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Emmanuel Church of England Primary School
Mathematics Calculation Policy
Updated March 2016
Emmanuel Church of England Primary School Calculation Policy
Nursery
Addition Subtraction Multiplication Division
Children are expected to work with
numbers 0 – 10.
Children are encouraged to use stories,
songs and rhymes to recall simple addition
facts e.g. One, two, buckle my shoe.
Children begin to use appropriate language
to compare two sets of objects (e.g. more).
Children find one more by counting on using
concrete resources e.g. bead strings.
Children are expected to work with
numbers 0 - 10.
Children are encouraged to use stories,
songs and rhymes to support their initial
understanding of subtraction e.g. Ten Green
Bottles, Five Little Ducks.
Children begin to use appropriate language
to compare two sets of objects (e.g. fewer)
using counting on.
Children should relate subtraction to
finding one less than a number by counting
back on a number track.
There
are
four balloons. How many are left if one is
removed?
Emmanuel Church of England Primary School Calculation Policy
Reception
Addition Subtraction Multiplication Division
Children are expected to work with
numbers 0 – 20.
Children are encouraged to use stories,
songs and rhymes to recall simple addition
facts e.g. One, two, buckle my shoe.
Children use moveable objects when finding
totals.
Children can add two groups of objects
together by starting with the biggest
number.
Children begin to develop ways of recording
calculations using pictures and objects.
Children
begin to
develop ways of recording calculations using
number lines, pictures and objects.
Children are expected to work with
numbers 0 - 20.
Children are encouraged to use stories,
songs and rhymes to support their initial
understanding of subtraction e.g. Ten Green
Bottles, Five Little Ducks.
Children begin to use appropriate language
to compare two sets of objects (e.g. fewer)
using counting on.
Children should relate subtraction to
finding one less than a number by counting
back on a number track.
There are four balloons. How many are left
if one is removed?
Children begin to develop ways of recording
calculations using number lines, pictures and
objects.
Children are expected to work with
numbers 0 - 20.
Objects/Pictures/Symbols
Children count equal groups of objects in
real-life contexts.
Count in 2s using socks or shoes.
Count in 5s and 10s using hands.
Count in multiplies using numicon.
Children will work on practical problem-
solving activities involving equal sets or
groups.
Doubling
Multiplication is related to doubling and
counting groups of the same size up to 10.
Children are expected to work with
numbers 0 - 20.
Objects/Pictures/Symbols
Children will understand equal groups and
will share objects equally into groups.
Grouping and sharing
Children will understand that sharing and
grouping means the same amount in each
group.
Children will work on practical problem-
solving activities involving sharing and
grouping e.g. 8 cars are shared between 4 children. How many cars does each child get?
There are 8 cars. How many children can have two cars each?
Halving
Children to understand
halving as having two equal
groups.
Emmanuel Church of England Primary School Calculation Policy
Year 1
Addition Subtraction Multiplication Division
Children are expected to work with
numbers 0 – 20.
Objects/Pictures/Symbols
Children are encouraged to use a variety of
visual resources and pictorial
representations to support their learning of
number facts.
Number Lines:
Record drawing jumps on prepared lines or
constructing own lines. Children should be
encouraged to add units to bridge through
ten.
Children are expected to work with
numbers 0 – 20.
Objects/Pictures/Symbols
Encourage the use of concrete objects and
pictorial representations. e.g. bead strings,
counters, dienes, number lines, number
tracks and hundred squares.
Number Lines:
Children begin to develop ways of recording
calculations using number lines, pictures and
objects.
Children are expected to work with
numbers 0 – 20.
Objects/Pictures/Symbols
Children will continue to participate in
practical activities that involve counting in
2s, 5s and 10s e.g. counting socks, shoes,
animals’ legs, fingers, gloves, numicon and
coins.
Children to begin to understand
multiplication as repeated addition using
bead strings.
Children to understand multiplication as
groups of objects.
Children are expected to work with
numbers 0 – 20.
Objects/Pictures/Symbols
Children will continue to participate in
practical activities that involve sharing and
grouping e.g. sharing cards when playing a
game, putting objects into cups, hoops and
onto plates.
Grouping and sharing
Children will understand that sharing and
grouping means the same amount in each
group.
Grouping
Children will sort a number of objects into
equal sized groups e.g.
There are 6 coins, how can I group them equally?
Halving
Children to understand halving as having
two equal groups and that it is the inverse
of doubling.
Emmanuel Church of England Primary School Calculation Policy
Establish that addition and subtraction are
related operations.
Encourage estimation and the use of inverse
to check answers.
Ensure that children know that addition is
commutative (can be done in any order).
Counting back:
End of Year 1:
Move on to bridging through 10 in
preparation for Year 2.
Counting on:
Establish that addition and subtraction are
related operations.
Encourage estimation and the use of inverse
to check answers.
Ensure that children know that subtraction
is non-commutative (cannot be done in any
order).
Children will describe pictorial arrays and
make their own arrays, counting in groups of
2s, 5s and 10s.
How many rows? How many columns? How many in each row? How many in each column?
Children will solve counting problems e.g.
How many 2s make 8?
Doubling
Multiplication is related to doubling and
counting groups of the same size up to 10.
Encourage estimation and the use of inverse
to check answers.
Ensure that children understand that
multiplication is commutative (can be done
in any order).
Encourage estimation and the use of inverse
to check answers.
Emmanuel Church of England Primary School Calculation Policy
Year 2
Addition Subtraction Multiplication Division
Children are expected to work and calculate
with numbers up to 100.
Objects/Pictures/Symbols
Blank Number Lines:
Children should be encouraged to use empty
number lines to add tens and ones and then
use these to bridge through ten using
number bond knowledge.
When adding two digit numbers, children
should partition the second number into
tens and ones.
Encourage moving on from adding single 10s
and single 1s.
Children are expected to work and calculate
with numbers up to 100.
Objects/Pictures/Symbols
Blank Number Lines:
Counting Back:
Subtracting a single digit:
42 – 6 =
Children are expected to recall and use the
multiplication facts for 2, 5 and 10
multiplication tables.
Objects/Pictures/Symbols
Missing box calculations:
Children to understand multiplication as
repeated addition.
Number lines
5 + 5 + 5 = 15
5 x 3 = 15
Children to understand multiplication as
grouping.
Children are expected to recall and use the
division facts for 2, 5 and 10 multiplication
tables.
Objects/Pictures/Symbols
Missing box calculations:
Children should understand division as
repeated subtraction.
Blank Number lines
How many 3s in 15? Children to count forward in 3s to 15 or
count back in 3s from 15.
15 ÷ 3 = 5
Grouping and sharing
Children will understand that sharing and
grouping means the same amount in each
group.
Emmanuel Church of England Primary School Calculation Policy
100 square
When modelling addition as partitioning and
recombining using a hundred square,
children should be reminded to move down
to count in tens and across to count in ones.
Partitioning in Spring Term
Use place value counters to support
children.
+
68
26
94
60 8
20 6
80 14
Fact Families:
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that addition is
Subtracting a 2 digit number:
Count back on an empty number line by
partitioning the number they are
subtracting into tens and ones.
Counting On:
46 – 33 =
100 square
Children should be encouraged to subtract
9 and 11 on a hundred square by subtracting
10 and then compensating with
understanding.
Partitioning in Spring Term
Use place value counters to support
children.
-
86
33
53
80 6
30 3
50 3
Arrays
Use arrays to group using the multiplication
sign.
2 x 3 = 6
3 x 2 = 6
Children will solve one-step multiplication
problems e.g.
There are 5 oranges in a net. How many oranges in 6 nets?
Doubling
Children to know that doubling is the same
as multiplying by 2.
Children to know the doubles to 30 + 30 and
doubles of multiples of 10 up to 100.
Encourage estimation and the use of
inverse to check answers.
Ensure that children understand that
multiplication is commutative (can be done
in any order).
Grouping
There are 30 balloons. How can I group them equally?
Sharing
15 sweets are shared between 3 children. How many sweets do they each get?
Arrays
Use arrays to find division facts for 2, 5
and 10 multiplication tables.
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
We divided 30 into rows of 5. We use the ÷ sign for divide. We write 30 ÷ 5 = 6 as 30 can be divided into 6 rows of 5.
Emmanuel Church of England Primary School Calculation Policy
commutative (can be done in any order). Encourage estimation and the use of
inverse to check answers.
Ensure that children understand that
subtraction is non-commutative (cannot be
done in any order).
Children will solve one-step division
problems e.g.
Oranges are in nets of 5. How many nets will 30 oranges fill?
Halving
Children to understand halving as the same
as dividing by 2.
Children should be able to halve numbers up
to 30 and halve multiples of 10 up to 100.
Encourage estimation and the use of inverse
to check answers.
Ensure that children understand that
division is non-commutative (cannot be done
in any order).
Emmanuel Church of England Primary School Calculation Policy
Year 3
Addition Subtraction Multiplication Division
Children are expected to work with
numbers up to 1000.
Blank number lines
Children continue to use number lines to
add HTU.
Children continue to use expanded method
to add HTU.
Progressing from:
Using place value counters to support
children.
+
68
26
94
60 8
20 6
80 14
To
Vertical expanded method
Use Place Value Counters to demonstrate
exchange. Record calculations on the right.
235
+ 28
______
13 (5+8)
50 (30+20)
200 (200)
_____
263
Children are expected to work with
numbers up to 1000.
Blank number lines
Counting back:
Children should be able to use an empty
number line to ‘take away’ by partitioning
the second number.
Counting on:
Children are encouraged to use empty
number lines to ‘find the difference’
113-87=
Compensating:
Children need to understand that 178 – 49
is the same as 178 – 50 + 1.
Progressing from:
Using place value counters to support
children.
-
86
33
53
80 6
30 3
50 3
Children are expected to recall and use the
multiplication facts for 3, 4, 6 and 8
multiplication tables.
Pictures/Symbols/Objects:
Missing box calculations:
Children to understand multiplication as
repeated addition.
Grid Method
46x
4
40x4
6x4
160
24
46x
4 = 184
Children to understand multiplication as
grouping.
Arrays (until children are secure in using
multiplication facts)
8 x 4 = 32
or
4 x 8 = 32
or
8 + 8 + 8 + 8
or
Children are expected to recall and use the
division facts for 3, 4, 6 and 8
multiplication tables.
Pictures/Symbols/Objects:
Missing box calculations:
Children to divide two digit numbers by one
digit.
Number lines
How many 6s are there in 32? How many left over? Groups of 6 are taken away and the
numbers of groups are counted. Any left
over are remainders.
32 ÷ 6 = 5 r2
Arrays (until children are secure in using
multiplication facts)
How many 4s in 32?
Emmanuel Church of England Primary School Calculation Policy
Compact column addition if children are
ready (modelled side by side)
H T O
2 6 8
3 5 6
1 4
1 1 0
5 0 0
6 2 4
H T 0
2 6 8
3 5 6 1 1 0
6 2 4
+ +
+
Children to leave a line after they have
written the column addition.
“8+6=14. We write 4 in the ones column and a 1 under the numbers we’re adding in the tens column for the 10. We add the 10s, put the 2 in the Tens and 1 for a 100 under the numbers we’re adding in the Hundreds column. Then we add the 100s to get 624”
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that addition is
commutative (can be done in any order).
To
Vertical Expanded Method:
Use Place Value Counters to demonstrate
exchange. Record calculations on the right.
287
- 145
_____
2 (7-5)
40 (80-40)
100 (200-100)
______
142
Progressing to:
Compact column subtraction if children are
ready (modelled side by side.)
287
- 145
_____
2 (7-5)
40 (80-40)
100 (200-100)
______
142
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that subtraction
is non-commutative (cannot be done in any
order).
4 + 4 + 4 + 4 + 4 + 4 + 4+ 4
Use known facts and place value
“If I know 3 x 8 = 24, I also know 30 x 8 = 240 3 x 80= 240”
Fact families
4 x 6 = 24
6 x 4 = 24
24 ÷ 4 = 6
24 ÷ 4 = 6
Children will solve multiplication problems.
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that
multiplication is commutative (can be done
in any order).
Use known facts and place value
÷ 4 is the same as halving and halving again.
Fact families
4 x 6 = 24
6 x 4 = 24
24 ÷ 4 = 6
24 ÷ 4 = 6
Children will solve division problems.
Encourage estimation and the use of inverse
to check answers.
Ensure that children know that division is
non-commutative (cannot be done in any
order).
287
-145
142
Emmanuel Church of England Primary School Calculation Policy
Year 4
Addition Subtraction Multiplication Division
Children are expected to work with
numbers up to 10 000.
Children to continue to use the vertical
expanded method to add larger numbers.
Children to leave a line after they have
written the column addition.
Column Addition:
Begin by explicitly writing the carrying of
tens and hundreds.
637
+ 894 1 1 0
______ 1531
Before moving to a more compact method
when appropriate.
637
+ 894 1 1
_____
1531
Further develop this method using decimals
within context e.g. money.
16.37
+ 28.94 1 1 1
_______
45.31
Encourage estimation and the use of
inverse to check answers.
Children are expected to work with
numbers up to 10 000.
Number Lines:
Children should first be encouraged to use
empty number lines to subtract by counting
up (Complementary Addition).
Expanded Method:
Place Value Counters should be used to
reinforce children’s understanding of place
value.
4287 4000 + 200 + 80 + 7
- 2145 2000 + 100 + 40 + 5
_______________________
2000 + 100 + 40 + 2 = 2142
Progressing to:
4287
- 2145
_____
2 (7-5)
40 (80-40)
100 (200-100)
2000 (4000-2000)
______
2142
Children are expected to recall and use the
multiplication facts up to 12 x 12.
Pictures/Symbols/Objects:
Missing box calculations:
Children to multiply two and three digit
numbers by a one digit and two digit
number.
Grid method
Progressing from:
46x4 40x4
6x4
160
24
46x4 = 184
351x9 300x9
50x9
1x9
2700
450
9
351x9 = 3159
To:
Children are expected to recall and use the
division facts up to 12 x 12.
Pictures/Symbols/Objects:
Missing box calculations:
Short division
Supported by use of place value counters.
Dividing by 10 and 100
Move the digits once to the right for ÷10
and twice to the right for ÷100. Use zero
as a placeholder where there are gaps.
Emmanuel Church of England Primary School Calculation Policy
Ensure that children know that addition is
commutative (can be done in any order).
Compact Column Subtraction
After exchange, the ‘13’ should be written
next to each other in the square above the
3 in this example.
Further develop this method using decimals
within context e.g. money.
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that subtraction
is non-commutative (cannot be done in any
order).
Column multiplication (expanded)
T U
5 7
0 x 3
2 1 (3 x 7)
1 5 0 (3 x 150)
1 7 1
When moving to formal methods:
-begin with calculations that children can do
mentally.
-demonstrate informal and formal methods
side by side and step by step e.g.
partitioning and grid method, grid method
and column method.
Multiplying by 10 and 100
Move the digits once to the left for x10
and twice to the left for x100. Use zero as
a placeholder where there are gaps.
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that
multiplication is commutative (can be done
in any order).
Encourage estimation and the use of inverse
to check answers.
Ensure that children know that division is
non-commutative (cannot be done in any
order).
Emmanuel Church of England Primary School Calculation Policy
Year 5
Addition Subtraction Multiplication Division
Children are expected to work with
numbers up to 10 000, including decimals
with up to two places.
Column Method:
1637
+ 2894
1 1 1
_______
4531
Consolidate method using decimals.
16.37
+ 28.94 1 1 1
_______
45.31
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that addition is
commutative (can be done in any order).
Children are expected to work with
numbers up to 10 000, including decimals
with up to three places.
Expanded Method:
4287
- 2145
_____
2 (7-5)
40 (80-40)
100 (200-100)
2000 (4000-2000)
______
2142
Progressing to:
Compact Column Subtraction
After exchange, the ‘16’ should be written
next to each other in the square above the
7 in this example.
Children are expected to multiply numbers
using known facts e.g.
7 x 6 = 42 Therefore 70 x 60 = 4200
Children are expected to multiply numbers
up to four digits by a one or two digit
number using an efficient written method,
including long multiplication for two digit
numbers.
Column method (expanded)
3 2 6 8
x 4
3 2 (4 x 8)
2 4 0 (4 x 60)
8 0 0 (4 x 200)
1 2 0 0 0 (4 x 3000)
1 3 0 7 2
1
Column method
Short multiplication to multiply by a 1-digit
number
Children to leave a line after writing the
calculation:
T O HT O ThHT O
8 7 3 8 7 4 3 8 7
x 6 x 6 x 6 4 5 4 2 5 4
5 2 2 2 3 2 2 2 6 3 2 2
Children are expected to divide numbers
using known facts e.g.
63 ÷ 9 = 7 Therefore 630 ÷ 9 = 70
Children are expected to divide up to 4
digit numbers by a 1 or 2 digit number using
the efficient written method of short
division and interpret remainders
appropriately.
Dividing by a 1-digit number
Progressing from:
T O
2 9
3 8 7
6 0
2 7
2 7
0
To work out 87÷3 we start with the 80. Count out the 10s of 3:10x3=30; 20x3=60; 30x3=90. Stop too big. We can only get 20 3s out of 80. We write the 2 in the tens column, as 20 is 2 tens, and write the 60 below the 87 and subtract. This leaves 27. Now we find the 3s in 27 which is 9. This means 87÷3=29.
Emmanuel Church of England Primary School Calculation Policy
Children should identify when numbers are
close together and finding the difference
on an empty number line would be more
appropriate.
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that subtraction
is non-commutative (cannot be done in any
order).
Long multiplication to multiply by a 2-digit
number
Children to leave a line after writing the
calculation.
Children to cross out smaller numbers so
they know not to add them.
8 7 3 8 7
x 3 4 x 3 4 2 3 2 3 4 8 1 5 4 8 2 2 2 2 6 1 0 1 1 6 1 0
2 9 5 8 1 3 1 5 8 1
Multiply 3 and 4-digit whole numbers and
decimals with up to 2 decimal places, by 1-
and 2-digit numbers
9 .5 3 8. 7 6
x 2 7 x 3 4____ 3 3 3 2
6 6. 5 1 5 5. 0 4 1 2 2 1
1 9 0. 0 1 1 6 2. 8 0
2 5 6. 5 1 3 1 7. 8 4 1
Children to solve multi-step word problems.
To:
Instead of writing down each step we do an
extra calculation in our heads.
T U
1 4
7 9 28
We work out how many 7s will go into the 9; a short cut to working out how many 10s of 7 go into 90. The answer is 1. The 2 left over we carry over to the 8 to make 28. 7s into 28? 4, we put the 4 in the 1s column.
Dividing by a 2-digit number
H T U
5 4 r 11
1 6 8 7 5
8 0
7 5
6 4
1 1
Since we are dividing by 16 it is useful to derive the 16 times table we can refer to as we do the division.
1x16 16
2x16 32
3x16 48
4x16 64
5x16 80
6x16 96
: :
Emmanuel Church of England Primary School Calculation Policy
Multiplying by 10, 100 and 1000
Children will multiply whole and decimal
numbers by 10, 100 and 1000 where answers
have up to 3 decimal places
Move the digits once to the left for x10,
twice to the left for x100 and three times
to the left for x1000. Use zero as a
placeholder where there are gaps.
Fractions
Multiply proper fractions and mixed
numbers by whole numbers, supported by
materials and diagrams.
e.g. 2/3 x 3 = 6/3 = 2
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that
multiplication is commutative (can be done
in any order).
We cannot divide the 8 by 16 so we work out how many 16s will go into 87. There are 5 as 5x16=80 so we write 5 in the 10s and subtract to get 7. Now we involve the 5 and use 75. There are 4 16s in 75 as 4x16=64 and 5x16=80 is too big. We write 4 in the 1s column and subtract; the remainder is 11, we write: 874÷16 = 54 r
11, or 54 11
16.
Children are expected to divide whole and
decimal numbers by 10, 100 and 1000 where
answers have up to 3 decimal places
Dividing by 10 and 100 and 1000
Move the digits once to the right for ÷10,
twice to the right for ÷100 and three times
to the right for ÷ 100. Use zero as a
placeholder where there are gaps.
Encourage estimation and the use of inverse
to check answers.
Ensure that children know that division is
non-commutative (cannot be done in any
order).
Emmanuel Church of England Primary School Calculation Policy
Year 6
Addition Subtraction Multiplication Division
Children are expected to work with
numbers with more than four digits.
Column Addition:
Extend to numbers with any number of
digits and decimals with 1 and 2 decimal
places. E.g. 124.9 + 117.25 = 242.15.
124.9
+ 117.25 1 1
________
242.15
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that addition is
commutative (can be done in any order).
Children are expected to work with
numbers with more than four digits.
Compact Column Subtraction:
After exchange, the ‘16’ should be written
next to each other in the square above the
7 in this example.
Extend to numbers with any number of
digits and decimals with 1, 2 and 3 decimal
places. e.g. 327.67 – 143.25 =
Children should identify when numbers are
close together and finding the difference
on an empty number line would be more
appropriate.
Find the difference by counting up. e.g. 0.5
Children are expected to identify common
factors, common multiples and prime
numbers.
Children are expected to multiply multi-
digit numbers, up to 4 digits, by a two digit
whole number using the efficient written
method of long multiplication.
Children to leave a line after writing the
calculation.
Children to cross out smaller numbers so
they know not to add them.
3 8 7 5
x 3 4 3 3 2
1 5 5 0 0 2 2 1
1 1 6 2 5 0
1 3 1 7 5 0 1
3 8. 7 6
x 3 4____ 3 3 2
1 5 5. 0 4 2 2 1
1 1 6 2. 8 0
1 3 1 7. 8 4 1
Fractions
Children are expected to identify common
factors, common multiples and prime
numbers.
Children are expected to divide up to 4
digit numbers by 1 and 2 digit numbers
using the efficient written method of long
division and interpret remainders as whole
number remainders, fractions or decimals.
Dividing by a 1-digit number
T U
1 4
7 9 28
We work out how many 7s will go into the 9; a short cut to working out how many 10s of 7 go into 90. The answer is 1. The 2 left over we carry over to the 8 to make 28. 7s into 28? 4, we put the 4 in the 1s column.
Dividing by a 2-digit number
H T U
5 4 r 11
1 6 8 7 5
8 0
7 5
6 4
1 1
Emmanuel Church of England Primary School Calculation Policy
– 0.31 = 0.19
This can be modelled on an empty number
line.
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that subtraction
is non-commutative (cannot be done in any
order).
Multiply simple pairs of proper fractions,
writing the answer in its simplest form e.g.
¼ x ½ = 1/8
Encourage estimation and the use of
inverse to check answers.
Ensure that children know that
multiplication is commutative (can be done
in any order).
Since we are dividing by 16 it is useful to derive the 16 times table we can refer to as we do the division.
We cannot divide the 8 by 16 so we work out how many 16s will go into 87. There are 5 as 5x16=80 so we write 5 in the 10s and subtract to get 7. Now we involve the 5 and use 75. There are 4 16s in 75 as 4x16=64 and 5x16=80 is too big. We write 4 in the 1s column and subtract; the remainder is 11, we write: 874÷16 = 54 r
11, or 54 11
16.
Fractions
Children are expected to divide proper
fractions by whole numbers e.g.
1/3 ÷ 2 = 1/6
Encourage estimation and the use of inverse
to check answers.
Ensure that children know that division is
non-commutative (cannot be done in any
order).
1x16 16
2x16 32
3x16 48
4x16 64
5x16 80
6x16 96
: :
Emmanuel Church of England Primary School Calculation Policy
Children are expected to divide whole and
decimal numbers by 10, 100 and 1000 where
answers have up to 3 decimal places
Dividing by 10 and 100 and 1000
Move the digits once to the right for ÷10,
twice to the right for ÷100 and three times
to the right for ÷ 100. Use zero as a
placeholder where there are gaps.
Encourage estimation and the use of inverse
to check answers.
Ensure that children know that division is
non-commutative (cannot be done in any
order).