PROGRESSION THROUGH CALCULATIONS FOR ADDITION AND SUBTRACTION
MENTAL CALCULATIONS FOR ADDITION (ongoing) These are a selection of mental calculation strategies: Mental recall of number bonds. This should continue throughout the school. 6 + 4 = 10 9 + 7 = 16 + 3 = 10 25 + 75 = 100 46 + 28 + 74= 80 + 40+ 28 = 148 19 + = 100 Children should initially be encouraged to bridge through 10s and 100s boundaries. Those who find mental recall of facts difficult can continue to use this as an alternative strategy. Recognise and use near doubles 6 + 7 = double 6 + 1 = 13 38 + 36 = double 37 = 74 Addition using partitioning and recombining 34 + 45 = (30 + 40) + (4 + 5) = 79 Counting on or back in repeated steps of 1, 10, 100, 1000 and recognising when to count on and when to count back. By Year 2, children should be able to add 10s onto any given 2 digit number. 86 + 50 = 136 (by counting on in tens) Children should be able to use number bonds to 20 to add two digit numbers. Add the nearest multiple of 10, 100 and 1000 and adjust 24 + 19 = 24 + 20 – 1 = 43 458 + 71 = 458 + 70 + 1 = 529 Use recombining strategies to identify easier combinations 458 + 71 = 460 + 69 = 529 or 460 + 40 + 29 = 529 Use the relationship between addition and subtraction 36 + 19 = 55 19 + 36 = 55 55 – 19 = 36 55 – 36 = 19
MENTAL CALCULATIONS FOR SUBTRACTION (ongoing) These are a selection of mental calculation strategies: Mental recall of subtraction facts need to be taught alongside addition facts. This should continue throughout the school, with emphasis on ‘missing number’ recall. 10 – 6 = 4 11 = 17 - 18 – 15 = 3 10 - = 2 Find a small difference by counting up 82 – 79 = 3 Counting on or back in repeated steps of 1, 10, 100, 1000 86 - 52 = 34 (by counting back in tens and then in ones) 460 - 300 = 160 (by counting back in hundreds) Subtract the nearest multiple of 10, 100 and 1000 and adjust 24 - 19 = 24 - 20 + 1 = 5 458 - 71 = 458 - 70 - 1 = 387 Use the relationship between addition and subtraction 36 + 19 = 55 19 + 36 = 55 55 – 19 = 36 55 – 36 = 19
MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED THROUGHOUT ALL YEAR GROUPS. THEY ARE NOT REPLACED BY WRITTEN METHODS.
L. Coe / R. Brown Page 1 of 14 September 2014 Key Stage One
L. Coe / R. Brown Page 2 of 14 September 2014 Year 1
In most lessons, calculations should be taught through problems, missing numbers or puzzles.
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.
Children should be able to add and subtract 1 to/from any number to 100. In Year 1, children are expected to add or subtract one and two digit numbers to 20, including zero. They are also expected to solve missing number problems, within 20. They use number lines alongside concrete objects and pictures to support calculation and teachers demonstrate the use of the number line.
3 + 2 = 5 +1 +1
______0 1 2 3 4 5
Children then begin to use numbered lines to support their own calculations using a numbered line to count on in ones, twos fives and tens, where appropriate.
8 + 5 = 13 +1 +1 +1 +1 +1
0 1 2 3 4 5 6 7 8 9 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.
L. Coe / R. Brown Page 3 of 14 September 2014 By the end of Year 1 children should be beginning to use empty number lines.
8 + 3 = 11
+1 +1 +1
8 9 10 11
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 alongside concrete objects to support calculation and teachers demonstrate the use of the number line.
6 – 3 = 3 -1 -1 -1
______0 1 2 3 4 5 6
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. This should be introduced alongside the above demonstration.
Children then begin to use numbered lines to support their own calculations - using a number line to count back in ones.
13 – 5 = 8 -1 -1 -1 -1 -1
0 1 2 3 4 5 6 7 8 9 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. Children can use concrete objects to begin to identify when to count on and when to count back.
L. Coe / R. Brown Page 4 of 14 September 2014 13 – 5 = 8
Number lines can be used alongside concrete and pictorial representations, as required.
Children should interpret mathematical statements involving +, -, = (7 = 5 + 2)
Year 2
In Year 2, children build on their understanding of addition and subtraction, applying methods to adding and subtracting two digit numbers and ones, two digit numbers and tens, two two digit numbers and three one digit numbers. They should be able to add or subtract 10 to/from any given 2 digit number.
Children will continue to use concrete objects, pictorial representations and ‘empty number lines’. For addition, they develop understanding the commutative law to start with the larger number and count on. For subtraction they should be taught to identify the number that is to be subtracted (they should never be told to subtract the smaller number from the larger number as this causes confusion when using column methods or when they move on to negative numbers).
Number lines (used with concrete objects and pictorial representations)
Addition
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
L. Coe / R. Brown Page 5 of 14 September 2014 Followed by adding the tens in one jump and the units in one jump.
34 + 23 = 57 +20 +3
34 54 57
Bridging through ten can help children become more efficient.
37+15=52
+10 +3 +2
37 47 50 52
Children should also begin to use Base 10 representations (using equipment – Deines) e.g.34 + 23 =
34 + 23 = 57
When children are confident with place value for two digit numbers, they can begin to use partitioned column addition.
37 + 15 = 52 30 + 7 10 + 5 40 + 12 = 52
Subtraction Counting back
First counting back in tens and ones.
47 – 23 = 24 -1 -1 -1 - 10 - 10
24 25 26 27 37 47
L. Coe / R. Brown Page 6 of 14 September 2014 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 -3 -10
24 27 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 -20 -3 -2
17 20 22 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. Children should be taught using practical equipment to recognise when to count on and when to count back.
Count up from 47 to 82 in jumps of 10 and jumps of 1.
The number line should begin with the smaller number and end with the larger.
82 - 47
+1 +10 +10 +10 +1 +1 +1 +1
47 48 49 50 60 70 80 81 82
Help children to become more efficient with counting on by:
Subtracting or counting on the units in one jump; Subtracting the tens in one jump and the units in one jump; Bridging through ten. Recognising when to count on and to count back
When children are confident with place value for two digit numbers, they can begin to use partitioned column addition.
57- 15 = 42 50 + 7 10 + 5 40 + 2 = 42
L. Coe / R. Brown Page 7 of 14 September 2014 Years Three and Four
L. Coe / R. Brown Page 8 of 14 September 2014 YEAR 3
Addition
In Year 3, children work within numbers to 1000. They mentally add three digit numbers and ones, tens and hundreds. Using a more formal written method, they add three digit numbers. They develop their skills of estimation and use their addition strategies to check subtraction calculations, introducing the language of inverse. Children should be expected to check their answers after calculation using an appropriate alternative strategy or the inverse. Children should be expected to consider if a mental calculation would be appropriate (efficient and accurate) before using written methods.
Children should continue to develop their use of partitioned column addition, expanding to three digit numbers.
37 + 15 = 52 30 + 7 10 + 5 40 + 12 = 52
Once children are confident in using partitioned column addition, they should move to expanded column addition. They should add the smallest place value first, working left.
2 6 7 + 8 5 1 2 1 4 0 2 0 0 3 5 2
Some children may move on to formal written methods. They should be able to explain what is happening in terms of place value (sixty plus eighty plus ten equals one hundred and fifty. 5 tens and 1 hundred added to the hundreds column) Please note – a ‘carried’ number is placed at the top of the written method.
1 1 2 6 7 + 8 5 3 5 2
Subtraction
Children will continue to use empty number lines with increasingly large numbers. Children will use estimation when working with larger numbers.
Children will begin to use informal pencil and paper methods (jottings) to support, record and explain partial mental methods building on existing mental strategies. These jottings should increasingly be used to replace number lines (mental number lines).
L. Coe / R. Brown Page 9 of 14 September 2014 Partitioning and decomposition
This process should be demonstrated using arrow cards to show the partitioning and base 10 materials to show the decomposition of the number. Note: When solving calculations such as 389 – 57, children should recognise that 57 is what is being subtracted and does not exist as an amount and therefore when using practical equipment such as base 10 materials children need only to count out 89 and not the 57.
389 = 300 + 80 + 9 - 57 50 + 7 300 + 30 + 2 = 332
Initially, the children will be taught using examples that do not need the children to exchange. They will continue to use practical equipment, such as cubes and bead strings, to develop partitioning techniques. A heavy emphasis will need to be placed on the understanding that the lower digit is subtracted from the upper digit not the smaller digit from the larger digit.
From this the children will begin to exchange.
371 = - 46
Step 1 300 + 70 + 1 - 40 + 6 The calculation should be read as e.g. take 6 from 1, Step 2 300 + 60 + 11 take 40 from 70. - 40 + 6 300 + 20 + 5 = 325
This would be recorded by the children as 60 300 + 70 + 11 - 40 + 6 300 + 20 + 5 = 325
Base 10 equipment should be used to support exchanging in the first instance. Children should know that units line up under units, tens under tens, and so on.
Children should be expected to approximate their answers before calculating. Children should be expected to check their answers after calculation using an appropriate alternative strategy or the inverse. Children should be expected to consider if a mental calculation would be appropriate (efficient and accurate) before using written methods.
L. Coe / R. Brown Page 10 of 14 September 2014 Year 4
Addition
Children should continue to use expanded column method. Some may then move on to formal columnar method up to 4 digits. By the end of the year all children should be able to use and explain formal columnar method. Children should be expected to approximate their answers before calculating. They should be expected to check their answers after calculation using an appropriate alternative strategy or the inverse. They should be expected to consider if a mental calculation (efficient and accurate ) would be appropriate before using written methods. From this, children will begin to carry above the line. 1 7625 + 348 7973 Using similar methods, children will: add several numbers with different numbers of digits; begin to add two or more four-digit sums of money, with adjustment from the pence to the pounds; know that the decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g. £3.59 + 78p.
Some children may also use an ‘imaginary’ number line with jottings, when appropriate.
625 + 48 =673
625 665 673
Subtraction
Partitioning and decomposition
Partitioning and decomposition continues into Year 4 with children manipulating four digit numbers and completing multiple exchanges.
8754 = - 386
Step 1 8000 + 700 + 50 + 4 - 300 + 80 + 6
Step 2 8000 + 700 + 40 + 14 (adjust from T to O) - 300 + 80 + 6
Step 2 8000 + 600 + 140 + 14 (adjust from H to T) - 300 + 80 + 6
This would be recorded by the children as
L. Coe / R. Brown Page 11 of 14 September 2014 600 140 8000 + 700 + 50 + 14 - 300 80 + 6 8000 + 300 + 60 + 8 = 8368
Decomposition (repartition) To increase efficiency in calculation, children should be encouraged to shorten the written method of exchange. This should only be introduced when children are ready and confident. It should be taught alongside the exchange method. The children should be able to explain the process. Children should be encouraged to estimate their answer.
6 14 14 8 / 7 / 5 / 4 - 3 8 6 8 3 6 8
If your children have reached the concise stage they will then continue this method through into years 5 and 6. They will not go back to using the expanded methods for larger numbers.
Children should: be able to subtract numbers with different numbers of digits; using this method, children should also begin to find the difference between two three- digit sums of money, with or without ‘adjustment’ from the pence to the pounds; know that decimal points should line up under each other.
For example: £8.95 – 4.38
8 + 0.9 + 0.05 - 4 + 0.3 + 0.08
= 8 + 0.8 + 0.15 (adjust from T to U) - 4 + 0.3 + 0.08 - 4 + 0.5 + 0.07
= £4.57
When ready, children can use decomposition for this.
L. Coe / R. Brown Page 12 of 14 September 2014 Years Five and Six
Year 5
L. Coe / R. Brown Page 13 of 14 September 2014 In Year 5, children should: work with numbers with more than four digits. This includes decimal numbers up to three decimal places in context e.g. money and measures. They should also add several numbers with different numbers of digits. They should develop their use of formal columnar method with increasing accuracy. Children should be expected to estimate using rounding before working out answers. Children should be expected to check their answers after calculation using an appropriate alternative strategy or the inverse. Children should be expected to consider if a mental calculation would be appropriate before using written methods, even with large numbers. (eg. 12453 + or – 2455). They begin to add and subtract fractions with the same denominator or multiples of the same number.
Year 6
In Year 6, children should: continue to work with larger numbers and decimal numbers up to three places, using the formal columnar method, within the context of multi-step problems, including other operations. estimate using rounding before working out answers. check their answers after calculation using an appropriate alternative strategy or the inverse. consider if a mental calculation would be appropriate(efficient and accurate) before using written methods. add and subtract fractions with mixed numbers and different denominators, using understanding of equivalent fractions.
2 + 3 = 8 + 18 = 26 = 2 = 1 6 4 24 24 24 1 24 1 12
+ - + - + - + - + - + - +
By the end of year 6, children will have a range of calculation methods, mental and written. Selection will depend upon the numbers involved.
We understand that some children will not progress to the more formal methods for addition however we expect that the majority of children will achieve the required method by the end of the Summer term. Intervention strategies, coordinated by the Mathematics and SEND team, are in place for children not reaching expectations.
L. Coe / R. Brown Page 14 of 14 September 2014