St Nicholas Catholic Primary School

St Nicholas Catholic Primary School
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. Parents can help their children at home using the same methods

Addition and Subtraction
Reception EYFS Framework: Numbers: children count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer.

Reception Make own marks or tallies to record numbers. Begin to record numbers. and Select two groups of objects to make a given total. Find own way of recording for subtraction e.g cross-outs. 7 – 2 = 5 Solve practical problems in a real or role play context.

Year 1 Programme of Study Pupils should be taught to: read, write and interpret mathematical statements involving addition (+), subtraction (−) and equals (=) signs represent and use number bonds and related subtraction facts within 20 add and subtract one-digit and two-digit numbers to 20, including 0 solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7 = ? − 9

Year 1 Methods: Be able to complete number sentences where a missing number is shown by a symbol e.g. 5 + 2 = ∆ ∆ = 7 ∆ - 2 = = 10 - ∆ Record addition by: showing jumps on number lines. Use a number line to bridge through 10 e.g 8 + 5 =13 2 3 Use number lines for subtraction: count back (take away); count on (find the difference). Begin to understand when it is sensible to count back or to count on. Explain methods and reasoning orally Solve practical problems in a real or role play context.

Year 2 Programme of study Pupils should be taught to: solve problems with addition and subtraction: using concrete objects and pictorial representations, including those involving numbers, quantities and measures applying their increasing knowledge of mental and written methods recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100 add and subtract numbers using concrete objects, pictorial representations, and mentally, including: a two-digit number and 1s a two-digit number and 10s 2 two-digit numbers adding 3 one-digit numbers show that addition of 2 numbers can be done in any order (commutative) and subtraction of 1 number from another cannot recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems

Year 2 Methods: Use of symbols to stand for unknown numbers and complete number sentences e.g. 9 + ∆ = ∆+ ◊= 13 ∆ - ◊= = ∆ - 10 Extend to 3 numbers: 5 + ∆ + 4 = 13 Using number lines: count on in tens (e.g ) Partition 2 digit numbers to add number line (e.g = ) not using number line e.g. 35 +23: = = 8 = 58 using drawing = 50 e.g = 8

Year 2 Add 9 or 11 by adding 10 and adjusting by 1 – extend to adding 19 or 21. Bridge through a multiple of 10 to add Use a number line or record as: = Use number lines for subtraction: count back (take away); count on (find the difference). Bridge through a multiple of 10 Understand when it is sensible to count back and when to count on Use known number facts and place value to subtract, using - partitioning (second number only) e.g. 71 – 25 71 – 25 = 71 – 20 – 5 = 51 – 5 = 46 Number line - counting on Subtract 9 or 11 by subtracting 10 and adjusting by 1 or use a 100 square or a number line Begin to subtract 19 or 21 Explain methods and reasoning orally and by using any of recording methods shown above

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

Year 3 Methods: Recognise the use of symbols such as    to stand for unknown numbers and complete number sentences. 19 + ∆ = ∆+ 14 = ∆ + 50 = 100 ∆ =  =  = 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

Year 3 Bridge through a multiple of 10 to add, explaining method e.g. 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 = = =24 Subtract a near multiple of 10 from a 2-digit number, explaining the method used e.g. 96 – 39 = 96 – or use a number line Solve ‘real life’ problems involving more complex addition and subtraction problems

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

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

Year 4 Progress to compact recording – including ’carrying’ carried numbers 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 – 4996 =7. (empty number line or jottings) Teach expanded decomposition then compact decomposition. 754 = = '4 '4 = 668 Extend to decimals as appropriate. e.g. money - decimal points should line up under each other

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

Methods: Develop methods from Y1, 2, 3 and 4. Addition: Use compact (‘carrying’) method. ‘carried’ numbers 3 digit then 4 digit then 5 digit numbers Addition of decimals ‘line up’ the decimal points particularly when adding mixed amounts e.g m m. ‘carried’ numbers m.

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 ' ' 0 Solve multi-step, ‘real-life’ word problems involving 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

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 = = 0.19 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).

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.

Multiplication and division
EYFS Framework: Children solve problems, including doubling, halving and sharing. Solve practical problems in a real or role play context.

Methods: Oral counting in twos and tens. Examples of problems: How many pairs of socks are there in the drawer? Can you cut the cake in half? How many pieces are there? How many cakes are there in the box? Take half of them out.

Year 1 Programme of Study Pupils should be taught to: solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher

Year 1 Methods: Oral counting on and back in small steps 2s and 10s. Group and share small quantities. Double and halve simple numbers using resources / concrete objects

Year 1 Know the doubles of all numbers up to 10 and work out the equivalent halves. Combine groups of two and five and share a group of objects into two equal parts. Example problems: How many shoe lace holes are there on this shoe? We need to put 12 cakes into boxes of 3 or 4. How many boxes will we have? How many wheels do we need to make 3 cars?

Year 2 Programme of Study Pupils should be taught to: recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs show that multiplication of 2 numbers can be done in any order (commutative) and division of 1 number by another cannot solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts

Year 2 Methods: Oral counting on and back in small steps: 2s, 5s and 10s and know these times tables. Understand multiplication is repeated addition: lllll lllll lllll lllll = or 4 lots of 5 or 4 x 5 Or repeated jumps on a number line.

Year 2 Understand multiplication as an array. ● ● ● ● ● ● ● ● ● ● x 4 = 20 4 x 5 = 20 Recognise the use of symbols to stand for unknown numbers and the symbols x  and = 6 x Δ = Δ x 2 = 12 12  2 = = 12   2 = 6 Interpret situations as multiplication calculations and explain reasoning. e.g. How many wheels are there on 3 cars? Katy’s box is 5 cm wide. Mary’s box is twice as wide as Katy’s box. How wide is Mary’s box? Understand the operation of division as: sharing equally, grouping or repeated subtraction 6 sweets are shared equally between 2 people. How many sweets does each one get?

Year 2 How many groups of 5 can you make from 15? Use pictures or a number line to show sharing and grouping Use ÷ to show sharing/grouping i.e. 20 shared between ÷5= interpret division number sentences 20  4 ‘If £20 is shared between 4 people how much would each get?’ Understand the relationship between multiplication and division facts for 2x and 10x tables: e.g. 5 x 10 =50 so  10 = 5 etc.

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

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.

Year 3 Recognise the use of symbols such as Δ or Ο to stand for unknown numbers e.g. 6 x Δ = Δ x 3 =  = 7 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 by partitioning and recombining 17x5 10 x 5 = x 5 = 35 17 x 5 = 85 Begin to use the Grid method: x =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?

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.

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

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 = x 8 = 24 23 x 8 = = 184

Year 4 Then Grid method x = 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. x 21 leading to x 7 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)=

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  (10 x 5) or (10 groups of 5) 22 (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.

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

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 ‘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 = 2160 2736

Year 5 progress to compact method 72 x 72 x 72 x ‘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) = = 14.7 Leading to 4.9 x 3 ‘carried’ number 14.7

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  (20 x 7) 116 (10 x 7) 46 (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) 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.

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

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 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)= = 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.

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 (10 x 36) 617 (10 x 36) 257 (5 x 36) 77 (2 x 36) 5 Answer: 27 remainder 5 Develop to more compact methods for long division _____ 36) 972 (20 x 36) 252 (7 x 36) Extend to decimals with up to 2 decimal places using ‘chunking’ or compact short 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

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