Transcript Maths Calculations - Northolmes Junior School

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.
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.
Magnitude of Calculations
Reception– Children count reliably with numbers from one 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.
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 twodecimal 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
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
Structuring Learning
Children must have concrete
experiences that enable them to
create visual images. They
should be encouraged to
articulate their learning and to
become pattern spotters.
Symbols
Language
Pictures
Haylock and
Cockburn (2008)
Concrete Experiences
Active/concrete
Building visual images
13 - 8
Abstract
12 + 19
bead string
count stick
Multilink
place value apparatus
0.1
10
place value
counters
10
1
1
0.1
100
Cuisenaire
Numicon
number line
double sided
counters
number
grids
100 and 200
Structures of Addition (Haylock and Cockburn 2008)
Children should experience problems with all the different addition structures in a
range of practical and relevant contexts e.g. money and measurement
Aggregation
Union of two sets
How many/much altogether?
The total
Augmentation
Start at and count on
Increase by
Go up by
+1
+1
+1
+1
+1
6
7
8
9 10
+1
11
+1
12
+1
13
+1
14
15
Commutative law
Understand addition can be done in any order
Start with bigger number when counting on
(Explain to children that subtraction does not have this
property)
is the same as/equal to (=)
Addition
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Possible Concrete and
Visual Representations
NC End of Year Expectations
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.
They solve problems,
including doubling, halving
and sharing.
0
1
2
3
4
5
6
7
8
9
‘’one more
than three is
four. One
less than
four is three’
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 groups.
10
Children’s Recording
Fluency
If using Numicon, children could use printed
Numicon icons and stick these in - progressing
to recording number sentences alongside
Count forward in ones,
1
+
+
=
2
=
Read digits up to 20
3
Example
Children may record
pictorially progressing
to recording number
sentences alongside
5+4
Be able to add one more
Match written numbers to
number of objects
Order concurrent
numbers upto 20
Recognise and use the +
symbol
Order non-concurrent
numbers eg: 1, 3, 5, 9
Addition
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Possible Concrete and
Visual Representations
NC End of Year Expectations
Year 1
Read, write and interpret
mathematical statements
involving addition (+),
subtraction (–) and equals (=)
signs
Represent and use number
bonds and related
subtraction facts within 20
Add and subtract one-digit
and two-digit numbers to 20,
including zero
Solve one-step problems that
involve addition and
subtraction, using concrete
objects and pictorial
representations, and missing
number problems such as 8 +
__ = 13
Children must experience
combining two, and then more
than two, groups of objects using
counting on and the language
of addition e.g. add, plus
Children must experience
increasing numbers e.g.
what is two more than
seven ?
Compare quantities to say
how many less and/or how
many more
0
1
2
3
4
+2
4+2
2
4
5
6
7
8
9
two more than four
6
8
10
12
14
16
10
Children’s Recording
Fluency
If using Numicon, children could use printed
Numicon icons and stick these in - progressing
to recording number sentences alongside
Represent and use
number bonds to 5
1
‘two more
than three is
five or two
less than five
is three’
=
+
+
18
2
Add using doubles
=
Add numbers mentally by
counting on TU + U or
U + U + U (not crossing 10
barrier)
3
Children may record
pictorially progressing
to recording number
sentences alongside
Count forwards, to and
across 100, beginning with
0 or 1 or from any given
number
Switch count between
tens and ones e.g. 10, 20,
30, 31, 32, 33 …
9+6
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 groups rather than ones
6
7
Represent and use
number bonds to ten
8
9
10
11
12
13
14
15
16
Represent and use
number bonds up to 20
(establish addition and
subtraction as related
operations)
Count, read and write
numerals to 100
Read and write numbers
to 20 in numerals or words
Find one more than a
number
Find ten more than a
number
Count in multiples of 2s, 5s
and 10s starting on
multiples to highlight
pattern recognition
Addition
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Possible Concrete and
Visual Representations
NC End of Year Expectations
Children’s Recording
Fluency
10
Year 2
Cuisenaire
Solve problems with addition
and subtraction:
- using concrete objects and
pictorial representations,
including those involving
numbers, quantities , measures,
money and real life contexts
-applying their increasing
knowledge of mental and
written methods
Recall and use addition and
subtraction facts to 20 fluently,
and derive and use related
facts up to 100
Add and subtract numbers
using concrete objects, pictorial
representations, and mentally,
including: a two-digit number
and ones, a two-digit number
and tens, two two-digit
numbers, adding three onedigit numbers
Show that addition of two
numbers can be done in any
order (commutative) and
subtraction of one number from
another cannot
??
Children apply, develop and secure their
understanding of place value
Use jottings and record number sentences
7
?
20
0
5
7
2
10
15
20
3
41
Bar Model
25
30
35
40
+
45
28
Numbered and partially numbered number lines
Use Numicon, number grids, place value apparatus/Dienes,
place value grids, place value cards, Encourage children to
partition numbers rather than counting in ones.
40
1
+ 20
+8
= 60
=9
60 + 9 = 69
Children use blank number lines for TU + TU or
HTU + TU
ENSURE CHILDREN HAVE THE
OPPORTUNITY TO ADD MORE
THAN TWO NUMBERS
Add numbers mentally
by counting on TU + U
or U + U + U or TU + TU
(crossing 10’s barrier)
Count in steps of 2, 3,
and 5 from 0, and in
tens from any number,
forward and
backward
Recognise the place
value of each digit in a
two-digit number (tens,
ones)
Identify, represent and
estimate numbers
using different
representations,
including the number
line
Compare and order
numbers from 0 up to
100; use <, > and =
signs
Read and write
numbers to at least 100
in numerals and in
words
Recognise and use the inverse
relationship between addition
and subtraction and use this to
check calculations and solve
missing number problems.
Children should understand the
language of sum
Represent and use
numberbonds to 20
Use place value and
number facts to solve
problems.
Children should be able to partition
numbers in different ways e.g. as
2+2+2+1 or 5+3 or 23 as 20 +3 or
10+13
Use knowledge to
begin to derive and
use number facts up to
100 (multiples of 10)
Addition
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. They should be taught together.
NC End of Year Expectations
Possible Concrete and
Visual Representations
Teacher Modelling/Children’s Recording
Year 3
10
Concrete/visual representatives SHOULD be used alongside
algortihms
Add and subtract numbers
mentally, including:
a three-digit number and
ones
a three-digit number and
tens
a three-digit number and
hundreds
Add and subtract numbers
with up to three digits,
leading to using formal
written methods of
columnar addition and
subtraction
Children should partition
numbers, up to 1000, in
different ways
e.g. 100 + 40 + 6 or 100 + 30
+ 16
ENSURE CHILDREN HAVE THE
OPPORTUNITY TO ADD MORE
THAN TWO NUMBERS WITH
DIFFERING NUMBERS OF
DIGITS
43 + 28
40 + 20 + 1 + 8 = 69
40 + 20 + 3 + 8
Or
Or
41 + 20 + 8 = 69
43 + 20 + 8
Concrete/Visual representatives SHOULD be used
alongside algortihms
Column addition (no exchanging) with up to three-digits
40 +
1
40 +
3
+ 20 +
8
20 +
8
9 = 69
70 +
1
60 +
100
10
1
1
1
100
10
3
Expanded recording
without exchange
100 + 40 + 1
+ 100 + 20 + 8
200 + 60 + 9 = 2 6 9
Expanded recording
Partially numbered and blank number lines
?
70
7
Cuisenaire
30
Bar Model
Represent and use
numberbonds to 100
Count from 0 in multiples
of 4, 8, 50 and 100;
Find 10 or 100 more or
less than a given number
E.g. 41 + 28
Children apply, develop and secure their understanding
of place value and begin to record in columns
Estimate the answer to a
calculation and use inverse
operations to check
answers
Solve problems, including
missing number problems,
using number facts, place
value, and more complex
addition and subtraction.
Children apply, develop and secure their understanding of place
value and use partitioning to add horizontally
Fluency
Mentally add HTU + ones,
HTU + tens, HTU +
hundreds
Count in ones, tens and
hundreds maintaining
fluency through varied
and frequent practice
Recognise the place
value of each digit in a
three-digit number
(hundreds, tens, ones)
Compare and order
numbers up to 1000
= 71
10
Expanded recording
with exchange
Identify, represent and
estimate numbers using
different representations
Read and write numbers
up to 1000 in numerals
and in words
Solve number problems
and practical problems
involving these ideas.
End of Year
Expectations
Possible Concrete and
Visual Representations
Teacher Modelling/Children’s Recording
Children apply, develop and secure their understanding
of place value and begin to record in columns
Year 4
Concrete/visual representatives SHOULD be used
alongside algortihms
Column addition (no exchanging) with up to three-digits
Add and subtract
numbers with up to 4 digits
using the formal written
methods of columnar
addition and subtraction
where appropriate
40 +
1
40 +
3
+ 20 +
8
20 +
8
9 = 69
70 +
1
60 +
= 71
10
Add together numbers
with up to two decimal
places in the context of
money
Expanded recording
without exchange
Estimate and use inverse
operations to check
answers to a calculation
Expanded recording
with exchange
HTU
141
+ 128
100 + 40 + 1
+ 100 + 20 + 8
Solve addition and
subtraction two-step
problems in contexts,
(including missing number
problems)deciding which
operations and methods
to use and why.
200 + 60 + 9 = 2 6 9
269
Expanded recording
Compact (column) recording
10
100
ENSURE CHILDREN HAVE
THE OPPORTUNITY TO ADD
MORE THAN TWO NUMBERS
INCLUDING DECIMALS,
WITH DIFFERING NUMBERS
OF DIGITS
11
10
100
11
1
143
+ 128
Column addition (with exchanging)
271
10
20
100
200
0
10
20
40
40
30
50
Partially numbered and blank number lines
HTU
789
+ 642
£ 7. 8 9
+ £ 6. 4 2
£ 1 4. 3 1
1431
1 1
?
?
70
7
Cuisenaire
Compact (column) recording
30
Bar Model
Count in multiples of 6,
7, 9, 25 and 1000
Find 1000 more or less
than a given number
Count forwards through
zero starting with
negative numbers
Recognise the place
value of each digit in a
four-digit number
(thousands, hundreds,
tens, and ones)
Order and compare
numbers beyond 1000
Identify, represent and
estimate numbers using
different representations
Add numbers mentally by
partitioning (TU + TU)
Use adjusting to add
mentally (45 + 19)
Perform mental
calculations with twodigit numbers, the
answer could exceed
100 and increasingly
large numbers
Round any number to
the nearest 10, 100 or
1000
1
3
3
Fluency
1 1
Add decimals in the
context of money
Solve number and
practical problems that
involve all of the above
and with increasingly
large positive numbers
Read Roman numerals
to 100 (I to C) and know
that over time, the
numeral system
changed to include the
concept of zero and
place value.
Addition
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Possible Concrete and
Visual Representations
End of Year Expectations
Year 5
0.01
0.1
1
0.02
0.2
2
Add and subtract whole numbers
with more than 4 digits, and
decimals up to 3 decimal places,
including using formal written
methods (columnar addition and
subtraction)
0.03
0.3
3
0.04
0.4
4
0.05
0.5
5
0.06
0.6
6
1/10
U
0.07
0.7
7
0.08
0.8
8
Teacher Modelling/Children’s Recording
0.09
0.9
9
2141
+ 1128
Add and subtract numbers mentally
with increasingly large numbers
1
Use rounding and estimation to
check answers to calculations and
determine, in the context of a
problem, levels of accuracy
0.1
3269
0.01
Read, write, order and
compare numbers to at
least 1 000 000 and
determine the value of
each digit
Concrete/Visual
representations could be used
alongside algorithms
1/100
+
Count forwards or
backwards in steps of
powers of 10 for any given
number up to 1 000 000
2 1. 4 1
1. 1 2
0. 3 5
Interpret negative numbers
in context, count forwards
and backwards with
positive and negative
whole numbers, including
through zero
2 2. 8 8
Column addition (no exchanging)
1
Solve addition and subtraction
multi-step problems in contexts,
deciding which operations and
methods to use and why.
0.1
0.01
?
Bar Model
0.7
?
5189
+ 3128
5 1. 8 9
+ 3. 1 2 8
8317
5 5. 0 1 8
11
0.3
1
1
Column addition (with exchanging)
0
0.1
0.2
0.3
Round any number up to 1
000 000 to the nearest 10,
100, 1000, 10 000 and
100 000
Begin to round decimal
numbers to the nearest
whole number
Cuisenaire
N.B. ENSURE CHILDREN HAVE THE
OPPORTUNITY TO ADD MORE THAN
TWO NUMBERS INCLUDING
DECIMALS, WITH DIFFERING
NUMBERS OF DIGITS
Fluency
0.4
0.5
Partially numbered and blank number lines
Addition with decimals up to three
decimal places including in different
contexts e.g. money and measures
Solve number problems
and practical problems
that involve all of the
above
Practise mental
calculations with
increasingly large numbers
Read Roman numerals to
1000 (M) and recognise
years written in Roman
numerals.
Addition
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Possible Concrete and
Visual Representations
End of Year Expectations
Year 6
0.01
0.1
1
0.02
0.2
2
0.03
0.3
3
0.04
0.4
4
0.05
0.5
5
0.06
0.6
6
0.07
0.7
7
0.08
0.8
8
Teacher Modelling/Children’s Recording
0.09
0.9
9
Perform mental calculations,
including with mixed operations
and large numbers
Solve addition and subtraction
multi-step problems in contexts,
deciding which operations and
methods to use and why
1/10
U
Concrete/Visual
representations could be used
alongside algorithms
1/100
2141
+ 1128
Solve problems involving
addition, subtraction,
multiplication and division using
their knowledge of the order of
operations to carry out
calculations involving the four
operations
1
0.1
3269
0.01
+
Read, write, order and
compare numbers up to
0 000 000 and determine
the value of each digit
Count in tens and hundreds
increasing fluency of order
and place value
2 1. 4 1
1. 1 2
0. 3 5
Round any whole number
to a required degree of
accuracy
2 2. 8 8
Column addition (no exchanging)
1
Use estimation to check
answers to calculations and
determine, in the context of a
problem, an appropriate
degree of accuracy.
0.1
0.01
Add numbers mentally
(HTU+ HTU) or (TH.th + TU.th)
?
Bar Model
N.B. ENSURE CHILDREN HAVE
THE OPPORTUNITY TO ADD
MORE THAN TWO NUMBERS,
INCLUDING DECIMALS, WITH
DIFFERING NUMBERS OF DIGITS
0.7
?
5189
+ 3128
5 1. 8 9
+ 3. 1 2 8
8317
5 5. 0 1 8
11
0.3
1
1
Column addition (with exchanging)
0
0.1
0.2
0.3
Round decimal numbers to
the nearest whole number
and to one or two decimal
places
Use negative numbers in
context, and calculate
intervals across zero
Cuisenaire
Add numbers with more than
four-digits and decimals up to
three places (formal written
column method)
Fluency
0.4
0.5
Partially numbered and blank number lines
Addition with decimals up to three
decimal places including in different
contexts e.g. money and measures
Solve number and practical
problems that involve all of
the above.
Use number-bond
knowledge to derive
decimal number-bonds
(0.6 + 0.4 = 1) or (0.63 + 0.37
= 1)
Structures of Subtraction (Haylock and Cockburn 2008)
Children should experience problems with all the different subtraction structures in a
range of practical and relevant contexts e.g. money and measurement
Partitioning
Take away
… how many left?
How many are not?
How many do not?
Inverse-of-addition
What must be added?
How many (much) more needed?
There are ten pegs
on the hanger –
how many are covered?
Comparison
What is the difference?
How many more?
How many less (fewer)?
How much greater?
How much smaller?
Reduction
Start at and reduce by
Count back by
Go down by
-1 -1
‘two more than three
is five or two less than
five is three’
1
2
3
4
5
6
7
8
9 10
16
Addition
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Possible Concrete and
Visual Representations
NC End of Year Expectations
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.
They solve problems,
including doubling, halving
and sharing.
0
1
2
3
4
5
6
7
8
Children’s Recording
9
‘’one more
than three is
four. One
less than
four is three’
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 back in ones and then groups.
10
If using Numicon, children could use printed
Numicon icons and stick these in - progressing
to recording number sentences alongside
Fluency
Count backward in ones
from 5 (five little ducks)
etc,
Count backward in ones
from 10.
Count backward in ones
from any number less
than 20.
Children may record
pictorially progressing
to recording number
sentences alongside
Find one less than a
number
Read digits up to 20
Match written numbers to
number of objects
Order concurrent
numbers upto 20 from
largest to smallest
Recognise and use the symbol
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Subtraction
Possible Concrete and
Visual Representations
End of Year Expectations
Year 1
0
1
2
3
4
5
6
7
8
9
10
Children may begin recording
pictorially progressing to recording
number sentences alongside
5 -3
Read, write and interpret
mathematical statements
involving addition (+),
subtraction (–) and equals
(=) signs
Represent and use number
bonds and related
subtraction facts within 20
0
1
2
3
4
5
6
7
8
9
Represent and use
number bonds to ten
Use numberbonds to ten
to derive subtraction
facts
Exam
ple
-1
‘two less
than five is
three’
Use practical resources such as bears, counters,
cubes and number lines/hundred grids and progress
to a resource such as Numicon to encourage counting
back in groups rather than ones
0
1
2
3
4
Count backwards
(including crossing 100)
any given number
-1
5
6
7
8
9 10
Children could use printed
Numicon icons and stick these in,
again progressing to recording
number sentences alongside
Switch count between
ones and tens e.g. 33, 32,
31, 30, 20, 10
Represent and use
number bonds up to 20
(establish addition and
subtraction as related
operations)
Count, read and write
numerals to 100
Read and write numbers
to 20 in numerals or
words
Compare quantities to say
how many less and/or how
many more
Understand subtraction as
taking away
What is … less than …?)
Represent and use
number bonds to 5
10
Add and subtract one-digit
and two-digit numbers to 20,
including zero
Solve one-step problems that
involve addition and
subtraction, using concrete
objects and pictorial
representations, and missing
number problems such as 7 =
___ – 9.
Fluency
Children’s Recording
Find one less than a
number
Children should use numberlines
to count back/take away by
counting back in ones
Find ten less than a
number
identify and represent
numbers using objects
and pictorial
representations including
the number line, and use
the language of: equal
to, more than, less than
(fewer), most, least
Count back in multiples
of 2s, 5s and 10s starting
on multiples to highlight
pattern
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Subtraction
Possible Concrete and
Visual Representations
End of Year Expectations
10
Year 2
10 - 4
Solve problems with addition and
subtraction:
- using concrete objects and
pictorial representations, including
those involving numbers,
quantities , measures, money and
real life contexts
-applying their increasing
knowledge of mental and written
methods
Recall and use addition and
subtraction facts to 20 fluently,
and derive and use related facts
up to 100
Subtract numbers using concrete
objects, pictorial representations,
and mentally, including: a twodigit number and ones, a twodigit number and tens, two twodigit numbers, adding three onedigit numbers
Show that addition of two
numbers can be done in any
order (commutative) and
subtraction of one number from
another cannot
20
0
Practise addition and
subtraction facts to 20
Finding the difference
Show increasing fluency in
deriving subtraction facts for
numbers up to 10 and then
up to 20
Children should
use concrete
materials and
pictorial
representations,
and use numbers
in different
contexts
e,g, money and
measures,
Encourage
children to
partition
numbers rather
than counting in
ones.
2
Subtract numbers mentally
by counting back TU - U or U
- U or TU - TU
(crossing 10’s barrier)
Children should use
numberlines to count
back or on and find the
difference, developing
into subtracting bigger
chunks than one
10 20 30 40 50 60 70 80 90
Numbered and partially numbered number lines
Cuisenaire
?
Bar Model
Children apply,
develop and secure
their understanding
of place value and
begin to record
using jottings and
number sentences
10
7
?
Recognise and use the inverse
relationship between addition
and subtraction and use this to
check calculations and solve
missing number problems.
16 - 3
no exchanging
Understand subtraction as taking
away and finding the difference
Be able to partition numbers in
different ways
Fluency
Children’s Recording
26 - 8
exchange
ten for
ten ones
exchanging
Count backward in steps of
2, 3, and 5 from 0, and in
tens from any number.
Recognise the place value
of each digit in a two-digit
number (tens, ones)
Identify, represent and
estimate numbers using
different representations,
including the number line
Compare and order
numbers from 0 up to 100;
use <, > and = signs
Read and write numbers to
at least 100 in numerals and
in words
Use place value and
number facts to solve
problems.
Use known facts to 20 to
derive new facts e.g. 10 - 7
/100 – 70
Use knowledge to derive
and use subtraction number
facts up to 100 (multiples of
10)
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Subtraction
End of Year
Expectations
Possible Concrete and
Visual Representations
Teacher Modelling/ Children’s Recording
Consolidate
confidence with
numberlines, finding
difference by
counting on in larger
chunks
Cuisenaire
Year 3
Add and subtract
numbers mentally,
including:
a three-digit number
and ones
a three-digit number
and tens
a three-digit number
and hundreds
?
Bar Model
100
7?
30
Children apply,
develop and secure
their understanding of
place value and begin
to record in columns
Use dienes alongside to begin to develop standard methods of
subtraction (starting with partitioning to complete the
expanded method)
no exchange
with exchange
68 - 23
Add and subtract
numbers with up to
three digits, using
formal written methods
of columnar
subtraction
60
8
20
3
63 - 28
50
4 0 + 5 = 45
Estimate the answer to
a calculation and use
inverse operations to
check answers
100
1
1
10
100
100
100
3
10
40
20
3
20
8
30 + 5 = 35
Column subtraction
(no exchange)
148 -121
1
Solve problems,
including missing
number problems,
using number facts,
place value, and more
complex addition and
subtraction.
10 +
60
8
1
148
- 121
27
0 + 20 + 7 =27
Children SHOULD use manipulatives alongside algorithms
to transition between practical and abstract
Ensure children can solve calculations where zero is a place holder
0
10
20
30
40
50
Fluency
Count back in ones,
tens and hundreds
maintaining fluency
through varied and
frequent practice
Switch count between
hundreds, tens and
ones e.g 500, 400, 300,
290, 280, 270, 269, 268,
267
Find 10 or 100 less than
a given number with
up to three digits
Recognise the place
value of each digit in a
three-digit number
(hundreds, tens, ones)
Mentally subtract HTU ones, HTU - tens,
HTU - hundreds
Perform mental
calculations with twodigit numbers
Compare and order
numbers up to 1000
Identify, represent and
estimate numbers
using different
representations
Read and write
numbers up to 1000 in
numerals and in words
Solve number
problems and
practical problems
involving these ideas.
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Subtraction
End of Year
Expectations
Possible Concrete and
Visual Representations
Teacher Modelling/ Children’s Recording
Children SHOULD use manipulatives alongside algorithms
to transition between practical and abstract
Cuisenaire
Year 4
no exchange
?
Subtract numbers with
up to 4 digits using the
formal written methods
of columnar
subtraction where
appropriate
with exchange
68 - 23
7?
Estimate and use
inverse operations to
check answers to a
calculation
30
60
8
20
3
50 6 0 10 + 3
20
4 0 + 5 = 45
Solve addition and
subtraction two-step
problems in contexts,
deciding which
operations and
methods to use and
justifying why .
100
100
Understand
subtraction as the
inverse of addition
40
20
8
30 + 5 = 35
Column subtraction
(no exchange)
148 -121
148
- 121
27
8
1
0 + 20 + 7 =27
100
100
10
200
10
10
1
1
1
30
1
1
7 2 3
- 3 1 7
723 -317
4 0 6
6
723 -367
11 1
7 2 3
- 3 6 7
3 5 6
0
10
20
30
40
Find 1000 less than a
given number
Subtract numbers
mentally by partitioning
(TU - TU)
Use adjusting to subtract
mentally (45 - 19)
Continue to practise
mental subtraction
calculations with
increasingly large
numbers to aid fluency
Recognise the place
value of each digit in a
four-digit number
(thousands, hundreds,
tens, and ones)
Order (in descending
order) and compare
numbers beyond 1000
Column subtraction (with exchange)
3
Count back in multiples of
6, 7, 9, 25 and 1000
Count backwards through
zero to include negative
numbers
63 - 28
Bar Model
100
Fluency
50
Ensure children can solve calculations where zero is a place holder
Identify, represent and
estimate numbers using
different representations
Round any number to the
nearest 10, 100 or 1000
Solve number and
practical problems that
involve all of the above
and with increasingly
large positive numbers
Read Roman numerals to
100 (I to C) and know that
over time, the numeral
system changed to
include the concept of
zero and place value.
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Subtraction
Possible Concrete and
Visual Representations
End of Year Expectations
Year 5
0.01
0.1
1
0.02
0.2
2
Subtract whole numbers with more
than 4 digits and decimals, including
using formal written methods
(columnar subtraction)
0.03
0.3
3
0.04
0.4
4
0.05
0.5
5
0.06
0.6
6
1/10
U
0.07
0.7
7
0.08
0.8
8
Teacher Modelling/ Children’s Recording
0.09
0.9
9
Pupils to
understand
the value of
numberline
subtraction
for contextual
problems
(time/money/
measures etc)
1/100
Subtract numbers mentally with
increasingly large numbers
Fluency
Read, write, order and
compare numbers to at
least 1 000 000 and
determine the value of
each digit
Count backwards in steps
of powers of 10 for any
given number up to 1 000
000
Subtract numbers
mentally by partitioning
(HTU- TU or HTU - HTU)
Use rounding to check answers to
calculations and determine, in the
context of a problem, levels of
accuracy
1
0.1
0.01
Children might use manipulatives alongside algorithms
Column subtraction (no exchanging)
Solve addition and subtraction multistep problems in contexts, deciding
which operations and methods to
use and why. (this may include
number lines)
1
0.1
0.01
13548
- 12128
2
1420
Cuisenaire
Column subtraction
(with exchanging)
?
7 4 5
Ensure children can solve calculations
where zero is a place holder
1
?
0.3
1.4 8
- 1.2 1
Bar Model
Column subtraction
(no exchanging)
0. 2 7
0
0.1
0.2
13 11 1
13 4 2 3
- 1 2 6 7 8
0.3
0.4
0.5
Column subtraction
(with exchanging)
6
11 1
7. 2 3
- 3. 6 7
3. 5 6
Subtraction with decimals up to three decimal
places including in different contexts e.g. money
and measures
Use adjusting to subtract
mentally (45 - 19)
Interpret negative
numbers in context, count
backwards with positive
and negative whole
numbers, including
through zero
Round any number up to
1 000 000 to the nearest
10, 100, 1000, 10 000 and
100 000
Begin to round decimal
numbers to the nearest
whole number
Practise mental
calculations with
increasingly large
numbers
Solve number problems
and practical problems
that involve all of the
above
Read Roman numerals to
1000 (M) and recognise
years written in Roman
numerals.
Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly.
Addition and subtraction should be taught together.
Subtraction
Possible Concrete and
Visual Representations
End of Year Expectations
Year 6
0.01
0.1
1
0.02
0.2
2
0.03
0.3
3
0.04
0.4
4
0.05
0.5
5
0.06
0.6
6
0.07
0.7
7
0.08
0.8
8
Teacher Modelling/ Children’s Recording
Pupils to
understand
the value of
numberline
subtraction
for contextual
problems
(time/money/
measures etc)
0.09
0.9
9
Perform mental calculations,
including with mixed operations
and large numbers
Solve addition and subtraction
multi-step problems in contexts,
deciding which operations and
methods to use and why (this
could include number lines)
U
Solve problems involving
addition, subtraction,
multiplication and division using
their knowledge of the order of
operations to carry out
calculations involving the four
operations
1
1/10
1/100
13548
- 12128
0.1
1
0.01
0.1
0.01
Column subtraction
(no exchanging)
2
13 11 1
13 4 2 3
- 1 2 6 7 8
1420
Column subtraction
(with exchanging)
7 4 5
Children might use manipulatives alongside algorithms
Use estimation to check answers
to calculations and determine, in
the context of a problem, an
appropriate degree of accuracy.
Ensure children can solve calculations
where zero is a place holder
Cuisenaire
1.4 8
- 1.2 1
?
Subtract numbers with more than
four-digits and decimals up to
three places (formal written
column method)
1
0. 2 7
?
N.B. ENSURE CHILDREN HAVE THE
OPPORTUNITY TO SUBTRACT
DECIMALS, WITH DIFFERING
NUMBERS OF DIGITS
Column subtraction
(no exchanging)
0.3
6
11 1
7. 2 3
Column subtraction - 3 . 6 7
Bar Model
(with exchanging)
3. 5 6
0
0.1
0.2
0.3
0.4
0.5
Column subtraction
(with differing numbers
of digits-adding place
holder zero)
1.6 87 0
- 1.2 146
Subtraction with decimals up to three decimal places
including in different contexts e.g. money and measures
Fluency
Read, write, order and
compare numbers up to
0 000 000 and determine
the value of each digit
Count back in tens and
hundreds increasing fluency
of order and place value
Round any whole number
to a required degree of
accuracy
Round decimal numbers to
the nearest whole number
and to one or two decimal
places
Use negative numbers in
context, and calculate
intervals across zero
Subtract numbers mentally
(HTU - HTU) or (TH.th - TU.th)
Solve number and practical
problems that involve all of
the above.
Use number-bond
knowledge to derive
decimal number-bonds
(1 - 0.4 = 0.6) or (1 - 0.37 =
0.63)