A guide to calculation in maths
Food for thought…
“Mathematics is essential for everyday life and understanding our world. It is also essential to science, technology and engineering, and the advances in these fields on which our economic future depends. It is therefore fundamentally important to ensure that all pupils have the best possible mathematics education. They need to understand the mathematics they learn so that they can be creative in solving problems, as well as being confident and fluent in developing and using the mathematical skills so valued by the world of industry and higher education… We must all play our part to ensure that all of our pupils receive the best possible mathematics education.”
Extract: Foreword from Her Majesty’s Chief Inspector
‘Mathematics: made to measure’ OfSTED
May 2012, No. 110159
Contents
Introduction…………………. 4
Addition ……………………….. 5 - 10
Subtraction………………….. 11 - 17
Multiplication…………….. 18 - 24
Division………………………….. 25 – 30
Hundred square………… 31
Multiplication square….32
Useful websites……………….33
Introduction
Some parents have asked about the methods we teach as they seem different to the maths that they learned at school. We hope this guide will make it easier to support your child in mathematics.
Being able to calculate mentally (‘in your head’) is important to solve maths problems. So, effective mental methods are regularly taught, used and applied in lessons. Children build up counting strategies and develop a secure understanding of place value (what a digit is ‘worth’) and number facts. Practical activities are used and the use of jottings is encouraged. As the numbers used get bigger, children use number lines and hundred squares to help them. Once children use mental strategies confidently, they start to look at more formal written methods. Children are taught a number of strategies, initially developing their mental strategies, moving towards efficient and more compact written methods which you may remember from school.
The aim is to make sure that every child can confidently use a reliable method to solve maths problems when they cannot work out the calculation in their heads or do not have access to a calculator. It is important to remember that children may prefer to use different methods. A wide range of calculation strategies are taught so that children are given the opportunity to choose one that they can use confidently and accurately.
Encourage your child to use the method with which they feel most comfortable and which they are using reliably at school.
Ask your child to explain the method they are using as it helps them achieve greater understanding.
Useful knowledge and skills
To add well, children should be able to:
· quickly recall all addition pairs to 9 + 9 eg. 7 + 8 = 15 and which pairs of numbers add to make 10; eg. 8 + 2 = 10
· add multiples of 10 (such as 40 + 80) or of 100 (such as 400 + 800) by using the related addition fact (4 + 8 = 12) and their knowledge of place value;
· partition (or split) two-digit and three-digit numbers in different ways eg. 23 is 20 +3; 132 is 100 + 30 + 2
Some useful mental strategies to solve problems:
· Add tens first (e.g. 52 + 25 = 52 + 20 + 5)
· Use near doubles (e.g. 35 + 36 = 35 + 35 + 1)
· Round and adjust (e.g. 27 + 99 = 27 + 100 – 1)
· Find number bonds (‘number pairs’) that make 10 first
(e.g. 16 + 19 + 34 = 20 + 19 + 30 = 69)
For calculations that children cannot solve in their heads, they need to be able to use an effective written method. So, read on…
Picture this
It can help if children are able to ‘see’ a picture of the calculation in their head or are able to draw their own diagram.
Number lines
Number-lines and practical resources (objects such as pasta or Lego or hundred squares) support addition by helping children count on.
3 + 2 = 5
______
0 1 2 3 4 5 6 7 8 9
8 + 5 = 13
Empty Number Lines
Children then use ‘empty’ number lines. Draw a line. Only show the numbers needed to solve the problem. Start with the larger number and count on.
First: Count on in tens and ones.
34 + 23 = 57
34 44 54 55 56 57
Then: Be more efficient by adding the units in one jump.
34 + 23 = 57
34 44 54 57
Next: Try adding the tens in one jump and the units in one jump.
34 + 23 = 57
34 54 57
Finally: Learning to ‘bridge’ to ten can make life easier
(encourage children to look out for a small jump to the next ten)
37 + 15 = 52
37 47 50 52
Empty number lines can even be used with larger numbers.
Remember, always count on from the largest number.
38 + 86 = 124
86 116 120 124
Be prepared to round (to the nearest 10) and then adjust if it helps!
49 + 73 = 122
(this can be read as 73 + 50 – 1 = 122)
73 122 123
Informal pencil and paper methods (‘jottings’)
Being able to make jottings help children to record and explain their mental strategies.
Children learn to partition (pull apart) numbers, often into tens and units, and learn to add the most significant digits first.
Eg.
57 + 22 = (50 + 20) + (7 + 2)
= 70 + 9
= 79
The children are then encouraged to record this in columns:
57 50 + 7
+ 22 20 + 2
79 70 + 9
This method stresses the value of each digit and ensures that children understand what they are doing and why.
Expanded column method of addition
The children can now move towards a more formal and familiar written method. This layout still uses the addition of tens to tens and separately;
Eg. 67 + 24 =
Emphasise the addition of the tens as 60 + 20 not 6 + 2
67+ 24
1 1 ( 7 + 4)
80 (60 + 20)
9 1 /
267
+ 85
12 ( 7 + 5)
140 (60 + 80)
200
352
Getting ready for ‘carrying’ – Standard Compact Method
Children then learn to add the units first so that they are ready to ‘carry’ below the line.
67+ 24
1 1
80
9 1 / becomes / 67
+ 24
9 1
1
Children will then progress to solve addition using this method for 2, 3 and 4-digit numbers, money, measures and decimals.
Eg.
587 3587
+ 475 + 675
1062 4262
1 1 1 1 1
Useful knowledge and skills
To subtract (‘take-away’) well, children should be able to:
· recall all addition and subtraction facts up to 20 (eg 7 + 13;
20 – 5);
· subtract multiples of 10 (such as 160 – 70 = 90 because 16 – 7 = 9);
· partition or split numbers into hundreds, tens and units in different ways (e.g. partition 74 into 70 + 4 or 60 + 14).
Some mental strategies
· Subtract tens first then units
(e.g. 67 – 35 = 67 – 30 – 5 = 32)
· Use near doubles (e.g. 54 – 28 = 54 – 27 – 1 = 26)
· Round and adjust (e.g. 60 – 19 = 60 – 20 + 1 = 41)
· Count on from the smaller number
(e.g. 2005 – 1996 count forward from 1996 to 2000 (4)
then to 2005 (5) 5 + 4 = 9)
The aim, as for addition, is that children may use mental methods when appropriate, but as they move through the school, they begin to use written methods for calculations that they cannot solve in their heads. Each stage builds on the previous one.
Number lines
Number lines and practical resources (objects or hundred squares) support subtraction by helping children count back.
6 – 3 = 3
______
0 1 2 3 4 5 6 7 8 9 10
13 – 5 = 8
Difference between
Number lines should also be used to show that 6 - 3 also means the ‘difference between’ 6 and 3, the ‘difference between’ 3 and 6 and how many jumps they are apart.
The difference between 6 and 3 is…
The difference between 3 and 6 is…
3 jumps
0 1 2 3 4 5 6 7 8 9 10
Empty Number Lines
Children then use ‘empty’ number lines. Draw a line. Only show the numbers needed to solve the problem. Often start with the larger number and count back.
Counting back
First: count back in tens and ones.
47 – 23 = 24
Then: children become more efficient by subtracting the units in one jump (by using the known fact 7 – 3 = 4).
47 – 23 = 24
Next: subtract the tens in one jump and the units in one jump.
47 – 23 = 24
Finally: Learning to ‘bridge’ to ten can make life easier
(encourage children to look for a small jump to the next ten)
42 – 25 = 17
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 (remember how shopkeepers used to give change?)
Count up from 47 to 82 in jumps of 10 and jumps of 1.
It is important to remind children that they are still subtracting (finding the difference).
82 – 47 = 35
Children become more efficient with counting on by:
· Counting on the units in one jump;
· Counting on the tens in one jump and the units in one jump;
· Bridging to ten (see the example above – very useful again!)
Informal pencil and paper methods (jottings)
Being able to make jottings helps children to record and explain their mental strategies. Children may still use number lines with bigger numbers but will move on to use the more familiar column method with exchanging or ‘borrowing’.
Expanded method (partitioning and decomposition)
Children begin to work with examples in which there is no need to exchange or to enable the subtraction to continue. It is important to stress that the process starts from the lowest value digits then progresses from left to right. Also, children should know that units line up under units, tens under tens, and so on.
89 – 57 =
89 = 80 + 9
- 57 = 50 + 7
30 + 2 = 32
Exchanging
Next: try exchanging – encourage children to show each step to begin with…
71 - 46 =
Step 1 70 à 1
- 40 à 6
Step 2 60 à 11
- 40 à 6
20 + 5 = 25
Then: it can simply be recorded as…
60
70 + 11
- 40 + 6
20 + 5 = 25
Another example showing each step…
755 700 50 5
- 86 - 80 6
700 40 15
- 80 6
600 140 15
- 80 6
600 60 9 = 669
Standard Compact Method
Finally: confident children will set it out like this familiar method. It is important that children are reminded to clearly show the numbers that are exchanged and any new values. Also, remind children that this final subtraction is, in order, 15 - 6; 140 – 80 and 600 – 0 . This reinforces the place value of each digit – for example, it is not 14 – 8 and 6 – 0!
6 14 1
7 5 5
- 8 6
6 6 9
Extend to four-digit numbers and decimals up to two places.
It is important that children know that all the decimal points must line up under each other.
6 14 1
1 7 5 4
- 2 8 6
1 4 6 8
324.90
- 7.25
Useful knowledge and skills
To multiply well, children should be able to:
· Count on in 10s, 2s and 5s confidently
· recall all times tables to 10 × 10;
· partition, or split, numbers into multiples of one hundred, ten and one;
· know and use rules for multiplying by 10, 100 and 1000.
· work out products such as 70 × 5, 70 × 50, 700 × 5 or 700 × 50 using the related fact 7 × 5;
· add two or more 1-digit numbers mentally.
Some mental strategies:
· Multiplying by 10, 100 etc. (‘jump’ numbers left to a new place value)
· Using doubling (e.g. 37 x 4 is double 37 (74) then double 74 (148)
· Using halving (e.g. 84 x 5 = Half of (87 x 10) = Half of 870 = 420 e.g. 54 x 25 ( (54 x 100) then halve twice = 1350)
· Using doubling and halving (e.g. 35 x 14 = 70 x 7 = 490 (double the odd number, halve the even number)
· Using rounding and adjusting (e.g. 26 x 99 = (26 x 100) - 26 = 2574 e.g. £1.99 x 3 = ( £2 x 3 ) – 3p = £5.97
As with the other operations (+, -, x and ÷) the aim is that children use mental methods when appropriate, but they use an efficient written method accurately for calculations that they cannot rapidly solve in their heads.
Grouping, sharing and arrays
Children learn about grouping and share items out in play and problem solving and learn to count on rapidly and confidently in 2s and 10s and later in 5s. They are encouraged to use jottings to support this process.
Repeated addition
Children develop their understanding of multiplication by thinking of it as repeated addition. Pictures, diagrams, sharing out objects and arrays can support this understanding.
3 times 5 is 5 + 5 + 5 = 15 or 3 lots of 5 or 5 x 3
Number lines
Repeated addition can be shown easily on a number line: