Coppermill Primary School
Calculations Policy
Ratified by the Curriculum and Achievement Committee on:
16 October 2014
To be reviewed: Every two years
Next review: Autumn 2016
Calculation at Coppermill
At Coppermill, we believe that children should enjoy and be engaged by Maths, in order
to equip them with the skills necessary in later life. Children will be taught Maths in a
practical way, enabling them to use and apply their mathematical knowledge in every
day contexts and situations. As such, the majority of our teaching will use concrete
resources, but will also increasingly be taught through worded problems, progressing to
the majority of children’s calculation work based on worded problems by the end of
Year 2.
Children will also be taught to make informed choices about the most reliable, efficient
method for performing a particular calculation and be encouraged to estimate the
answer first, as a means of checking if the answer makes sense. In the autumn term, we
focus primarily on number and calculation skills to ensure that children’s ability with
number is embedded, allowing them to progress to the next stage. This policy includes
both mental and written calculation strategies. Mental calculation includes instant
recall, but also children using apparatus and informal jottings to support their
calculation. On the other hand, written calculations refer to formal/standard written
methods (often more traditional) of recording.
From time-to-time, we will provide support through workshops, to enable parents to
use these resources confidently with children at home.
REGARDLESS OF THE YEAR LEVEL ATTACHED TO METHOD, CHILDREN MUST ONLY
PROGRESS TO THE NEXT METHOD IF THEY ARE SECURE IN THE PREVIOUS METHOD
AND KNOWLEDGE.
Addition (+)
E
Y
F
S
Subtraction (-)
Counting on
Children use multilink or
counting bears to add on.
They will practise this
practically to understand
number correspondence.
They will then learn one
more with numbers to 20.
Counting back using objects
In the same way that children learn
how to add on using objects, they are
taught how to count backwards using
objects. Children learn that less means
taking away.
E.g. One more than 4 is 5
Children are then taught how to write
take away using number sentences.
20 – 1 = 19
Multiplication (x)
Counting in 2s, 5s and 10s
Children are taught multiples of 2s, 5s
and 10s
E.g. 2, 4, 6, 8, 10, 12, 14, 16
5, 10, 15, 20, 25, 30, 35, 40
E.g. one less than 20 is 19
Division (÷)
Sharing equally
Children are taught to share a
quantity equally into a given
number of portions, and work out
how many are in each portion.
e.g. 6 ÷ 2 (share 6 sweets
between 2 children)
10, 20, 30, 40, 50, 60
One more than 19 is 20.
Y1
–
Y2
Counting on using a
number line
Children are taught how to
use a number line to add
on. Initially they count in
unit, moving to counting in
tens, then units when
adding.
Counting backwards using the number
line
Children are taught how to count
backwards on the number line when
taking away.
Counting in 2s, 5s and 10s
Children then move onto understanding
groupings of 2s, 5s and 10s using arrays
E.g. 2 groups of 2 is 4.
Grouping (or repeated
subtraction)
Used when we are asked to find
how many groups of a given size
are equivalent to the original
quantity.
E.g. 10 – 3 = 7
Doubling numbers
Double 2 = 4
Double 12 = 24
For example how many groups of 2
marbles are in a set of 6 marbles,
the calculation 6 ÷ 2 = , therefore
how many 2s in 6?
Children will add to
number bonds to 10 and
later to 20
e.g. 7 + 3 = 10
3 + 7 = 10
17 + 3 = 20
Finding the difference
Children are then taught how to find
the difference between two numbers.
Partitioning
Where the tens and units
are added separately.
Initially carried out with
jottings, then mentally.
e.g.
TU TU
47 + 76 = (40 + 70) + (7 + 6)
= 110 + 13
= 123
(Quite a difficult method
when crossing tens or
hundreds mentally. Needs
to be supported by a
numberline or 100 square)
Shown/calculated on a number
line:
This is then shown through arrays
Stage 1
12 – 8 =
Start on 8 and count onto 12 using a
numberline.
12 – 8 = 4
3 + 17 = 20
Children are taught the
inverse of their number
sentences to embed the
understanding that the total
will remain the same.
Repeated addition
2+2+2+2=8
Stage 2
Children are taught to count on to the
nearest 10 when counting on.
19 – 7 =
Start at 7, count to 10 and then to 19
19 – 7 = 11
Finding 10 less
Children use their tens and units
knowledge to partition numbers
T U
Children count in 2s to arrive at the
answer 8.
Repeated addition using the number
line
IT IS VITAL THAT WE TEACH ALL THE
ABOVE METHODS OF DIVISION,
INCLUDING REPEATED SUBTRACTION,
TO ENABLE CHILDREN TO
UNDERSTAND METHODS TAUGHT
LATER ON.
5 + 5 + 5 = 15
Halving numbers
Half of 12 = 6
Half of 20 = 10
Writing multiplication number
sentence
1 9 = 10 + 9
The 3 represents
They are then taught to find 10 less by
taking away the 10s
19 – 10 = 9
5 x 3 = 15
3 represents how many groups.
5 represents how many in each group.
Y3
–
Y4
Column method
The next step is to show
children the vertical format
(units under units, tens
under tens, etc.) and link it
to the mental method. They
first practise this method
with calculations they can
do mentally, and then
extend to three-digit
numbers.
Finding the difference using a
numberline
Show the children the vertical layout
for a calculation so they can do it
mentally. Link the steps to those on an
empty number line.
e.g. 326 - 178
*Children should be able to
describe what they are
doing by referring to the
actual values of the digits in
the columns i.e. ’20 + 50’ or
‘2 tens + 5 tens’, never ‘2 +
5’.
e.g. 47 + 76
326
- 178
2 ( 180)
20 ( 200)
100 ( 300)
26 ( 326)
148
You can reduce the number of stages
further, by using knowledge of pairs of
numbers that total 100.
+ 22
+ 126
With particular emphasis on
the column values,
hundreds, tens and unit.
Units (or right hand column)
must be added first.
1
47
+ 76
123
+ 100
+2
178
Vertical repeated addition
30 x 5 =
30
30
30
30
+30
150
+ 26
+ 20
180
200
300
326
Vertical or short multiplication
The method is made more compact by
combining steps. If after practise,
children cannot use the compact
method without making errors, as with
all cases they should return to the
previous step.
5
38
 7
266
Informal written methods –
subtracting multiples of the divisor
On a number line:
Standard written methods:
a) Easy - 844 divided by 2
422
2(844
b) Combining the first two
digits – 150 divided by 5
030
5 (150
c) Remainder at end of sum –
247 divided by 6
041 r 1
6 (247
178
200
326
- 178
22 ( 200)
126 ( 326)
148
326
d) Remainder in sum 75
divided by 3
25
3(75
1
Column method/decomposition
Children should be able to explain the
link and appreciate that the compact
method saves time recording their
working. If, with a little practise, they
cannot use the compact method
without making errors, they should
return to counting up on a number
line.
Grid method can be used to assist
understanding:
The mental method from which written
methods are developed involves
partitioning, and then multiplying the
tens and ones separately. A useful way
of recording intermediate steps is the
‘grid’ method.
*Again, when describing each step,
children should refer to the actual
values of the digits.
e.g. 725 – 367
7 2 15
-3 6 7
3 5 8
210 + 56 = 266
6 11
Extended to bigger numbers
e.g. 56  27
Estimate: 1800 because 60  30 = 1800
56  27 = (50 + 6)  (20 + 7)
1000 + 350 + 120 + 42 = 1512
Mental methods:
a) Children should be able
to mentally count on in
10s, 100s and 1000s
from any starting point
(256, 356, 456 etc).
b) This should then be
extended to mental
addition of multiples of
10, 100 and 1000. (59 +
40 = 99, 133 + 400 =
533, 5,700 + 2000 =
9,700).
c)
Y5
Y6
Mental methods: the following would
support and simplify ‘numberline’
subtraction:
a) Rapid recall of all pairs of
numbers that make 10.
b) Knowledge of how to quickly
calculate pairs of numbers that
make 100 (65 + 35 the tens
add up to 90 and the units
make the other 10)
Mental method, using partitioning:
a) Rapid recall of ALL
multiplication tables
b) Knowledge of how to X by
multiples of 10
(i.e. 3 x 7 = 21, 3 x 70 = 210, 3 x
70 = 210)
38  7 = (30  7) + (8  7)
= (3  7 (then add a zero)) + (8  7)
= 210
= 266
+ 56
A
lso use partitioning
mental maths methods
to add.
Extend to bigger numbers
and decimals
As above, remembering to
line up the decimal points.
11
35.6
+16.6
52.2
Either decomposition or the
numberline method can be used
Numberline: Extend to bigger
numbers and decimals
e.g. 22.4 – 17.8
+ 0.2
+4
Vertical or short multiplication with
decimals
The amount of digits after the decimal
point in the answer must be equal to
the number of digits after the decimal
points in the whole number sentence –
in this case 3.
+ 0.4
7 5
________________________________
17.8 18
22
22.4
3.86
0.9
3.474
Mental methods:
a) Rapid recall of ALL
multiplication tables, as
division facts (i.e. how
many 8s in 56? How many
7s in 49?)
b) Rapid mental calculation of
remainders
(i.e. what is the remainder
when we divide 47 by 5)
Timesaving mental methods:
a) Dividing by 4: halve and
halve again
b) Dividing by 10: moving
number one place to the
right
c) Dividing by 5: dividing by
10 and doubling
Dividing by bigger numbers
Dividing by a two-digit number –
546 divided by 13
As most children don’t know the 13
times table, the easiest thing is for
them to quickly jot it on their
whiteboard (13, 26, 39, 52, 65, 78,
91 etc)
Then, the same method as previous
is applied:
42
Show the children the vertical layout
for a calculation so they can do it
mentally. Link the steps to those on an
empty number line.
22.4
- 17.8
0.2 ( 18)
4.0 ( 22)
0.4 ( 22.4)
4.6
Long multiplication
2
56
Dividing exactly
Often, problems require an exact
answer, rather than one with a
remainder. For example 90 divided
by 4 as 22.5, not 22 remainder 2.
x 27
1
392
1120
1512
(56 x 7)
(56 x 20)
Remember to add the 0 as this row is
multiplying tens and tens.
Decomposition: Extend to bigger
numbers and decimals
As above, remembering to line up the
decimal points.
e.g. 725.8 – 367.4
6 11
Mental methods: by this
stage, children should be
using a range of useful
strategies for mental
addition.
a) When adding 18 and 14,
they should mentally
add the tens (20), the
units (12) and then find
the total (32)
b) When adding two
consecutive numbers
This can be done by extending the
number divided with zeros, after a
decimal point.
17.5
4(70.0
3 2
The decimal point in the answer
must appear above the decimal
point in the dividend (the number
being divided.)
1
7 2 5.8
- 3 6 7 .4
3 5 8 .4
Mental methods: the following would
support and simplify ‘numberline’
subtraction at this level:
a) Rapid recall of all decimal pairs
that add up to 1 (i.e. 0.8 + 0.2)
and 0.1 (0.07 + 0.03)
b) Doing simple ‘counting up’
subtraction, with jotting only
(i.e. 72 – 37:
3 + 30 + 2 = 35
c) Doing simple decimal ‘counting
up’ subtraction, with jotting
13(546
1
Mental method, using partitioning:
a) Rapid recall of ALL
multiplication tables
b) Knowledge of how to X by
multiples of 10
(i.e. 3 x 7 = 21, 3 x 70 = 210, 3 x
70 = 210)
38  7 = (30  7) + (8  7)
= (3  7 (then add a zero)) + (8  7)
= 210
= 266
+ 56
Mental methods:
a) Rapid recall of ALL
multiplication tables, as
division facts (i.e. how
many 8s in 56? How many
7s in 49?)
b) Rapid mental calculation of
remainders
(i.e. what is the remainder
when we divide 47 by 5)
(26 and 27), they should
use doubling (2 x 26 =
52) and then add the
extra unit (52 + 1 = 53)
c) When adding three two
digit number (17 + 13 +
17) children should look
for good ‘starting
places’. For example,
they could begin by
adding 7 and 3 (a pair
that makes 10) or 7 and
7 (a double).
only (i.e. 10 – 3.7: 0.3 + 6 =
6.3)
VOCABULARY
Vocabulary should be taught and expectations of use made explicit through series of lessons. The vocabulary should be built up as
a display with the children so that it has meaning and use for them and they are able to apply the vocabulary in context.
EYFS
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
add,
more,
and
+
__
plus,
near
addition,
one
hundreds
Increase,
units
integer
Addition and
make, sum, total
double,
hundred more,
boundary
decrease,
boundary,
subtraction
altogether
how much more
subtraction, one
inverse
tenths
Multiplication
and division
score
double
one more, two more, ten more…
how many more to make… ?
how many more is… than…?
take (away), leave
how many are left/left over?
how many have gone?
one less, two less… ten less…
how many fewer is… than…?
difference between
is the same as
count in 2s,5s,10s
is…?
- , subtract, minus
how much less
is…?
half, halve
=, __equals
Tens, ones
hundred less,
tens boundary,
ones, units, tens
partition
1 digit
2 digit
3 digit
inverse
double, halve
share, left, left over
lots of, groups of
multiplication
multiplication,
product
half
x, times, multiply,
multiplied by
multiple of
once, twice, three times… ten
times…
times as (big, long,
wide… and so on)
repeated addition
array
row, column
share equally
one each, two each, three
each…
group in pairs, threes… tens
equal groups of
÷, divide, divided by, divided
into
left
left over
remainder
round up/down
grid
row
column
boundary
Thousands
4 digits
decimal point
decimal place
tenths/hundre
dths
negative
ten
thousands
million
Count up
Number sentence
Digit
near double
count in 2s,5s,10s
groups of
same thing, lots
of times
factor,
quotient,
divisible by
inverse
divisor
dividend