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1 St Catherine’s R C Primary School Calculations Policy Rationale This policy is designed to outline our approach to mental and written calculations in a progressive and consistent format. We recognise the importance of mental strategies being introduced and taught systematically from reception classes onwards and pupils being given regular opportunities to develop and practise these skills. We aim to ensure that mental and written calculations are there to complement each other. Every written method has elements of mental operations whole written recording helps pupils clarify their thinking and supports the development of more fluent mental strategies. It is important that children do not abandon jotting and empty number line approaches once paper and pencil methods have been introduced. Pupils of all ages and abilities should be encouraged to: To examine the calculation or problems and the numbers involved as the size of or complexity of the numbers involved often influences the chosen methods of calculation. Decide upon the best method to use. This may be mental calculation with or without jottings, an empty number line, expanded written method, compact written method, using a calculator or another chosen method that the child is comfortable and accurate in using. Be aware of the importance of approximating and estimating an answer before carrying out the calculation and then complete th e calculation and check the appropriateness of their answer. When faced with a calculation, pupils are able to select an efficient method of their choice that is appropriate for a given task. They should always ask themselves. Have I estimated my answer? BT 2011 2 Can I do this in my head? Do I need jottings such as an empty number line? Do I need a pencil and paper method? Should I use a calculator? At whatever stage in their learning, and whatever method being used, it must still be underpinned by a secure and appropriate knowledge of number facts, along with those mental skills that are needed to carry out the process and judge if it was successful. Our overall aim is that when our children leave primary school they: have a secure knowledge of number facts and a good understanding of the four operations; are able to use this knowledge to carry out calculations mentally and to apply general strategies involving bigger numbers; make use of diagrams and jottings to help record steps and part answers when using mental methods that generate more information that they can keep in their heads; have an efficient and reliable written method of calculation for each operation that children can apply with confidence ; use a calculator effectively, using their mental skills to monitor the process, check the steps involved and decide if the numbers displayed make sense. Teaching children to calculate mentally – (DFE publication 2010) The four chapters of the booklet cover: 1. Progression in mental calculation skills This describes the progression in the number facts that children should derive and recall, the calculations that they are expected to do mentally and the range of calculation strategies or methods that they can draw on. 2. Principles of teaching mental calculation This promotes a broad interpretation of mental calculation and identifies principles that underpin teaching: for example, encouraging children to share their mental methods, to choose efficient strategies and to use informal jottings to keep track of the information they need when calculating. It also looks at the role of tests and questioning. 3. Addition and subtraction strategies BT 2011 3 This sets out the main strategies for adding and subtracting mentally. It describes activities to support teaching of these strategies and typical problems. 4. Multiplication and division strategies This sets out the main strategies for multiplying and dividing mentally. Again, it describes activities to support teaching of these strategies and typical problems. Written methods for addition of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence.. To add successfully, children need to be able to: • recall all addition pairs to 9 + 9 and complements in 10; • add mentally a series of one-digit numbers, such as 5 + 8 + 4; • add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value; • partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways. Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for addition. Written methods for subtraction of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence These notes show the stages in building up to using an efficient method for subtraction of two-digit and three-digit whole numbers by the end of Year 4. To subtract successfully, children need to be able to: • recall all addition and subtraction facts to 20; • subtract multiples of 10 (such as 160 – 70) using the related subtraction fact,16 – 7, and their knowledge of place value; • partition two-digit and three-digit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14). BT 2011 4 Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for subtraction Progression in Teaching Written Methods for Multiplication and Division It is important to teach both of the above operations together to ensure that pupils understand the relationship between them and they can use the inverse to check their calculations. We also need to be aware that when introducing the next stage the links between methods are made explicit by showing methods side by side. It is also necessary to reinforce the importance of estimating and approximating so children can identify if their answers make sense. Written methods for multiplication of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. These notes show the stages in building up to using an efficient method for two-digit by one-digit multiplication by the end of Year 4, two-digit by two-digit multiplication by the end of Year 5, and three-digit by two-digit multiplication by the end of Year 6. To multiply successfully, children need to be able to: • recall all multiplication facts to 10 × 10; • partition number into multiples of one hundred, ten and one; • work out products such as 70 × 5, 70 × 50, 700 × 5 or 700 × 50 using the related fact 7 × 5 and their knowledge of place value; • add two or more single-digit numbers mentally; • add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value; • add combinations of whole numbers using the column method BT 2011 5 Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for multiplication. Written methods for division of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence.. These notes show the stages in building up to long division through Years 4 to 6 – first long division TU ÷ U, extending to HTU ÷ U, then HTU ÷ TU, and then short division HTU ÷ U. To divide successfully in their heads, children need to be able to: • 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; • partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways; • recall multiplication and division facts to 10 × 10, recognise multiples of one-digit numbers and divide multiples of 10 or 100 by a single-digit number using their knowledge of division facts and place value; • 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. Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for division. To carry out written methods of division successful, children also need to be able to: • understand division as repeated subtraction; • estimate how many times one number divides into another – for example, how many sixes there are in 47, or how many 23s there are in 92; • multiply a two-digit number by a single-digit number mentally; • subtract numbers using the column method. BT 2011 6 As teachers we may need to consider how we present calculations to pupils. Calculations that are presented horizontally are more likely to prompt a pupil to ask him or herself the above questions. As pupils gain confidence at using expanded written methods of calculation we can encourage them to move towards a more compact form of recording. If errors appear at this stage teachers are advised to return to the expanded method stage until full understanding is consolidated. Pupils needs to be given regular opportunities to use and apply calculation methods efficiently to solve a range of problems including word problems and investigations. Addition Early stage s 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 Multiplication 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. Children will experience equal groups of objects. They will count in 1s, 2s and 10s and begin to count in 5s. Division Children will understand equal groups and share items out in play and problem solving. They will count in1s, 2s and 10s and later in 5s. They will work on practical problem solving activities involving equal sets or groups. Bead strings or bead bars can be used to illustrate addition 8+2=10 They use numberlines and practical resources to support calculation and teachers demonstrate the use of the numberline. Bead strings or bead bars can be used to illustrate subtraction including bridging through ten by counting back 3 then counting back 2. 6-2=4 They use numberlines and practical resources to support calculation. Teachers demonstrate the BT 2011 7 Addition Subtraction Multiplication Division use of the numberline. KS1 Continue using pictures to record Continue using pictures to record Children will experience equal groups of objects. Bead strings or bead bars can be used to illustrate addition including bridging through ten by counting on 2 then counting on 3. Bead strings or bead bars can be used to illustrate subtraction including bridging through ten by counting back 3 then counting back 2. They will count in 2s and 10s and begin to count in 5s. Children will understand equal groups and share items out in play and problem solving. They will count in 1s, 2s and 10s and later in 5s. They will work on practical problem solving activities involving equal sets or groups. 13-5=8 They use numberlines and practical resources to support calculation and teachers demonstrate the use of the numberline. Children then begin to use numbered lines to support their own calculations using a numbered line to count on in ones Children then begin to use numbered lines to support their own calculations - using a numbered line to count back in ones. The numberline 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. BT 2011 8 Addition Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on. First counting on in tens and ones. Subtraction Children will begin to use empty number lines to support calculations. Counting back: First counting back in tens and ones. Multiplication Division 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 x3 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? Repeated addition can be shown easily on a number line: Then helping children to become more efficient by adding the units in one jump (by using the known fact 4 + 3 = 7). Followed by adding the tens in one jump and the units in one jump. Bridging through ten can help children become more efficient. Then helping children to become more efficient by subtracting the units in one jump (by using the known fact 7 – 3 = 4). Subtracting the tens in one jump and the units in one jump. Bridging through ten can help children become more efficient. and on a bead bar: Grouping or repeated subtraction There are 6 sweets, how many people can have 2 sweets each? 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. Repeated subtraction using a number line or bead bar 12 ÷ 3 = 4 Arrays Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method. BT 2011 9 Addition Subtraction Multiplication Division Using symbols to stand for unknown numbers to complete equations using inverse operations ÷2=4 20 ÷ = 4 BT 2011 ÷=4 10 Addition 1. Empty number lines KS 2 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 Subtraction Children will continue to use empty number lines with increasingly large numbers. Children will begin to use informal pencil and paper methods (jottings). Partitioning 74 -27 = 74 – 20 – 7 = 54 -7 = 47 Expanded method Partitioning and decomposition Partitioning – demonstrated using arrow cards Decomposition - base 10 materials NOTE When solving the calculation 89 – 57, children should know that 57 does NOT EXIST AS AN AMOUNT it is what you are subtracting from the other number. Therefore, when using base 10 materials, children would need to count out only the 89. Multiplication Children will continue to use: Repeated addition 4 times 6 is 6 + 6 + 6 + 6 = 24 or 4 lots of 6 or 6 x 4 Children should use number lines or bead bars to support their understanding. Ensure that the emphasis in Y3 is on grouping rather than sharing. Children will continue to use: Repeated subtraction using a 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. Scaling e.g. Find a ribbon that is 4 times as long as the blue ribbon Children will begin to use informal pencil and paper methods (jottings) to support, record and explain partial mental methods building on existing mental strategies. Division Using symbols to stand for unknown numbers to complete equations using inverse operations x 5 = 20 3 x = 18 x= 32 Children should also move onto calculations involving remainders. Using symbols to stand for unknown numbers to complete equations using inverse operations The + sign may be taken out but leave a clear space between numbers 26 ÷ 2 = 24 ÷ = 12 ÷ 10 = 8 Partitioning 38 x 5 = (30 x 5) + (8 x 5) BT 2011 11 Addition 2. Partitioning Partitioning both numbers into tens and units mirrors the column method Using partitioning 43 + 65 = 40 + 60 + 3 + 5 = 100 + 8 =108 Subtraction Multiplication = 150 + 40 = 190 Division Begin to exchange. 1–6 Partitioned numbers are then written under on another 40 3 60 5 100 8 = 108 Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc counting on using a number line should be used. Expanded method in columns Write the numbers in columns adding units first Brackets can be removed when more children more confident BT 2011 12 Addition Column Method Subtraction Partitioning and decomposition Multiplication Children will continue to use arrays where appropriate leading into the grid method of multiplication. In this method the recording is reduced further Carry below the line under the correct column 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. £3.59 + 78p. Division Children will 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. 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 Decomposition Then onto the vertical method: Short division TU ÷ U 27 3 8 21 The carry digit ‘2’ represents the 2 tens that have been exchanged for 20 ones. In the first recording above it is written in front of the 1 to show that 21 is to be divided by 3. The 27 written above the line represents the answer: 20 + 7, or 2 tens and 7 ones. BT 2011 13 Addition Subtraction Multiplication Division Chunking method This method is based on subtracting multiples 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 threedigit sums of money, with or without ‘adjustment’ from the pence to the pounds; know that decimal points should line up under each other. 6 196 60 6 10 136 60 6 10 76 60 6 10 16 12 6 2 4 32 Answer: 32 R 4 Any remainders should be shown as integers, i.e. 14 remainder 2 or 14 r 2. Children need to be able to decide what to do after division and round up or down accordingly. They should make sensible decisions about rounding up or down after division to solve a word problem BT 2011 14 Addition Children should extend the carrying method to numbers with at least four digits. Subtraction Partitioning and decomposition 3 digit - 3 digit Multiplication 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 Division Children will continue to use written methods to solve short division TU ÷ U. Children can start to subtract larger multiples of the divisor, e.g. 30x Short division HTU ÷ U 6 196 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. 3.2 m – 280 cm. Decomposition 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 Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc counting on using a number line should be used. 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 180 6 30 16 12 6 2 4 32 Answer: 32 R 4 Using similar methods, they will be able to multiply decimals with one decimal place by a single digit number, approximating first. They should know that the decimal points line up under each other. e.g. 4.9 x 3 Children will approximate first 4.9 x 3 is approximately 5 x 3 = 15 BT 2011 15 Addition Subtraction Multiplication Division BT 2011 16 Addition Children should extend the carrying method to number with any number of digits. Subtraction Decomposition 4 digit – 4 digit Multiplication 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 Division Children will continue to use written methods to solve short division TU ÷ U and HTU ÷ U. Long division HTU ÷ TU 23 560 480 80 72 8 Answer: 23 R 8 24 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. 401.2 + 26.85 + 0.71. 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. Where the numbers are involved in the calculation are close together or near to multiples of 10, 100 etc counting on using a number line should be used. 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 = 10000 Extend to decimals with up to two decimal places. Children should know that decimal points line up under each other. 87.5 ÷ 7 1 2 . 5 7 Using similar methods, they will be able to multiply decimals with up to two decimal places by a single digit number and then two digit numbers, approximating first. They should know that the decimal points line up under each other. For example: 8 - 7 1 - 1 7 . 5 0. 0 7. 5 4. 0 3. 5 3.5 7 x10 7x2 7 x 0.5 0 4.92 x 3 Children will approximate first 4.92 x 3 is approximately 5 x 3 = 15 BT 2011 17 Addition Subtraction Multiplication Division Children who are already secure with multiplication should have little difficulty in using more compact method 286 x 29 X 286 29 2574 5720 286 x 9 286 x20 8294 1 1) 2) By the end of year 6, children will have a range of calculation methods, mental and written. Selection will depend upon the numbers involved. Children should not be made to go onto the next stage if: they are not ready. they are not confident. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods. BT 2011 18 Objectives The objectives in the revised Framework show the progression in children’s use of written methods of calculation in the strands ‘Using and applying mathematics’ and ‘Calculating’. Using and applying mathematics Calculating Year 1 • Solve problems involving counting, adding, subtracting, doubling or halving in the context of numbers, measures or money, for example to ‘pay’ and ‘give change’ • Describe a puzzle or problem using numbers, practical materials and diagrams; use these to solve the problem and set the solution in the original context Year 1 • Relate addition to counting on; recognise that addition can be done in any order; use practical and informal written methods to support the addition of a one-digit number or a multiple of 10 to a one-digit or two-digit number • Understand subtraction as ‘take away’ and find a ‘difference’ by counting up; use practical and informal written methods to support the subtraction of a one-digit number from a one-digit or two-digit number and a multiple of 10 from a two-digit number • Use the vocabulary related to addition and subtraction and symbols to describe and record addition and subtraction number sentences BT 2011 19 Using and applying mathematics Calculating Year 2 • Solve problems involving addition, subtraction, multiplication or division in contexts of numbers, measures or pounds and pence • Identify and record the information or calculation needed to solve a puzzle or problem; carry out the steps or calculations and check the solution in the context of the problem Year 2 • Represent repeated addition and arrays as multiplication, and sharing and repeated subtraction (grouping) as division; use practical and informal written methods and related vocabulary to support multiplication and division, including calculations with remainders • Use the symbols +, –, ×, ÷ and = to record and interpret number sentences involving all four operations; calculate the value of an unknown in a number sentence (e.g. ÷ 2 = 6, 30 – = 24) Year 3 Year 3 • Solve one-step and two-step problems involving numbers, money or measures, including time, choosing and carrying out appropriate calculations • Represent the information in a puzzle or problem using numbers, images or diagrams; use these to find a solution and present it in context, where appropriate using £.p notation or units of measure • Develop and use written methods to record, support or explain addition and subtraction of two-digit and three-digit numbers • Use practical and informal written methods to multiply and divide two-digit numbers (e.g. 13 × 3, 50 ÷ 4); round remainders up or down, depending on the context • Understand that division is the inverse of multiplication and vice versa; use this to derive and record related multiplication and division number sentences BT 2011 20 Using and applying mathematics Calculating Year 4 • Solve one-step and two-step problems involving numbers, money or measures, including time; choose and carry out appropriate calculations, using calculator methods where appropriate • Represent a puzzle or problem using number sentences, statements or diagrams; use these to solve the problem; present and interpret the solution in the context of the problem Year 4 • Refine and use efficient written methods to add and subtract two-digit and three-digit whole numbers and £.p • Develop and use written methods to record, support and explain multiplication and division of two-digit numbers by a one-digit number, including division with remainders (e.g. 15 × 9, 98 ÷ 6) Year 5 Year 5 • Solve one-step and two-step problems involving whole numbers and decimals and all four operations, choosing and using appropriate calculation strategies, including calculator use • Represent a puzzle or problem by identifying and recording the information or calculations needed to solve it; find possible solutions and confirm them in the context of the problem • Use efficient written methods to add and subtract whole numbers and decimals with up to two places • Use understanding of place value to multiply and divide whole numbers and decimals by 10, 100 or 1000 • Refine and use efficient written methods to multiply and divide HTU × U, TU × TU, U.t × U and HTU ÷ U BT 2011 21 Using and applying mathematics Calculating Year 6 • Solve multi-step problems, and problems involving fractions, decimals and percentages; choose and use appropriate calculation strategies at each stage, including calculator use • Represent and interpret sequences, patterns and relationships involving numbers and shapes; suggest and test hypotheses; construct and use simple expressions and formulae in words then symbols (e.g. the cost of c pens at 15 pence each is 15c pence) Year 6 • Use efficient written methods to add and subtract integers and decimals, to multiply and divide integers and decimals by a onedigit integer, and to multiply two-digit and three-digit integers by a two-digit integer BT 2011 22 BT 2011