
calculations with a partner and talk about how you would solve the problem.
 recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including
recognising odd and even numbers
Children should be able to: Recognise a multiple of 2, 5 or 10 and use their knowledge of multiplication facts to
find corresponding division facts. They can say which numbers are odd and which are even. e.g. 2 x 5 = 10, show
me three more number facts using these numbers. Is 34 an odd number? How do you know?
NCETM activity A:

Multiplication tables - pelmanism style matching cards. Find the pairs of cards showing the product and
multiplication fact in a timed activity.

Counting stick: using a metre stick with ten divisions use sticky notes to mark multiples of 2, 5 or 10. Practise
counting on and back from different numbers. As the children become more fluent, remove more sticky notes
and see how they can recall multiples starting at different points counting on and back.
BBC KS1 Bitesize: Camel Times Tables and Division Mine. These two games are a super resource for the children
to practise their recall of multiplication and division facts independently
Taken from: Non-statutory guidance: Development Matters 30 to 50 months A unique
child
• Uses some number names and number language spontaneously.
• Uses some number names accurately in play.
• Recites numbers in order to 10.
• Knows that numbers identify how many objects are in a set.
• Beginning to represent numbers using fingers, marks on paper or pictures.
• Sometimes matches numeral and quantity correctly.
• Shows curiosity about numbers by offering comments or asking questions.
• Compares two groups of objects, saying when they have the same number.
• Shows an interest in number problems.
• Separates a group of three or four objects in different ways, beginning to recognise that the
total is still the same.
• Shows an interest in numerals in the environment.
• Shows an interest in representing numbers.
• Realises not only objects, but anything can be counted, including steps, claps or jumps.
Taken from: Non-statutory guidance: Development Matters 40 to 60+ months A unique child
• Recognise some numerals of personal significance.
• Recognises numerals 1 to 5.
• Counts up to three or four objects by saying one number name for each item.
• Counts actions or objects which cannot be moved.
• Counts objects to 10, and beginning to count beyond 10.
• Counts out up to six objects from a larger group.
Selects the correct numeral to represent 1 to 5, then 1 to 10 objects.
• Counts an irregular arrangement of up to ten objects.
• Estimates how many objects they can see and checks by counting them.
• Uses the language of ‘more’ and ‘fewer’ to compare two sets of objects.
• Finds the total number of items in two groups by counting all of them.
• Says the number that is one more than a given number.
• Finds one more or one less from a group of up to five objects, then ten objects.
• In practical activities & discussion, beginning to use the vocabulary involved in adding &
subtracting.
• Records, using marks that they can interpret and explain.
• Begins to identify own mathematical problems based on own interests and fascinations.
Multiplication and division structures: synonymous at this stage combining two or more
groups of the same size and counting to establish total ( repeated addition or repeated
aggregation);
Also division as uequal sharing and also as grouping
Multiplication and division structures: synonymous at this stage combining two or more groups
of the same size and counting to establish total ( repeated addition or repeated aggregation);
Also division as equal sharing and also as grouping
Recording: teacher demonstration of calculation of pictorial recording. Children begin using
pictorial recording. See children’s-mathematics link below.
VOCABULARY: more, and, make, sum, total, altogether, take away, leave,
Equipment: every day objects, counters, fingers
http://www.childrens-mathematics.net/
see ‘galleries’
http://www.foundati onyears.org.uk/wp-c ontent/upl oads/2011/10/Numbers_and_Patter ns.pdf
Recording: teacher demonstration of calculation to match pictorial recording using numberlines
Children begin using pictorial recording and numberlines. See children’s-mathematics link below.
VOCABULARY: double; half, halve; pair; group; divide into groups of; count out, share out; left,
left over
Equipment and resources: every day resources but also ‘maths’ resources such as number
tracks, number lines, counters, fingers, straws organised into groups of 10
http://www.childrens-mathematics.net/ see ‘galleries’
http://www.foundati onyears.org.uk/wp-c ontent/upl oads/2011/10/Numbers_and_Patter ns.pdf
Non-statutory guidance
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.
Non-statutory guidance
Pupils use a variety of language to describe multiplication and 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 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 to relate to grouping and
sharing discrete and continuous quantities, and relating these to fractions and measures (e.g. 40 ÷ 2 = 20, 20 is a
half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (e.g. 4 x 5 = 20 and 20
÷ 5= 4).
NB: In Y1 most strategies are based on counting but mental strategies are developing
Multiplication and division structures: repeated aggregation (addition) of groups of the same
size and recorded as an array; begin to recognise sharing equally; also repeated addition of
groups of the same size i.e. divided into groups of ….
Recording: Following teacher demonstration, children continue to develop pictorial recording
including an array; beginning to record on their own empty number line.
VOCABULARY: double, groups of, sets of, lots of; groups, share, left over, half, each
Equipment and resources : number lines, counters, fingers, straws, Base 10 equipment, every
day objects, counters, fingers,
ITPs: Multiplication Array ; Grouping;
NCETM video examples: ‘Multiple Representations’ and ‘ Reinforcing tables facts
The commutative law of multiplication at KS1 Sharing and grouping
NB: In Y2 mental and written strategies are still largely interchangeable
Multiplication structures: to recognise repetitive addition of groups of the same size; as an array; counting in
steps of 2, 5 & 10
Division structures: recognise division as sharing equally; and, also repeated addition or subtraction of groups of
the same size i.e. grouping
Recording: Following teacher demonstration of calculation to match pictorial recording (number line & array),
children to continue to develop recording especially using pictures and their own number lines. They begin to use
standard notation (including the symbols x ÷ and =)
VOCABULARY: lots of, groups of; times, multiply, multiplied by; multiple of; once, twice, three times… ten
times…; times as (big, long, wide… and so on); repeated addition; array; row, column; double, halve; share, share
equally; one each, two each, three each…; group in pairs, threes… tens; equal groups of; ¸, divide, divided by,
divided into; left, left over
Equipment and resources : number lines, hundred squares, counters, array, pegboards, straws, squared paper,
cubes etc. & everyday objects e.g. bars of chocolate, egg boxes,
ITPs: Multiplication Array; Multiplication facts; Number dials; Grouping
NCETM video examples: ‘Multiple Representations’ and Reinforcing tables facts and The commutative law of
multiplication at KS1
Sharing and grouping
Also, BBC Learning Zone Class Clips
©
Shropshire Council
[email protected]
C. Hardy Calculation policy version 29/09/2014
1
Progression in calculation Year N to Year 2: MULTIPLICATION AND DIVISION
 National Curriculum statements 2014 (statutory) in blue and bold
Nursery
Exemplification and activities taken from NCETM National Curriculum toolkit
Reception
Year 1
Year 2
School examples of multiplication
School examples of multiplication
Multiplication: example from exemplification (NCETM)
Multiplication: example
Example from exemplification
Example from exemplification
School examples of multiplication
School examples of multiplication
 Ben had 5 football stickers on each page. He has two pages of stickers. How many does he
have altogether? ARRAY and
Example from exemplification: Anna has 3 boxes of cakes. Each box contains 5 cakes. How many cakes does
she have altogether? Show how you worked this out.
How many feet have these three teddy bears got altogether?
How many wheels do we need for these three lego cars?
For example, ‘practically’ and counting
in 5s.
4 + 4 + 4 = 12
0
Strategy: begin to recognise repetitive addition of groups of the same size
Recording: teacher demonstration of appropriate pictorial recording where appropriate.
Strategy: begin to recognise repetitive addition of groups of the same size; counting in steps of 10
or 2
Vocabulary: groups, sets
Equipment: every day objects, counters, fingers
Recording: teacher demonstration of calculation to match pictorial recording using standard
notation of + and =. Demonstrate on number line.
1
2
3
4
5
6
7
8
9
Model on a bead bar/string as ‘repeated’ addition i.e. 5 + 5 + 5 = 15
AND ‘5 three times equals 15’ so 5 x 3 = 15 or ‘five multiplied or times by three equals 15’.
Demonstrate recording on a number line.
10
Show two groups of 5 football stickers. Model this on a bead bar/string.
Demonstrate recording of two groups of 5 stickers on a number line.
Also show as a rectangular ‘array’ i.e. two groups of 5 or 5 + 5 = 10
x5
1
0
Vocabulary: double, groups of, sets of, lots of
1
2
1
3
4
5
6
7
1
8
9
Equipment: every day objects: e.g. cars, chairs, bears, children, fingers, gloves, toy cars, pairs of
socks. Also ‘maths’ objects e.g. counters,
10
11
12
13
14
15
Model/demonstrate as a rectangular ‘array’
using the cakes or beads (fold bead string) or
counters (left). Also show as a rectangular
‘array’ (right) as ‘three rows of 5’ i.e. 5 x 3 =15
Use the ITP Multiplication Facts to
demonstrate these processes and
also commutativity.
School examples of division
School examples of division
Example from division exemplification (NCETM)
Example from division exemplification (NCETM)
If we share out these cakes so everyone has one each, how many will be left over? If everyone
has two cakes, how many children will be able to have cakes today?
Can we share out these cakes fairly? How shall we do it?
 Share 12 sweets between two children. How many do they each have?
Other example: How many sticks of 4 cubes can you make from 20 cubes? OR If 20 cubes are shared equally
between 4 people, how many cubes do they each get?
If we put two cakes on each plate, how many plates do we need?
0
Strategy: begin to recognise sharing equally; also repetitive addition or subtraction of groups of
the same size
Strategy: begin to recognise sharing equally; also repetitive addition of ‘groups’ of the same size
i.e. grouping
Recording: teacher demonstration of appropriate pictorial recording where appropriate.
Vocabulary: groups, share
Recording: teacher demonstration of calculation to match pictorial recording using standard
notation of + and =. Demonstrate on number line.
Equipment: every day objects, counters, fingers
Vocabulary: groups, share, left over, half
Equipment: every day objects, counters, fingers
2
4
6
8
10
12
GROUPING Show 20 cubes. Divide the cubes into groups of 4 i.e. one group of 4, two groups of 4 and so on…..
Model this on a bead bar/string and demonstrate recording of one group of 4 cubes on a number line….. And
another group of 4 cubes etc. So, 4 + 4 + 4 + 4 + 4 = 4 x 5 = 20 so 20 ÷ 4 = 5 i.e. 20 divided into groups of 4
equals 5.
2 + 2 + 2 + 2 + 2 + 2 = 12
Or ‘sharing’ model to create an array, as above.
EQUAL SHARING Show 12 sweets. Share the sweets equally between two i.e. one for you and
one for me – we’ve used 2 of the sweets. Model this on a bead bar/string and demonstrate
recording of one group of 2 sweets on a number line….. Another one for you and one for me –
we’ve used 2 more of the sweets. Model this on a bead bar/string and demonstrate recording of
the second group of 2 sweets on a number line….. etc.
ALSO: use as above with GROUPING e.g. We have 12 sweets. If each child has 2 sweets, how
many children will have 2 sweets? i.e. two for you – that’s one group of two – that’s one person model on bead bar and record on a number line as above.
÷4
1
0
1
4
1
8
1
12
1
16
20
EQUAL SHARING use same model as above. e.g. We have 20 cubes. Share the cubes equally between each
person – we’ve used 4 of the cubes. Model this on a bead bar/string and demonstrate recording of one group of 4
cubes on a number line….. Another one for each person etc. So, 4 + 4 + 4 + 4 + 4 = 4 x 5 = 20 so 20 ÷ 4 = 5 i.e. 20
shared equally between 4 equals 5.
Find missing numbers or symbols in a calculation:  ÷ 10 = 3 Represent e.g. using the Singapore bar:
The bar above represents the unknown quantity.
If we divide the bar into 10 equal sections and we know that each section
has a value of 3.
3
3
3
3
3
3
3
3
3
3
Grouping ITP
Opportunities for links with other mathematics domains:
Opportunities for links with other mathematics domains:
Cross-curricular and real life connections
Cross-curricular and real life connections
Learners will encounter multiplication and division in:
Learners will encounter multiplication and division in:
Money - when shopping and recognising prices of items, ordering items by price, finding
quantities in multiple purchases, sales prices, sharing costs.
Money – shopping: finding quantities in multiple purchases, sales prices, sharing costs.
Measurement - calculating area and perimeter, finding journey distances, reading and calculating
scales, adjusting recipe quantities.
Data - interpreting and evaluating data, calculating amounts from pie charts and pictograms.
©
Shropshire Council
[email protected]
C. Hardy Calculation policy version 29/09/2014
So, 3 +3 +3 +3 +3 +3 +3 +3 +3 +3 = 30 OR 3 x 10 = 30
So the value of the empty box here  ÷ 10 = 3 must be 30
because 30 ÷ 10 = 3
OR as above but divided into GROUPS of 10.
2
Measurement - calculating area and perimeter, finding journey distances, reading and calculating scales, adjusting
recipe quantities.
Data – interpreting and evaluating data, calculating amounts from pie charts and pictograms
Progression in calculation Year 3 to Year 6: MULTIPLICATION AND DIVISION
National Curriculum statements 2014 (statutory) in blue and bold
Year 3
Exemplification and activities taken from NCETM National Curriculum toolkit
Year 4
Programme of Study Pupils should be taught to:
 recall and use multiplication and division facts for the 3,4 and 8 times tables.
 multiply seven by three; what is four multiplied by nine? Etc.
 Circle three numbers that add to make a multiple of 4: 11 12 13 14 15 16 17 18 19
 Leila puts 4 seeds in each of her pots. She uses 6 pots and has 1 seed left over. How many seeds
did she start with?
 At Christmas, there are 49 chocolates in a tin and Tim shares them between himself and 7 other
members of the family. How many chocolates will each person get?
 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.
 One orange costs nineteen pence. How much will three oranges cost?
 Mark drives 19 miles to work every day and 19 miles back. He does this on Mondays, Tuesdays,
Wednesdays, Thursdays and Fridays. How many miles does he travel to work and back in one
week?
 solve problems, including missing number problems, involving multiplication and division,
including integer scaling problems and correspondence problems in which n objects are
connected to m objects.
 Miss West needs 28 paper cups. She has to buy them in packs of 6 How many packs does she
have to buy?
NCETM ACTIVITES
Year 5
Programme of Study Pupils should be taught to:
 recall multiplication and division facts for multiplication tables up to 12 × 12
Children should be able to:
 Pupils continue to practise recalling and using multiplication tables and related division facts to aid
fluency. e.g.
o One orange costs nineteen pence. How much will three oranges cost?
o What is twenty-one multiplied by nine?
o How many twos are there in four hundred and forty?
Year 6
Programme of Study Pupils should be taught to:
 identify multiples and factors, including finding all factor pairs of a number, and common factors of 2
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
 Use the vocabulary factor, multiple and product. They identify all the factors of a given number; eg, the factors
of 20 are 1, 2, 4, 5, 10 and 20. They answer questions such as:
o Find some numbers that have a factor of 4 and a factor of 5. What do you notice?
o My age is a multiple of 8. Next year my age will be a multiple of 7. How old am I?
 They recognise that numbers with only two factors are prime numbers and can apply their knowledge of
multiples and tests of divisibility to identify the prime numbers less than 100. They explain that 73 children can
only be organised as 1 group of 73 or 73 groups of 1, whereas 44 children could be organised as 1 group of 44,
2 groups of 22, 4 groups of 11, 11 groups of 4, 22 groups of 2 or 44 groups of 1. They explore the pattern of
primes on a 100-square, explaining why there’ll never be a prime number in the 10th column & the 4th column.
 use place value, known and derived facts to multiply and divide mentally, including:
multiplying by 0 and 1; dividing by 1; multiplying together 3 numbers
 Children should be able to:
 Pupils practise mental methods and extend this to three-digit numbers to derive facts, for example
200 × 3 = 600 into 600 ÷ 3 = 200. e.g.
o Divide thirty-one point five by ten.
o Ten times a number is eighty-six. What is the number?
 recognise and use factor pairs and commutativity in mental calculations
 multiply and divide numbers mentally, drawing upon known facts
 Rehearse multiplication facts and use these to derive division facts, to find factors of two-digit numbers and to
multiply multiples of 10 and 100, e.g. 40 × 50. They use and discuss mental strategies for special cases of
harder types of calculations, for example to work out 274 + 96,< 8006 – 2993, 35 × 11, 72 ÷ 3, 50 × 900. They
use factors to work out a calculation such as 16 × 6 by thinking of it as 16 × 2 × 3. They record their methods
using diagrams (such as number lines) or jottings and explain their methods to each other. They compare
alternative methods for the same calculation & discuss any merits &disadvantages.
Children should be able to:
 Pupils write statements about the equality of expressions (e.g. 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 e.g. 2 x 6 x 5 = 10 x 6.
 e.g. Understand and use when appropriate the principles (but not the names) of the commutativ e,
associative and distributive laws as they apply to multiplication:
 Example of commutativ e law 8 × 15 = 15 × 8
 Example of associative law 6 × 15 = 6 × (5 × 3) = (6 × 5) × 3 = 30 × 3 = 90
 Example of distributive law 18 × 5 = (10 + 8) × 5 = (10 × 5) + (8 × 5) = 50 + 40 = 90
 multiply two-digit and three-digit numbers by a one-digit number using formal written layout
 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
 Develop and refine written methods for multiplication. They move from expanded layouts (such as the grid
method) towards a compact layout for HTU × U and TU × TU calculations. They suggest what they expect the
approximate answer to be before starting a calculation and use this to check that their answer sounds sensible.
For example, 56 × 27 is approximately 60 × 30 = 1800.
 solve problems involving multiplying and adding, including using the distributive law to
multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence
problems such as n objects are connected to m objects
Children should be able to:
 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, or three cakes shared equally between 10 children. e.g.
o 185 people go to the school concert. They pay £l.35 each. How much ticket money is
collected?
o Programmes cost 15p each. Selling programmes raises £12.30. How many programmes are
sold?
 multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000
 Recall quickly multiplication facts up to 10 × 10 and use them to multiply pairs of multiples of 10 and 100. They
should be able to answer problems such as: the product is 400. At least one of the numbers is a multiple of 10.
What two numbers could have been multiplied together? Are there any other possibilities?
 recognise and use square numbers and cube numbers, and the notation for squared (²) & cubed (³)
 solve problems involving multiplication and division, including using their knowledge of factors and multiples,
squares and cubes
 use knowledge of multiplication facts to derive quickly squares of numbers to 12 × 12 & the corresponding
squares of multiples of 10. They should be able to answer problems such as:
o tell me how to work out the area of the cardboard with dimensions 30cm by 30cm
o find two square numbers that total 45
NCETM ACTIVITES
 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
Extend written methods for division to include HTU ÷ U, including calculations with remainders. They suggest what
they expect the approximate answer to be before starting a calculation and use this to check that their answer
sounds sensible. They increase the efficiency of the methods that they are using. For example: 196 ÷ 6 is
approximately 200 ÷ 5 = 40
4
2
32
r 4 or or
6
Non-statutory guidance
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 (e.g. 4 x 12 x 5 = 4 x 5 x
12= 20 x 12 = 240) and multiplication and division facts (e.g. using 3 x 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷3)
to derive related facts (30 x 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).
Pupils develop reliable written methods for multiplication and 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,
including measuring and scaling contexts, and correspondence problems in which m objects are
connected to n objects (e.g. 3 hats and 4 coats, how many different outfits?; 12 sweets shared
equally between 4 children; 4 cakes shared equally between 8 children).
Non-statutory guidance
Pupils continue to practise recalling and using multiplication tables and related division facts to aid
fluency.
Pupils practise mental methods and extend this to 3-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 (see Mathematics Appendix 1).
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, or 3 cakes shared equally between 10 children.
Multiplication structures: to recognise repeated aggregation (addition of groups of the same size);
and scaling (increase/decrease a number of times/by a scale factor of) – both structures should be
represented as an array.
Division structures: recognise division as equal sharing; the inverse of repeated aggregation
(divided into groups of…) i.e. grouping
Multiplication structures: to recognise repeated aggregation (addition of groups of the same size);
and scaling (increase/decrease a number of times/by a scale factor of) – both structures should be
represented as an array.
Division structures: recognise division as equal sharing; the inverse of repeated aggregation
(divided into groups of…) i.e. grouping
Recording: Following teacher demonstration to link pictorial recording (number line & array), children
continue to develop recording especially using pictures and their own number lines. They use
standard notation (including the symbols x ÷ and =)
Recording: Following teacher demonstration to link pictorial recording (number line & array), children
continue to develop recording especially their own number lines, grid multiplication to represent arrays,
and vertical recording of expanded vertical layouts of multiplication & division. They use standard
notation (including the symbols x ÷ and =)
VOCABULARY: lots of, groups of; ´, times, multiply, multiplication, multiplied by; multiple of, product,
repeated addition, array; row, column;, halve; share, share equally; one each, two each, three each…
group in pairs, threes… tens; equal groups of, divide, division, divided by, divided into, left, left over,
remainder
Language of scaling: once, twice, three times… ten times…; times as (big, long, wide… and so on);
double, scaling, scale, factor, doubling, trebling, so many times bigger, than (longer than, heavier
than, and so on), so many times as much as (or as many as).
Equipment and resources : number lines, hundred squares, counters, array, pegboards, straws,
squared paper, cubes etc. & everyday objects e.g. bars of chocolate, egg boxes,
ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials; Grouping;
Multiplication Board (tables square); Multiplication grid (grid multiplication)
NCETM video examples: Times Tables in Ten Minutes;
©
Shropshire Council
[email protected]
Grid multiplication as an interim step
Recording: Following teacher demonstration to link pictorial recording (number line & array), children continue to
develop recording especially grid multiplication to represent arrays, and vertical recording of expanded vertical
layouts of multiplication & division. They use begin to use standard compact layouts for multiplication & division.
VOCABULARY: lots of, groups of, times, multiply, multiplication, multiplied by, multiple of, product, once, twice,
three times… ten times…, times as (big, long, wide… and so on), repeated addition, array, row, column, double,
halve, share, share equally one each, two each, three each…, group in pairs, threes… tens equal groups of,
divide, division, divided by, divided into, remainder, factor, quotient, divisible by, inverse
VOCABULARY: lots of, groups of’ times, multiply, multiplication, multiplied by, multiple of, product,
once, twice, three times… ten times…times as (big, long, wide… and so on), repeated addition, array,
row, column, double, halve, share, share equally, one each, two each, three each…, group in pairs,
threes… tens, equal groups of, divide, division, divided by, divided into, remainder, factor, quotient,
divisible by
Inverse
Equipment and resources : number lines, hundred squares, counters, array, pegboards, straws,
squared paper, cubes etc. & everyday objects e.g. bars of chocolate, egg boxes,
Equipment and resources : a range of mathematics equipment as well as everyday situations especially those
that produce arrays e.g. number of seats in an auditorium
ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials; Multiplication Board (tables
square); Multiplication grid (grid multiplication)
Moving Digits
http://www.wmnet.org.uk/resour ces/gordon/Chunking.swf
ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials; Multiplication
Board (tables square); Multiplication grid (grid multiplication)
Moving Digits
http://www.wmnet.org.uk/resour ces/gordon/Chunking.swf
NCETM video examples: Grid multiplication as an interim step
Times Tables in Ten Minutes
C. Hardy Calculation policy version 29/09/2014
3
6
196
Children know that, depending on the context, answers to division questions may need to be rounded up or
rounded down. They explain how they decided whether to round up or down to answer problems eg.: Egg boxes
hold 6 eggs. A farmer collects 439 eggs. How many boxes can he fill? OR Egg boxes hold 6 eggs. How many
boxes must a restaurant buy to have 200 eggs?
 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
NCETM ACTIVITES
Non-statutory guidance
Pupils practise and extend their use of the formal written methods of short multiplication and short division (see
Mathematics Appendix 1). They apply all the multiplication tables and related division facts frequently, commit
them to memory and use them confidently to make larger calculations.
They use and understand the terms factor, multiple and prime, square and cube numbers.
Pupils interpret non-integer answers to division by expressing results in different ways according to the context,
98
including with remainders, as fractions, as decimals or by rounding (eg, 98 ÷ 4 = 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 (eg, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 92 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 ).
Multiplication structures: to recognise repeated aggregation (addition of groups of the same size); and scaling
(increase/decrease a number of times/by a scale factor of) – both structures should be represented as an array.
Division structures: recognise division as equal sharing; the inverse of repeated aggregation (divided into
groups of…) i.e. grouping; and, ratio structure for division i.e. comparison of two quantities – how many times
less than or more than …
NCETM video examples: Moving from grid to a column method
counters
rapid recall of multiplication facts
Representing division with place value
Programme of Study Pupils 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
 Look at long-multiplication calculations containing errors, identify the errors and
determine how they should be corrected.
 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
 Every day a machine makes 100 000 paper clips, which go into boxes. A full box has
120 paper clips. How many full boxes can be made from 100 000 paper clips?
 Each paper clip is made from 9.2 centimetres of wire. What is the greatest number of
paper clips that can be made from 10 metres of wire?
 A DJ has two different sized storage boxes for her CDs. Small boxes hold 15 CDs.
Large boxes hold 28 CDs. The DJ has 411 CDs. How could the DJ pack her CDs?
 Solve problems such as: Printing charges for a book are 3p per page and 75p for the
cover. I paid £4.35 to get this book printed. How many pages are there in the book?
Write down the calculations that you did. Seeds are £1.45 for a packet. I have £10 to
spend on seeds. What is the greatest number of packets I can buy?
 solve problems involving multiplication and division
 use estimation to check answers to calculations and determine, in the context of
a problem, levels of accuracy
Children should be able to:
 Give the best approximation to work out 4.4 × 18.6 and explain why. Answer
questions such as: roughly, what answer do you expect to get? How did you arrive at
that estimate? Do you expect your answer to be greater or less than your estimate?
Why?
 perform mental calculations, including with mixed operations and large numbers
 Use mental strategies to calculate in their heads, using jottings and/or diagrams where
appropriate. For example, to calculate 24 × 15, they multiply 24 × 10 and then halve
this to get 24 × 5, adding these two results together. They record their method as (24
× 10) + (24 × 5). Alternatively, they work out 24 × 5 = 120 (half of 24 × 10), then
multiply 120 by 3 to get 360.
 identify common factors, common multiples and prime numbers
 Children should be able to answer questions such as:
o How can you use factors to multiply 17 by 12?
o Start from a two-digit number with at least six factors, e.g. 72. How many
different multiplication and division facts can you make using what you know
about 72? What facts involving decimals can you derive?
o What if you started with 7.2? What about 0.72?
o Which three prime numbers multiply to make 231?
o use their knowledge of the order of operations to carry out calculations involving
the four operations
 Children should be able to find answers to calculations such as 5.6 ⬜ = 0.7 or 3 × 0.6,
drawing on their knowledge of number facts and understanding of place value. They
should be able to approximate, use inverses and apply tests of divisibility to check
their results.
 Children should know the square numbers up to 12 × 12 and derive the corresponding
squares of multiples of 10, for example 80 × 80 = 6400.
NCETM ACTIVITES
Non-statutory guidance
Pupils use the whole number system, including saying, reading and writing numbers
accurately Pupils practise addition, subtraction, multiplication and division for larger
numbers, using the formal written methods of columnar addition and subtraction, short
and long multiplication, and short and long division (see Mathematics Appendix 1).
They undertake mental calculations with increasingly large numbers and more complex
calculations.
Pupils continue to use all the multiplication tables to calculate mathematic al 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.
Multiplication structures: to recognise repeated aggregation (addition of groups of the
same size); and scaling (increase/decrease a number of times/by a scale factor of) – both
structures should be represented as an array.
Division structures: recognise division as equal sharing; the inverse of repeated
aggregation (divided into groups of…) i.e. grouping; and, ratio structure for division i.e.
comparison of two quantities – how many times less than or more than …
Recording: Following teacher demonstration to link pictorial recording (number line &
array), children continue to develop recording especially grid multiplication to represent
arrays, and vertical recording of expanded vertical layouts of multiplication & division.
They use begin to use standard compact layouts for multiplication & division.
VOCABULARY: lots of, groups of, times, multiply, multiplication, multiplied by, multiple
of, product, once, twice, three times… ten times…, times as (big, long, wide… and so
on), repeated addition, array, row, column, double, halve, share, share equally one each,
two each, three each…, group in pairs, threes… tens, equal groups of, divide, division,
divided by, divided into, remainder, factor, quotient, divisible by, inverse
Equipment and resources : a range of mathematics equipment as well as everyday
situations especially those that produce arrays e.g. number of seats in an auditorium
ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials;
Multiplication Board (tables square); Multiplication grid (grid multiplication)
Moving
Digits http://www.wmnet.org.uk/resources/gordon/Chunking.swf
NCETM video examples:
Moving from grid to a column method
Representing
division with place value counters
rapid recall of multiplication facts
3
Progression in calculation Year 3 to Year 6: MULTIPLICATION AND DIVISION
National Curriculum statements 2014 (statutory) in blue and bold
Year 3
Exemplification and activities taken from NCETM National Curriculum toolkit
Year 4
Year 5
Multiplication: example from exemplification (NCETM)
Multiplication: example
Mark drives 19 miles to work every day. He does this on Mondays, Tuesdays, Wednesdays,
Thursdays and Fridays. How many miles does he travel to work in one week?
The class wants to make 235 spiders for a display. How many legs do they need to make?
235 x 8 is approximately 235 x 10 = 2350
Demonstrate using a numberline as ‘repeated’ addition using straws or Base 10 as a model i.e.
Model using Base 10 equipment or straws. Use the
equipment to demonstrate partitioning i.e. 235 x 8 =
200 x 8 + 30 x 8 + 5 x 8 = (200 + 30 + 5) x 8
9
10
9
10
9
10
9
10
0
0
Use the equipment e.g. Base 10 materials, to show partitioning i.e. 1328 x 43 =
1000 x 43 + 300 x 43 + 20 x 43 + 8 x 43 = (1000 + 300 + 20 + 8) x 43
Leading to formal recording of ‘grid multiplication’ as a representation of the rectangular array. Possible to
record on a number line – see Y4 example.
Leading to recording using ‘grid multiplication’ as a representation of a rectangular
array. Possible use of Excel to generate array – see Y5 example.
Possible use of Excel to generate array to represent 56 x 27
1840
1880
Leading to formal recording of ‘grid multiplication’
as a representation of the rectangular array.
Then ‘regrouped’ as 10s and 1s:
10 + 10 + 10 + 10 + 10 + 9 + 9 + 9 + 9 + 9 =
10
10
10
10
10
9
9
9
9
x
200
30
5
8
1600
240
40
27
1
Leading to an
expanded
‘vertical’ layout.
Either begin with
least or most
significant figure.
1600 + 240 + 40 = 1880 so, 235 x 8 = 1880
9
This can be rewritten as: 10 x 5 + 9 x 5 = 50 + 45 = 95
x5
10
9
45
0
50
95
x
200
30
5
As 235 x 8 = 8 x 235 (commutative), the
layout for grid multiplication can also be
presented ‘vertically’ as a step towards
‘long multiplication’.
Use informal mental methods of
calculation i.e. PARTITIONING
19 x 5 = (10 x 5) + (9 x 5)
2
Leading to an expanded ‘vertical’ layout:.
x
19
3
5
8
0
0
0
0
5
10 x 5 = 50
4
4
0
8
2
6
8
1
1
As 56 x 27 = 27 x 56
(commutative), the layout for
grid multiplication can also be
presented ‘vertically’ (largest
number on the vertical axis) as
a step towards ‘long
multiplication’.
8
1600
240
40
1880
x
x
40
3
1000
40 000
3000
So,
300
12 000
900
53 120
20
800
60
+ 3 984
8
320
24
57 104
53 120
+ 3984
3
2
4
2
6
0
0
2
0
0
0
0
x
0
ITP: Multiplication array
Leading to formal recording of ‘grid
multiplication’ as a representation of
the rectangular array. (Link with
‘area’)
50 + 45 = 95 so, 19 x 5 = 95
As 1328 x 43 = 43 x
1328 (commutative),
the layout for grid
multiplication can also
be presented
‘vertically’ as a step
towards ‘long
multiplication’.
40
1600
1328 x 43 is approximately 1200 × 40 = 48000
Where possible, model using Base 10 equipment. See Y4 example. Use the equipment to show partitioning
i.e. 56 x 27 = 50 x 20 + 50 x 7 + 6 x 20 + 6 x 7 = (50 + 6) x 27
5
240
9
Multiplication: example 1328 x 43
56
Record on a number line.
x8
200
30
19 + 19 + 19 + 19 + 19 = partitioned into 10s and 1s as:
10 + 9 + 10 + 9 + 10 + 9 + 10 + 9 + 10 + 9 =
10
Year 6
Multiplication: example from exemplification (NCETM) 56 × 27 is approximately 60 × 30 = 1800.
x
20
7
50
1000
350
So,
6
120
42
1120
392
+ 392
1120
1512
1
2
5
4
2
5
0
1
x
8 multiplied by 5 is 40
8 multiplied by 30 is 240
8 multiplied by 200 is 1600
To give a total of 1880
Leading to an expanded ‘vertical’
layout. Either begin with least or
most significant figure.
1
1
9 x 5 = 45
Leading to a compact ‘vertical’
layout:
OR COMPENSATION
i.e. 19 x 5 = 20 x 5 – 5
+
1
1
x
1
3
5
1
1
3
0
5
1
2
5
6
5
1
7
6
2
0
0
0
2
7
6
2
0
2
9
0
3
8
0
0
1
1
3
1
4
5
Leading to a compact
‘vertical’ layout. Either
begin with least or most
significant figure.
2
0
7
2
8
3
4
0
0
0
0
0
0
0
4
1
3
3
9
2
4
8
3
1
2
x
5
3 multiplied by 8 is 24
3 multiplied by 20 is 60
3 multiplied by 300 is 900
3 multiplied by 1000 is 3000
40 multiplied 8 is 320
40 multiplied by 20 is 800
40 multiplied by 300 is 12 000
40 multiplied by 1000 is 40 000
To give a total of 57 104
8
3
4
0
3 multiplied by 1328 is 39
40 multiplied by 1328 is
53 120
To give a total of 1880
5
7
1
0
4
1
1
Also include examples that involve decimals, money and other measures e.g. 18·6 x 4·4
6 multiplied by 7 is 42
6 multiplied by 20 is 120
50 multiplied by 7 is 350
50 multiplied by 20 is 1000
To give a total of 1512
6 multiplied by 27 is 162
50 multiplied by 27 is 1350
To give a total of 1512
As 18·6 x 4·4 = 4·4 x
18·6 (commutative),
the layout for grid
multiplication can also
be presented ‘vertically’
as a step towards ‘long
multiplication’.
x
4
0·4
10
40·0
4·00
So,
8
32·0
3·20
74.40
2·4
0·24
+ 7.44
74·4
+ 7·44
81·84
0.6
Leading to an expanded ‘vertical’ layout. Either begin with least or most significant
figure and then a compact ‘vertical’ layout – see above.
Division: example from exemplification (NCETM)
Division example: There are 87 shopping days to Christmas. How many weeks is that? 87  7
Division: example from exemplification (NCETM)
Miss West needs 28 paper cups. She has to buy them in packs of 6 How many packs does she have
to buy?
So, ‘87 divided into groups of 7’ is approximately 70 ÷ 7 = 10.
196 ÷ 6
÷6
1
1
1
1
r. 4
0
6
12
18
24 28
GROUPING i.e. 28 ÷ 6 in this
context: ‘28 cups divided into
groups of 6’.Modelled using
cups then a bead string or bar
and recorded on a number
line as above. So, 28 ÷ 6 = 4
remainder 4
Division: example from exemplification (NCETM)
i.e. ‘196 divided into groups of 6’ is approximately 180 ÷ 6 = 30
Each paper clip is made from 9·2 centimetres of wire. What is the greatest number of
paper clips that can be made from 10 metres of wire?
Leading on from Year 4, divide into ‘groups of the divisor’ using multiplies of groups of 10, 5 and 2
wherever possible and record on a number line.
GROUPING: Model using bead bar or string. Demonstrate partitioning into groups of 7 using the
interpretation of 87 ÷ 7 as ‘ 87 divided into groups of 7’
÷6
30
12
0
7
14
21
28
35
42
49
56
63
180
Leading on from Year 4 and Year 5, divide into ‘groups of the divisor’ using multiplies
of groups of 10, 5 and 2 wherever possible and record on a number line.
r. 4
771192
196
4
2
6
3
So, ’196 divided into groups of 6 equals 32 remainder 4’ i.e. 196  6 = 32 r.4 or 32 or 32
÷ 9·2
÷7
1
1
1
1
1
1
1
0
1
1
1
1
1
7
14
21
28
35
42
49
56
63
70
77
So, ’87 divided into groups of 7 equals 12 remainder 3’ i.e. 87  7 = 12 r.3 or 12
84
87
3
Where necessary
dividing by ‘groups of
the divisor’ using
groups of 10, 5 and 2
(see left) and then
using ‘multiple groups
of 10, 5 and 2 (see
right)
7
Then begin to divide into ‘groups of the divisor’ using groups of 10, 5 and 2 wherever possible:
EQUAL SHARING e.g. There are 28 paper cups to share equally between 6 people. So if everyone
has one paper cup, that’s one group of 6 paper cups. Demonstrate recording of one group of 6 paper
cups on a number line….. Another one for each person etc.
÷7
10
2
14
0
7
14
21
28
35
42
49
56
63
70
r. 3
77
84
87
So, 6 + 6 + 6 + 6 + 4 = 6 x 4 + 1 = 28 OR so 28 ÷ 6 = 4 r.4 i.e. 28 shared equally between 6 equals 4
remainder 4.
If children’s understanding is secure, introduce
vertical recording e.g. during summer term.
Demonstrate vertical recording alongside
number line recording. Begin to record
information vertically introducing ‘repeated
subtraction/chunking’ (or ‘repeated
addition/additive chunking’.
©
Shropshire Council
[email protected]
C. Hardy Calculation policy version 29/09/2014
-
8
7
÷7
7
1
1
0
7
4
3
10
2
so, 87  7 = 12 r.3 or 12
3
7
4
-
1
9
6
÷6
1
8
0
30
1
6
1
2
-
2
4
4
so, 196  6 = 32 r.4 or 32 6 or 32
8
7
14
21
28
35
42
49
920
r. 6·4
993·6
1000
So, ’1000 divided into lengths of 9·2 equals 108 paper clips with 6.4cm of wire
remaining.
When children’s understanding is secure, use vertical recording. Demonstrate vertical recording
alongside number line recording. Begin to record information vertically introducing ‘repeated
subtraction/chunking’ (or ‘repeated addition/additive chunking’) lead
r. 3
0
100
73·6
Record on a number line alongside bead bar or string.
Also use Grouping ITP:
10m  9·2cm = 1000  9·2 i.e. ‘10m or 1000cm of wire divided into lengths (groups) of
9·2cm’ is approximately 1000 ÷ 10 = 100
2
2
3
When children’s understanding is
secure, use vertical recording.
Demonstrate vertical recording
alongside number line recording. Begin
to record information vertically
introducing ‘repeated
subtraction/chunking’ (or ‘repeated
addition/additive chunking’).
1
-
0
9
7
-
0
2
8
4
3
2
0
0
0
6
3
4
7
6
 9·2
100
1
5
· 10
·6
·4
so, 1000  9·2 = 108 r.6·4
3
Progression in calculation Year 3 to Year 6: MULTIPLICATION AND DIVISION
National Curriculum statements 2014 (statutory) in blue and bold
Year 3
Exemplification and activities taken from NCETM National Curriculum toolkit
Year 4
Year 5
Year 6
For Year 5 and Year 6: When children are secure with ‘chunking’ and tables facts, introduce short division for single digit divisors using Base 10 equipment or place value counters to secure children’s
understanding of the process. For example, 196 ÷ 6 i.e. ‘196 divided into groups of 6’ is approximately 180 ÷ 6 = 30
Opportunities for links with other mathematics domains:
Cross-curricular and real life connections
Learners will encounter aspects of multiplication and division when working on area, relating to
arrays. Problem solving work involving finding all possibilities and combinations also draws on
knowledge of multiplication tables facts.
Fractions work within other curriculum areas and in real life links naturally to multiplication and
division work.
The notion of equal groups can emerge in many different activities and contexts, e.g. when packing
boxes, purchasing quantities of items for several people etc.
Opportunities for links with other mathematics domains:
Making connections to other topics within this year group
Measurement
 Convert between different units of measure [for example, kilometre to metre; hour to
minute]
 solve problems involving converting from hours to minutes; minutes to seconds; years to
months; weeks to days.
Fractions
 recognise and show, using diagrams, families of common equivalent fractions
 count up and down in hundredths; recognise that hundredths arise when dividing an object
by one hundred and dividing tenths by ten
 recognise and write decimal equivalents to 1/4 , 1/2 , 3/4
 find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of
the digits in the answer as ones, tenths and hundredths
Cross-curricular and real life connections
Learners will encounter multiplication and division in:
Counting – Calculating totals by counting small amounts or a proportion and then scaling up e.g.
standing against a tree and using your known height to work out ‘How many of me are equal to the
height of the tree?’ or counting people on one part of a stadium and multiplying to calculate the total
number of spectators.
Money – shopping: adding multiple products of the same price, adding coins of same value, working
out fraction/percentage discounts and special offers, sharing bills.
Measurement – Scaling quantities (e.g. recipes) to cater for more and less people, reading scales and
unlabelled increments on measuring apparatus, calculating area for carpets, decorating etc., scaling
shapes to scale geometric artwork e.g. How would you make this triangle three times its size/half its
size? Comparing river lengths/building heights e.g. the River Nile is x times longer than the River X.
The height of Snowdon is (fraction) of the height of Everest.
Statistics – Reading scales and determining appropriate scales for different types of graph relating to
weather, temperature, sound etc., Working with proportion, fractions and percentages using pie charts,
comparing data using ratio, fractions and scaling such as proportion of children missing breakfast or 1
in 7 children under 10 now has a mobile phone etc.

a. Set out 1 hundred, 9 tens and 6 ones.
b. ‘One hundred divided into (groups of) six is difficult using the
counters, so exchange 1 hundred for 10 tens. Set out counters
as indicated and amend written layout.
d. ‘One ten and six ones divided into (groups of) six is difficult
using the counters, so exchange 1 ten for 10 ones. Set out
counters as indicated and amend written layout.
e. ‘Sixteen ones divided into (groups of) six equals two (groups of
six ‘ones counters’) with four ones remaining.’ Set out
counters as indicated and amend written layout.
Opportunities for links with other mathematics domains:
Making connections to other topics within this year group
Learners will encounter multiplication and division in:
Fractions (including decimals and percentages)
Requirements include:
 multiply proper fractions and mixed numbers by whole numbers, supported by materials and
diagrams
 solve problems involving number up to 3 decimal places
When working on multiplication and division and/or fractions (including decimals and percentages), there are
opportunities to make connections between them, for example:
You could give the children strips of paper and ask them to fold them to show you different proper and mixed
fractions, for example, 5⁄8, 1 3⁄4. Next ask them to multiply these fractions by single digit numbers. They could use
the strips to help them: 1 5⁄8 x 6
5
1x 6 = 6
⁄8 x 6 = 30 ⁄8 or 3 6⁄8
1 5⁄8 x 6 = 6 + 3 6⁄8 = 9 6⁄8 or 9 3⁄4
Numbers with decimals are frequently seen in real life, for example when using money, so give the children
opportunities to multiply these in context. For example, you could give them take-away menus and ask them to
find out how much it would cost to buy four of a meal deal or a particular course. You could give them the total
cost of six of the same dish and ask to work out which dish you chose.
You could ask the children problems that involve multiplying numbers up to 3 decimal places and link to
measures, such as:
 Jessie had eight lengths of rope. Each was1m 36cm. If he put them side by side what would the total length
be?
 Paddy had 12 cartons of orange juice. Each carton contained 0.750l. How much juice did he have
altogether?
 Suzie, the baker, was making 14 loaves of bread for the local supermarket. For each loaf she needed
1.275kg of flour. What is the total amount of flour that she needed?
 India took part in a sponsored bike ride at her school. She cycled 25 times around the perimeter of the
school playground. The perimeter is 105.34m. How far did she travel?
Measurement
When working on multiplication and division and/or measurement there are opportunities to make connections
between them, for example:
You could give the children opportunities to rehearse multiplying by 10, 100 and1000 by converting, for example,
millimetres to centimetres, centimetres to metres, metres to kilometres. They could then multiply lengths, masses
and capacities of different sizes, for example, 14.75kg by 8. You could then put these into problem format, eg:
 Benji, a party organiser, was going to make a fruit punch. For each guest he needed 0.250ml of orange juice
and 0.250l of mango juice. If there are 25 guests coming to the party, what is the total amount of juice Benji
needs?
You could give the children an approximate equivalence between miles and kilometres, for example1.6km is
approximately 1 mile. Then they multiply this amount to find approximate equivalences for other miles, for
example 5 miles, 8 miles, 10 miles, 14 miles. The children could make a spider diagram for this and other
equivalences.
You could give the children lengths of one side of different regular polygons, for example, pentagon, octagon,
decagon, dodecagon and ask them to find their perimeters by multiplying each length by the number of sides the
polygon has.
You could also give the children the lengths of different sized rectangles and ask them to find their areas, for
example, a rectangle 28cm by 12cm.
Set problems involving time and money for the children to use, for example:
 Samir spent 45 minutes completing his homework. It took Pete three times as long. How long did it take Pete
to complete his homework?
 It took Carol 1 ½ hours to drive from Oxford to London. It took Lorna a third of that time. How long did it take
Lorna to travel to London?
 Harry is given £3.75 a week as pocket money. He is saving it to buy a computer game. How much will he
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c. ‘Nineteen tens divided into (groups of) six equals three (groups
of six ‘tens counters’) with one ten remaining’. Set out
counters as indicated and amend written layout.
Opportunities for links with other mathematics domains:
Making connections to other topics within this year group
Fractions
 identify the value of each digit in numbers given to three decimal places and
multiply and divide numbers by 10, 100 and 1000 giving answers up to three
decimal places
 multiply one-digit numbers with up to two decimal places by whole numbers
 use written division methods in cases where the answer has up to two decimal
places
 divide proper fractions by whole numbers [e.g. ⅓ ÷ 2 = ⅙ ]
 multiply simple pairs of proper fractions, writing the answer in its simplest form
When working on multiplication and division and/or fractions there are opportunities to
make connections between them, for example:
Multiply numbers such as:
 245.25 by 10, 100 and 1000
 1.35 by 8
 ¼x½
Divide numbers such as:
 12 578 by 10, 100 and 1000
 237 by 5
 ⅓÷2
Ratio and proportion
 solve problems involving the relative sizes of two quantities where missing
values can be found by using integer multiplication and division facts
When working on multiplication and division and/or ratio and proportion there are
opportunities to make connections between them, for example:
Convert the ingredients in this lasagne recipe for 4 people so that it will serve 12:
 350g minced beef
 1 onion
 1 clove garlic
 600g tin of tomatoes
 2 tablespoons tomato puree
 175g lasagne sheets
Algebra
 express missing number problems algebraically
 use simple formulae
When working on multiplication and division and/or algebra there are opportunities to make
connections between them, for example:
Solve missing number problems, e.g. 6(a + 12)
= 144
multiply out the equation:
6a + 72
= 144
balance by -72:
6a + 72 – 72 = 144 – 72
6a
= 72
Use known division facts:
a
= 72 ÷ 6
a
= 12
Find perimeters and areas of rectangles using the appropriate formulae, e.g. a square field
has sides of 24.75m. What is its perimeter? What is its area?
Measurement
 solve problems involving the calculation and conversion of units of measure,
using decimal notation up to three decimal places where appropriate
 convert between miles and kilometres
When working on multiplication and division and/or measurement there are opportunities to
make connections between them by solving problems such as;
 1 pint = 0.57 litres, how many litres in 8 pints? How many pints in 12 litres?
 Dan was driving between two cities in France. The sign said the distance was 185km.
He wanted to know what that was in miles. How can he find out? How many miles is
it?
Statistics
 calculate and interpret the mean as an average
When working on multiplication and division and/or statistics there are opportunities to
make connections between them, for example:
Solve problems, e.g. find the mean monthly temperature for Reykjavik, Iceland
Progression in calculation Year 3 to Year 6: MULTIPLICATION AND DIVISION
National Curriculum statements 2014 (statutory) in blue and bold
Year 3
Exemplification and activities taken from NCETM National Curriculum toolkit
Year 4
Year 5
have saved over 8 weeks? What about 12 weeks?
 Georgie saved £2.25 of her pocket money each week. How much will she have saved over 9 weeks?
 Penny had saved £75 over a period of 12 weeks. She saved an equal amount every week. How much did
she save each week?
Cross-curricular and real life connections
Learners will encounter number and place value in:
Within the geography curriculum there are opportunities to connect with multiplication and division, for example in
the introduction of the Key Stage 2 Programme of Study it states that pupils should extend their knowledge and
understanding beyond the local area to include the United Kingdom and Europe, North and South America. This
will include the location and characteristics of a range of the world’s most significant human and physical
features. Children could, for example, find out about the currencies used in a selection of countries. They could
then make up a currency converter using mental calculation strategies and then check using multiplication, for
example:
£1= 1.20 Euros
£2 = 2.40 Euros
£3 = 3.60 Euros
£4 = 4.80 Euros
£5 = 6 Euros
Year 6
Monthly temperatures for Reykjavik
Jan Feb March April May June July Aug Sept Oct Nov Dec
-2°C -1°C 3°C 6°C 10°C 13°C 14°C 14°C 11°C 7°C 5°C -2°C
Cross-curricular and real life connections
Learners will encounter multiplication and division in:
Art & Design
Within the art and design curriculum there are opportunities to connect with multiplication
and division, for example in the introduction of the Key Stage 2 Programme of Study it
states that pupils should be taught to develop their techniques, including their control and
their use of materials, with creativity, experimentation and an increasing awareness of
different kinds of art, craft and design. This could include designing and creating life size
models of, for example a Barbara Hepworth sculpture or a Van Gogh painting where the
children need to find realistic measurements and then scale them down using division.
Geography
Within the geography curriculum there are opportunities to connect with multiplication and
division, for example in the introduction of the Key Stage 2 Programme of Study it states
that pupils should extend their knowledge and understanding beyond the local area to
include the United Kingdom and Europe, North and South America. This will include the
location and characteristics of a range of the world’s most significant human and physical
features. Work on multiplication and division could include converting between miles and
kilometres and vice versa when looking at distances between countries or famous
locations, making currency converters for pounds stirling and the currency in the country
they are investigating.
See, for example:

Mathematics and geography
History
Within the history curriculum, there are opportunities to connect with multiplication and
division, for example in the introduction of the Key Stage 2 Programme of Study it states
that ‘in planning to ensure the progression described above through teaching the British,
local and world history outlined below, teachers should combine overview and depth
studies to help pupils understand both the long arc of development and the complexity of
specific aspects of the content’. The history curriculum requires that pupils should
‘compare aspects of life in different periods’, suggesting comparisons between Tudor and
Victorian periods, for example. Scale models could be one way of learning about life in
different periods.
See, for example:




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The Tudors
The Victorians
The Ancient Egyptians
The Ancient Greeks