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Calculations policy

Calculations policy. St Nicholas Catholic Primary School. outline the preferred calculation methods to be used when teaching the calculation elements of the Numeracy curriculum a consistent programme of teaching throughout the school with progression from year to year clearly shown. 

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Calculations policy

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  1. Calculations policy St Nicholas Catholic Primary School

  2. outline the preferred calculation methods to be used when teaching the calculation elements of the Numeracy curriculum • a consistent programme of teaching throughout the school with progression from year to year clearly shown.  • Parents can help their children at home using the same methods

  3. Addition and Subtraction Year 3 Programme of Study Pupils should be taught to: • add and subtract numbers mentally, including: • a three-digit number and 1s • a three-digit number and 10s • a three-digit number and 100s • add and subtract numbers with up to 3 digits, using formal written methods of columnar addition and subtraction • estimate the answer to a calculation and use inverse operations to check answers • solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction

  4. Addition and Subtraction Year 3 Methods: Recognise the use of symbols such as  to stand for unknown numbers and complete number sentences. 19 + ∆ = 33 ∆+ 14 = 33 10 + ∆ + 50 = 100 ∆ - 15 = 19 347 +  = 447 520 -  = 345 Addition: Develop partitioning numbers and empty number line; include partitioning second number only. Add a near multiple of 10 to a two digit number and show on a number line

  5. Addition and Subtraction Year 3 Bridge through a multiple of 10 to add, explaining method e.g. 67 83 24 + 42 + 80 120 11 5 91 125 Subtraction: Develop counting on or counting back with an empty number line Subtract by counting on with 2 and 3-digit numbers. Expanded decomposition (‘borrowing’) e.g 81 –57 = 81 80 1 = 70 11 - 57 -50 7 -50 7 20 4 =24 Subtract a near multiple of 10 from a 2-digit number, explaining the method used e.g. 96 – 39 = 96 – 40 +1 or use a number line Solve ‘real life’ problems involving more complex addition and subtraction problems

  6. Addition and Subtraction Year 4 Programme of Study Pupils should be taught to: • add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate • estimate and use inverse operations to check answers to a calculation • solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why

  7. Addition and Subtraction Year 4 Methods: Develop use of methods developed in Y1,2 and 3 to support and explain calculations where appropriate Addition: Begin expanded method, adding most significant digit first HTU + TU then HTU + HTU Add mentally from the top 625 205 + 48 + 176 600 300 60 70 13 11 673 381

  8. Addition and Subtraction Year 4 Progress to compact recording – including ’carrying’ 358 358 + 73 + 73 11 11 carried numbers 120 431 300 Solve problems explaining methods and reasoning orally and in writing. Subtraction: Continue to use counting up method, with informal jottings, when appropriate • When subtracting from multiples of 100 or 1000 • Finding a small difference by counting up e.g. 5003 – 4996 =7. (empty number line or jottings) Teach expanded decomposition then compact decomposition. 754 = 700 50 4 700 40 14 - 86- 80 6 - 80 6 = 600 140 14 6 '4 '4 - 80 6 - 8 6 600 60 8 6 6 8 = 668 Extend to decimals as appropriate. e.g. money - decimal points should line up under each other

  9. Addition and Subtraction Year 5 Programme of Study Pupils should be taught to: • add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction) • add and subtract numbers mentally with increasingly large numbers • use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy • solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

  10. Addition and Subtraction Methods: Develop methods from Y1, 2, 3 and 4. Addition: Use compact (‘carrying’) method. 587 3587 + 475 + 675 11 111 ‘carried’ numbers 10624262 3 digit then 4 digit then 5 digit numbers Addition of decimals ‘line up’ the decimal points particularly when adding mixed amounts e.g. 16.4 m. + 7.68 m. 1 6 . 4 + 7 . 6 8 1 1 ‘carried’ numbers 2 4 . 0 8 m.

  11. Addition and Subtraction Subtraction: Continue to use counting up method • When subtracting from multiples of 100 or 1000 • Finding a small difference by counting up e.g 8006 – 2993 = 5013. • Using known number facts and place value to subtract e.g 4.1 – 1.8 = 2.3 • To support or explain mental calculations • To support or explain the subtraction of the nearest multiple of 10 or 100 then adjust e.g Continue to develop compact decomposition with different numbers of digits and decimals. Understand the importance of lining up 5 '7 6 4 .' 0 8 2 1 . 6 4 9 4 2 . 4 Solve multi-step, ‘real-life’ word problems involving addition and subtraction.

  12. Addition and Subtraction Year 6 Programme of Study Pupils should be taught to: • compare and order fractions, including fractions >1 • add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions • solve problems which require answers to be rounded to specified degrees of accuracy

  13. Addition and Subtraction Methods: Develop methods from previous years (including decimals). Extend method to any number of digits and decimal places use complimentary addition; informal jottings and number lines: • 0.5 – 0.31 = 0.1 + 0.09 = 0.19 • Subtracting the nearest multiple of 10,100, 1000 • Subtracting from any multiple of 1000, 10,000 etc i.e. where using decomposition would be very complicated. Adding and subtracting fractions: • Make sure the denominators are the same – find a common denominator if necessary; • Add or subtract the numerators; • put the answer over the denominator; • Simplify the fraction (if needed).

  14. Addition and Subtraction To compare fractions – calculate a common denominator first, then compare the numerators For mixed numbers – convert to improper fractions first then proceed with the above steps. Continue to develop compact decomposition with different numbers of digits and decimals. Understand the importance of lining up Solve multi-step, ‘real-life’ word problems involving addition and subtraction – using different numbers of digits and decimals.

  15. Multiplication and division Year 3 Programme of Study Pupils should be taught to: • recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables • write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods • solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects

  16. Multiplication and division Year 3 Methods: Understand multiplication as: repeated addition describing an array Scaling Consolidate 2x, 5x and 10x tables and learn 3x, 4x and 8x tables. Be able to count in steps of 2,3,4,5,8 and 10 Record multiplications and divisions in a number sentence using the x,  and = signs.

  17. Multiplication and division Year 3 Recognise the use of symbols such as Δ or Ο to stand for unknown numbers e.g. 6 x Δ = 18 Δ x 3 = 18 35 = 7 8  = 2 8 = 16   Use knowledge of number facts and place value to multiply or divide mentally: • Multiply a single digit by 1,10 or 100. • Divide a three digit multiple of 100 by 10 or 100. • Double any multiple of 5 up to 50. • Halve any multiple of 10 to 100. • Multiply a 2-digit multiple of 10 up to 50, by 2, 3, 4, 5, 8 or 10. • Multiply a 2-digit number by 2, 3, 4, 5, 8 or 10 without crossing the tens boundary (e.g 23 x 3). Solve multiplication bypartitioning and recombining 17x5 10 x 5 = 50 7 x 5 = 35 17 x 5 = 85 Begin to use the Grid method: x 20 3 8 160 24 =184 23 x 8 = 184 Interpret situations as multiplication calculations. E.g. A baker puts 6 buns in each of 4 rows. How many buns does she make?

  18. Multiplication and division Year 3 Understand division as: Sharing equally, grouping and the inverse of multiplication. Show grouping using number lines. Interpret division number sentences e.g. 24  4 could mean: ‘If 24 tulips are shared equally between 4 plant pots, how many will be in each pot?’ Be able to round up or down after division, according to the context. Understand the concept of a remainder. Understand the relationship between multiplication and division; derive division facts for 2, 3, 4, 5, 8 x and 10x tables.

  19. Multiplication and division Year 4 Programme of Study Pupils should be taught to: • recall multiplication and division facts for multiplication tables up to 12 × 12 • use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together 3 numbers • recognise and use factor pairs and commutativity in mental calculations • multiply two-digit and three-digit numbers by a one-digit number using formal written layout • solve problems involving multiplying and adding, including using the distributive law to multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects

  20. Multiplication and division Year 4 Methods: Know by heart multiplication facts up to 12x table (including multiplication by 0 and 1) Understand that division is the inverse of multiplication and use this to check results. Develop partitioning (approximating first) 23 x 8 20 x 8 = 160 3 x 8 = 24 23 x 8 = 160 + 24 = 184

  21. Multiplication and division Year 4 Then Grid method x 300 20 3 8 2400 160 24 = 2584 323 x 8 = 2584 Progress to vertical expanded method (‘ladder’); most significant digit first. Expanded short multiplication 23 x 7 140 (20 x 7) 21 (3 x 7) 161 then the least significant digit first, to prepare for ‘Compact Standard Method’ i.e. 23 x 7 23 21 leading to x 7 140161 161 2 Interpret situations as multiplication calculations e.g. There are 4 stacks of plates. Three stacks have 15 plates each. One stack has 5 plates. How many plates are there altogether? (multi- step problem) (3x15)+(1x5)=

  22. Multiplication and division Year 4 Understand the operation of division as: • Grouping • Sharing • Repeated subtraction • The inverse of multiplication (and use this to check results) Start with modelling on a number line This leads on to ‘chunking’ / repeated subtraction i.e. 72 5 72 - 50 (10 x 5) or (10 groups of 5) 22 - 20 (4 x 5) or (4 groups of 5) 2 Answer: 14 r.2 Reason (through ‘real-life’ problems) whether to round up or down after division (involving remainders) depending on the context.

  23. Multiplication and division Year 5 Programme of Study Pupils should be taught to: • identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers • know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers • establish whether a number up to 100 is prime and recall prime numbers up to 19 • multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers • multiply and divide numbers mentally, drawing upon known facts • divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context • multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000 • recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³) • solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes • solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign • solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates

  24. Multiplication and division Year 5 Methods: Know by heart all multiplication facts up to 12 x 12, including multiplication by 0 and 1 and the effect of multiplying and dividing integers and decimals by 10, 100 and 1000 Understand that division is the inverse of multiplication and use this to check results. Recap grid method and ladder method from y3 and y4 Short Multiplication ‘Compact Standard Method’ (ThHTU x U). 2346 x 9 345 ‘carried’ numbers 21114 Approximate answers first. Long multiplication – begin with the ‘grid’ method. e.g. 72 x 38 (ans. approx. 70 x 40 = 2800) x 70 2 302100 60 = 2160 • 560 16 576 + 2736

  25. Multiplication and division Year 5 progress to compact method 72 x 38 72 x 30 2160 72 x 8 576 1 ‘carried’ number 2736 Interpret situations as multiplication calculations e.g. • I think of a number then divide it by 15. The answer is 20. What was my number? • There are 8 shelves of books. Six of the shelves hold 25 books each. Two of the shelves have 35 books each. How many books are there altogether on the shelves? Extend to simple decimals, with one decimal place, multiplied by a single digit. Approximate first. e.g. 4.9 x 3 is approx. 5x3 = 15 4.9 x 3 = (4.0 x 3) + (0.9 x 3) = 12 + 2.7 = 14.7 Leading to 4.9 x 3 2 ‘carried’ number 14.7

  26. Multiplication and division Year 5 Use ‘chunking’ for division (ThHTU by U) 20x and 30x the divisor, where appropriate. modelled on a blank number line Then as repeated subtraction E.g. 256  7 256 - 140 (20 x 7) 116 - 70 (10 x 7) 46 - 42 (6 x 7) 4 Answer: 36 r.4 approximating first to gain a sensible idea Progress to standard compact division recording e.g 197  6 3 2 r 5 6) 1 917 Answer: 32 r.5 Approximating first and lining up digits in the correct columns Work on round up or down after division (involving remainders) depending on the context. Solve more complex multi-step, ‘real-life’ word problems involving multiplication and division of larger numbers.

  27. Multiplication and division Year 6 Programme of Study Pupils should be taught to: • use common factors to simplify fractions; use common multiples to express fractions in the same denomination • multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 1/4 × 1/2 = 1/8  ] • divide proper fractions by whole numbers [for example, 1/3 ÷ 2 = 1/6] • associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8 ] • identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places • multiply one-digit numbers with up to 2 decimal places by whole numbers • use written division methods in cases where the answer has up to 2 decimal places • recall and use equivalences between simple fractions, decimals and percentages, including in different contexts

  28. Multiplication and division Year 6 Methods: Know by heart all multiplication facts to 12 x 12, including multiplication by 0 and 1 and the effect of multiplying and dividing integers and decimals by multiples of 10 e.g. 0.734 x 20 = Understand that division is the inverse of multiplication and use this to check results Short multiplication – recap the compact method (from Y5) Children should be able to multiply ThHTU x U Approximate the answers first. Multiply numbers with up to 2 decimal places by 1-digit numbers 8.62 x 7 = (8 x7) + (0.6 x7) + (0.02x7)= 56 + 4.2 + 0.14= 60.34 Long multiplication – recap grid method but focus on compact method extending to HTU x TU. Interpret situations as multiplication calculations e.g.: • There are 35 rows of chairs. There are 28 chairs in each row. How many chairs are there altogether? • 960 marbles are put into 16 bags. There is the same number of marbles in each bag. How many marbles are there in 3 of these bags? Continue to develop ‘chunking’ method using multiples of 10x the divisor (20/30x etc) – see year 5 examples.

  29. Multiplication and division Year 6 Develop the compact method for short division (bus stop) (see Y5). Teach long division (HTU  TU) using ‘chunking’ method that school prefers i.e. ‘repeated subtraction’ or ‘counting on’ method. Children should approximate answers first. Chunking: 977  36 is approximately 1000  40 = 25 977 - 360 (10 x 36) 617 - 360 (10 x 36) 257 - 180 (5 x 36) 77 - 72 (2 x 36) 5 Answer: 27 remainder 5 Develop to more compact methods for long division _____ 36) 972 - 720 (20 x 36) 252 - 252 (7 x 36) 0 Extend to decimals with up to 2 decimal places using ‘chunking’ or compact short division.

  30. Multiplication and division Year 6 Decide whether to round up or down after division (involving remainders) depending on the context. Fractions: • compare fractions with unlike, but related, denominators; • correctly use the terms fraction, denominator and numerator; understand what improper fractions and mixed numbers are and add fractions with the same denominator, writing the answer as a mixed number • express a remainder as a fraction, simplifying where possible. • Add and subtract unit fractions with different denominators including mixed numbers • Multiply fractions less than 1 by whole numbers, converting improper fractions to whole numbers; • use commutativity to efficiently multiply fractions by whole numbers; divide unit and non-unit fractions by whole numbers; • solve word problems involving fractions • multiply pairs of unit fractions and multiply unit fractions by non-unit fractions revise how brackets can be used in calculation problems, revise the order of operations for calculations involving the four operations 

  31. Online homework and resources • www.activelearnprimary.co.uk • www.mymaths.co.uk

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