Revised
and Agreed by Staff
2009
Calculation
Strategies
Herne Calculation
Strategies
Evidence (March 2008)
shows that our children are getting confused by all the different ways
to work out an answer to a calculation. With the emphasis on understanding
and explaining the maths behind an answer it is clear that we need to
give children strong strategies that are based in mental calculation.
Key Principles of Calculation
The first thing anyone should
ask when faced with a calculation is “Can I do this in my head?”
Many calculations which can’t
be done mentally can often be performed very efficiently with the use
of jottings.
Calculation methods must be
appropriate, reliable and efficient.
Children must understand and
be able to explain their methods.
Opportunities to apply their
calculation strategies to solving problem should be integrated into
all lessons.
If children encounter persistent
difficulties when being introduced to a new calculation strategy they
should go back to the previous step.
To ensure children’s
understanding and progression of the four operations
(+ - x and ÷) the staff
of Herne Junior School have agreed to teach the same calculation progression
of strategies throughout the school. The strategies chosen are
detailed in the National Numeracy Strategy and have also been recommended
by Hampshire’s Maths Advisory Team. Children need to see that
they can perform the four operations on the number line however the
expectations shown in this booklet detail the main teaching strategy
for each operation. A quick reference guide has also been put together
for class teachers.
We hope you find this booklet useful.
Addition:
Year 3 (level
2A)
Objective:
To progress from recording only on an empty number line to being able
to record vertically starting the year with most significant digits
first and by the end of the summer term adding vertically with least
significant number first.
Expectation:
Initially children working with numbers up to 101 will use a number
line. In parallel with this they shall be encouraged to partition numbers
and add in the horizontal method.- Pupils will develop from horizontal
method to vertical. By the end of the year the majority of pupils should
be able to record in this way:
83 + 42=
Horizontal
method
Vertical method
H
T
U
8
3
+
4
2
5
]
add mentally from
1
2
0
]
bottom
1
2
5
Add the units (3 + 2 = 5)
Add the tens (80 + 40 = 120)
Add together the units and tens totals
(5 + 120 = 125)
Final answer 125.
This will link with knowledge of partitioning
(breaking a number into parts relating to its place value (e.g. 34 =
30, 4)) while retaining understanding of value of digits.
Year 4 (level 3b)
Objective:
To calculate Hundreds Tens
Units + Tens Units, then Hundreds Tens
Units + Hundreds Tens Units progressing from
the vertical expanded example to the ‘carrying’ addition method.
Expectation: By the end of the
year the majority of Y4 pupils should be able to carry out column addition
of two three digit integers. In Year 4 more able pupils (level 4c) will
move to ‘carrying’ below the line. Some children may need to expand
this calculation to see what is going on more clearly.
H
T
U
3
5
8
+
7
3
1
1
]
add mentally from
1
2
0
]
bottom
3
0
0
]
4
3
1
358 + 73 =
Expanded addition example
Add the units 8 + 3 = 11
Add the tens 50 + 70 = 120
Add the hundreds 300 + 0 = 300
Add the totals together (11 + 120 + 300
= 431)
Although some children will need the
expanded method in the majority of cases it will not be lingered upon.
358 + 73 =
H
T
U
3
5
8
+
7
3
4
3
1
1
1
Leading to: ‘carrying’ addition
method
Add the units 8 + 3 = 11
Carry the 1 ten
Add the tens 50 + 70 + 10 = 130
Carry the 1 hundred
Add the hundreds 300 + 100 = 400
Total = 431
Year
5 (level
4b)
Objective:
To make increasingly efficient jumps and to record column addition of
numbers up to 10, 000. Adding least significant digit first.
Expectation:
Children should be writing more efficient vertical recordings. Hundreds
Tens Units + Hundreds Tens Units, then
Thousands Hundreds Tens Units + Thousands
Hundreds Tens Units.
Teachers will remind children of the
expanded method to ensure understanding and develop to use ‘carrying’
once their understanding of the expanded method is secure. Expanded
methods will not be lingered upon as children who are ready to carry
can make careless mistakes even when they are more than capable of the
calculation.
4587+3475
Expanded addition method
‘Carrying’ addition method
Th
H
T
U
leading to
Th
H
T
U
4
5
8
7
4
5
8
7
+
3
4
7
5
+
3
4
7
5
1
2
]
add mentally
8
0
6
2
1
5
0
]
from
1
1
1
9
0
0
]
bottom
7
0
0
0
8
0
6
2
Add units 7 + 5 = 12
Add units 7 + 5 = 12
Add tens 80 + 70 = 150
Carry 1 ten
Add hundreds 500 + 400 = 900
Add tens 80 + 70 + 10 = 160
Add thousands 4000+3000 = 7000
Carry 1 hundred
Add hundreds 500 + 400 + 100 = 1000
Carry 1000
Add totals 12 + 150 + 900 + 7000 = 8062
Add thousands 4000+3000 + 1000= 8000
Final total = 8062
Year 6
Objective:
To continue to increase the efficiency of calculations, record as column
addition and increase the context to numbers with up to 2 decimal places.
Expectation:
Children should be able to apply the addition strategy for decimal numbers.
Expanded addition method
‘Carrying’ addition method
T
U
ths
hths
T
U
ths
hths
7
6
.
4
8
7
6
.
4
8
+
1
4
.
8
6
leading to
+
1
4
.
8
6
.
1
4
]
add
9
1
.
3
4
1
.
2
0
]
mentally
1
1
.
1
1
0
.
0
0
]
from
8
0
.
0
0
]
bottom
9
1
.
3
4
For Level 5 Pupils a variety of addition
strategies to broaden learning can be employed.
Subtraction:
Year
3 (Level 2A)
Objectives: 1)
To understand the concept and vocabulary of subtraction 2)
To use an empty number line to solve subtractions. 3) To encourage
children to jump to the next multiple of 10 then 100 and count on
in 10’s or multiple of 10’s and 1’s to the target number.
Expectation:
By the end of the year the majority of Y3 pupils should record in this
way. Pupils will calculate Tens Units – Tens
Units, developing to Hundreds Tens Units –
Tens Units or Hundreds Tens Units –
Hundreds Tens Units as appropriate:
84 - 56
+4
+20
+4
56
60
80
84
milestone
milestone
Appendix I
– Subtraction
Yr 3
Some children will struggle with the idea of difference and the notion
that this can be calculated by counting both forwards and backwards.
It may be necessary for some children to use counting back on the number
line to solve subtraction problems. Eg.
-2
-3
= 5
62-57
57
60
62
The use of language is very important
children will be taught that all subtractions can be solved by
counting on from the smallest number.
Year 4 (Level 3)
Objective:
To progress from recording every jump to making more efficient jumps
on an empty number line.
Expectation: By the end of the year the majority of Y4 pupils
should make more efficient jumps. Pupils will calculate Hundreds Tens
Units – Tens Units, then Hundreds Tens Units – Hundreds Tens Units:
723 -356
+23
+300
+40
+4
400
723
700
356
360
Y4 See Appendix 1
Year
5 (Level 3a
– 4)
Objective:
To make increasingly efficient jumps on an empty number line and to
record vertically.
Expectation:
Children should be writing more efficient vertical recordings.
+7
+200
607-389
+11
607
600
400
389
All children should be able to explain
this method and have a good understanding of it as well as be able to
use it efficiently for age-appropriate calculations. Decomposition will
not be taught at this stage.
Once the understanding of use of number
lines are secure (LEVEL 4) then the children may move to decomposition
and the more traditional vertical method of subtraction. These methods
do not help with the understanding of the concept they are merely a
method that gives an answer.
Year
6 (level 4a-5)
Objective:
To continue to increase the efficiency of jumps on an empty number line
especially working with time or mixed units. To record vertically and
develop a repertoire of strategies which could include decomposition.
Expectation:
Pupils will calculate Thousands Hundreds Tens
Units – Thousands Hundreds Tens Units
and decimals to 2 decimal places. They should also be able to apply
the subtraction strategy for a mix of integers and decimals to one and
two places decimal numbers. Eg. 7.24
-5.6
Y6 cont.
For example,
1.12 - 0.84
0.06 (0.90) rounding
to the nearest 0.1
0.10 (1.00) rounding
to the nearest 1.0
0.12 (1.12) Target number
0.28
1.12-0.84
+0.12
+0.10
+0.06
1.00
0.90
0.84
1.12
Year 6 Cont
Once the understanding of use of number
lines are secure then move the children towards the traditional method
of subtraction with reference to a vertical expanded method of subtraction.
Children should all be able to use methods
confidently for subtractions. Decomposition will only be taught where
children have a good understanding of the above method and are able
to explain their working.
To develop the children’s repertoire,
children should be able to explain the expanded method for decomposition
as shown below leading to the compact method.
Important note For all
the above to be effective it is crucial that children learn and have
secure knowledge of their number bonds. These methods are based on mental
strategies and links need to be made as appropriate. It is also important
that children have a good foundation of knowledge in order to progress
to the next stage of learning. We want to create high expectations and
levels of achievement and this is best accomplished by creating a good
understanding at each level of development rather than pushing onto
the next too soon.
Multiplication
Children
must know their tables at Herne we aspire to children
being able to instantly recall all multiplication facts up to 10x10.
This is what we call “knowing tables” and without this children’s
progress along written methods is severely hampered.
Year
3 (level 2a)
Objectives: To understand
multiplication as repeated addition. They will use equipment (cubes),
pictorial arrays, an empty number line and partitioning to develop visual
images to aid the children’s understanding of multiplication or ‘groups
of’.
Expectation:
To describe multiplication as an array 2 x 4 = 8
4 x 2 = 8
When children in year 3 are secure with
arrays and know what happens when a number is multiplied by 10 they
will be moved onto the grid method. (See below)
Year 4
(level 3)
Objectives: To be able to use
the grid method to record or explain multiplication.
Expectation:
Children should be able to estimate and record multiplications of a
two digit integer by a single digit. (lay out as ITP) eg
8
x 23
T
U
x
20
3
U
8
160
24
160
+
24
184
Children should be secure with what happens
to place value when multiplying by 10.
Further development of this would lead
to 2 digit by 2 digit multiplication.
Year
5 (level 3a-4)
Objectives: To be able to approximate
and explain through informal written methods multiplication of a two
digit number by a two digit number and a three digit number and a two
digit number.
Expectation:
Children should be able to explain orally how grid method works for
two digit numbers.
84
x 23
T
U
x
20
3
T
80
1600
240
1600
U
4
80
12
240
+
80
12
1932
1
.
Only when children are secure with their
understanding of what happens to numbers multiplied by 10 and the grid
method will other methods be taught. (Level 4)
Long multiplication:
Tens Units x Tens
Units
72 x 38
Estimate 72 x 38 is approximately 70
x 40 = 2800.
Expanded method of long multiplication
Th
H
T
U
7
2
x
3
8
1
6
(8
x
2)
5
6
0
(8
x
70)
6
0
(30
x
2)
2
1
0
0
(30
x
70)
2
7
3
6
1
Check answer of 2736 with estimate of
2800.
Year 6 (level 4-5)
Objectives: To develop a range
of pencil and paper methods to support, record and explain calculations
achieving consistent accuracy
Expectation:
Pupils will be able to use the grid method for ThHTUx U and HTU and
TU (lay out as ITP) They will broaden their range of strategies calculating
using least significant digit first for short multiplication: Thousands
Hundreds Tens Units x Units and long multiplication:
Hundreds Tens Units x Tens Units.
84
x 237
H
T
U
x
200
30
7
T
80
16000
2400
560
16000
U
4
800
120
28
2400
560
+
800
120
28
19908
1 1
.
Children will be encouraged to count
the 0 place holders to aid with multiplication by 10 and 100’s.
Development strategies will only be taught when children’s understanding
of place value is secure (eg Level 4). Development of strategies may
include:
Long multiplication:
Th
H
T
U
2
3
5
2
x
2
7
1
6
4
6
4
(7
x
2
3
5
2)
4
7
0
4
0
(2
0
X 2
3
5
2)
6
3
5
0
4
1
1
THTU x TU
For example,
Check answer
In Year 6 all pupils will be reminded
of the expanded method to reinforce their understanding of place value
as well the strategy. More able should use ‘carrying’ below the
line with confidence.
Important note
For all the above to be effective it
is crucial that children learn and have a secure recall of their tables
knowledge. These methods are based on mental strategies and links need
to be made as appropriate. It is also important that children have a
good foundation of knowledge in order to progress to the next stage
of learning. We want to create high expectations and levels of achievement
and this is best accomplished by creating a good understanding at each
level of development rather than pushing onto the next too soon.
Division
Year 3 (level
2a)
Objectives: To use equipment (cubes),
arrays, pictorial diagrams, an empty number line and repeated subtraction
to aid the children’s understanding of division or ‘groups of or
sharing’. Formal written methods will not be introduced until
children are secure in this knowledge.
Eg
8÷ 2 = 4 groups
Expectation:
Children should be able to draw out the
sharing or groupings using a Dividing Line or arrays.
8÷2=4 groups
1group 2groups
3 groups
4 groups
4 jumps of 2
0
2
4
6
8
Year 4 (level 3b)
Objectives: To use informal written
methods to support, record or explain calculation of TU divided by U.
The Dividing line will be used with a focus on ‘using multiples
of the divisor’ otherwise known as ‘chunking’. This method relies
upon using knowledge of multiples to group and then finding the difference
subtract from a target number. It may be useful for children
to record key multiplication facts about the divisor before beginning
to make chunking jumps (e.g. in the example below the child could write
key multiplication facts they know about the 5 x table to help them
2x5 5x5 10x5 20x5 etc, this also builds their multiplication and doubling
and halving skills.) Pupils will calculate Tens Units
÷ Units.
72 ÷ 5 =
73 ÷ 5 =
(10 + 4 )
14 Remainder 2
( 4x5)
= 20
( 10 x 5) = 50
70
72
Need 2 more = 72
50
0
Year 4 continued:
Children with strength in subtraction
may develop onto vertical the chunking strategy
Expectation:
By the end of the year children should be able to make more efficient
jumps of chunks on a number line. Circling the groupings on both the
dividing line and the vertical method. Children should be able to deal
with remainders in context.
Year
5 (level 3-4)
Objectives: To continue to use
the Dividing Line and develop standard written methods for HTU
divided by U by collecting all the facts to get as close as possible
to the target number before approaching the problem.
For example
Estimate 256 ÷ 7 is less that the approximation of 250 ÷ 5 = 50
-(10x7)
186
-(20x7)
46
-(6x7)
Remainder 4
256
2
5
6
÷
7
2
5
6
-
7
0
1
0
x
7
1
8
6
-
1
4
0
2
0
x
7
4
6
-
4
2
6
x
7
4
2
5
6
÷
7
=
3
6
r.
4
Start with target number (number to be
divided e.g. 256)
Using key multiplication facts about
the divisor (e.g. 7) take away suitable ‘chunks’.
Keep subtracting chunks until you number
is too small to be divided again.
Check answer of 36 r.4 with estimate.
Children should be able to explain this
method and have a good understanding of it as well as being able to
use it efficiently for age-appropriate calculations. By approximating
first, children should be able to check the sense of their answer.
They should also be able to express any remainder in an appropriate
context.
Year 6(level 4-5)
Objectives: To extend the range
of methods for division and/or make more efficient jumps by collecting
all the facts to get as close as possible to the target number before
approaching the problem. HTU divided by TU extended to decimals.
899 ÷ 23 = 39
23
899
-
690
(30 x 23)
10 x 23
= 230
-
209
115
(5 x 23)
20 x 23
30 x 23
= 460
= 690
-
94
92
(4 x 23)
5 x 23
= 115
2
Answer
30+5+4 = 39 remainder 2
Expectation:
Pupils should be able to estimate and explain working for division of
3 digit numbers by two digit numbers including remainders. They should
also be able to express remainders as a decimal.
Development of strategies
for very secure level 4 pupils
may include: short division eg
3
5
Remainder 5
23
8
9
189
23
into 89(0)
= 3(0)
remainder18
23
Into 189
= 8
remainder 5
Children will also be able taught how
to express remainders as a decimal.
Important note
For all the above to be effective it
is crucial that children learn and have a secure recall of their tables
knowledge. These methods are based on mental strategies and links need
to be made as appropriate. It is also important that children have a
good foundation of knowledge in order to progress to the next stage
of learning. We want to create high expectations and levels of achievement
and this is best accomplished by creating a good understanding at each
level of development rather than pushing onto the next too soon.
/13 AL Updated
May 2009