Year One
lWe read, write and
interpret mathematical
statements involving
the concept of
addition.
lWe add one digit and
two digit numbers up
to twenty.
Progress is from
concrete and pictorial
representations of
addition to mental
methods
lWe can solve
statements with
missing numbers
lBoth the
‘augmentation’ and
‘aggregation’ models of
addition are covered
(adding to and
combining).
Year Two
Year Three
Year Four
lWe focus on
concrete objects,
pictorial
representations
and mental
methods.
Remember to
‘ENGAGE BRAIN
BEFORE PENCIL!’ – ask
‘What is the most
appropriate method to
solve this problem?’
lWe use the
Empty Number
Line, ensuring
that we use the
vocabulary of the
Empty Number
Line.
lAdd up to three digits
using written methods
based on
understanding of place
value.
lWe understand
the value of the
digits in three
digit numbers.
482 + 64 =
400 80
2
60
4
____________
400 140 6
lWe can
partition
numbers in to H
T U.
Progress is from
mental
representations
to
representations
on a number line
starting with the
first number
unpartitioned.
Example:
=546
Remember to ‘ENGAGE
BRAIN BEFORE PENCIL!’ – ask
‘What is the most appropriate
method to solve this
problem?’
Remember to ‘ENGAGE BRAIN
BEFORE PENCIL!’ – ask ‘What is
the most appropriate method to
solve this problem?’
lAdd numbers up to four
digits using formal written
methods.
lAdd whole numbers of more
than four digits using formal
written methods.
This is a key year for progress!
We transfer from using the
‘horizontal expanded
method’ (as in Abacus) to a
more compact column
method.
We start with one exchange
and then progress to more
than one exchange.
lMost year fives will be able to
use compact column addition with
numbers of more than four digits.
Example:
Note:
The Abacus scheme only asks for
addition problems to be solved
using numbers with two decimal
places – We need to extend this.
or
482
64
_________
6 (4+2)
1 4 0 (80 + 60)
4 0 0 (400+0)
_________
546
Year Five
We start with:
482 + 64 =
400 80
2
60
4
____________
400 140 6 =546
We finish the year with:
3 4 8 2 +
1 4 6 4
1
_____________
4 9 4 6
_____________
(note: the ‘1’ is smaller
and above the line.)
lAdd decimals; including a mix of
whole numbers and decimals and
including where the numbers have
a different number of decimal
places.
We need to do additions with:
lfive digits
ldifferent numbers of decimal
places
lmore than two numbers to total
Year Six
Remember
to ‘ENGAGE
BRAIN
BEFORE
PENCIL!’ –
ask ‘What is
the most
appropriate
method to
solve this
problem?’
This is a year
for
consolidation
and
extension.
lWe will
practise
addition
problems for
larger
numbers.
lWe will
ensure that
pupils have
practice
adding more
than two
large
numbers.
Year Two
Year One
•
•
•
•
•
•
Objects and pictorial
representations
Count backwards from any
given number
Language - less than fewer
than
Represent and use number
bonds and related
subtraction facts within 20
Subtract one digit and two
digit numbers to 20
Solve one step problems
involving subtraction
(concrete objects, pictorial
representations, missing
number problems)
•
•
•
•
•
Solve problems with
subtraction
Derive and use related
subtraction facts up to 100
Use the inverse
relationship with addition
to solve missing number
problems and check
calculations
Language – difference
Record subtraction in
columns to prepare for
formal written methods
Year Three
Remember to ‘ENGAGE BRAIN BEFORE
PENCIL!’ – ask ‘What is the most appropriate
method to solve this problem?’
l Subtract up to 3 digits using formal written
methods.
l The Empty Number Line (including the
appropriate vocabulary) will be an
important representation.
Note: The ‘Abacus’ scheme does not cover
formal written methods at this stage.
l Use understanding of place value.
Expanded column subtraction example:
482 – 64 =
70 12
400 80 2
60 4
____________
400 10 8
____________
= 418
Year Three
Year Four
Remember to ‘ENGAGE
BRAIN BEFORE PENCIL!’ – ask
‘What is the most appropriate
method to solve this
problem?’
l Subtract up to 3 digits
using formal written
methods.
l The Empty Number Line
(including the appropriate
vocabulary) will be an
important representation.
Note: The ‘Abacus’ scheme
does not cover formal written
methods at this stage.
Year Five
Remember to ‘ENGAGE BRAIN
BEFORE PENCIL!’ – ask ‘What is
the most appropriate method to
solve this problem?’
Remember to ‘ENGAGE BRAIN
BEFORE PENCIL!’ – ask ‘What is the
most appropriate method to solve
this problem?’
Remember to ‘ENGAGE BRAIN
BEFORE PENCIL!’ – ask ‘What is the
most appropriate method to solve
this problem?’
l Subtract up to 4 digits using
formal written methods.
l Subtract up to and including 5
digits using formal written
methods.
l This is a year for consolidation
and extension.
This is a key year for progress!
We transfer from using the
expanded method to a compact
column method.
We progress from one exchange
to more than one exchange
when confident.
Note:
This is not covered in Abacus.
l Subtract decimals; including a
mix of whole numbers and decimals.
Expanded column subtraction
example:
482 – 64 =
£12.52 - £7.48
4
70 12
400 80
2
60 4
____________
400 10 8
____________
= 418
70 12
400 80 2
60 4
____________
400 10 8
____________
We finish the year with:
71
= 418
l Use the formal written method
of compact column
decomposition.
(Except when it is clearly not the
most appropriate method
e.g. 2001 – 199)
Example (in the context of money):
Example:
482 – 64 =
l Use understanding of
place value.
Year Six
482
64
418
l Use numbers with more than 4
digits and becoming increasingly
large.
1
1 2 .5 2–
7. 4 8
__________
5 . 0 4
__________
l Pupils should be confident in
the use of the compact column
decomposition method.
l Consolidate Year 5 work on
decimals including numbers with
different numbers of decimal
places.
Example:
17.3 – 5.02
2 1
1 7 . 3 0
5 . 0 2
____________
2 . 2 8
____________
Key Stage One
l Use practical methods –
concrete objects, pictorial
representations, arrays.
l Use mental methods.
l Moving to counting jumps of
the same size on a number line.
l Arranging arrays.
2x3
ll
ll
ll
lll
lll
l Pupils should know their two,
five and ten times tables
including the inverse.
Year Three
Remember to ‘ENGAGE BRAIN BEFORE PENCIL!’ – ask ‘What is the most appropriate
method to solve this problem?’
l Write and calculate for multiplication and division using the 2, 5, 10, 3, 4 and 8
multiplication tables.
l Pupils to be able to do two digit multiplied by one digit calculations.
l Pupils progress from using mental methods to a formal written layout.
l Pupils to use the grid method – with the teacher making sure that each
calculation is acceptable for the times tables known.
l Pupils to link the grid representation to the array representation by using a
‘proportional grid’.
l Pupils will be eased into the grid method by briefly doing
Example:
5 x 27 =
10
5
llllllllll
llllllllll
llllllllll
llllllllll
llllllllll
10
Progressing to:
20
5
7
llllllllll
llllllllll
llllllllll
llllllllll
llllllllll
llllllllllllllllllll
llllllllllllllllllll
llllllllllllllllllll
llllllllllllllllllll
llllllllllllllllllll
lllllll
lllllll
lllllll
lllllll
lllllll
7
lllllll
lllllll
lllllll
lllllll
lllllll
Note:
(Array knowledge from previous years must be strong to underpin this.)
Year Four
Remember to ‘ENGAGE BRAIN
BEFORE PENCIL!’ – ask ‘What is
the most appropriate method to
solve this problem?’
l Pupils to be able to multiply
two or three digits by one
digit using formal written
layout = GRID METHOD.
l Pupils to be able to recall all
multiplication facts up to
12 x 12
(including inverse
knowledge.)
Year Six
Year Five
Remember to ‘ENGAGE BRAIN BEFORE PENCIL!’ – ask ‘What is the most appropriate
method to solve this problem?’
Remember to ‘ENGAGE BRAIN BEFORE PENCIL!’ – ask ‘What is the most
appropriate method to solve this problem?’
l Pupils will be able to multiply multi – digit numbers up to 4 x 2 digits.
l Consolidation of multiplying multi-digit numbers up to 4 digits by 2
digits. Use formal written method of long multiplication.
l Pupils will progress from Grid Method to formal long multiplication.
Example:
Examples:
1. 7 x 346 =
From:
x
300
2100
7
40
280
1 2
2
2 4 8
1 2
7 4
1 1
3 2 2
6
42
4
6 x
0
4
4
l PLUS: Numbers between zero and ten with up to two decimal places
by whole numbers.
= 2422
To:
3 4 6
7x
Example:
Method 1 – We can say that the answer will have two decimal places and
be very aware and careful with place value.
3 4
2 4 2 2
3
2. 124 x 26 =
From:
l
3
100
20
4
20
2000
400
80
= 2480
6
600
120
24
= 744
4 6 x
7
4
2 4l 2 2
To:
1 2 4
2 6 x
2 4 8 0
1 2
7 4 4
1 1
3 2 2 4
l Pupils should attempt example 2. type questions only when they are confident
with example 1. type questions.
A quick approximation check: 3 x 7 = 21 - so 21 is in the region of 24.22 –
not 2.422 or 242.2
Method 2 – Get rid of decimals to start with to get whole numbers – if
we had to multiply by 100 WE MUST REMEMBER TO DIVIDE ANSWER BY
100 AT THE END – AND double check with approximation as above.
This can be seen in a practical context by turning pounds into pence and
back etc.
Year Three
Key Stage One
Year Four
l Solve problems practically :
- Concrete objects
- Arrays
- Pictorial Representations
Remember to ‘ENGAGE BRAIN BEFORE PENCIL!’ – ask ‘What is the most
appropriate method to solve this problem?’
l Use the two different
representations of sharing sweets
and grouping.
l The unit (divisor) will be either 2, 3, 4, 6 or 8 (Division by 10 also possible)
Examples:
(Six sweets are equally shared
between two people, how many do
they each get? Six sweets are put into
groups of three, how many groups are
there?)
l Year two will be expected to use
repeated subtraction along a
number line
(small numbers).
Remember to ‘ENGAGE BRAIN
BEFORE PENCIL!’ – ask ‘What is the
most appropriate method to solve
this problem?’
l Problems will be solved involving T U ÷ U – with no remainders involved
l TU÷U
l Method: repeated subtraction on number line (with pupils putting some of
the jumps together)
Example: 48 ÷ 4
10 x 4 = 40
2x4=8
____________________________________________
0
40
48
Or
2x4=8
l Pupils to use the chunking
method of written long division.
Example (75 ÷ 5)
_____
5) 76
- 5 0 (10 x 5)
26
-2 5 (5 x 5)
0
l Pupils to have sound knowledge
of times tables.
10 x 4 = 40
____________________________________________
0
8
48
l Remainders are possible.
l Arrays should be used to link division to multiplication
l RELATE multiplication grid to
arrays and inverse.
Example: 75 ÷ 5 = 15
Representation: 76 ÷ 5
5
10
5
50
25
5
10
5
50
25
75
76
l Pupils will use the number facts for multiplication and division; solving
mathematical statements using the tables they know (empty box questions for
example).
Examples:
6÷
l
=3
so
6÷2=3
60 ÷ 2 =
Answer = 15 rem 1
Year Five
Remember to ‘ENGAGE BRAIN BEFORE PENCIL!’ –
ask ‘What is the most appropriate method to solve
this problem?’
Year Six
Remember to ‘ENGAGE BRAIN BEFORE PENCIL!’ – ask ‘What is the most appropriate method to solve this
problem?’
l Pupils need to be confident using both long and short division.
l Problems will be solved involving H T U ÷ U
l Pupils need to be confident using the chunking
method of long division then progressing onto
short division.
(see example in year six column )
l Pupils will be introduced to remainders being
expressed as a fraction or decimal
(or using rounding in a contextual problem i.e.
children in buses or CD’s in CD racks).
Example: 76 ÷ 5
5) 76
- 5 0 (10 x 5)
26
-2 5 (5 x 5)
rem 1
Answer: 15 rem 1 = 15 ⅕ = 15.2
l Pupils will be expected to be able to solve problems with three or four digit numbers divided by one or
two digit numbers.
l Pupils will be expected to be able to select the ‘short’ or ‘long’ method to use by considering the divisor.
Where the divisor is less than or equal to twelve, or the divisor is a multiple of five, pupils should consider
using ‘short’ division.
l For the written method of ‘long’ division, pupils will use the ‘chunking’ method (as shown).
Example:
496 ÷ 27 (Looking at the divisor – we choose ‘long’ division.)
It often helps to begin with a selection of multiplication facts connected to the divisor, a VIB (Very Important
Box) or Jolly Jottings.
1x27=27
2x27=54
3x27=81
4x27=108
5x27=135
10x27=270
20x27=540
etc
27) 496
- 270 (10 x 27)
226
- 135 (5 x 27)
91
-81 (3 x 27)
10
Answer: 18 rem 10
l For single digit divisors (or divisors of 12 or less) and for divisors that are multiples of 5 (15, 20, etc) –
consider short division.
3 3 rem 1
15) 4 9 4 6
l Pupils will solve problems involving division of a decimal using written methods in cases where an answer
has up to two decimal places.
l Remember: the remainder needs to be expressed as a fraction or decimal.
Plus: Division of decimal numbers by one digit, whole numbers in a practical context.
i.e. pounds/pence, metres/cm – Turn pounds into pence at the start and then apply the
appropriate ‘short’ method.