MANY MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED. THEY ARE NOT REPLACED BY WRITTEN METHODS.

Transcription

1 Calculation Policy

2 + PROGRESSION THROUGH CALCULATIONS FOR ADDITION Mental Calculations (ongoing) These are a selection of mental calculation strategies: Mental recall of number bonds = = = = 20 Use near doubles = double = 13 Addition using partitioning and recombining = ( ) + (4 + 5) = 79 Counting on or back in repeated steps of 1, 10, 100, = 143 (by counting on in tens and then in ones) = 160 (by counting back in hundreds) Add the nearest multiple of 10, 100, and 1000 and adjust = = = = 529 Use the relationship between addition and subtraction = = = = 19 MANY MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED. THEY ARE NOT REPLACED BY WRITTEN METHODS.

3 STAGE 1 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. Children then begin to use numbered lines to support their own calculations using a numbered line to count on in ones. Bead strings or bead bars can be used to illustrate addition including bridging through ten by counting on 2 then counting on 3.

4 STAGE 2 Children will begin to use empty number lines themselves starting with the larger number and counting on. First counting on in tens and ones. Then helping children to become more efficient by adding the units in one jump (by using the known fact = 7). Followed by adding the tens in one jump and the units in one jump. Bridging through ten can help children become more efficient.

5 STAGE 3 Children will continue to use empty number lines with increasingly large numbers, including compensation where appropriate. Count on from the largest number irrespective of the order of the calculation. Compensation Children will begin to use informal pencil and paper methods (jottings) to support, record and explain partial mental methods building on existing mental strategies. STAGE 4 Children will then progress to partitioning without the use of a number line = ( ) + (7 + 4) = = 91

6 STAGE 5 Moving to adding the least significant digits first in preparation for carrying. STAGE 6 From this, children will begin to carry below the line. NB WHEN USING THIS METHOD, IT IS REALLY IMPORTANT THAT THE TEACHER MODELS THE CALCULATION VERBALLY USING THE CORRECT PLACE VALUE OF THE DIGITS. E.G. IN THE FIRST EXAMPLE 5 AND 8 MAKES 13, WHICH IS 3 AND 1 TEN. THE TEN IS CARRIED INTO THE TENS COLUMN, THAT IS 1 TEN. THE THREE IS RECORDED IN THE UNITS COLUMN. TWENTY AND FORTY IS SIXTY, ADD ON THE CARRIED TEN MAKES SEVENTY WHICH WE RECORD AS A 7 IN THE TENS COLUMN. 7 TENS IS 70 ETC. Using similar methods, children will: add several numbers with different numbers of digits; begin to add two or more three-digit sums of money, with or without 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 p.

7 STAGE 7 (i) Children should extend the carrying method to numbers with at least four digits. Using similar methods, children will: add several numbers with different numbers of digits; begin to add two or more decimal fractions with up to three digits and the same number of decimal places; know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g. 3.2m cm. STAGE 7(ii) Children should extend the carrying method to numbers with any number of digits. Using similar methods, children will: add several numbers with different numbers of digits; begin to add two or more decimal fractions with up to four digits and either one or two decimal places; know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g

8 By the end of Year 6, children will have a range of calculation methods, both mental and a preferred written method. Selection will depend upon the numbers involved. Children should not be made to go onto the next stage if they are not ready or if they are not confident. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to check their answers after calculation using an appropriate strategy. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods.

9 - PROGRESSION THROUGH CALCULATIONS FOR SUBTRACTION Mental Calculations (ongoing) These are a selection of mental calculation strategies: Mental recall of addition and subtraction facts 10 6 = = = = 2 Find a small difference by counting up = 3 Counting on or back in repeated steps of 1, 10, 100, = 34 (by counting back in tens and then in ones) = 160 (by counting back in hundreds) Subtract the nearest multiple of 10, 100 and 1000 and adjust = = = = 387 Use the relationship between addition and subtraction = = = = 19 MANY MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED. THEY ARE NOT REPLACED BY WRITTEN METHODS.

10 STAGE 1 Children are encouraged to develop a mental picture of the number system in their heads to use for children. They develop ways of recording calculations using pictures etc. They use number lines and practical resources to support calculation. 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. Children then begin to use numbered lines to support their own calculations using a numbered line to count back in ones.

11 Bead strings or bead bars can be used to illustrate subtraction including bridging through ten by counting back 3 then counting back = 8 STAGE 2 Children will begin to use empty number lines to support calculations. Counting Back First counting back in tens and ones = 24 Then helping children to become more efficient by subtracting the units in one jump (by using the known number fact 7 3 = 4) = 24

12 Subtracting the tens in one jump and the units in one jump = 24 Bridging through ten can help children become more efficient = 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. 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 Help children improve at counting on by making more efficient jumps and by bridging through ten. Children will continue to use empty number lines, extended to increasingly large numbers by using complements to 100.

13 Subtraction of decimals is just as simple using the number line. The aim is for the children to be secure in a method for subtraction. There should be no pressure for children to have to progress through all the methods too quickly. Some children may leave Year 6 confidently working in subtraction on a number line. STAGE 3 (i) Decomposition

14 Children should: be able to subtract numbers with different numbers of digits; begin to find the difference between two decimal fractions with up to three digits and the same number of decimal places; know that decimal points should line up under each other. (ii) Decomposition Children should: be able to subtract numbers with different numbers of digits; be able to subtract two or more decimal fractions with up to three digits and either one or two decimal places. know that decimal points should line up under each other. By the end of Year 6, children will have a range of calculation methods, both mental and a preferred written method. Selection will depend upon the numbers involved. Children should not be made to go onto the next stage if they are not ready or if they are not confident. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to check their answers after calculation using an appropriate strategy. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods.

15 X PROGRESSION THROUGH CALCULATIONS FOR MULTIPLICATION Mental Calculations (ongoing) These are a selection of mental calculation strategies: Doubling and halving Applying the knowledge of doubles and halves to known facts. e.g. 8 x 4 is double 4 x 4 Deriving and recalling division facts Tables should be taught and practised regularly from Year 1 onwards, either as part of the mental oral starter or other times as appropriate within the maths lesson. Year 3 Derive and recall multiplication facts for the 2, 3, 4, 5, 6 and 10 times tables and the corresponding division facts; recognise multiples of 2, 5 or 10 up to Use knowledge of number operations and corresponding inverses, including doubling and halving, to estimate and check calculations. Year 4 Derive and recall multiplication facts up to 10 x 10, the corresponding division facts and multiples of numbers to 10 up to the tenth multiple. Year 5 Recall quickly multiplication facts up to 10 x 10 and use them to multiply pairs of multiples of 10 and 100; derive quickly corresponding division facts. Identify pairs of factors of two-digit whole numbers and find common multiples (e.g. for 6 and 9).

16 Year 6 Use knowledge of place value and multiplication facts to 10 x 10 to derive related multiplication and division facts involving decimals (e.g. 0.8 x 7, 4.8 6). Use knowledge of multiplication facts to derive quickly squares of numbers to 12 x 12. Recognise that prime numbers have only two factors and identify prime numbers less than 100; find the prime factors of two-digit numbers. Use approximations, inverse operations and tests of divisibility to estimate and check results. Using and applying division facts Children should be able to utilise their tables knowledge to derive other facts. e.g. If I know 3 x 7 = 21, what else do I know? 30 x 7 = 210, 300 x 7 = 2100, 3000 x 7 = 21000, 0.3 x 7 = 2.1 etc. Use closely related facts already known 13 x 11 = (13 x 10) + (13 x 1) = = 143 Multiplying by 10 or 100 Knowing that the effect of multiplying by 10 is a shift in the digits one place to the left. Knowing that the effect of multiplying by 100 is a shift in the digits two places to the left. Partitioning 23 x 4 = (20 x 4) + (3 x 4) = = 102 Use of factors 8 x 12 = 8 x 4 x 3

17 MANY MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED. THEY ARE NOT REPLACED BY WRITTEN METHODS. STAGE 1 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. It is really important that children experience multiplication in context. i.e. pairs of socks, egg boxes, fingers and toes etc. STAGE 2 Children will develop their understanding of multiplication and use jottings to support calculation: Repeated addition 3 times 5 is = 15 or 3 lots of 5 or 5 x 3 Repeated addition can be shown easily on a number line: 5 x 3 =

18 and on a bead bar: 5 x 3 = 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. Arrays Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method.

19 STAGE 3 Children will continue to use repeated addition and arrays. Children will also develop an understanding of: Scaling e.g. Find a ribbon that is 4 times as long as the blue ribbon. Using symbols to stand for unknown numbers to complete equations using inverse operations x 5 = 20 3 x = 18 x = 32 Partitioning 38 x 5 = (30 x 5) + (8 x 5) = = 190 STAGE 4 Children will continue to use arrays where appropriate, leading into the grid method of multiplication.

20 Grid method TU x U (Short multiplication multiplication by a single digit.) 23 x 8 Children will approximate first: 23 x 8 is approximately 25 x 8 = 200 STAGE 5 Grid method HTU x U (Short multiplication multiplication by a single digit.) 346 x 9 Children will approximate first: 346 x 9 is approximately 350 x 10 = 3500 Encourage the children to add up mentally at first, but then they may need a suitable written method if they cannot add mentally.

21 TU x TU (Long multiplication multiplication by more than a single digit.) 72 x 38 Children will approximate first: 72 x 38 is approximately 70 x 40 = 2800 Using similar methods, they will be able to multiply decimals with one decimal place by a single digit number. e.g. 4.9 x 3 Children will approximate first: 4.9 x 3 is approximately 5 x 3 = 15 STAGE 6 ThHTU x U (Short multiplication multiplication by a single digit.) 4346 x 8 Children will approximate first: 4346 x 8 is approximately 4346 x 10 = 43460

22 HTU x TU (Long multiplication multiplication by more than a single digit.) 372 x 24 Children will approximate first: 372 x 24 is approximately 400 x 25 = Using similar methods, they will also be able to multiply decimals with up to two decimal places by a single digit number and then two digit numbers. e.g x 3 Children will approximate first: 4.92 x 3 is approximately 5 x 3 = 15

23 STAGE 7 The next step is to represent the method of recording in a column format, but showing the working. Draw attention to the links with the grid method. Children should describe what they do by referring to the actual values of the digits in the columns. For example, the first step in 38 x 7 is thirty multiplied by seven, not three times seven, although the relationship 3 x 7 should be stressed. 56 x 27 is approximately 60 x 30 = 1800 By the end of Year 6, children will have a range of calculation methods, both mental and a preferred written method. Selection will depend upon the numbers involved. Children should not be made to go onto the next stage if they are not ready or if they are not confident. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to check their answers after calculation using an appropriate strategy. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods.

24 PROGRESSION THROUGH CALCULATIONS FOR DIVISION Mental Calculations (ongoing) These are a selection of mental calculation strategies: Doubling and halving Knowing that halving is dividing by 2 Deriving and recalling division facts Tables should be taught and practised regularly from Year 1 onwards, either as part of the mental oral starter or other times as appropriate within the maths lesson. Year 3 Derive and recall multiplication facts for the 2, 3, 4, 5, 6 and 10 times tables and the corresponding division facts; recognise multiples of 2, 5 or 10 up to Use knowledge of number operations and corresponding inverses, including doubling and halving, to estimate and check calculations. Year 4 Derive and recall multiplication facts up to 10 x 10, the corresponding division facts and multiples of numbers to 10 up to the tenth multiple. Year 5 Recall quickly multiplication facts up to 10 x 10 and use them to multiply pairs of multiples of 10 and 100; derive quickly corresponding division facts. Identify pairs of factors of two-digit whole numbers and find common multiples (e.g. for 6 and 9).

25 Year 6 Use knowledge of place value and multiplication facts to 10 x 10 to derive related multiplication and division facts involving decimals (e.g. 0.8 x 7, 4.8 6). Use knowledge of multiplication facts to derive quickly squares of numbers to 12 x 12. Recognise that prime numbers have only two factors and identify prime numbers less than 100; find the prime factors of two-digit numbers. Use approximations, inverse operations and tests of divisibility to estimate and check results. Using and applying division facts Children should be able to utilise their tables knowledge to derive other facts. e.g. If I know 3 x 7 = 21, what else do I know? 30 x 7 = 210, 300 x 7 = 2100, 3000 x 7 = 21000, 0.3 x 7 = 2.1 etc. Dividing by 10 or 100 Knowing that the effect of dividing by 10 is a shift in the digits one place to the right. Knowing that the effect of dividing by 100 is a shift in the digits two places to the right. Use of factors = = 18 Use related facts = 18 Given that 1.4 x 1.1 = 1.54 What is or ? MANY MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED. THEY ARE NOT REPLACED BY WRITTEN METHODS.

26 Please note that it is important for children to recognise the similarities but also the difference between two division calculations involving the same numbers such as: 18 3 = 6 and 18 6 = 3. Although 6 x 3 and 3 x 6 both have the answer 18, the meanings of 18 3 and 18 6 are different, especially when applied to real life problems. An example is 18 eggs in 3 boxes, each box containing 6 eggs. 18 eggs 6 eggs gives the number of boxes (3), whereas 18 eggs 3 boxes gives the number of eggs in each box (6). STAGE 1 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. STAGE 2 Children will develop their understanding of division and use jottings to support calculation. It is really important that children have lots of practical experience of sharing and grouping. Sharing equally 6 sweets shared between 2 people. How many do they each get?

27 Grouping or repeated subtraction There are 6 sweets. How many people can have 2 sweets each? Repeated addition or subtraction using number line or bead bar 12 3 = 4 The bead bar will help children with interpreting division calculations such as 12 3 as How may 3s make 12? Using symbols to stand for unknown numbers to complete equations using inverse operations 2 = 4 20 = 4 = 4 STAGE 3 Ensure that the emphasis in stage 3 is on grouping rather than sharing. Children will continue to use: Repeated addition or subtraction using a number line Children will continue to use an empty number line to support their calculation.

28 A context for the calculation is important. E.g. I have 24 sweets 24 4 = 6 Children should also move onto calculations involving remainders = 3 r 1 Using symbols to stand for unknown numbers to complete equations using inverse operations 26 2 = 24 = = 8 STAGE 4 Children will develop their use of repeated addition and subtraction to be able to add and subtract multiples of the divisor. Initially, these should be multiples of 10s, 5s 2s and 1s numbers with which the children are more familiar (The model below shows subtracting groups.)

29 STAGE 5 Bus stop method This more formal method is taught when children are ready to move on to the next stage. On the whole, this is not until Year 5 or 6. By the end of Year 6, children will have a range of calculation methods, both mental and a preferred written method. The written method may still be a number line for some children. Selection will depend upon the numbers involved. Children should not be made to go onto the next stage if they are not ready or if they are not confident. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to check their answers after calculation using an appropriate strategy. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods.