Transcript Maths Calculations - multiply and divide

Greenfields Federation
Littlehaven Infant School
Northolmes Junior School
Aims
The national curriculum for mathematics aims to ensure that all
pupils:
• become fluent in the fundamentals of mathematics, including
through varied and frequent practice with increasingly
complex problems over time, so that pupils develop
conceptual understanding and the ability to recall and apply
knowledge rapidly and accurately.
• reason mathematically by following a line of enquiry,
conjecturing relationships and generalisations, and
developing an argument, justification or proof using
mathematical language
• can solve problems by applying their mathematics to a
variety of routine and non-routine problems with increasing
sophistication, including breaking down problems into a series
of simpler steps and persevering in seeking solutions.
2
Introduction
Written methods of calculations are based on mental strategies. Each
of the four operations builds on mental skills which provide the
foundation for jottings and informal written methods of recording. Skills
need to be taught, practised and reviewed constantly. These skills lead
on to more formal written methods of calculation.
Strategies for calculation need to be represented by models and
images to support, develop and secure understanding. This, in turn,
builds fluency. When teaching a new strategy it is important to start
with numbers that the child can easily manipulate so that they can
understand the methodology.
The transition between stages should not be hurried as not all children
will be ready to move on to the next stage at the same time, therefore
the progression in this document is outlined in stages. Previous stages
may need to be revisited to consolidate understanding when
introducing a new strategy.
A sound understanding of the number system is essential for children to
carry out calculations efficiently and accurately.
3
Magnitude of Calculations
Year 1 – U + U, U + TU (numbers up to 20) including adding zero, U – U, TU – U (numbers up to 20)
including subtracting zero, U x U, U ÷ U
Year 2 - TU + U, TU + multiples of 10, TU + TU, U + U + U, TU - U, TU – tens, TU – TU, TU x U, U ÷ U
Year 3 – add numbers with up to three-digits, HTU + multiples of 10, HTU + multiples of 100, subtract
numbers up to three-digits, HTU – U, HTU – multiples of 10, HTU – multiples of 100, HTU – HTU,
TU x U, TU ÷ U
Year 4 - add and subtract numbers with up to four-digits, ThHTU + ThHTU, ThHTU - ThHTU, add and
subtract decimals with up to two decimal places in the context of money, multiply three numbers
together, TU x U, HTU x U, TU x U, multiply by zero and one, TU ÷ U, HTU ÷ U
Year 5 – add and subtract numbers with more than four-digits, add and subtract decimals with
up to three decimal places, ThHTU x U, ThHTU x TU, HTU x TU, multiply whole numbers and decimals
with up to three-decimal places by 10, 100 and 1000, divide numbers with up to four-digits by U
(including remainders as fractions and decimals and rounding according to the context)
Year 6 - add and subtract numbers with more than four-digits, add and subtract decimals with up to
three decimal places, multiply numbers with up to four-digits by TU, multiply numbers with up to
two-decimal places by a whole number, divide numbers up to four-digits by TU (interpreting remainder
according to the context), divide decimals up to two-decimal places by U or TU
4
Mathematics is an interconnected subject in
which pupils need to be able to move fluently
between representations of mathematical
ideas. … pupils should make rich connections
across mathematical ideas to develop
fluency, mathematical reasoning and
competence in solving increasingly
sophisticated problems. They should also
apply their mathematical knowledge to
science and other subjects.
National Curriculum 2014
Structures of Multiplication (Haylock and Cockburn 2008)
Children should experience problems with all the different multiplication structures in a
range of practical and relevant contexts e.g. money and measurement
Repeated addition
10
So many lots (sets) of so many
How many (how much) altogether
Per, each
Scaling
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)
Commutative law
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)
3
4
I’m 3 times as
tall as you.
I’m 3 metres
tall
I’m only
1 metre
tall
scaling with Cuisenaire
a x b and b x a are equal
4 x 2 is the same as/equal to 2 x 4
6
Multiplication
End of Year Expectations
Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly.
Multiplication and division should be taught together.
Possible concrete and visual
representation
Children’s Recording
Practical only e.g. link to small world
Year R
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.
Using concrete objects, pictorial
representations and arrays with the support of
an adult – take photographs/draw pictures – if
using Numicon small icons could be stuck in
counting
in twos
arrays- Numicon,
Cuisenaire, counters
They solve problems, including
doubling, halving and sharing
two lots of
two is four
Recognise dots/objects that are the
same or have the same value
Fluency
Count forward in
ones,
Be able to add one
more
Read digits up to 20
Match written
numbers to number
of objects
Order concurrent
numbers upto 20
Recognise and use
the + symbol
Order nonconcurrent numbers
eg: 1, 3, 5, 9
Understand what a number looks like.. Eg what is
6? 6 bears. 6 [pencils. 6 children etc.
Use practical resources such as bears, counters,
cubes and number lines/hundred grids and
progress to a resource such as Numicon to
encourage counting in ones and then twos and
fives..
Use objects to count
in twos and fives
Multiplication
End of Year Expectations
Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly.
Multiplication and division should be taught together.
Possible concrete and visual
representation
Children’s Recording
Practical only e.g. link to small world
Year 1
Using concrete objects, pictorial
representations and arrays with the support of
an adult – take photographs/draw pictures – if
using Numicon small icons could be stuck in
Solve single step practical
problems involving multiplication
and division
Use concrete objects, pictorial
representations and arrays to
explore grouping
Make connections between
arrays, number patterns and
counting in twos, fives and tens
four lots of
two is eight
counting
in twos
two lots of
four is eight
arrays- Numicon,
Cuisenaire, counters
0
2
4
6
8
Double numbers and quantities
track with cuisenaire
Through grouping and
sharing small quantities,
pupils begin to understand:
multiplication and division;
doubling numbers and
quantities
flexible array
Fluency
Count in multiples of
2s, 5s and 10s
starting on multiples
to highlight pattern
recognition
e.g. 6, 8, 10, 12 etc
Emphasise number
patterns
Double numbers and
quantities
Add using doubles
Switch count
between tens and
ones e.g. 10, 20, 30,
31, 32, 33 …
Multiplication
End of Year Expectations
Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly.
Multiplication and division should be taught together.
Possible concrete and visual
representation
Children’s Recording
Year 2
Understand multiplication
as repeated addition
Calculate mathematical
statements for multiplication
and division within the tables
and write them using symbols
(×), (÷) (=)
Record practical work as number sentences, this
may go alongside arrays/photos of physical
multiplication in their books/whiteboards
2 + 2 + 2 + 2 = 4 x 2
2
4
6
4x2=8
2x4=8
8
Understand and solve
problems in contexts
using arrays, physical
objects, repeated
addition or
multiplication facts to
solve
Ensure children understand
that multiplication is
commutative (can be done
in any order)
Begin to understand that
multiplication and division are
inverse operations
flexible array
5
10
Emphasise number
patterns
Introduction to
multiplication tables.
Practise to become
fluent in multiplication
facts for 2, 5 and 10
Pupils use a variety of
language to describe
multiplication and division.
Understand and solve
problems involving
arrays
Count in twos, threes,
fives from zero and tens
from any number
forward and begin to
do it backwards
e.g. 6, 8, 10, 12 etc
two add two add two add two add two
= four lots of two
0
Fluency
15
0
2
4
6
8
Solve multiplication
problems mentally using
arrays, contexts and
materials to help them,
Begin to use know
multiplication facts to
help them to find new
facts, including related
division facts
Connect 10 times
table to place value.
Connect 5 times
table to divisions on
clock face
Pupils use a variety of
language to
describe
multiplication and
division.
Multiplication – multiplication and division should be taught together– refer to structures of multiplication
End of Year Expectations
Year 3
Possible concrete and
visual representation
Cuisenaire to
Statue is 3 times as
represent scaling
tall: 3 metres
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
Use mental methods
including partitioning,
progressing to reliable (more
formal) written methods
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.
Understand and solve
problems in context
including, missing number
problems, measuring
and scaling problems
Solve problems
involving multiplication
including
correspondence in
which n objects is
connected to m
objects
Teacher Modelling/Children’s Recording
Fluency
Children must use manipulatives alongside algorithms
Count from 0 in multiples
of 4, 8, 50 and 100
4 x 13
I am 1 metre
tall
‘four lots of thirteen’
10
3
4
Use multiples of 2, 3, 4, 5,
8, 10, 50 and 100
Expanded methods – grid and partitioning
flexible array
1
10
10
10
1
10
1
1
3
4
40
12
Expanded/
ladder method
of
multiplication
20
?
1
1
1
1
1
1
Some children MAY Progress
to developing fluency in short
multiplication
1
1
1 3
x 4
5 2
?
20
10
18 X 5 =
10 X 5 = 50
8 X 5 = 40
90
4 x 13
place value
counters
x
40 + 12 = 52
arrays
20
bar models
20
Practise, Recall and use
multiplication table facts
for the 3,4 and 8 times
tables
1
Start with digits that are below five so children
can practise method without encountering
difficulty with multiplication tables
Connect the 2, 4 and 8
times tables using
doubling
Develop efficient
mental methods using
commutativity and
multiplication facts to
derive related facts e.g.
4 x 4 x 12 = 12 x 4 x 5 =
12 x 20
Pupils continue to count
in ones, tens and
hundreds
Multiply and divide
whole numbers by ten
and hundred(not
decimals)
Count up and down in
tenths; recognise that
tenths arise from dividing
an object into 10 equal
parts and in dividing onedigit numbers or quantities
by 10
Multiply by 20 by x 2 and x
10, understand that this
works as 20 is two lots of
ten
Multiplication – multiplication and division should be taught together– refer to structures of multiplication
End of Year Expectations
Year 4
Possible concrete and
visual representation
Teacher Modelling/Children’s Recording
Children must use manipulatives alongside algorithms
Cuisenaire to
Statue is 3 times as
represent scaling
tall: 3 metres
Multiplying three numbers
together eg 2 x 3 x 4 –
associative law –(2 x 3) x 4 = 2
x (3 x 4)
4 x 13
I am 1 metre
tall
4
Multiplying by 0 and by 1
Use mental methods including
partitioning, ladder and grid
method progressing to reliable
(more formal) written methods
For those who are confident,
develop fluency in short
multiplication using formal
written layout (2 and 3 digit
numbers by a 1 digit number).
Solve problems involving
multiplication including using
the distributive law to multiply
a 2 digit number by a 1 digit
number (24 x 7 = 20 x 7 + 4
x7), integer scaling problems (I
am two times bigger/I am five
times bigger)and harder
correspondence problems
such as n objects are
connected to m objects
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.
‘four lots of thirteen’
10
3
x
10
3
4
40
12
40 + 12 = 52
place value
counters
1
10
10
10
1
10
1
1
1
1
1
1
1
1
1
1
MOST children SHOULD progress to developing fluency in
short multiplication
1 3
x 4
5 2
?
20
20
?
20
bar models
20
Recall and use
multiplication facts and
related division facts up to
12 x 12 with increasing
fluency
Recognise and use factor
pairs and commutativity in
mental calculations
Use the distributive law
(multiplying a number by a
group of numbers added
together is the same as
doing each multiplication
separately
Eg: 3 × (2 + 4) = 3×2 + 3×4)
Combine knowledge of
number facts and rules of
arithmetic to solve mental
and written calculations
e.g. 2 x 6 x 5 = 10 x 6
18 X 5 =
10 X 5 = 50
8 X 5 = 40
90
4 x 13
Count in multiples of 6, 7,
9, 25 and 1000
Derive multiplication facts
with up to three-digits (for
example 600 ÷ 3 = 200 can
be derived from 2 x 3 = 6).
Expanded methods – grid, ladder and partitioning
flexible array
Fluency
1 3 3
x
4
5 3 2
1
1 1
Higher ability children could look at using this
method involving decimals
Multiply and divide whole
numbers by ten and a
hundred
(including decimal s)
Count up and down in
hundredths; recognise that
hundredths arise when
dividing an object by one
hundred and dividing tenths
by ten.
They begin to extend their
knowledge of the number
system to include the
decimal numbers and
fractions that they have
met so far.
Multiplication - multiplication and division should be taught together– refer to structures of multiplication
Possible concrete and
visual representation
End of Year Expectations
Year 5
Cuisenaire to
represent scaling
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
Teacher Modelling/Children’s Recording
Statue is 3
times as tall:
3 metres
Children might use manipulatives alongside algorithms
I am 1
metre
tall
Identify multiples and factors
including finding all factor pairs of a
number, and common factors of two
numbers
3. 2 4
1 3 2 4
flexible array
x
Multiply whole numbers and those
involving decimals by 10, 100 & 1000
1
1
x
6
7 9 4 4
4 x 13
Understand and use multiplication
and division as inverses including in
problems involving missing numbers
and balancing equations
6
1 9. 4 4
2
1
2
4 x 23
Solve problems involving
multiplication and division including
scaling by simple fractions
Know and use the vocabulary of
prime numbers, prime factors and
composite (non-prime), recall prime
numbers to 19 and work out if any
number up to 100 is prime
recognise and use square numbers
and cube numbers, and the
notation for squared (2) and cubed
(3)
0.1
0.01
0.01
0.1
0.1
0.01
0.01
place value
counters
0.01
0.01
1 3 2 4
3. 2 4
x 2 6
x 2 6
7 9 4 4
1 9. 4 4
2 6 4 8 0
6 4. 8 0
0.01
0.01 0.01 0.01
Multiply decimals with up to three
decimal places by a one digit
number
?
0.2
Long multiplication
0.01
0.01
0.2
?
0.2
bar models
0.2
Multiply numbers
mentally drawing
upon known facts
Practise and develop
knowledge of factors,
multiples, square
numbers and prime
numbers
Multiply whole
numbers and those
involving decimals by
10, 100 & 1000
Continue to use
number in context,
including
measurement.
arrays
0.1
Count forwards in
steps of powers of 10
from any given
number up to
1 000 000
Apply all
multiplication tables
frequently. Commit
them to memory and
use them confidently
to make larger
calculations
Short
multiplication
Solve problems involving all four
operations where larger numbers are
used, using knowledge of multiples,
factors, squares and cubes.
Fluency
1
1
2
3 4 4 2 4
1
1
1
1
2
8 4. 2 4
1
1
Extend and apply
their understanding of
the number system to
the decimal numbers
and fractions that
they have met so far.
Recognise and
describe linear
number sequences,
including those
involving fractions
and decimals, and
find the term-to-term
rule.
Multiplication - multiplication and division should be taught together– refer to structures of multiplication
Possible concrete and
visual representation
End of Year Expectations
Year 6
Cuisenaire to
represent scaling
Teacher Modelling/Children’s Recording
Statue is 3
times as tall:
3 metres
Children might use manipulatives alongside algorithms
I am 1
metre
tall
Multiply numbers up to 4-digit x TU
using formal methods of short and
long multiplication
Multiply numbers with up to two
decimal places x whole number
Short
multiplication
Solve contextual problems
involving all four operations,
deciding which operations and
methods to use and why
3. 2 4
1 3 2 4
flexible array
Use estimation to check answers
to calculations and determine, in
the context of a problem, an
appropriate degree of accuracy.
x
7 9 4 4
4 x 13
Identify common factors,
common multiples and prime
numbers
1
1
x
6
6
1 9. 4 4
2
1
2
4 x 23
Multiply simple pairs of proper
fractions, writing the answer in its
simplest form [for example, ¼ x
½ = 1/8)
arrays
0.1
0.01
0.01
0.1
0.1
0.1
0.01
0.01
place value
counters
0.01
0.01
1 3 2 4
3. 2 4
x 2 6
x 2 6
7 9 4 4
1 9. 4 4
2 6 4 8 0
6 4. 8 0
0.01
0.01 0.01 0.01
?
0.2
Long multiplication
0.01
0.01
0.2
?
0.2
bar models
0.2
Undertake mental
calculations with
increasingly large
numbers and more
complex
calculations
Continue to use all
multiplication
tables to calculate
mathematical
statements in order
to maintain fluency
Practise and
develop
knowledge of
factors, multiples,
square numbers
and prime
numbers
Multiply and divide
numbers by 10, 100
and 1000 giving
answers up to three
decimal places
Use their knowledge of the
order of operations to carry out
calculations involving the four
operations
Use common factors to simplify
fractions; use common multiples
to express fractions in the same
denomination
Fluency
1
1
2
3 4 4 2 4
1
1
1
1
2
8 4. 2 4
1
1
Recall and use
equivalences
between simple
fractions, decimals
and percentages,
including in
different contexts.
Pupils explore the
order of operations
using brackets; for
example, 2 + 1 x 3 =
5 and (2 + 1) x 3 = 9.
Structures for Division (Haylock and Cockburn 2008)
Children should experience problems with the different division structures in a range of
practical and relevant contexts e.g. money and measurement
Equal-sharing
Sharing equally between
How many (much) each?
Inverse of multiplication
(Grouping)
So many lots (sets/groups) of so many
Share equally in to groups of …
Divide twelve into equal
groups of four
=3
Make 12
Ratio structure
comparison
inverse of scaling structure of multiplication
scale factor (decrease)
Overlay
groups of
four
Barney earns three times more than Fred. If
Barney earns £900 how much does Fred earn?
Jo’s journey to school is three times as
long as Ella’s. If Jo walks to school in
30 minutes how long does it take Ella?
Multiplication
End of Year Expectations
Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly.
Multiplication and division should be taught together.
Possible concrete and visual
representation
Practical only e.g. link to small world
Year R
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.
Children’s Recording
Using concrete objects, pictorial
representations and arrays with the support of
an adult – take photographs/draw pictures – if
using Numicon small icons could be stuck in
Group numicon into twos to see
there are four groups
They solve problems, including
doubling, halving and sharing
Recognise dots/objects that are the
Understand what a number looks like.. Eg what is 6?
6 bears. 6 [pencils. 6 children etc.
Use practical resources such as bears, counters,
cubes to share objects between a set number of
people. Use objects and put them into groups of 5
etc
Count forward in
ones,
Be able to add one
more
Read digits up to 20
Match written
numbers to number
of objects
counting
in twos
arrays- Numicon,
Cuisenaire, counters
Fluency
same or have the same value
Order concurrent
numbers upto 20
Recognise and use
the + symbol
Order nonconcurrent numbers
eg: 1, 3, 5, 9
Use objects to count
in twos and fives
Division
End of Year Expectations
Year 1
Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly.
Multiplication and division should be taught together.
Possible concrete and visual
representation
counting in groups of twos
Solve single step practical
problems involving division
Use concrete objects,
pictorial representations
and arrays
Understand division as
grouping and sharing
Recognise, find and name a
quarter as one of four equal
parts of an object, shape or
quantity.
Connect halves and quarters
to equal sharing and
quantities of sets
Through grouping and
sharing small quantities,
pupils begin to understand:
multiplication and division;
doubling numbers and
quantities
Using concrete objects, pictorial
representations and arrays with the support of
an adult – take photographs/draw pictures – if
using Numicon small icons could be stuck in
Eight can be
divided into
four equal
groups of
two or two
equal
groups of
four
Half and quarters of shapes and quantities
straw bundles
Use the language of
‘sharing equally
between’
Recognise, find and name a
half as one of two equal parts
of an object, shape or
quantity
Teacher Modelling/Children’s Recording
four lots
of two
four lots
of two
Record as number sentences using ÷ and =
8÷4
Eight divided into four equal
groups = two in each group
two lots
of four
doubling
flexible array
?
?
?
?
bar models
?
8
?
?
?
Count in twos, fives
and tens from different
multiples
e.g. 6, 8, 10, 12 etc
Emphasise patterns
Find simple fractions eg
half and quarter, of
objects, numbers and
quantities
Read and write
numbers from 1 to 20 in
numerals and words.
Numicon and counter arrays
Cuisenaire
Fluency
Eight can be
divided into
four equal
groups of
two or two
equal
groups of
four
Division
End of Year Expectations
Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly.
Multiplication and division should be taught together.
Possible concrete and visual
representation
counting in groups of twos
Year 2
Understand division as grouping
or sharing
Calculate mathematical statements
for multiplication and division within
the tables and write them using
symbols (×), (÷) (=)
straw bundles
Understand and solve single steps
practical problems involving
division
Teacher Modelling/Children’s Recording
Using concrete objects, pictorial representations
and arrays with the support of an adult – take
photographs/draw pictures – if using Numicon
small icons could be stuck in
Eight can be
divided into
four equal
groups of
two or two
equal
groups of
four
Use concrete objects, pictorial
representations
Show that multiplication of two
numbers can be done in any order
(commutative) and division of one
number by another cannot
Work with a range of materials and
contexts in which multiplication and
division relate to grouping and
sharing discrete quantities and to
arrays
Begin to understand that
multiplication and division are inverse
operations and 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).
Recognise, find, name and write
fractions 1/3, ¼,2/4 and ¾ of a
length, shape, set of objects or
quantity
Write simple fractions for example, 2
1 of 6 = 3 and recognise the
equivalence of 2/4 and 1/2.
four lots
of two
Fractions of
quantities
Numicon and counter arrays
Cuisenaire
two lots
of four
1/3 of 9 = 3
Record as number sentences using ÷ and =
Eight divided into four equal
groups = two in each group
doubling
flexible array
?
?
?
?
bar models
?
8
?
?
?
Count back in twos,
threes, fives TO zero and
tens e.g. 12, 10, 8, 6 etc
Emphasise number
patterns
Introduction to
multiplication tables.
Practise to become
fluent in multiplication
facts for 2, 5 and 10 –
make links between
multiplication facts and
known division facts
Begin to use know
multiplication facts to
help them to find new
facts, including related
division facts
8÷4
four lots
of two
Fluency
Eight can be
divided into
four equal
groups of
two or two
equal
groups of
four
Solve division problems
involving grouping and
sharing
Connect 10 times
table to place value.
Connect 5 times
table to divisions on
clock face
Pupils use a variety of
language to
describe
multiplication and
division.
Find halves and then
quarters
Division - multiplication and division should be taught together– refer to structures of division
Possible concrete and visual
representation
End of Year Expectations
Year 3
Cuisenaire to
represent scaling
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
Teacher Modelling/Children’s Recording
Use a numberline to see how many ‘lots’ of a number fit into
another number (develop to 10 lots/5 lots jumps)
10 lots of 3 10 lots of 310 lots of 3 10 lots of 3 10 lots of 3
10 lots of 3
Statue is 3 metres
I am 3 times
smaller
0
Understand and solve
problems in context
including, missing number
problems, measuring
and scaling problems
Connect 1/10 to
division by 10
Count in tenths
60
90
120
150
180
Repeated subtraction - chunking
Ensure children see/understand the link
between grouping on a number line either
horizontally or vertically recording for chunking
(NOT ALL CHILDREN will use vertical chunking)
15
12
95 ÷ 5 = 19
- 50 ( 10 x 5 )
arrays
- 25
2
÷
2
(5x5)
10
4
10
1
10
10
?
?
10
?
?
?
80
1
1
1
10
10
?
bar models
6
3
10 x 5 = 50
0
20
- 20 ( 4 x 5 )
10
2 x 5 = 10
5 x 5 = 25
45
88 ÷ 4
9
Fact Box
95
Recognise, find and
name ½ and ¼ of an
object, shape or
quantity
Understand the link
between unit fractions
and division
30
18
Use mental methods
including numberlines,
progressing to reliable (more
formal) written methods
Pupils develop reliable
written methods for division,
starting with calculations of
two-digit numbers by onedigit numbers with no
remainders.
Fluency
Children should use manipulatives alongside algorithms
1
?
?
0
1
Use place value counters to
introduce bus stop method, use
place value grids to support
÷
1
1
1 2 1
3 3 6 3
321 ÷ 3
Short divisionno remainders
560 ÷ 4
÷
1 4 0
4 5 16 0
Count from 0 in multiples
of 4, 8, 50 and 100 and
backwards
Practise, recall and use
multiplication and
division facts for the 3,4
and 8 times tables
Use multiples of 2, 3, 4, 5,
8, 10, 50 and 100
Connect the 2, 4 and 8
times tables using
doubling
Develop efficient
mental methods using
commutativity and
multiplication and
division facts to derive
related facts e.g. 30 × 2
= 60, 60 ÷ 3 = 20 and 20
= 60 ÷ 3
Write and calculate
mathematical
statements for division
using what is known
Use division facts to
derive related division
facts e.g. using 6 ÷ 3 = 2
to work out 60 ÷ 3 = 20
Count up and down in
tenths; recognise that
tenths arise from dividing
an object into 10 equal
parts and in dividing onedigit numbers or quantities
by 10
Division - multiplication and division should be taught together– refer to structures of division
Possible concrete and visual
representation
End of Year Expectations
Year 4
Cuisenaire to
represent scaling
Use mental methods including
numberlines, progressing to
reliable (more formal) written
methods then to become fluent
in the formal written method of
short division with exact answers
when dividing by a one-digit
number
Teacher Modelling/Children’s Recording
Use a numberline to see how many ‘lots’ of a number fit into
another number (develop to 10 lots/5 lots jumps)
10 lots of 3 10 lots of 310 lots of 3 10 lots of 3 10 lots of 3
10 lots of 3
Statue is 3 metres
I am 3 times
smaller
0
60
90
120
150
180
Repeated subtraction - chunking
Ensure children see/understand the link
between grouping on a number line either
horizontally or vertically recording for chunking
(NOT ALL CHILDREN will use vertical chunking)
Solve two-step problems in
different contexts, choosing the
appropriate operation, working
with increasingly harder numbers
including correspondence
questions e.g. three cakes shared
equally between 10 children
15
12
95 ÷ 5 = 19
- 50 ( 10 x 5 )
arrays
- 25
2
÷
2
(5x5)
10
4
10
10
?
?
10
?
?
?
80
1
1
1
10
10
Pupils understand the relation
between fractions and
multiplication and division of
quantities,
1
10
?
bar models
6
3
10 x 5 = 50
0
20
- 20 ( 4 x 5 )
10
2 x 5 = 10
5 x 5 = 25
45
88 ÷ 4
9
Fact Box
95
Solve problems involving division
including integer scaling problems
(I am two times smaller/I am five
times smaller)
Recognise and show, using
diagrams, families of common
equivalent fractions and
understand link to multiplication
and division
30
18
Divide one- or two-digit numbers
by 10 or 100, identifying value of
digits as tenths or hundredths
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.
Fluency
Children should use manipulatives alongside algorithms
1
?
?
0
1
Use place value counters to
introduce bus stop method, use
place value grids to support
÷
1
1
1 2 1
3 3 6 3
321 ÷ 3
Short divisionno remainders
560 ÷ 4
÷
1 4 0
4 5 16 0
Count in multiples of 6, 7, 9,
25 and 1000
Recall and use
multiplication facts and
related division facts up to 12
x 12 with increasing fluency
Derive multiplication facts
with up to three-digits (for
example 600 ÷ 3 = 200 can
be derived from 2 x 3 = 6).
Recognise and use factor
pairs and commutativity in
mental calculations
Use place value, known and
derived facts to divide
mentally, including dividing
by 1
Use the distributive law
(multiplying a number by a
group of numbers added
together is the same as
doing each multiplication
separately
Eg: 3 × (2 + 4) = 3×2 + 3×4)
Combine knowledge of
number facts and rules of
arithmetic to solve mental
and written calculations e.g.
2 x 6 x 5 = 10 x 6
Multiply and divide whole
numbers by ten and a
hundred
(including decimal s)
Count up and down in
hundredths; recognise that
hundredths arise when dividing
an object by one hundred
and dividing tenths by ten.
They begin to extend their
knowledge of the number
system to include the
decimal numbers and
fractions that they have met
so far.
Division - multiplication and division should be taught together– refer to structures of division
End of Year Expectations
Year 5
Practise and extend the formal
written method of short division:
numbers up to four-digits by a onedigit number
Possible concrete and visual
representation
Children might use manipulatives alongside algorithms
Cuisenaire to Statue is 3 metres
represent scaling
1
564 ÷ 5
1 1 2.8
remainder as a
fraction or as a
decimal
1
4
5 5 6 4.0
1 1 2
5 5 6 4
Higher ability children may be
introduced to long division
flexible arrays
2 3 r8
-4 8
4.8 ÷ 4
8 0
1
÷
1
8
0.1
0.1
0.1
0.1
Divide whole and decimal numbers
by 10,100 and 1000
Fact boxes for long
division may still be
useful eg-
0.1
0.1
(20 x 15)
0.8
?
Fact Box
2 x 24 = 48
(8 x 15)
?
?
3 x 24 = 72
4 x 24= 96
?
?
0.8
?
bar models
?
Apply all multiplication
tables frequently.
Commit them to
memory and use them
confidently to make
larger calculations
Count backwards with
positive and negative
whole numbers through
zero
Multiply and divide
numbers mentally
drawing upon known
facts
Multiply and divide
whole numbers and
those involving decimals
by 10, 100 & 1000
0.1
1
4
?
remainder
as a whole
number
-7 2
0.1
1
1
4
2
.
2/5
Count backwards in
steps of powers of 10
from any given number
up to 1 000 000
Practise and develop
knowledge of factors,
multiples, square
numbers, cube numbers
and prime numbers
24 5 6 0
recognise and use square numbers
and cube numbers, and the notation
for squared (2) and cubed (3)
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
short division
1
Solve problems involving all four
operations where larger numbers are
used, using knowledge of multiples,
factors, squares and cubes.
Know and use the vocabulary of
prime numbers, prime factors and
composite (non-prime), recall prime
numbers to 19 and work out if any
number up to 100 is prime
1 4 0
4 5 6 0
Identify multiples and factors
including finding all factor pairs of a
number, and common factors of two
numbers
Solve problems involving
multiplication and division including
scaling by simple fractions
without
remainder
560 ÷ 4
I am 3 times
smaller
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 as appropriate for the
context
Understand and use multiplication
and division as inverses including in
problems involving missing numbers
and balancing equations
Fluency
Teacher Modelling/Children’s Recording
5 x 24 = 120
Continue to use number
in context, including
measurement.
Extend and apply their
understanding of the
number system to the
decimal numbers and
fractions that they have
met so far.
Recognise and describe
linear number
sequences, including
those involving fractions
and decimals, and find
the term-to-term rule.
Division - multiplication and division should be taught together– refer to structures of division
End of Year Expectations
Possible concrete and visual
representation
Children might use manipulatives alongside algorithms
Year 6
Divide numbers up to 4-digit bu a 2
digit using formal methods of short
and long division interpret
remainders as whole numbers,
fractions or by rounding, as
appropriate for the context
Cuisenaire to
represent scaling
Statue is 3 metres
1 4 0
1
4 5 6 0
long division
flexible arrays
Solve contextual problems involving
all four operations, deciding which
operations and methods to use and
why
-4 8
4.8 ÷ 4
1
÷
1
4
0.1
1
1
remainder 8
as a whole
number - 7
2
.
0.1
0.1
1
0.1
4
0.1
0.1
Use common factors to simplify
fractions; use common multiples to
express fractions in the same
denomination
Associate a fraction with division
and calculate decimal fraction
equivalents [for example, 0.375] for
a simple fraction [for example, 3/8)
2 3 r8
24 5 6 0
Solve problems involving division
Divide proper fractions by whole
numbers [for example, 1/3 ÷ 2 =1/6)
564 ÷ 5
1 1 2.8
remainder as a
fraction or as a
decimal
4
1
5 5 6 4.0
1 1 2
2/5
1
Recognise division calculations as
the inverse of multiplication
Use their knowledge of the order of
operations to carry out calculations
involving the four operations
short division
5 5 6 4
Understand the relationship
between unit fractions and division
Identify common factors, common
multiples and prime numbers
without
remainder
560 ÷ 4
I am 3 times
smaller
Divide numbers with up to 2 decimal
places by 1-digit and 2-digit whole
numbers, initially in practical
contexts involving money and
measures
Use estimation to check answers to
calculations and determine, in the
context of a problem, an
appropriate degree of accuracy.
Fluency
Teacher Modelling/Children’s Recording
0.1
0.1
0.8
?
?
?
560 ÷ 24
2 3 8/24 (1/3)
24 5 6 0
-4 8
0
8 0
2
7 2
8
8
remainder as a
fraction in its
lowest form
remainder
as a decimal
2 3.3
24 5 6 0.0
-4 8
8 0
7 2
?
8 0
7 2
?
?
0.8
?
bar models
?
DIVISION SHOULD BE LINKED TO
OTHER AREAS SUCH AS
PERCENTAGES, RATIO AND
PROPORTION
Undertake mental
calculations with
increasingly large
numbers and more
complex calculations
Continue to use all
multiplication tables
and division facts to
calculate mathematical
statements in order to
maintain fluency
Practise and develop
knowledge of factors,
multiples, square
numbers and prime
numbers
Multiply and divide
numbers by 10, 100 and
1000 giving answers up to
three decimal places
Recall and use
equivalences between
simple fractions,
decimals and
percentages, including in
different contexts.
Pupils explore the
order of operations
using brackets; for
example, 2 + 1 x 3 = 5
and (2 + 1) x 3 = 9.
Perform mental
calculations, including
with mixed operations
and larger numbers