Download Maths Calculation Policy - The Batt C of E Primary School

Calculation Objectives
(Year 1 – 6)
Year 1
Addition and subtraction
Pupils should be taught to:
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read, interpret and practise writing mathematical statements involving addition (+),
subtraction (-) and equals (=) signs accurately.
add and subtract 1-digit and 2-digit numbers to 20 (9 + 9, 18 - 9), including zero.
add three 1-digit numbers.
recall and use number bonds and related subtraction facts within 20.
solve simple word problems that involve addition and subtraction.
Multiplication and division
Pupils should be taught to:
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recognise and write the multiplication symbol (x) and the division symbol (÷) in mathematical
statements, calculating the answer with the teacher using concrete objects.
solve word problems involving simple multiplication and division, with teacher support.
Year 2
Addition and subtraction
Pupils should be taught to:
 rapidly recall and use addition and subtraction facts to 20.
 add and subtract numbers with up to two 2-digits including using column addition without
carrying and column subtraction without borrowing.
 add and subtract numbers mentally including:
- a 2-digit number and ones
- a 2-digit number and tens
- two 2-digit numbers
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use subtraction in ‘take away’ and ‘find the difference’ problems.
recognise and show that addition can be done in any order (commutative) and subtraction
cannot.
recognise and use addition and subtraction as inverse operations including to check
calculations.
solve word problems with addition and subtraction of numbers with up to 2-digits.
The Batt School Calculation Policy 2015
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Multiplication and division
Pupils should be taught to:
 recall multiplication and division facts for the 2, 5 and 10 multiplication tables.
 use the multiplication (x), division (÷) and equals (=) signs to read and write mathematical
statements.
 write and calculate mathematical statements for multiplication and division within the
multiplication tables.
 recognise and use the inverse relationship between multiplication and division to check
calculations.
 ensure pupils can recognise and show that multiplication can be done in any order
(commutative) and division cannot.
 solve word problems involving multiplication and division.
Year 3
Addition and subtraction
Pupils should be taught to:
 add and subtract numbers with up to 3 digits, including using columnar addition and
subtraction.
 accurately add and subtract numbers mentally including: pairs of one- and 2-digit numbers; 3digit numbers and ones; 3-digit numbers and tens; 3-digit numbers and hundreds.
 solve word problems including missing number problems, using number facts, place value, and
more complex addition and subtraction.
Multiplication and division
Pupils should be taught to:
 recall and use multiplication and division facts for the 2, 3, 4, 5, 8 and 10 multiplication tables.
 write and calculate mathematical statements for multiplication and division within the
multiplication tables; and for 2-digit numbers x 1-digit numbers, using mental and written
methods.
 solve word problems involving the four operations, including missing number problems.
Year 4
Addition and subtraction
Pupils should be taught to:
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add and subtract numbers using formal written methods with up to 4 digits.
accurately add and subtract numbers mentally including two 2-digit numbers.
estimate, within a range, the answer to a calculation and use inverse operations to check
answers.
Multiplication and division
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Pupils should be taught to:
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recall multiplication and division facts for multiplication tables up to 12 x 12.
mentally perform multiplication and division calculations quickly and accurately, including
multiplying by 0 and dividing by 1.
multiply or divide 2-digit and 3-digit numbers by a 1-digit number using formal written
methods; interpret remainders appropriately as integers.
recognise and use factor pairs within 144.
solve word problems involving the four operations.
find the effect of dividing a 2-digit number by 10 and 100, identifying the value of the digits
in the answer as units, tenths and hundredths.
Year 5
Addition and subtraction
Pupils should be taught to:
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add and subtract whole numbers with up to 5 digits, including using formal written methods.
add and subtract numbers mentally with increasingly large numbers.
add and subtract numbers with up to three decimal places.
Multiplication and division
Pupils should be taught to:
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identify multiples including common multiples, and factors including common factors.
know and use the vocabulary of prime numbers, prime factors and composite (non-prime)
numbers.
establish whether a number up to 100 is prime and recall the prime numbers up to 19.
multiply numbers up to 4-digits by a 1 or 2-digit number using a formal written method,
including long multiplication.
accurately multiply and divide numbers mentally drawing upon known facts.
divide numbers up to 4 digits by a 1-digit number and 10 and interpret remainders
appropriately.
multiply and divide numbers by 10, 100 and 1000.
recognise and use square numbers and square roots and the notation for square (2) and square
root (√)
solve word problems involving addition and subtraction, multiplication and division.
Year 6
Addition, subtraction, multiplication and division
Pupils should be taught to:
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add and subtract negative integers.
multiply numbers with at least 4-digits by 2-digits of whole number using long multiplication.
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divide numbers up to 4-digits by a 2-digit whole number using long division, and interpret
remainders as whole number remainders, fractions, decimals or by rounding.
perform mental calculations, including with mixed operations and large numbers.
use estimation to check answers to calculations and determine in the context of a problem
whether an answer should be rounded, or written as a fraction or a decimal.
carry out combined operations involving the four operations accurately and state the order of
operations.
solve word problems involving addition, subtraction, multiplication and division.
identify the value of each digit to three decimal places and multiply and divide numbers up to
three decimal place by 10, 100 and 1000.
multiply and divide numbers with up to two decimal places by 1-digit and 2-digit whole numbers
recognise and use division in the context of fractions, percentages and ratio.
solve linear missing number problems, including those involving decimals and fractions, and find
pairs of number that satisfy number sentences involving two unknowns.
The following pages show our written calculation methods for maths. They are organised by
number operations – Addition (+), Subtraction (-), Division () and Multiplication (x). Each
section shows the progression of written calculation methods for each operation starting from
simple methods and progressing to more advanced ones. Children will progress through the stages
according to their mathematical knowledge and understanding rather than year group.
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Addition
Calculation Method
.
Explanation
Step 1
Initially, children will work with
pictures and objects.
2 + 3 = 5
At a party, I eat 2 cakes and my
friend eats 3.
How many cakes did we eat
altogether?
Children draw a picture to help
them work out the answer.
7 + 4 = 11
Children use dots or symbols to
represent objects (quicker than
drawing a picture)
7 people are on the bus. 4 more get
on at the next stop. How many
people are on the bus now?
……. ….
We would then progress to;
7 +
….
We would then progress to
counting on from 7.
= 11
Children use a number line to
record jumps made. Using single
jumps…
5 + 3 = 8
+1
5 6
+1
+1
7
8
9
10 11
6 + 4 + 3 =13
+ 4
6
or larger jumps.
+ 3
10
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5
Step 2
Draw an empty number line.
Children will be taught to place
the largest number at the left
side of the line and jump forward
in appropriate steps.
7+9=16
+1
9
+6
10
16
13 + 19=
+1
19
+10
20
This method is also used when
adding 19, 29, 39 etc
+2
30 32
7 + 9 = (7 + 10) – 1= 16
+ 10
7
-1
16
17
Children use a sound
17
– 1 = 16
or…..
understanding
of adding 10 to
adapt calculations. Instead of
+ 10
adding 9, they add 10 then adjust
- 1
the calculation by subtracting 1.
7
This is also used for adding other
numbers e.g. 15 + 19 (15 + 20, - 1)
This is a mental strategy.
127 + 74 = 201
There are 127 boys and 74 girls in a Children will move to partitioning
school. How many children are there using a number line.
altogether?
Children will start their number
line at 127. Add a jump of 70 to
197. Either add 3 then 1 or simply
+70
+4
a jump of 4 to land on 201.
127
197
201
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Step 3
57 + 25 =
50 + 20 = 70
7 + 5 = 12
70 + 12 = 82
Or
57 + 25 =
57 + 20 = 77
77 + 5 = 82
The next stage is to record mental
methods partitioning tens and ones
separately. This can be done by
adding the tens and then the ones
and finding the total or by
partitioning only the second
number.
This can also be done working with
the ones first to prepare for more
formal written methods.
Step 4
24
+ 35
9
50
59
Children will be introduced to a
formal written method. No
regrouping is involved at this early
stage and the expanded method
should be taught first to ensure
children understand the place
value. It is important that the
correct language is used. They are
adding 2 tens to 3 tens and 4
ones to 5 ones.
24
+ 35
59
Step 5
298
+358
6
1
298
+3 5 8
56
1 1
298
+3 5 8
656
1 1
The Batt School Calculation Policy 2015
When children are confident using
the expanded method, this can be
shortened into the traditional
compact method using regrouping,
initially using two digit numbers.
7
Step 6
2786 + 2568 =
2 786 people visited the museum
last month. The numbers increased
by 2 568 this month. How many
people altogether visited this
month?
Children continue to use the
traditional compact method using
regrouping to solve problems with
larger numbers.
2786
+ 2568
5354
111
Step 7
24.566
39 . 7 0 0
0.560
Children will use the compact
method to add larger numbers and
decimals up to 3 places.
64.826
1 1 1
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Subtraction
Calculation Method
Explanation
Step 1
Take away
5 - 2 = 3
Initially, children will work with
pictures and objects, removing a
quantity and understanding there is
a smaller number left.
I had five balloons. Two burst. How Drawing a picture helps children to
many did I have left?
visualise the problem.
Using dots or tally marks is quicker
Mum baked 7 biscuits. I ate 3. How than drawing a detailed picture.
many were left?
7 - 3 = 4
Take away
Step 2
Children start counting
backwards on a marked
number line in order to
subtract.
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The empty number line helps to
record or explain the steps in
mental subtraction by counting
back from the highest number in
appropriate steps which enable the
children to bridge through a
multiple of ten.
Counting Back
15 – 7 = 8
74 – 27 = 47
The steps may be recorded in a
different order
Step 3
Lisa has 7 felt tip pens and Tim
has 3. How many more does Lisa
have?


Find the
difference
The Batt School Calculation Policy 2015
Once children have a secure
understanding of how to subtract
by counting backwards, the
concept of ‘finding the difference’
by counting on can be taught using
pictures or objects and then a
number line.
10
Step 4
834 - 378 =
The library owns 834 books. 378
are out on loan. How many are on the
shelves?
-8
456
-70
464
-300
534
834
John has £5.60. If he spends
£3.99 on a game, how much will
he have left?
Once children have a secure
understanding of how to subtract
by counting backwards and by
finding the difference, both are
practised using larger numbers.
Children should be taught how to
choose which method is more
efficient for a particular
calculation.
Children count up (from the
smallest number to the biggest)
using an empty number line. It is
easiest to count up to a multiple
of 10 or 100 (a friendly number).
Step 5
7 4 – 2 7 =
7 4 – 2 0 = 5 4
5 4 – 7 = 4 7
Children can progress to recording
their subtraction using partitioning.
For 74 – 27 this involves
partitioning the 27 into 20 and 7,
and then subtracting from 74, the
20 and the 7 in turn.
This can also be done subtracting
the ones first.
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Step 6
Children will be introduced to a
formal written method. No
regrouping (decomposition) from
other columns is introduced until
the children have a secure
understanding of place value
through regrouping in column
addition.
8 9
- 5 5
3 4
Step 7
Decomposition
6 1
5 7 2
- 2 4 5
3 2 7
Once the children are secure with
place value and regrouping in
addition, they can move onto
regrouping (decomposition).
_____
2 1
3
3
The children will use the same
method to work with; column and
decomposition but the value of the
digits should not exceed 4 digits.
1
3 5 4 7
- 2 6 2 8
0 9 1 9
Step 8
The children will then progress to
decomposition using numbers which
contain zero as a place holder.
Step 9
2
1
3
1
1 7 1
3 5 4 7.885
- 2 6 2 8.278
0 9 1 0.507
The Batt School Calculation Policy 2015
The children will use the same
method to work with larger numbers
up to 5 digits and decimals to three
decimal places.
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Division
Calculation Method
Explanation
Step 1
Sharing spots between two
parts of a ladybird
More pictures! Drawing often gives
children a way into solving the
problem.
6 ÷ 2 = 3
6 Easter eggs are shared between 2
children. How many eggs do they
get each?
Sharing
between 2
There are 6 Easter eggs. How many
children can have two each?
Grouping
in twos
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Step 2
12 ÷ 4 = 3
4 apples are packed in a basket.
How many baskets can you fill with
12 apples?
Grouping
in fours
Dots or tally marks can either be
shared out one at a time or split up
into groups.
During practical activities, use the
language of sharing and grouping.
6 ÷ 3 = 2
6 is shared into 3 groups.
15 ÷ 5 = 3
Step 3
15 ÷ 3 = 5
5
Number lines can be used to show
groupings and divisions.
5
5
The above method would progress
into a number line. When you
3
2
1
= 3 lots of/sets of 4 subtract ‘lots of 4’ or ‘sets of 4’
-4
-4
-4
from 12 until you reach 0. You then
0
4
8
12
count up how many lots of 4 you
have subtracted to get the answer
3.
12 ÷ 4= 3
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Children should also be aware that
not all numbers are exactly divisible
= 3 lots of/sets of 4 and that dividing some numbers will
with 3 left over
leave a remainder.
15 ÷ 4= 3 r 3
3
-4
3
2
-4
7
1
-4
11
15
Step 4
84 ÷ 6= 14
4 lots of6
0
It would take a long time for the
children to jump in sixes from 84 so
children can jump back in bigger
‘jumps’.
10 lots of 6
24
A jump of 10 groups of 6 takes you
to 24. Then you need another 4 lots
of 6 which is 24 will take you to 0.
Altogether this is 14 sixes.
84
142 ÷ 6= 23 r 4
3 lots of6 10 lots of 6 10 lots of6
4
22
82
142
Step 5
142 ÷ 6= 23 r 4
Children should then be able to
divide the number by grouping
without the use of a number line.
4
10 3
10
10 3
10
10 3
10
10 3
10
This should also be done with
numbers that will leave a remainder.
10 3
10
10 3
10
The Batt School Calculation Policy 2015
Initially, the child will recognise
that they can divide 60 into 6
groups by placing 10 into each
group. This is then repeated,
meaning there is now 20 in each
group leaving 22 to be divided by 6.
Using their knowledge of division
facts they should divide 18 by 6
placing an additional 3 in each group.
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Step 6
Chunking
65 ÷ 5 =
50 ÷ 5 = 10
15
15 ÷ 5 = 3
The children now use their
knowledge of the method above to
develop a more formal written
method without the use of
diagrams.
13
142 ÷ 6=
60 ÷ 6= 10
82
60 ÷ 6= 10
22
18 ÷ 6 = 3
4
23 r 4
142 ÷ 6=
120 ÷ 6= 20
22
18 ÷ 6 = 3
4
23 r 4
The Batt School Calculation Policy 2015
As children become more confident
with this method they will be able
to work with larger groupings.
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Step 7
Extended Short Division
(600 ÷ 3)
(60 ÷ 3)
Children with a secure knowledge of
the Chunking Method will be able to
progress to the more formal
Extended Short Division which uses
the same principle.
It is important that the children
are aware of the value of each digit
E.g. They know they are dividing
600 by 3 and not 6 by 3.
(12 ÷ 3)
Step 8
Short Division
73 ÷ 3 =
Divide 7 tens
by 3 which
will give an
answer of 2
tens and 1
ten left over.
Children now use their knowledge of
regrouping to progress to standard
Short Division.
Exhange the
1 ten for 10
ones
Divide the 13
ones by 3
which will
give an
answer of 4
with 1
remaining.
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Step 9
Translating remainders into
decimals
Long Division
033r3
15 4 6 8
45
018
15
03
The Batt School Calculation Policy 2015
Once children are confident using
the Short Division with regrouping
and remainders, they will progress
to converting remainders to
decimals.
Long division requires the children
to be competent and confident with
all of the above before they can use
it as a division strategy.
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Multiplication
Calculation Method
Explanation
Step 1
4 x 2 =
Each child has two feet. How many
feet do four children have?
2 + 2
A picture can also be useful for
early multiplication.
We say “four lots of 2 feet.”
“four groups of 2 feet.”
“four sets of 2 feet.”
Dots or tally marks are often drawn
in groups. This shows 3 groups of 5.
+ 2 + 2
3 x 5 =
There are 5 cakes in a pack. How
many cakes in 3 packs?
5
+
5
+
5
Step 2
3 x 4 =
A chew costs 4p. How much do 3
chews cost?
3 lots of 4
3 sets of 4
3 groups of 4
3 x 4
Drawing an array showing 3 columns
of 4, gives children an image of the
answer.
Children can now start to use a
marked number line to record 3 lots
of 4 or 4 lots of 3.
4 x 3
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Step 3
3 x 4 =
A chew costs 4p. How much do 3
chews cost?
Drawing an array (4 rows of 3 or 3
columns of 4) gives children an
image of the answer. It also helps
develop the understanding that
4x3 is equivalent to 3x4.
3 lots of 4
3 sets of 4
4 x 3 =
A chew costs 3p. How much do 4
chews cost?
4 lots of 3
4 sets of 3
Step 4
13 x 7 = 91
There are 13 biscuits in a packet.
How many biscuits in 7 packets?
+70
+21
10x7
0
When numbers get bigger, it is
inefficient to do lots of small jumps.
Partition 13 (10 and 3). This gives
you two jumps (10x7 and 3x7). The
answer is the number you land on 91.
3x7
70
Children will partition using dienes
apparatus as well as a number line.
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Step 5
Children use partitioning to multiply
mentally and when using the Grid
Method.
7 x 13 =
Partitioning.
7 x 10 = 70
7 x 3= 21
70 + 21 = 91
Grid multiplication
54 x 7 =
X 50
4
7 350
28
= 3 78
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Step 6
34 x 6
Children can progress to using use
short multiplication to multiply by a
single digit number. Some children
may require use of the extended
method whilst developing their
knowledge of regrouping.
4 x 6 = 24
30 x 6 = 180
204
1
Short Multiplication (extended)
100s 10s 1s
3
x
1
2
4
6
2 4
8 0
0 4
It is important that the children
understand and say the place value
of each digit as they work through
the calculation.
E.g The ‘3’ in the example would be
referred to as ‘3 tens’ rather than
‘3’.
Short Multiplication
100s 10s 1s
x
3
2
0
2
2
4
6
4
Short Multiplication
1000s
100s
3
10s
1s
2
5
X
Children continue to use short
multiplication to multiply larger
numbers by single digit numbers.
7
2
2
7
1
3
5
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Step 7
Children will return to using the
Grid Method to develop their
understanding of multiplying
numbers by more than one digit.
Step 8
Long Multiplication (extended)
Once secure with Step 7 children
will progress to Long Multiplication
with some children initially using the
extended method.
59 x 26
Th H T U
59
X
26
354
3 5
1180
1 1
1534
1
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