Calculation Policy
Overview of Calculation Methods and Strategies used
at Hove Junior School
Overview of Calculation Methods and Strategies used at Hove Junior
School: Guide for Parents/Carers 2014/15
This policy contains the key methods of calculation that are taught at Hove Junior School. It
has been written to ensure consistency and progression throughout the school.
The methods we are advocating are in line with the new National Curriculum. We hope this
will be helpful to you and that you will be able to support your child if they need help.
The methods that we use in school may or may not be familiar to you. Children can often
become confused when they ask parents for help at home and they try to teach the methods
that they themselves were taught. Knowing how the methods in this booklet work will help you
to help your children. All staff in school work from this document so that we can ensure the
consistency of our approach and can make sure that the children move onto the next step
when they are ready.
The National Curriculum for mathematics aims to ensure that young people become fluent in
the fundamentals of mathematics, reason mathematically and are able to problem-solve by
applying mathematical thinking. By the time they leave junior school they should:

have a secure knowledge of number facts and a good understanding of the four
operations.

estimate / mental methods.

make effective use of diagrams and informal notes to help record steps/part answers
when using mental methods.

have an efficient, reliable, formal, written method of calculation for each operation that
they can apply with confidence when undertaking calculations that they cannot carry out
mentally.
How calculation is taught is key. At Hove Junior School our children are taught a variety of
methods, both mental and written, so that they develop the skills required to select an efficient
method to carry out a calculation with confidence.
The four operations that are covered by this booklet are addition, subtraction, multiplication
and division.
Whichever operation is being taught the child needs to experience all of the following ‘steps’ to
completely master it. When they first start school, the first step children will begin by using
objects/manipulatives before any recording occurs.
They are then introduced to resources such as number lines. Number lines provide a mental
strategy for addition and subtraction and enable the child to clearly see the calculation that
they are working on.
2
As they progress, they will begin using more formal written methods (in an ‘expanded’ form)
before using a compact written method. Concrete apparatus are available for the child to
access at any point throughout a lesson. Children routinely use whiteboards and have
resources (such as number lines) at hand, should they wish to use them.
We hope this booklet demonstrates how we move towards using formal, compact and written
methods.
All terminology is outlined in our Glossary section and prefaces the step-by-step pictorial and
written description of methods and strategies.
In order to see the progression of methodology, we have also included descriptions/images of
strategies your child may have covered in infants.
(photo:
3
using ‘grid method’ to multiply)
Glossary of Key Vocabulary
We understand that terms and vocabulary may have changed since you were at school
yourselves. Here is a brief definition of some of the words that are used in calculation that your
child will be taught and use at school. It is not an exhaustive list of mathematical words. Links
can be found at the end of this document, to websites and videos that you may find informative
and useful.
Addition
Addition is finding the total, or sum, by combining two or more numbers.
Example: 5 + 11 + 3 = 19 is an addition.
Algebra The part of mathematics that deals with generalised arithmetic. Letters are used to
denote unknown numbers and to state general properties. (The explicit mention of algebra in
the primary NC happens in the Year 6 programme of study).
Arrays
Arrays are objects or shapes arranged in a rectangle. In the array, the answer is always the
same (e.g. 2 × 7 is the same as 7 × 2) even when you rotate.
Bridge to ten
A strategy when using numberlines. Adding a number that takes you to the next ‘tens’ number.
Calculation
Is the process of adding, subtracting, multiplying or dividing to get an answer e.g. 45-13=32.
Chunking
Using facts you already know, you can reduce the number of steps used to solve a calculation
e.g. instead of counting on in groups of 5 to reach 60, if you know 10x5 you can do one big
jump to 50. Then count on 2 more groups of 5 to reach 60. There have been 12 jumps in
groups of 5 to reach 60.
Column chunking
Method of division involving taking chunks or groups or the divisor away from the larger
number.
Compact Method
The shortest route to solve the calculation.
Concrete apparatus
If your child’s teacher refers to ‘concrete apparatus’ they mean any objects to help children
count – these are most often counters and cubes (multilink) but can be anything they can hold
and move. Dienes (thousands, hundreds, tens and units blocks), Numicon and Cuisenaire rods
are also examples of ‘concrete apparatus’.
Decimal numbers
A number that uses a decimal point followed by digits as a way of showing values less than
one. A decimal point separates the whole number from the part of the number which is less
than one.
Decomposition
Expanding the numbers and separating them into their components. Used in column addition
when ‘carrying’ and column subtraction; expanding to exchange.
4
Digit
A symbol used to make numerals.
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numbers.
Division
Division is splitting into equal parts or groups. It is the result of "fair sharing".
e.g. There are 12 chocolates and 4 friends want to share them. If you share the out, each will
get 3 chocolates.
Division can also be thought of as how many groups of a number are in the total number.
e.g. Eggs can be put in boxes of 6. There are 54 eggs altogether. How many boxes will be
needed?
Divisor
The number by which another is divided. Example: In the calculation 30 ‚ 6 = 5, the divisor is 6.
In this example, 30 is the dividend and 5 is the quotient.
Difference (find the difference)
The result of subtracting one number from another. How much one number differs from another.
Example: The difference between 8 and 3 is 5.
Estimate
To arrive at a rough or approximate answer by calculating with suitable approximations for terms
or, in measurement, by using previous experience. A rough or approximate answer.
Exchanging
Moving a ‘ten’ or a ‘hundred’ from its column into the next column and splitting it up into ten
‘ones’ (or ‘units’) or ten ‘tens’ and putting it into a different column.
Expanded Method
Showing each stage and then combining to solve the calculation. E.g. Expanded Multiplication
– a method for multiplication where each stage is written down and then added up at the end in a
column.
Formal written methods
Setting out working in columnar form. In multiplication, the formal methods are called short or
long multiplication depending on the size of the numbers involved. Similarly, in division the
formal processes are called short or long division.
Geometry
The aspect of mathematics concerned with the properties of space and figures or shapes in
space.
Grid Method
The grid method is a way of breaking up numbers into separate units to make multiplication
easier. We use an empty number square.
To calculate 35 × 7, the grid looks like this:
X
30
5
7
X
30
5
7
210
35
210 + 35 = 245
5
Groups
If children are counting in groups, they are counting up or down in the particular number,
instead of counting in 1s e.g. counting up in groups of 4 to see how many 4s are in 20.
Integer
Any of the positive or negative whole numbers and zero.
Inverse
The opposite operation. Addition is the inverse of subtraction. Multiplication is the inverse of
division.
Multiple
The result of multiplying a number by an integer (not a fraction).
Examples:
12 is a multiple of 3, because 4 × 3 = 12
30 is a multiple of 5, because 6 × 5 = 30
But 17 is NOT a multiple of 3
Multiplication
The basic idea of multiplication is repeated addition.
For example: 5 × 3 = 5 + 5 + 5 = 15
But as well as multiplying by whole numbers, you can also multiply by fractions or decimals.
For example 5 × 3½ = 5 + 5 + 5 + (half of 5) = 17.5
Numberline
A line with numbers placed in their correct position.
Useful for addition and subtraction, and for showing relations between numbers.
Numbers on the left are smaller than numbers on the right.
Number sentence
Writing out a calculation with just the numbers in a line e.g. 2+4=6 or 35 ‚7 = 5 or 12 x 3 =36 or
32 – 5 = 27.
Partition
Separating a number into its place value parts e.g. 157= 100 and 50 and 7.
Place Value
The value of where the digit is in the number, such as units, tens, hundreds, etc.
Example: In 352, the place value of the 5 is ‘tens’
Example: In 17.59, the place value of the place value of 1 is ‘tens’, the 9 is ‘hundredths’
Product
The answer when two or more numbers are multiplied together.
Regroup
For addition, once the numbers have been partitioned into hundreds, tens and units then add
hundreds together, then add the tens to that total, then add the units to that total.
6
Remainder
The amount left over after division.
Example: 19 cannot be divided exactly by 5.
The closest you can get without going over is 3 x 5 = 15, which is 4 less than 19.
So the answer of 19 ‚ 5 is 3 with a remainder of 4.
Rounding
In the context of a number, express to a required degree of accuracy. Example: 543 rounded to
the nearest 10 is 540.
Scaling
To enlarge or reduce a number, quantity or measurement by a given amount.
Statistics The collection, analysis, interpretation, presentation, and organisation of data
(previously ‘Data Handling’)
Subtraction
Taking one number away from another.
If you have 5 apples and you subtract 2, you will be left with 3.
The symbol of subtraction is –
Sum
The result of adding two or more numbers.
Example: 9 is the sum of 2, 4 and 3 (because 2 + 4 + 3 = 9).
Times Table facts
e.g. 3x4=12, 10x7=70
Children need to be able to recall multiplication and division facts up to 12 x 12.
7
ADDITION
Working towards a formal written method
By using visual images and objects, your child begins to make sense of combining two groups to
make a larger group, to add amounts. They develop mental reasoning and learn mathematical
vocabulary as they process what they are doing. Using resources is the best way for a child to
understand the concept of number. Manipulatives/objects can be used to support mental imagery
and conceptual understanding. Following on from this, number lines are used as an important
image to support thinking, and the use of informal jottings is encouraged. Examples using
resources can be seen below.
Using Numicon
to ‘bond’ to 10.
9 + 1 = 10
1 + 9 = 10
4 + 6 = 10
6 + 4 = 10
Using Numicon to see
that addition can be
done in any order.
Dienes or Base 10 are useful in
showing what a ‘one’ (unit), a
ten, a hundred and a thousand
look like and how they can be
added together and split up to
form smaller and larger numbers.
8
Cuisenaire Rods.
Although these rods that represent numbers from 1 to 10 can be
used as an addition tool, they can also be used with calculations
using other operations.
Using a variety of resources when dividing. Whiteboard,
number line, counters, 100 Square, marbles on plates,
beads and straws.
Using 'real' coins to make amounts in different ways.
Using bundles of straws to see the relationship
between 4 + 6 = 10 and 40 + 60 = 100.
Using a 100 bead string to see that
40 + 60 = 100
Further examples of resources:
Moveable counting objects, coat hangers and pegs, beads, number tracks and lines, number
fans, 100 square, Multilink, place value counters and place value cards (to help children move
towards regrouping units for tens) arrow cards, straws/objects grouped in tens, etc. resources
such as number lines and 100 squares are the next step.
Example vocabulary associated with addition.
Add, addition, more, plus, make, sum, total, altogether, score, regrouping
How many more to make?
How many more is … than…?
9
Moving on to a written method.
Below are the key methods your child will have used/be using in school. The updating of the
National Curriculum (2014) has meant that there are some changes to the objectives for each
year group. The methods we teach are outlined below. Your child may now be introduced to a
method of calculating earlier than they would have been previously.
Below is an example of using a numberline. Your child may be familiar with this method as it is
used in Key Stage 1. There are a variety of types of number lines. They are very useful and are
used in all year groups.
1 2
1
3
2
4 5
3
4 5
6 7
6 7
8 9
10 11 12 13 14 15
8 9 10 11 12 13 14 15
4+5=9
Using a number line.
55 + 37=
Estimate
60 + 40 = 100
+30
55
+7
85
92
Using a blank number line.
To begin with your child can count along on a number line to add two numbers together. Make
sure your child starts their number line with the largest number and then add the small number.
Partition the number into tens and units to help with adding.
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Partitioning
The next step is to partition to add two numbers together. Make sure your child partitions the
numbers into their correct place value, e.g.
47 = 40 + 7
246 = 200 + 40+ 6
47+56
T
U
40 + 7
+ 50 + 6
246+175
90 +13=103
H
200
+100
T
40
70
U
6
5
300 + 110 +11=421
Expanded Addition
The next step is the expanded method for addition. You child completes complete these
calculations in the same way as the previous method. They start with the units and then add the
tens.
Remember to start with the units
Remember to line up the columns
T U
H T U
5 6
2 4 6
+ 4 7
+ 1 7 5
1 3 (6 + 7 = 13)
1 1
(6+5=11)
9 0 (50 + 40 = 90)
1 1 0
(40+70=110)
3 0 0
(200+100=300)
4 2 1 (300+110+11=421)
Now we have the standard ‘compact method’ for addition. Make sure when you work through
your calculation that you carry the extra 10 or 100 underneath, in the appropriate column.
56 + 47
56
+ 47
103
1
246+175
246
+ 175
421
11
When adding decimal numbers make sure your child thinks about the place value of each digit
and lines up the appropriate columns.
Remember to start with the numbers on the right (tenths or hundredths)
3.54
+ 2.17
5.71
1
Remember to line up the decimal points!
11
54.70
+ 2.89
57.59
1
You can put
a zero here
to avoid
confusion.
SUBTRACTION
1. To begin with, a number line is used to count on or back to solve subtraction calculations.
The smallest number is on the left and the largest is on the right. Add the jumps together to
make the final answer, like getting change in a shop. This is called finding the difference.
Count on if the numbers are close together.
60 - 47 = 13
Estimate
60 - 50 =
10
+3
+10
47
50
60
Count back if the numbers are far apart.
60 - 13 = 47
47
50
60
-10
-3
Count on if the numbers are close together.
203 - 186 = 17
Estimate
200 - 180
= 20
+4
186
+3
+10
190
200
203
10+4+3=17
Count back if the numbers are far apart.
203 - 17 = 186
190
186
-4
200
-10
12
203
-3
2. The next step is to partition the numbers and write them vertically.
679-135=
Estimate
700 - 100
= 600
H
T
U
600
70
9
-100
30
5
500
+
First we teach children to begin with
numbers where no exchanges are
needed.
40 + 4
= 544
673 -135=
600
60
-100
500 +
Then once they are confident
they move onto exchanging one
ten for ten units.
1
70
3
30
5
30 +
8
= 538
673-185=
500
600
- 100
400 +
160
70
Then the next step is to exchange from
several columns.
1
3
80
5
80 +
8
= 488
13
3. This is the standard method for subtraction. As with the previous method, your child begins
with numbers where no exchanges are needed.
Estimate
700 - 100
= 600
Step 1
679
6 67 13
Step 2
Step 3
5
6 167 13
-135
-1 3 5
- 1 8 5
544
5 3 8
4 8 8
4. When subtracting decimal numbers make sure your child thinks about the place value of each
digit and line them up in the appropriate columns. Estimating is extremely important.
3.54 - 2.17
Estimate
4-2=2
3 . 45 14
54.7 - 2.89
A zero is helpful here
to avoid confusion.
Estimate
55 - 3 =
52
5 34 . 16710
-
- 2. 1 7
1. 3 7
14
2.
8 9
5 1.
8 1
MULTIPLICATION
Multiplication is repeated addition.
9 columns of 5 dots
1.
9 x 5 = 45
First use arrays to consolidate
understanding of the concept
of multiplication.
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











































5 x 9 = 45
5 rows of 9 dots
15
2. Then a number line is used to solve multiplication calculations by counting on until the correct
amount of multiples, e.g. 14 groups of 5 is reached. The answer is on the number line.
14 x 5 = 70
1
0
2
5
3
10
4
15
5
20
6
25
7
30
8
35
40
10
45
11
50
12
55
13
60
4 x 5=20
10 x 5=50
0
9
50
70
Using table facts to make bigger jumps is more efficient.
16
14
65
70
The next step is using the grid method.
3. You need to partition both numbers and use them to multiply each other. Then add the totals in
each box.
23 x 5 = 115
Estimate
20 x 5 =
100
23 x 52 = 1196
x
20
3
5
100
15
Estimate
20 x 50 =
1000
x
50
2
20
3
1000
150
40
6
5 x 20 = 100
50 x 20 = 1000
5 x 3 = 15
50 x 3 = 150
100 + 15 = 115
2 x 20 = 40
23 x 5 = 115
2x3=6
1000 + 150 +40 + 6 = 1196
17
The next step is using grid multiplication to introduce long multiplication: 18 x 13 =
Estimation: 20 x 10 = 200
We introduce long multiplication for multiplying by 2 digits:
Multiplication made easy [ish!] …
https://www.youtube.com/watch?v=t_bnlB2KRL4
18
4. The next step is the compact method for short multiplication. Your child needs to multiply the
single digit number by the units and multiply the single digit number by the tens (8 x 3 = 24, 20
x 8 = 160).
H T U
2 3
x
8
1 8 4
The tens are carried over and put into the tens column.
2
H T U
4 6
x
3 2
9 2
Here a two-digit number is multiplied by another two-digit number.
The method is the same as the short method but calculated in steps.
Start with 2 x 6 = 12 and carry the ten
Then it is 40 x 2 = 80 (+ 10)
1
138 0
1
147 2
A zero is placed in the units column as your child is now multiplying
the numbers by a multiple of ten, not a unit.
Finally add the two answers together.
1
30 x 6 = 180 (carry the hundred)
30 x 40 = 1200 (plus the hundred)
19
Multiplying with decimals
Estimate
700 - 100
= 600
3∙5 x 6∙7
First remove the decimal point (3∙5
35). Put
numbers into the grid and use grid method to
solve, as before.
x
30
5
6 0
1800
300
7
210
35
Then add all the answers together. Now look back
at the original question. In total, how many digits
are to the right of the decimal point?
60 x 30 = 1 8 0 0
60 x 5 =
300
7 x 30 =
210
7x5 =
35
+
2345
1
3∙5 x 6∙7 in this case there are 2 digits be the right of the decimal point. So there needs to be
2 digits to the right of the decimal in the answer.
2345 becomes 23∙45
Don’t forget to put the decimal point back. This is where estimating really helps.
20
DIVISION
To begin with your child will use on a number line to solve division calculations. You must start
at zero and count on in groups of the divisor e.g. 5, until you reach the target number, e.g. 35.
Take care as there could be a remainder! The answer is the number of jumps made.
Remember to estimate
Division using a Number line
35 ‚ 5 = 7
1
0
2
3
5
10
4
15
5
20
6
25
7
30
35
Here we are using chunking to solve division calculations using a number line starting at zero.
We count up in groups to see ‘how many lots’ go into the bigger number. We use our times
table knowledge to help us.
a)
72 ‚ 6 = 12
2x6=12
10 x 6 = 60
0
b)
60
72
256 ‚ 7 = 36 r4
10 x 7 = 70
0
10 x 7 = 70
70
c)
256 ‚ 7 = 36
10 x 7 = 70
140
210
252 256
r4
30 x 7 = 210
0
r 4
6 x 7 = 42
6 x 7 = 42
210
r4
252
256
For this calculation we have used the same method, but because we have used larger
numbers, we have made larger jumps along the number line, making larger chunks in our
Calculation.
21
Division using ‘Chunking’
Chunking is a method used for dividing larger numbers that cannot easily be divided
mentally.
Chunking is repeated subtraction of the divisor and multiples of the divisor. In other
words, working out how many groups of a number fit into another number.
The purpose of chunking is for children to be able to think about the relationship between
multiplication and division. It involves using rough estimates of how many times a number will
go into another number and then adjusting until the right answer is found. Once these skills
have been practised, teachers will encourage children to move onto the quicker 'bus stop'
division method and continue onto long division, when appropriate.
Each ‘chunk’ is an easy multiple (for example, 100x, 10x, 5x, 2x, etc) of the divisor, until the
large number has been reduced to zero or the remainder is less than the divisor.
73 ‚ 5 =
With larger divisors, it may be useful for the child to jot down the multiples of that number at the
side, for reference. Also, in the example below, the child has represented the remainder as a
fraction by using it as the numerator, over the divisor (denominator) before simplifying the
fraction.
386 ‚ 24
22
Short Division
The new National Curriculum states that children must be able to divide numbers up to 4 digits
by a one-digit number using the formal written method of short division and interpret
remainders appropriately for the context (as a remainder, fraction or decimal)
The next step is short division. This is sometimes known as the bus stop method.
471 ‚ 3 = 157
It is easier to think, ‘how many 3s are in 4?’ But you must
bear in mind this is actually 400.
3 4 71
First work out the largest number of hundreds that will divide exactly by 3. Carry the remainder
to the right.
1
3 4 17 1
Secondly work out the largest number of tens that will divide exactly by 3. Carry the remainder to
the right.
1 5
3 4 17 21
Lastly, work out the largest number of units that will divide exactly by 3.
1 5
7
3 4 17 21
21 divided by 3 = 7
Take care as there might be a remainder!
23
Long Division
Long Division [the tricky method!]
Following on from this, when appropriate, your child learns long division.
Long division breaks down a division problem into a series of easier steps. As in all division
problems, one number, called the dividend, is divided by another, called the divisor, producing
a result called the quotient. As stated, abbreviated form of long division is called short division,
which is almost always used instead of long division when the divisor has only one digit. Long
division is tricky as it relies upon knowledge of:

multiplication tables (including related multiplication tables
e.g. 3 x 12 = 36, 30 x 12 = 360, 30 x 120 = 3600, and so on)

the basic division concept, based on multiplication tables (for example 28 ‚ 7 or 56 ‚ 8)

basic division with remainders (for example 54 ‚ 7 or 23 ‚ 5)
Long division is an algorithm that repeats the basic steps of:
1) Divide 2) Multiply 3) Subtract 4) Drop down the next digit.
Of these steps, number 2 and 3 can become difficult and confusing to children because they
don't seemingly have anything to do with division—they have to do with finding the remainder. It
is worth bearing in mind that children will not be taught this method until they are ready i.e. until
they have the ability to apply what they know, relating to the three facts above.
Below is an expanded example of how to do long division. There are links to YouTube videos
and websites with further practical demonstrations, should you wish to view them. This
explanation is rather long-winded because, in order to fully understand how long division
‘works’, one must look at (and understand) each step. Firstly, an example is given where there
is a straightforward answer with no remainders. Children will not be taught this until they
are ready and can apply what they know.
24
Division example:
Key Information:

The first digit of the dividend is divided by the divisor

The whole number result is placed at the top

The answer from the first operation is multiplied by the divisor. The result is placed under
the number divided into.

The bottom number is subtracted from the top number.

Bring down the next digit of the dividend.

Divide this number by the divisor.

The whole number result is placed at the top. Any remainders are ignored at this point.

The answer from the above operation is multiplied by the divisor. The result is placed
under the number divided into.

Now subtract the bottom number from the top number.
Sometimes division problems will not come out evenly, and they will have another number (not
0) at the end. This leftover number is called a remainder and it is written as part of the quotient.
See the example above.
The red circled number at the bottom of the example is the remainder. This is recorded on top
of the division bar with a r in front (25r3) like in short division.
25
Here’s a new example:
Our answer to this problem is 23 r 1; note that we always write the remainder after the quotient,
on top of the division bar. Also notice that our remainder (1) is smaller than our divisor (6).
Long Division with Remainders as Fractions
The next step is writing remainders as fractions. Instead of writing r and then the number take
the remainder and make it the numerator (top number) of a fraction. The denominator (bottom
number) is the divisor.
Let’s look at the following example:
Notice that our r is not used at all in front of the remainder when turning it into a fraction. The
fraction is recorded instead.
26
Long Division with Remainders as Decimals
Another way your child may be asked to express a remainder is in the form of a decimal. When
they are asked to express the remainder as a decimal, first complete the division as usual, until
reaching the remainder stage. Follow along with this example:
The division calculation will keep going. Add a decimal point on
the top bar. After the decimal in the divided, add a zero and
continue to divide. Keep adding zeros until the subtraction step
results in an answer of zero as well.
Notice that we added a decimal after the 6 in the dividend, as well as a decimal after the 5 in
our quotient. Then, we started adding zeroes to the dividend. This time, it only took us one
added zero before our remainder was zero.
Now, let’s look at a problem where more than one zero is added to the dividend:
USEFUL WEBSITES:
http://www.mathsisfun.com/long_division.html
http://www.wyzant.com/resources/lessons/math/elementary_math/long_division/
long_division_with_remainders
https://www.gov.uk/government/publications/national-curriculum-in-england-mathematicsprogrammes-of-study
Models and Images documents – Primary National Strategy www.ncetm.org.uk – National
Centre for Excellence in Mathematics.
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