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Healey J. I. & N. School RENEWED POLICY FOR WRITTEN CALCULATION (September 2009) To be reviewed: September 2012 1 Progression towards a standard written method of calculation INTRODUCTION The National Numeracy Strategy provides a structured and systematic approach to teaching number. There is a considerable emphasis on teaching mental calculation strategies. Up to the age of 9 (Year 4) informal written recording should take place regularly and is an important part of learning and understanding. More formal written methods should follow only when the child is able to use a wide range of mental calculation strategies. REASONS FOR USING WRITTEN METHODS To aid mental calculation by writing down some of the numbers and answers involved To make clear a mental procedure for the pupil To help communicate methods and solutions To provide a record of work to be done To aid calculation when the problem is too difficult to be done mentally To develop and refine a set of rules for calculation A useful written method is one that helps children to carry out a calculation and can be understood by others. Written recording is needed to help us to keep track of where we are in our calculation and to help explain our method or thinking to someone else. It is important to encourage children to look at the problem and decide which is the best method to choose. Pictures Mental Mental with jottings Structured recording Calculator This policy shows a developmental path through written calculations. Not all children will follow the route at the same speed. Some children will need lots of reinforcement of the previous year group and some children may move quicker than expected onto the next year group. It is important to move at the pace of the child and not rush them through written methods without full understanding. Where possible written methods will be practised through real life problems and opened up to include problem solving activities. Children will be given opportunities to explain their reasoning and methods to each other. 2 A Whole School Approach We have developed a consistent approach to the teaching of written calculation methods. This will establish continuity and progression throughout the school. Mental methods will be established. These will be based on a solid understanding of place value in number and will include the following: i. ii. iii. iv. v. Remembering number facts and recalling them without hesitation e.g. pairs of numbers which make 10 Doubles & halves to 20 Using known facts to calculate unknown facts e.g. 6 + 6 = 12 therefore 6 + 7 = 13 24 + 10 = 34 therefore 24 + 9 = 33 Understanding and using relationships between addition & subtraction to find answers and check results e.g. 14 + 6 = 20 therefore 20 –6 = 14 Having a repertoire of mental strategies to solve calculations e.g. doubles / near doubles bridging 10 / bridging 20 adding 9 by +10 & -1 Making use of informal jottings such as blank number lines to assist in calculations with larger numbers e.g.83 – 18 = 65 + 60 +3 +2 18 vi. vii. viii. 20 80 83 Solving one-step word problems (either mentally or with jottings) by identifying which operation to use, drawing upon their knowledge of number bonds and explaining their reasoning Beginning to present calculations in a horizontal format and explain mental steps using numbers, symbols or words Learn to estimate/approximate first e.g. 29 + 30 (round up to nearest 10, the answer will be near to 60). 3 WHEN ARE CHILDREN READY FOR WRITTEN CALCULATIONS? Addition and subtraction Do they know addition and subtraction facts to 20? Do they understand place value and can they partition numbers? Can they add three single digit numbers mentally? Can they add and subtract any pair of two digit numbers mentally? Can they explain their mental strategies orally and record them using informal jottings? Do they use and apply the commutative and associative laws of addition? Multiplication and division Do they know the 2, 3, 4, 5 and 10 times table Do they know the result of multiplying by 0 and 1? Do they understand 0 as a placeholder? Can they multiply two and three digit numbers by 10 and 100? Can they double and halve two digit numbers mentally? Can they use multiplication facts they know to derive mentally other multiplication facts that they do not know? Can they explain their mental strategies orally and record them using informal jottings? Do they use the commutative and associative laws for multiplication and the distributive law of multiplication over addition and subtraction? Do they recognise that multiplication and division are inverse operations. The above lists are not exhaustive but are a guide for the teacher to judge when a child is ready to move from informal to formal methods of calculation. The Examples shown in this document demonstrate how informal jottings taught in KS1 lead into more formal methods in KS2. 4 Stages in 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. They use number lines and practical resources to support calculation and teachers demonstrate the use of the number line. 3+2=5 +1 +1 ___________________________________________ 0 1 2 3 4 5 6 7 8 9 Children then begin to use numbered lines to support their own calculations using a numbered line to count on in ones. 8 + 5 = 13 0 1 2 +1 +1 +1 +1 3 4 5 6 7 8 9 +1 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. 5 Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on. 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 Followed by adding the tens in one jump and the units in one jump. 34 + 23 = 57 +20 +3 34 54 6 5 Level 2 Bridging through ten can help children become more efficient. This should be generally aided by using apparatus such as arrow cards, number lines and bead strings etc. 37 + 15 = 52 +10 +3 37 47 +2 50 52 Mental methods (using jottings) Partitioning 47 + 76 = (40 + 70) + (7 + 6) or 47 + 76 = (47 + 70) + 6 Using number lines, for example bridging through 10. 27 + 8 = 27 + 3 + 5 = +3 +5 ______________________________________ 27 30 35 Developing other mental strategies and recall of number facts. 7 Vertical layout, expanded working, adding the most significant digit first: Level 3 368 + 493 700 150 11 861 Level 4 3547 + 4769 7000 1200 100 16 8316 Level 5 36.8 + 49.3 70.0 15.0 1.1 86.1 8 Make sure when adding quantities i.e. grams and kilograms that the quantities are written in the same unit. Children should have experience of adding more than two sets of numbers and calculations should be practised through word problems including the use of money, decimals and measure. Stages in 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 and practical resources to support calculation. Teachers demonstrate the use of the number line. 6–3=3 -1 -1 -1 ___________________________________ 0 1 2 3 4 5 6 7 8 9 10 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. 9 0 1 2 3 4 5 6 7 8 9 10 Children then begin to use numbered lines to support their own calculations using a numbered line to count back in ones. 13 – 5 = 8 -1 0 1 2 3 4 5 6 7 8 -1 -1 9 -1 -1 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. 13 – 5 = 8 Children will begin to use empty number lines to support calculations. Counting back First counting back in tens and ones. 47 – 23 = 24 -1 24 25 -1 26 -1 27 - 10 - 10 37 47 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 -10 -3 24 27 -10 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 -3 17 20 -2 22 -20 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. 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’. 11 82 - 47 +1 +1 +1 0 47 48 49 50 +10 +10 60 +10 70 +1 +1 80 81 82 Help children to become more efficient with counting on by: Subtracting the units in one jump; Subtracting the tens in one jump and the units in one jump; Bridging through ten Blank number line method - Finding the difference Level 3 563 - 281 +19 +200 +63 _____________________________________________________ 281 300 500 563 Answer 200 + 63 +19 = 200 +70 +12 = 282 Money £5 - £2.36 +64p +£2.00 Level 4 £2.36 ______________________________________________ £3.00 £5.00 Answer £2.00 + 64p = £2.64 12 Time Level 4 8:35 to 1:25 +25 m +4h +25m _____________________________________________________ 8:35 9:00 1:00 1:25 Answer 4h 50m Decimals Level 5 9.2 - 3.45 +0.55 3.45 +5.00 +0.20 ___________________________________________________ 4.00 9.00 9.20 Answer 5.00 + 0.55 + 0.20 = 5.75 13 Stages in Subtraction by Decomposition 563 - 241 - Level 3 500 60 3 200 40 1 300 20 2 = 322 563 - 278 - 500 60 3 400 150 13 200 70 8 - 200 70 8 200 80 5 = 285 leading to 4 5 15 6 13 - 2 7 8 2 8 5 Level 4 4 digit numbers Level 5 Decimal numbers Only secure Level 4 children in Y5 and Y6 will be taught and expected to use the decomposition method for subtraction. The method will only be taught if the teacher feels it is necessary. 14 Stages in Multiplication 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. Children will develop their understanding of multiplication and use jottings to support calculation: Repeated addition 3 times 5 is 5 + 5 + 5 = 15 or 3 lots of 5 or 5 x 3 Repeated addition can be shown easily on a number line: 5x3=5+5+5 5 0 1 2 5 3 4 5 6 7 5 8 9 10 11 12 13 14 15 15 and on a bead bar: 5x3=5+5+5 5 5 5 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. 5 0 1 2 3 3 5 4 5 3 6 7 5 8 9 3 10 11 12 13 14 15 3 3 Arrays Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method. 5 x 3 = 15 3 x 5 = 15 16 Mental method using partitioning multiplying tens first: 32 x 3 32 x 3 = (30 x3) + (2 x 3) = 90 + 6 = 96 Reinforcement of multiplying by 10 and 100 and the effect this has on the place value of the digits. 6 X 10 = 60 6 X 100 = 600 6 X 20 = 6 X 2 X 10 = 120 Arrays showing links to division Learning multiplication tables Grid layout 38 x 7 Level 3 x 30 8 7 210 56 266 Grid layout - extend to bigger numbers i.e. 238 x 7 Level 4 x 200 30 8 7 1400 210 56 1666 Learning multiplication tables Y3 and Y4 should have a strong focus on the learning of multiplication tables and related division facts as this underpins work in Y5 and Y6 17 Extend to bigger numbers: 56 x 27 56 x 27 = (50 + 6) x (20 + 7) Level 4 x 50 6 20 1000 120 1120 7 350 42 392 1512 Extend to decimals: 5.24 x 6 Level 5 x 5 .00 0.20 0.04 6 30.00 1.20 0.24 31.44 Grid method 3 digits x 2 digits Level 5 18 Stages in Division 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. Children will develop their understanding of division and use jottings to support calculation Sharing equally 6 sweets shared between 2 people, how many do they each get? 19 Grouping or repeated subtraction There are 6 sweets, how many people can have 2 sweets each? Repeated subtraction using a number line or bead bar 12 ÷ 3 = 4 0 1 3 2 3 4 5 6 7 3 8 9 10 11 12 3 3 The bead bar will help children with interpreting division calculations such as 10 ÷ 5 as ‘how many 5s make 10?’ Using symbols to stand for unknown numbers to complete equations using inverse operations ÷2=4 20 ÷ = 4 ÷=4 20 Number lines & grouping 2 0 2 2 2 8 2 shown as 8 – 2 – 2 – 2 – 2 0 2 4 2 2 2 6 2 4 8 2 6 8 Division by repeated subtraction – represented by counting back and counting up. 8 -2 - 2 - 2 -2 Informal methods using multiples of the divisor or ‘chunking’ TU U 10 + 4 = 14 r 2 72 5 Level 3 5 Answer 14 r 2 256 7 72 -50 22 - 20 2 (10 x 5) (4 x 5) Level 4 30 + 6 = 36 r 4 7 256 -210 (30 x 7) 46 -42 (6 x 7) 4 Answer: 36 r 4 21 Level 4/5 Decimal numbers 87.5 7 10 + 2 + 0.5 = 12.5 7 87.5 - 70.0 (10 x 7) 17.5 14.0 (2 x 7) 3.5 -3.5 (0.5 x 7) Extend to decimals with up to 2 decimal places Level 5 3 digits 2 digits Level 5 560 24 20 + 3 = 23 r 8 24 560 - 480 80 - 72 8 Summary children should always estimate first always check the answer, preferably using a different method eg. the inverse operation always decide first whether a mental method is appropriate pay attention to language - refer to the actual value of digits children who make persistent mistakes should return to the method that they can use accurately until ready to move on children need to know number and multiplication facts by heart (Y3/4) discuss errors and diagnose problem and then work through problem - do not simply re-teach the method when revising or extending to harder numbers, refer back to expanded methods. This helps reinforce understanding and reminds children that they have an alternative to fall back on if they are having difficulties. 22