Download Progression towards a standard written method of calculation

Healey J. I. & N. School
RENEWED POLICY FOR
WRITTEN CALCULATION
(September 2009)
To be reviewed: September 2012
1
Progression towards a standard written method of calculation
INTRODUCTION
The National Numeracy Strategy provides a structured and systematic approach to
teaching number. There is a considerable emphasis on teaching mental calculation
strategies. Up to the age of 9 (Year 4) informal written recording should take place
regularly and is an important part of learning and understanding. More formal
written methods should follow only when the child is able to use a wide range of
mental calculation strategies.
REASONS FOR USING WRITTEN METHODS






To aid mental calculation by writing down some of the numbers and answers
involved
To make clear a mental procedure for the pupil
To help communicate methods and solutions
To provide a record of work to be done
To aid calculation when the problem is too difficult to be done mentally
To develop and refine a set of rules for calculation
A useful written method is one that helps children to carry out a calculation and can
be understood by others.
Written recording is needed to help us to keep track of where we are in our
calculation and to help explain our method or thinking to someone else.
It is important to encourage children to look at the problem and decide which is the
best method to choose.
Pictures
Mental
Mental with jottings
Structured recording
Calculator
This policy shows a developmental path through written calculations. Not all children
will follow the route at the same speed. Some children will need lots of reinforcement
of the previous year group and some children may move quicker than expected onto
the next year group. It is important to move at the pace of the child and not rush them
through written methods without full understanding.
Where possible written methods will be practised through real life problems and
opened up to include problem solving activities. Children will be given opportunities
to explain their reasoning and methods to each other.
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A Whole School Approach
We have developed a consistent approach to the teaching of written calculation
methods. This will establish continuity and progression throughout the school.
Mental methods will be established. These will be based on a solid
understanding of place value in number and will include the following:
i.
ii.
iii.
iv.
v.
Remembering number facts and recalling them without hesitation
e.g. pairs of numbers which make 10
Doubles & halves to 20
Using known facts to calculate unknown facts
e.g. 6 + 6 = 12 therefore 6 + 7 = 13
24 + 10 = 34 therefore 24 + 9 = 33
Understanding and using relationships between addition & subtraction to
find answers and check results
e.g. 14 + 6 = 20 therefore 20 –6 = 14
Having a repertoire of mental strategies to solve calculations
e.g. doubles / near doubles
bridging 10 / bridging 20
adding 9 by +10 & -1
Making use of informal jottings such as blank number lines to assist in
calculations with larger numbers e.g.83 – 18 = 65
+ 60
+3
+2
18
vi.
vii.
viii.
20
80
83
Solving one-step word problems (either mentally or with jottings) by
identifying which operation to use, drawing upon their knowledge of
number bonds and explaining their reasoning
Beginning to present calculations in a horizontal format and explain mental
steps using numbers, symbols or words
Learn to estimate/approximate first e.g. 29 + 30 (round up to nearest 10,
the answer will be near to 60).
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WHEN ARE CHILDREN READY FOR WRITTEN CALCULATIONS?
Addition and subtraction
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



Do they know addition and subtraction facts to 20?
Do they understand place value and can they partition numbers?
Can they add three single digit numbers mentally?
Can they add and subtract any pair of two digit numbers mentally?
Can they explain their mental strategies orally and record them using informal
jottings?
Do they use and apply the commutative and associative laws of addition?
Multiplication and division
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
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Do they know the 2, 3, 4, 5 and 10 times table
Do they know the result of multiplying by 0 and 1?
Do they understand 0 as a placeholder?
Can they multiply two and three digit numbers by 10 and 100?
Can they double and halve two digit numbers mentally?
Can they use multiplication facts they know to derive mentally other
multiplication facts that they do not know?
Can they explain their mental strategies orally and record them using informal
jottings?
Do they use the commutative and associative laws for multiplication and the
distributive law of multiplication over addition and subtraction?
Do they recognise that multiplication and division are inverse operations.
The above lists are not exhaustive but are a guide for the teacher to judge
when a child is ready to move from informal to formal methods of calculation.
The Examples shown in this document demonstrate how informal jottings
taught in KS1 lead into more formal methods in KS2.
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Stages in Addition
Children are encouraged to develop a mental picture of the number system in
their heads to use for calculation. They develop ways of recording calculations
using pictures, etc.
They use number lines and practical resources to support calculation and
teachers demonstrate the use of the number line.
3+2=5
+1
+1
___________________________________________
0
1
2
3
4
5
6
7
8
9
Children then begin to use numbered lines to support their own calculations using
a numbered line to count on in ones.
8 + 5 = 13
0
1
2
+1 +1 +1 +1
3
4
5
6
7
8
9
+1
10 11 12 13 14 15
Bead strings or bead bars can be used to illustrate addition including bridging
through ten by counting on 2 then counting on 3.
5
Children will begin to use ‘empty number lines’ themselves starting with the
larger number and counting on.

First counting on in tens and ones.
34 + 23 = 57
+10
+10
+1 +1 +1
34

44
54 55 56 57
Then helping children to become more efficient by adding the units in one
jump (by using the known fact 4 + 3 = 7).
34 + 23 = 57
+10
+10
+3
34

44
54
57
Followed by adding the tens in one jump and the units in one jump.
34 + 23 = 57
+20
+3
34
54
6
5
Level 2

Bridging through ten can help children become more efficient.
This should be generally aided by using apparatus such as arrow cards, number
lines and bead strings etc.
37 + 15 = 52
+10
+3
37
47
+2
50
52
Mental methods (using jottings)
Partitioning
47 + 76 = (40 + 70) + (7 + 6)
or
47 + 76 = (47 + 70) + 6
Using number lines, for example bridging through 10.
27 + 8 = 27 + 3 + 5 =
+3
+5
______________________________________
27
30
35
Developing other mental strategies and recall of number facts.
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Vertical layout, expanded working, adding the most significant digit first:
Level 3
368
+ 493
700
150
11
861
Level 4
3547
+ 4769
7000
1200
100
16
8316
Level 5
36.8
+ 49.3
70.0
15.0
1.1
86.1
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Make sure when adding quantities i.e. grams and kilograms that the quantities are
written in the same unit.
Children should have experience of adding more than two sets of numbers and
calculations should be practised through word problems including the use of money,
decimals and measure.
Stages in Subtraction
Children are encouraged to develop a mental picture of the number system in
their heads to use for calculation. They develop ways of recording calculations
using pictures etc.
They use number lines and practical resources to support calculation. Teachers
demonstrate the use of the number line.
6–3=3
-1
-1
-1
___________________________________
0
1 2 3 4 5 6 7 8 9 10
The number line should also be used to show that 6 - 3 means the ‘difference
between
6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are apart.
9
0
1
2
3
4
5
6
7
8
9
10
Children then begin to use numbered lines to support their own calculations using a numbered line to count back in ones.
13 – 5 = 8
-1
0
1
2
3
4
5
6
7
8
-1 -1
9
-1 -1
10 11 12 13 14 15
Bead strings or bead bars can be used to illustrate subtraction including
bridging through ten by counting back 3 then counting back 2.
13 – 5 = 8
Children will begin to use empty number lines to support calculations.
Counting back

First counting back in tens and ones.
47 – 23 = 24
-1
24

25
-1
26
-1
27
- 10
- 10
37
47
Then helping children to become more efficient by subtracting the units in
one jump (by using the known fact 7 – 3 = 4).
47 – 23 = 24
10
-10
-3
24

27
-10
37
47
Subtracting the tens in one jump and the units in one jump.
47 – 23 = 24
-20
-3
24 27

47
Bridging through ten can help children become more efficient.
42 – 25 = 17
-3
17
20
-2
22
-20
42
Counting on
If the numbers involved in the calculation are close together or near to multiples
of 10, 100 etc, it can be more efficient to count on.
Count up from 47 to 82 in jumps of 10 and jumps of 1.
The number line should still show 0 so children can cross out the section from 0
to the smallest number. They then associate this method with ‘taking away’.
11
82 - 47
+1 +1 +1
0
47 48 49 50
+10
+10
60
+10
70
+1 +1
80 81 82
Help children to become more efficient with counting on by:

Subtracting the units in one jump;

Subtracting the tens in one jump and the units in one jump;
 Bridging through ten
Blank number line method - Finding the difference
Level 3
563 - 281
+19
+200
+63
_____________________________________________________
281
300
500
563
Answer 200 + 63 +19 = 200 +70 +12 = 282
Money
£5 - £2.36
+64p
+£2.00
Level 4
£2.36
______________________________________________
£3.00
£5.00
Answer £2.00 + 64p = £2.64
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Time
Level 4
8:35 to 1:25
+25 m
+4h
+25m
_____________________________________________________
8:35
9:00
1:00
1:25
Answer 4h 50m
Decimals
Level 5
9.2 - 3.45
+0.55
3.45
+5.00
+0.20
___________________________________________________
4.00
9.00
9.20
Answer 5.00 + 0.55 + 0.20 = 5.75
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Stages in Subtraction by Decomposition
563 - 241
-
Level 3
500
60
3
200
40
1
300
20
2 = 322
563 - 278
-
500
60
3

400
150
13
200
70
8

- 200
70
8
200
80
5
= 285
leading to
4
5
15 6
13
- 2
7
8
2
8
5
Level 4
4 digit numbers
Level 5
Decimal numbers
Only secure Level 4 children in Y5 and Y6 will be taught and expected to use
the decomposition method for subtraction. The method will only be taught if
the teacher feels it is necessary.
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Stages in Multiplication
Children will experience equal groups of objects and will count in 2s and 10s and
begin to count in 5s. They will work on practical problem solving activities
involving equal sets or groups.
Children will develop their understanding of multiplication and use jottings to
support calculation:
Repeated addition

3 times 5
is
5 + 5 + 5 = 15
or 3 lots of 5 or 5 x 3
Repeated addition can be shown easily on a number line:
5x3=5+5+5
5
0
1
2
5
3
4
5
6
7
5
8
9
10 11 12 13 14 15
15
and on a bead bar:
5x3=5+5+5
5
5
5
Commutativity

Children should know that 3 x 5 has the same answer as 5 x 3. This can also be
shown on the number line.
5
0
1
2
3
3

5
4
5
3
6
7
5
8
9
3
10 11 12 13 14 15
3
3
Arrays
Children should be able to model a multiplication calculation using an array. This
knowledge will support with the development of the grid method.
5 x 3 = 15
3 x 5 = 15
16
Mental method using partitioning multiplying tens first: 32 x 3
32 x 3 = (30 x3) + (2 x 3) = 90 + 6 = 96
Reinforcement of multiplying by 10 and 100 and the effect this has on the place
value of the digits.
6 X 10 = 60
6 X 100 = 600
6 X 20 = 6 X 2 X 10 = 120
Arrays showing links to division
Learning multiplication tables
Grid layout 38 x 7
Level 3
x
30
8
7
210
56
266
Grid layout - extend to bigger numbers
i.e. 238 x 7
Level 4
x
200
30
8
7
1400
210
56
1666
Learning multiplication tables
Y3 and Y4 should have a strong focus on the learning of multiplication tables
and related division facts as this underpins work in Y5 and Y6
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Extend to bigger numbers:
56 x 27
56 x 27 = (50 + 6) x (20 + 7)
Level 4
x
50
6
20
1000
120
1120
7
350
42
392
1512
Extend to decimals:
5.24 x 6
Level 5
x
5 .00
0.20
0.04
6
30.00
1.20
0.24
31.44
Grid method 3 digits x 2 digits
Level 5
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Stages in Division
Children will understand equal groups and share items out in play and problem
solving. They will count in 2s and 10s and later in 5s.
Children will develop their understanding of division and use jottings to support
calculation

Sharing equally
6 sweets shared between 2 people, how many do they each get?
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Grouping or repeated subtraction

There are 6 sweets, how many people can have 2 sweets each?
Repeated subtraction using a number line or bead bar

12 ÷ 3 = 4
0
1
3
2
3
4
5
6
7
3
8
9
10 11 12
3
3
The bead bar will help children with interpreting division calculations such as 10
÷ 5 as ‘how many 5s make 10?’

Using symbols to stand for unknown numbers to complete equations
using inverse operations
÷2=4
20 ÷  = 4
÷=4
20
Number lines & grouping
2
0
2
2
2
8  2 shown as 8 – 2 – 2 – 2 – 2
0
2
4
2
2
2
6
2
4
8
2
6
8
Division by repeated subtraction – represented by counting back and counting
up.
8
-2
- 2
- 2
-2
Informal methods using multiples of the divisor or ‘chunking’ TU  U
10 + 4 = 14 r 2
72  5
Level 3
5
Answer 14 r 2
256  7
72
-50
22
- 20
2
(10 x 5)
(4 x 5)
Level 4
30 + 6 = 36 r 4
7 256
-210
(30 x 7)
46
-42
(6 x 7)
4
Answer: 36 r 4
21
Level 4/5
Decimal numbers 87.5  7
10 + 2 + 0.5 = 12.5
7
87.5
- 70.0
(10 x 7)
17.5
14.0
(2 x 7)
3.5
-3.5
(0.5 x 7)
Extend to decimals with up to 2 decimal places
Level 5
3 digits  2 digits
Level 5
560  24
20 + 3 = 23 r 8
24
560
- 480
80
- 72
8
Summary

children should always estimate first

always check the answer, preferably using a different method eg. the inverse
operation

always decide first whether a mental method is appropriate

pay attention to language - refer to the actual value of digits

children who make persistent mistakes should return to the method that they
can use accurately until ready to move on

children need to know number and multiplication facts by heart (Y3/4)

discuss errors and diagnose problem and then work through problem - do not
simply re-teach the method

when revising or extending to harder numbers, refer back to expanded
methods. This helps reinforce understanding and reminds children that they
have an alternative to fall back on if they are having difficulties.
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