Download Mathematics Calculations for parents 2013

Helping you to help your child:
Calculation methods
From this booklet you will notice that as your child progresses through primary school
they will be required to use skills they have been taught in one operation to assist them
in another. Although the methods in this booklet are assigned to year groups it will
depend on the confidence of the individual child as to when they are ready to
adopt them.
When children leave Hermitage Primary school we aim that they:
 Have a secure knowledge of number facts and a good understanding of the four
operations (+ - x ÷)
 Can carry out mental calculations and apply appropriate strategies when working
with bigger numbers
 Use notes and diagrams to help record steps when solving more complex
problems
 Have efficient, reliable written methods for each operation that they can
confidently apply to calculations that they cannot carry out mentally
 Use a calculator effectively, using their mental skills to decide if the answer
make sense.
Mental calculation
The ability to calculate mentally forms the basis of all methods of calculation. This
involves:
 Instant recall of number facts (+ - x ÷)
 Have a secure understanding of place value and the number system
 Know the best strategy to apply to a calculation
 Understand the language and rules of Maths
Written calculation
Throughout their primary years children progress from informal jottings to efficient
written methods for each of the four operations.
Standard written methods are only introduced when a child has a secure knowledge and
understanding of the process involved and can clearly explain the strategies they have
used.
Children become secure with these methods when they have regular practice and
persevere!
What you can do to help your child at home
 Give them plenty of experience of maths in ‘real life’ contexts: board games,
shopping, cooking, reading calendars and timetables
 Regular practice of number facts: number bonds, times tables
 Supporting your child in using the strategies they have been taught at school.
 Encouraging children to explain what they are doing at different stages of a
calculation and to estimate answers first.
ADDITION
Mental strategies for addition
Secure mental addition requires children have the ability to:
 Recall key number facts instantly (number pairs to 10, 20 & 100, doubles etc)
and to apply these to similar calculations
 Recognise that addition can be done in any order and use this to add mentally
different combinations of numbers

Partition two-digit numbers in different ways, including adding the tens and
units separately before recombining
 Understand the language of addition including ‘more than’, ‘sum’, ‘plus’, ‘greater
than’, ‘total’, ‘altogether’ etc.
Written methods for addition
In their early years of school children will have many practical opportunities to secure
their understanding of addition. It is crucial that this is secure before they begin to
record their methods for addition.
Stage 1: The number track and number line
The number track and line helps to guide children in adding one number to another.
They will practise making jumps up the number line using objects and their finger.
Sparklebox
Stage 2: The hundred square:
The children learn to use a hundred square to help in adding multiples of 10 to numbers.
Stage 3: The ‘empty’ number line
The ‘empty’ number line helps to record the steps on the way to calculating the total.
The steps often bridge through a multiple of 10.
8 + 7 = 15
Stage 4: Partitioning
The next stage is to record mental methods using partitioning: breaking large numbers
down into tens and units before adding them together. This mirrors the column method
where ones are placed under ones and tens under tens.
Children will start by partitioning both numbers.
Next they will add the units, then add the tens. Finally they will combine both answers.
48 + 36
40+8
30+6
8 + 6 = 14
40 + 30 = 70
70 + 14 = 84
= 84
Pupils should be able to partition
numbers in different ways.
e.g. 23= 20+3 and 23= 10+13
Children continue to use partitioning with increasingly large numbers.
The children learn to count on from the largest number irrespective of the order of
the calculation.
Children are also taught to use ‘compensation’. This involves adding a near multiple of 10
and subtracting accordingly:
73 + 49 = 122
73 + 50 (near multiple of 10) = 123
123- 1= 122
Stage 5:Column addition
The next stage is to use their knowledge of adding by partitioning, to record addition
calculations in a column.
e.g.
47
134
+ 76
+ 251
13 Adding units
110
5 Add units
Adding tens
80 Add tens
123
300 Add hundreds
385
Stage 6:The column method is refined and children begin to ‘carry ten’ or ‘carry one
hundred’ NOT ‘carry one’.
eg.
47
+ 76
123
or
Add whole
numbers with more
than 4 digits
or
258
366
+ 87
+ 458
345
824
SUBTRACTION
Mental strategies for subtraction
Secure mental subtraction requires the ability to:

Recall key subtraction facts instantly (inverse of number pairs to 10, 20 & 100,
halves etc) and to apply these to similar calculations

Mentally subtract combinations of one and two digit numbers

Understand that subtraction is the inverse of addition and recognise that
subtraction can’t be done in any order (it has to start with the larger number)

Understand the language of subtraction including ‘less’, ‘minus’, ‘take away’,
‘difference between’ etc.
Written methods for subtraction
In their early years of school children will have many practical opportunities to secure
their understanding of subtraction. It is crucial that this is secure before they begin
to record their methods for subtraction.
Stage 1: The number track and number line
The number track and line helps to guide children in subtracting one number to another.
They will practise making jumps down the number line using objects and their finger.
Stage 2:The hundred square
Children learn to use a hundred square to help them in subtracting multiples of 10.
Stage 3: The empty number line
The empty number line helps to record the steps in mental subtraction.
Counting Back
15 – 7 = 8
A calculation such as 74 - 27 can be recorded by counting back 27 from 74 to reach 47
The steps can also be recorded in a different order:
or combined:
Stage 4: The steps can also be recorded by counting up from the smaller number to
find the difference. This mental method of counting up from the smaller to the larger
number can be recorded using either number lines or vertically in columns. The number
of rows (or steps) can be reduced by combining steps.
or
With three-digit numbers the number of steps can again be reduced.
or
With practice, children will need to record less information and decide whether to
count back or forward. It is useful to ask your child whether counting up or back is the
more efficient way for calculations such as 57 - 12, 86 - 77 or 43 - 28.
Stages 5 and 6: Partitioning
Subtraction can be recorded using partitioning to write equivalent calculations that are
easier to carry out mentally. The partitioned numbers are then written under one
another. Finally children will learn to use the shortened method.
74 – 27
741 – 367
The children progress to subtracting the ones, then the tens, then the hundreds.
563 – 241
This leads to:
Adjustment from the hundreds to the tens, or partitioning the hundreds:
Adjustment from the hundreds to the tens and the tens to the ones:
Dealing with zeros whe
Dealing with zeros when adjusting:
Subtract numbers
with more than 4
digits
MULTIPLICATION
Mental strategies for multiplication
To multiply successfully children need to be able to:

Recall all multiplication facts to 10 × 10

Use knowledge of place value to apply times tables facts to similar calculations
such as:
7x5
70 × 5
70 × 50
700 × 5
700 × 50

Partition numbers into multiples of Hundreds, Tens and Units

Add two or more single-digit numbers, multiples of 10 and 100 and combinations
of whole numbers using the column method

Understand the language of multiplication including ‘lots of’, ‘groups of’, ‘times’,
‘multiply’, ‘product’
Written methods for multiplication
Stage 1:
Multiplication is introduced as ‘lots of’ or ‘groups of’ real objects or pictures. Children
group objects themselves and learn to count objects in 2s, 10s and 5s.
Stage 2: Repeated addition
Children learn that 2 + 2 + 2 + 2 + 2 + 2 means the same as 6 lots of 2 and 6 times 2.
They understand that it can be written as 6 x 2.
This can also be recorded on a number line:
The children learn commutativitythat
2x6
has the same answer as
6x2
Stage 3: Arrays
Children learn that multiplication can be shown in picture form.
2 x 6 = 12
6 x 2 = 12
During this stage the children continue to use and consolidate repeated addition,
number lines and arrays to support their understanding.
They begin to use symbols to stand for unknown numbers:
x 5 = 20
3x
= 18
x
= 30
Stage 4: Grid Method
The partitioning that children have learnt to use to add and subtract is also used to
multiply.
First this requires partitioning the numbers to be multiplied. Then you multiply each
number in turn. Once the numbers are multiplied by one another the answers are added
to find a total.
Eg. 42 x 5
x
5
200 + 10 = 210
Eg. 273 x 8
1600
+
560
+
24
2184
40
2
200
10
An extra row is added to the grid to multiply HTU x TU.
eg.
146 x 28
2000
800
+
800
+
320
+
120
48
2920
1168
2920
+
1168
4088
Expanded written method
When children are secure enough with the value of each digit in the question they are
taught to condense the grid method into a column. They record each stage of the
multiplication.
eg.
HTU X U
273
x
8
24
3x8
560
70 x 8
1600
200 x 8
2184
eg.
HTU x TU
146
x
28
48
6x8
320
40 x 8
800
100 x 8
120
6 x 20
800
40 x 20
2000
100 x 20
4088
2
‘Standard’ written method:
When children show that they are secure with the above ‘expanded’ method they will
be taught to condense their recordings by carrying ones, tens and hundreds.
eg.
HTU x U
273
x
8
2 1 8 4
5 2
273 x 8
DIVISION
Mental strategies for division
To divide successfully children need to be able to:

Partition two-digit and three-digit numbers into multiples of 100, 10 and 1

Recall multiplication and division facts to 10 × 10 and recognise multiples of onedigit numbers

Know how to find a remainder working mentally - for example, find the
remainder when 48 is divided by 5;

Understand and use multiplication and division as inverse operations.

Understand and use the vocabulary of division - for example in 18 ÷ 3 = 6, the
18 is the dividend, the 3 is the divisor and the 6 is the quotient.
Written methods for division
Stage 1: Sharing objects or pictures
Initially division is introduced as ‘sharing’ and ‘equal groups’ using real objects or
pictures.
The children could be asked to share 10 apples equally between 2 children.
This progress to become
10 ÷ 2 = 5
Stage 2: Recording sharing
Sharing equally is recorded as:
6÷2=3
Stage 3: Grouping rather than sharing
The emphasis becomes grouping rather than sharing.
The children progress to using repeated subtraction on a number line. Children are
taught to look to see how many ‘equal groups’ of a number they can subtract before
they reach zero.
eg.
9÷3=3
Or children draw the total amount using circles, and group them using large circles
24 ÷ 4 = 6
Remainders
or
How many groups of 4 are there in 24?
Children move onto calculations involving remainders:
13 ÷ 4 = 3 r 1
or
Children develop their use of repeated subtraction to be able to subtract multiples of
the divisor. Initially, these should be multiples of 10s, 5s, 2s and 1s (numbers with
which the children are more familiar).
Stage 4: Chunking
Children are taught to identify a ‘chunk’ of the question to be dealt with at a time.
They are taught to determine how many equal groups of the divisor can be taken away
until zero is reached.
For example, in the question below the child has decided to take ten lots of three from
seventy two. They are now left with forty two to share equally by three. Again the
child has decided to take another ten lots of three leaving them with twelve. The child
feels comfortable with taking two lots of three from twelve leaving them with six. They
finally take another two lots of three from six until they reach zero. The children add
up how many ‘lots of’ three they shared 72 into to reach an answer.
72 ÷ 3 = 24
3
72
30
10 x 3
42
30
10
x3
2
x3
6
2
x3
0
=24 equal lots of 3 were taken from 72
12
6
6
Stage 5: Condensed chunking
The children build on the method taught in Year 4 and are encouraged to condense the
‘chunks’ they take away from the original number in the question.
They are taught that this method can be used to identify remainders.
eg.
196 ÷ 6 = 32 r 4
6
196
- 180
30
x6
2
x6
16
-
12
4
4 cannot be shared equally by 6 so there is a remainder of 4
=32 equal lots of 6 were taken away
Stage 6: Short division
72÷ 3=
2
3
4
7 12
Stage 7
The method taught in Years 4 and 5 continues to be used. However children are taught
to record remainders as fractions or decimals.
eg.
196 ÷ 6 = 32 r 4
196 ÷ 6 = 32 r
4
6
=
32 r 2
3
L. Street June. 2013