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Heswall Primary School
Written Calculation Policy
This policy reflects the values and philosophy of Heswall Primary School in relation to
the written calculation policy. It has been developed to ensure progression and
continuity in the recording of mathematics across all year groups. This will result in a
consistent approach throughout the school.
This document is intended for use by all teaching staff, and for information to governors,
inspectors and the Local Authority advisory service.
Each teacher will develop the mental strategies for their children appropriate to the year
group with modifications for the more able and less able. Less able children do not have
to progress through all levels.
Within the school there will be a development of recording mental strategies, which
allows the children to make their thinking visible. The work recorded in the children’s
books will only reflect a sample of the mathematical learning which they experience, as
some work is done practically and through investigative tasks. Children will record their
calculations on squared and plain paper, depending on the topic and year group.
Children in all year groups may also record on white boards.
In Foundation Stage Two the emphasis is placed on children working practically to
develop their emerging mathematical knowledge. They will begin to demonstrate their
thinking through their own drawings, pictures and marks.
Initially children’s learning will focus on counting skills and recognising the size of
numbers. E.g. a group of five teddies can be counted and labelled with the figure ‘5’ to
represent the total number of the group. In Foundation Two the children will be given
opportunities to recognise, write and compare numbers to ten.
Children will develop understanding of plus and minus signs by using appropriate
contexts, alongside the use of the equals sign. The equals sign will be used accurately:
representing ‘the same as’.
E.g. 3 + 4 =7 + 8 = 15 is not appropriate.
Children will have experience of a range of word sentences in a diversity of forms:
eg
3 + 4 + 5 = 12
7=4+3
3+ 4 = 5 + 2
In all calculations, ensure children:






Maintain, develop and refine their mental methods.
Understand and use an appropriate written method if necessary.
Recognise when and why methods are efficient.
Always begin by asking ‘Can I do this in my head?’
Record numbers using 1 digit per square with the exception of fractions where
the denominator is to be written in one square.
The decimal point is to written on the line between squares.
Policies: Curriculum – 3/10
ADDITION
Children will learn to use a full range of mathematical vocabulary as they progress
through the school.
Children will use a variety of addition methods, beginning with a pictorial representation
of what they have done practically, combining two groups of objects into one group and
saying how many altogether by counting all the objects.
Children will develop their ability to add two sets together by counting on from a given
number, usually the largest number. The children will develop their addition strategies
using a number line, initially practically, then using a marked number line, progressing
to an empty number line.
Empty number lines should not be introduced until children have a lot of experience
using numbered number lines and 100 squares to aid calculation.
Addition recorded on the empty number lines supports and improves pupils’ mental
strategies but does not progress into a standard written form.
Numbered number line example (Y1)
8 + 4 = 12
0
1
2
3
4
5
6
7
8
9
10
11
12
The empty number line helps the children to demonstrate their mental calculations.
Empty number line example (Y2/Y3)
15 + 7 =22
+5
15
+2
20
22
Children will then move on to ‘ partitioning ’ or splitting numbers up into tens and units,
then hundreds, tens and units and ‘recombining’ them to find the total. These
calculations can be set out horizontally.
Policies: Curriculum – 3/10
2
24 + 23 = 47
+10
24
+10
34
+3
44
+20
47
+3
24
44
47
Or partitioning, eg
24
+ 23
40
24
20
4
Tens
Units
Total
+ 7 = 47
+
+
+
23
20
=
3 =
40
7
47
Y3 example
238 + 123 = 361
+10
+100
238
338
+100
238
348
+20
338
+3
+10
358 361
+3
358
361
Or vertically, eg
Hundreds
Tens
Units
Total
238
200
30
8
+ 147 =
+ 100
= 300
+ 40
= 70
+ 7
= 15
385
It is important to keep the digits in the correct place value.
Policies: Curriculum – 3/10
3
Expanded Method
The next stage is an ‘expanded’ method of calculation. This method will only be taught
when the children can:
-
add and subtract two single digit numbers mentally
add and subtract multiples of ten mentally
partition numbers into tens and units mentally
add and subtract two, two digit numbers mentally and record the method
informally.
The following expanded form of algorithm will be used for addition. This serves as an
introductory stage to more condensed formats. Children can use brackets to show the
expanded format for each number.
Y4 example
H
2
2
+
T
3
2
5
0
5
4
4
U
2
5
7
0
0
7
(
(
2
(
3
0
2
0
0
+
+
+
5
2 0
0 0
2
)
)
)
Progressing to:
+
H
2
2
T
3
2
4
4
5
0
5
U
2
5
7
0
0
7
Children should only be working on numbers this large when they are able to add
multiples of 10 and 100 with confidence.
Standard Compact Method
The standard compact method will only be introduced to children who are already
confident using the expanded method. It is not expected that all children will be using
this method and they may choose a method of calculation in which they are confident
and accurate. Some children will continue to calculate using a number line throughout
the school.
+
H
2
1
3
T
3
4
8
U
8
7
5
1
Policies: Curriculum – 3/10
Th
7
8
H
6
3
0
T
5
4
0
1
1
1
+
U
8
6
4
4
Explicit language must be used in standard compact method.
+
H
2
3
6
T
4
9
4
1
1
U
7
5
2
7 plus 5 equals 12, 12 is one ten and two units.
Put the two in the units column and carry the ten in the tens column [underneath].
Forty plus ninety equals 130, plus the extra ten equals 140.
Put the 40 in the tens column and carry the 100 in the hundreds column [underneath].
200 plus 300 plus the extra 100 equals 600.
Put the 600 in the hundreds column.
The total equals 642.
Policies: Curriculum – 3/10
5
SUBTRACTION
Children will begin with a pictorial representation of what they have done practically,
taking away a number of objects.
8–2=6
Pop!
Pop!
Empty Number Lines [from Year 2]
Empty number lines should not be introduced until children have had a lot of experience
using numbered number lines to aid calculation.
The empty number line will be used to find the ‘difference’ between two numbers. The
difference can be found by counting on.
Count On Method
The term ‘finding the difference’ is introduced in Year 2:
Eg find the difference between 7 and 12
Finding the difference between 7 and 12 is 5
+3
7
+2
10
12
Counting on can be done in several ways.
Y3 example: counting on in tens or multiples of ten.
Policies: Curriculum – 3/10
6
Children will add numbers from the top of the number lines in their heads or with simple
jottings.
81 - 37 = 44
+10
37
+10
47
57
+10
67
+40
+3
37
+10
+3
77
+1
80 81
+1
40
80
81
Y4 example – Pupils continue to record on the number line by ‘counting on’. The
calculations should be extended to bigger numbers.
146 - 78 = 68
+20
+2
78
80
+40
100
+6
140 146
Children can record their jottings in this way:
40
6
+
+
20
2
Policies: Curriculum – 3/10
=
=
60
8
68
7
314 - 186 = 128
+10
+4
186 190
100
10
4
+
+
10
4
+100
0
200
=
=
+10
300
+4
310
314
100
20
8
128
NB Some children may not progress beyond this stage – this is ok!
Standard Compact Method
Explicit language must be used in Standard Compact Method.
-
H
T
7
12
8
5
2
U
3
8
4
1
4
6
8
4 minus 6 you can’t do without going into negative numbers, so I will rearrange the
numbers by borrowing a ten from 30 to make it 20.
14 minus 6 is 8. Put the 8 in the units column of the answer.
20 minus 80 you can’t do without going into negative numbers, so I will rearrange the
numbers again by borrowing a hundred from 800 to make it 700.
120 minus 80 is 40. Put the 40 in the tens column of the answer.
700 minus 500 is 200. Put the 200 in the hundreds column of the answer.
The answer is 248.
Check that I still have 834 as the first number –
700 + 120 + 14 is 834.
There is no need to teach decomposition at any stage. Many pupils will continue
to calculate using a number line to the end of Y6.
Policies: Curriculum – 3/10
8
MULTIPLICATION
The multiplication and division signs will be introduced in Year 2.
Children will use a pictorial representation of what they have done practically, drawing
pictures to record a number of sets with a number of objects in:
eg 3 sets of 3
Children will then start to use repeated addition and some multiplication facts, to solve
multiplication problems:
3
+
3
+
3
+
3
=
12
(4 x 3 = 12)
Number Lines
Number lines will be used to demonstrate repeated addition
+2
0
+2
2
+2
4
6
Arrays
Arrays will be used to link mental strategies to written calculation and to demonstrate
the distributive and commutative law
Y2 Example
Arrays
3x4=
4x3=
4
0000
3 0000
0000
Y3 Example
3 x 4 = 4 x 3 (commutative law)
Distributive law ie partitioning to aid mental calculation
(3 x 4) + (4 x 4) = 7 x 4
16 x 3 = (10 x 3) + (6 x 3)
Policies: Curriculum – 3/10
9
Grid Method
When children are secure in their understanding of arrays they will be gradually
introduced to the grid method.
The grid method involves partitioning numbers to multiply and encourages the children
to use known facts. It also provides a visual image of multiplication stemming from
arrays.
Y4 Example
13 x 7 = 91
x
7
1
7
0
0
3
1
2
 70 + 21 = 91
Year 5/6 Examples
137 x 3 = 411
x
3
1
3
0
0
0
0
3
9
0
0
2
7
1
 300 + 90 +21 = 411
The grid method can then be expanded to use TU x TU and progress to HTU x TU:
When multiplying by more than units, eg TU or HTU then add the answers vertically
rather than horizontally in order to maintain place values.
17 x 19 – 323
x
10
9
1
1
1
0
9
9
0
0
0
0
1
7
6
3
7
0
3
3
0
0
0
0
6
1
7
+

1 9
1 3
1 2
2 0
3 2
0
3
3
0
0
3
or
1
+ 1
3
9
3
2
0
3
3
1
724 x 35 = 25340
x
30 2
5
2
1
3
4
7
0
5
5
0
0
0
0
2
0
0
0
0
0
0
0
1
1
2
2
4
4
0
0
0
+

2
4
2
5
5
7
1
3
0
0
4
4
0
0
0
0
1
The grid method can also be used to multiply decimals in the same way:
34.5 x 6 = 207
x
6
30
180
4 0.5
24 3
Policies: Curriculum – 3/10
 180 + 24 + 3 = 207
10
At each stage the children can use the grid method to multiply 2, then 3, then 4 digit
numbers.
NB Not all children need to learn to multiply using traditional algorithms.
Expanded Method
Th
H
2
T
6
4
6
1
4
8
0
2
x
1
2
U
5
8
0
0
0
0
(
(
2
(
6
0
5
0
0
x
x
x
8
8
8
)
)
)
Progressing to
Th
H
2
T
6
4
6
1
4
8
0
2
x
1
2
U
5
8
0
0
0
0
As before, the grid method can help with understanding and can eventually lead to the
standard compact method of multiplication:
Th
x
2
H
2
T
6
1
2
5
4
U
5
8
0
Explicit language must be used in standard compact method.







8 times 5 equal 40, 4 tens and no units;
Put the zero in the units column and carry the four in the tens column (underneath);
8 times 60 equals 480, plus the extra 40 equals 520;
Put the 2 in the tens column and carry the 5 in the hundreds column (underneath);
8 times 200 equals 1600, plus the extra 500 equals 2100;
Put the one in the hundreds column and the 2 in the thousands column;
The total equals 2120.
It is important that the traditional standard compact method is seen as the
ultimate objective. Children will need to use an expanded form for such
calculations before steps in the method are meaningful and they can use this
efficient method with understanding. Children’s progression and readiness will
obviously vary considerably. Many children will leave school using the grid
method and this is valued by Maths teachers in Secondary schools.
Policies: Curriculum – 3/10
11
DIVISION
Children will initially meet division as equal sharing or grouping. Children will begin with
pictorial representation of what they have done practically and become familiar with the
language of sharing and grouping:
Sharing
I have 6 sweets to share between 3 children. How many will they each get? 2
Grouping
9 flowers are bundled in bunches of 3.
How many bunches can be made? 3
3
3
3
Arrays can be used to show the relationship between multiplication and division, eg
How many 3’s in 12?
000
000
000
000
How many 4’s in 12
0000
0000
0000
This can also be demonstrated on a number line by jumping forward in groups.
3
0
3
3
Policies: Curriculum – 3/10
3
3
6
9
12
There are 4 groups of 3 in 12
3+3+3+3=12
12
4
4
0
4
There are 3 groups of 4 in 12
4+4+4=12
4
8
12
Year3/ 4
Continue to use number lines (only to work forwards),
eg 36 ÷ 4 = 9
There are nine 4’s in 36
4
4
0
4
4
4
8
4
12
16
4
20
4
24
4
28
4
32
36
Extend to using number lines to show remainders,
Eg 17 ÷ 5 = 3 r2
There are three 5's in 17 and 2 left over
5
0
5
5
5
10
1
15
1
16
17
Years 4/5
As the children gain in confidence they will be able to make bigger jumps along the
number line using multiples,
Eg 74 ÷ 5 = 14 r4
10 x5
0
4 x5
50
1
1
1
1
70 71 72 73 74
= 14 r 4
Number lines can continue to be used to calculate without introducing the standard
compact method.
Policies: Curriculum – 3/10
13
STANDARD COMPACT METHOD
The traditional division algorithm presents a very contracted format for recording.
Eg 488 ÷ 4 = 122
4
1
4
2
8
2
8
Progressing to
168 ÷ 4 = 42
4
0
1
4
16
2
8
Progressing to
263÷ 4 = 65 r3
4
0
2
6 5
26 23
r3
Children should be reminded of the term quotient and the written abbreviation r for
remainder
Eg 263 divided by 4 equals a quotient of 65 with a remainder of 3
NB. Again, not all children will learn to divide using the compact method and may
continue to use the number line to the end of Y6.
NB. It is more important that children can find answers to calculations by any of
the methods outlined, than it is to learn the traditional methods or algorithms.
Policies: Curriculum – 3/10
14