St Swithun Wells Progression in Calculation Revised February 2015

Addition Year 1Addition Year 2Addition Year 3 Mental Methods Children need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’. 2 = = Missing numbers need to be placed in all possible places = = = 7 7 = + 4 Counting and Combining sets of Objects Combining two sets of which will progress onto adding on to a set (counting on). 8 5 = 13 Understanding of counting on with a numbertrack. Understanding of counting on with a numberline (supported by models and images) Mental methods Missing number problems e.g = = = It is valuable to use a range of representations (also see Y1). Continue to use numberlines to develop understanding of: Counting on in tens and ones = = = 35 Partitioning and bridging through 10. The steps in addition often bridge through a multiple of 10 e.g. Children should be able to partition the 7 to relate adding the 2 and then the = 15 Adding 9 or 11 by adding 10 and adjusting by 1 Add 9 by adding 10 and adjusting by = 44 Towards a Written Method Partitioning in different ways and recombine Leading to exchanging: 72 Partitioning = = 79 Mental methods Missing number problems using a range of equations as in Year 1 and 2 but with appropriate, larger numbers. Partition into hundreds tens and ones Using models and images from Year 2 Partition both numbers and recombine. Count on by partitioning the second number only e.g = = = = 372 Add near multiples of 10 and 100: , Number facts to 100 e.g Towards a Written Method Expanded column addition modelled with place value counters (Dienes could be used for those who need a less abstract representation) Leading to children understanding the exchange between tens and ones. Some children may begin to use a formal columnar algorithm Add like fractions, ½ + ¼ Using place value Counting on in tens is 45,55,65 Patterns using known facts = 7 so = 37 Know that addition is the inverse of subtraction Know that it is commutative

Addition Year 4Addition Year 5Addition Year 6 Mental methods Support by a range of models and images, including the number line and place value grid. Missing number/digit problems: __ = 2000 Add near multiples: Use number facts : Written methods (progressing to 4-digits) Expanded column addition modelled with place value counters, progressing to calculations with 4- digit numbers. Compact written method Extend to numbers with at least four digits. Extend to up to two places of decimals (same number of decimals places) and adding several numbers (with different numbers of digits). £ £ ___ £ Mental methods Support by a range of models and images, including the number line and place value grid. Missing number/digit problems: 6.00 = ____ Practise with increasingly large numbers to aid fluency: = Counting on: as 5.72 add 3 (8.72) then add 0.05 Add near multiples of 1: Adding to the next whole number: Adding to the next 10: Written methods (progressing to more than 4-digits) As year 4, progressing when understanding of the expanded method is secure, children will move on to the formal columnar method for whole numbers and decimal numbers as an efficient written algorithm. Add pairs of 5 digit numbers Add towers of larger numbers Place value counters/grids are used alongside the columnar method to develop understanding of addition with decimal numbers. Adding fractions with related denominators: 1/2 + 1/3 Mental methods Support by a range of models and images, including the number line and place value grid. Missing number/digit problems Written methods As year 5, progressing to larger numbers, aiming for both conceptual understanding and procedural fluency with columnar method to be secured. Continue calculating with decimals, including those with different numbers of decimal places Add fractions with unlike denominators: 2 ¾ + 1/3 = 3 1/12 1/3 + 2/4. Add like fractions 3/8 + 1/8 + 1/8 = 5/ ____ 64434

Subtraction Year 1Subtraction Year 2Subtraction Year 3 Mental methods Use concrete objects and pictorial representations. All number bonds to 20 e.g. to 9, to 17 Patterns using facts : 5-2, 15-2, 45-2 Missing number problems e.g. 7 = □ - 9; 20 - □ = 9; 15 – 9 = □; □ - □ = 11; 16 – 0 = □ Count back in tens: as 53,43,33 Understand subtraction as take-away: Understand subtraction as finding the difference: Bar model would be introduced with concrete objects which children can move (including cards with pictures) before progressing to pictorial representation. The use of other images is also valuable for modelling subtraction e.g. Numicon, bundles of straws, Dienes apparatus, multi-link cubes, bead strings Mental Methods Missing number problems e.g. 52 – 8 = □; □ – 20 = 25; 22 = □ – 21; 6 + □ + 3 = 11 Use partitioning e.g as 50 – 20 and 7- 2 (place value cards) It is valuable to use a range of representations (also see Y1). Continue to use number lines to model take-away and difference. E.g = Subtracts tens and multiples of 10: as 76,66,56,46,36 Subtract near multiples: or The bar model should continue to be used, as well as images in the context of measures. Towards written methods Recording addition and subtraction in expanded columns can support understanding of the quantity aspect of place value and prepare for efficient written methods with larger numbers. The numbers may be represented with Dienes apparatus. E.g. 75 – 42 Mental methods Support by a range of models and images, including the number line. Subtract multiples of 10, 100: £4.72-£2 Count back in 100s,10s and ones: as (663) (643) Find the difference by counting up: Using number facts = 65 Missing number problems e.g. □ = 43 – 27; 145 – □ = 138; 274 – 30 = □; 364 – 153 = □ Bar model: How many more pencils does Tom have? Children should make choices about whether to use complementary addition or counting back, depending on the numbers involved. Written methods (progressing to 3-digits) Introduce expanded column subtraction with no decomposition, modelled with place value counters (or Dienes see Year 2) For some children this will lead to exchanging, modelled using place value counters or dienes. A number line and expanded column method may be compared next to each other. Some children may begin to use a formal columnar algorithm, initially introduced alongside the expanded method. The formal method should be seen as a more streamlined version of the expanded method, not a new method. Subtract fractions from a whole: 1-1/4 = 3/4 4

Subtraction Year 4Subtraction Year 5Subtraction Year 6 Mental methods Support by a range of models and images, including the number line. Subtract multiples of 1000 or 0.1: Count back using partitioning 6482 – 1301 as 6482 – 1000, then – 300 then – 1 Find the difference by counting up : Missing number/digit problems: □ = 710; 1□7 + 6□ = 200; □ – 25 = 67; □ = 900 The bar model (see Year 3) should continue to be used to help with problem solving. Written methods (progressing to 4-digits) Expanded column subtraction with decomposition, modelled with place value counters, progressing to calculations with 4-digit numbers. If understanding of the expanded method is secure, children will move on to the formal method of decomposition, which again can be initially modelled with place value counters. Subtract like fractions: 3/8 – 1/8 = 2/8 Mental methods Support by a range of models and images, including the number line. Use place value to subtract decimals: Subtract multiples of powers of 10: 15, ; Subtract near multiples: 86,456 – 3999 or Count back e.g – 1051 or 5.72 – 2.01 Find the difference by counting up: Missing number/digit problems: 6.45 = □; □ = 86; □ = ; The bar model to help with problem solving. Written methods (progressing to more than 4-digits) Formal method of decomposition, which can be initially modelled with place value counters. Calculating with decimals, including those with different decimal places. Subtract fractions with like denominators: 1 ¼ - 3/8 as 1 2/8 - 3/8 Mental methods Use place value to subtract e.g – 0.08 Subtract powers of 10: 182,956 – 40,000 Count back: 3962 – 1051 Subtract near multiples: 360,987 – 99,998 Missing number/digit problems: □ and # each stand for a different number. # = 34. # + # = □ + □ + #. What is the value of □? What if # = 28? What if # = 21; = □ 7 – 2 x 3 = □; (7 – 2) x 3 = □; (□ - 2) x 3 = 15 The bar model to be used to help with problem solving (as year 5). Written methods Progressing to larger numbers, aiming for both conceptual understanding and procedural fluency with decomposition to be secured. Continue calculating with decimals, including those with different numbers of decimal places. Subtracting fractions with unlike denominators: 1 ¼ - 2/3 as 1 3/12 – 8/12 or 15/12 – 8/12 = 7/12

Multiplication Year 1Multiplication Year 2Multiplication Year 3 Mental methods Understand multiplication is related to doubling and combing groups of the same size (repeated addition) Washing line, and other practical resources for counting. Concrete objects. Numicon; bundles of straws, bead strings Problem solving with concrete objects (including money and measures Doubling Using fingers and objects to double : Double 5 Grouping Use visual and concrete arrays to find 3 lots of 5 or 5 lots of 3. Mental methods Count in 10s, 5s, 2s and 3s, Learn 10x, 5x, 2x tables. Develop understanding of multiplication using array and number lines. Begin to develop understanding of multiplication as scaling (3 times bigger/taller) Doubling Doubling numbers up to double 20 Doubles of multiples of 5 to 50 (double 35) Towards written methods Use jottings to develop an understanding of doubling two digit numbers double Mental methods Counting in 2s, 3s, 4s, 5s, 6s, 8s, 10s, 50s, 100s, tenths Learn the 3, 4, 6, 8 times tables Demonstrating multiplication on a number line – jumping in larger groups of amounts Recognising that multiplication can be done in any order Multiply by partitioning: 13 x 4 is 10 groups 4 and 3 groups of 4 Doubling Doubling 2 digit numbers using partitioning and jottings. Learn doubles of multiples of 5 to 100. Written methods Developing visual images into written methods The grid method 2 6 Developing a visual image

Multiplication Year 4Multiplication Year 5Multiplication Year 6 Mental methods Counting in multiples of 6, 7, 9, 11s, 12s 25s and 1000s, and steps of 1/100. Know times table up to 12 x 12 Doubling Double by partitioning: Double Double money: Double £3.50 Relate doubling to 2,4,8 times tables Grouping Multiply multiples of 100 linking to number facts: 400 x 8 Multiply near multiples by rounding 24 x 19 as (24 x 20) - 24 Relate multiplication to scaling : A 25cm sunflower grows 6 times taller Solve equations with missing digits: 2 x 5 = 160 Written methods Grid method Three digit by one digit numbers: Use a vertical algorithm to multiply 3 digit numbers by 1 digit numbers. Two digit times two digit numbers (Grid): Mental methods Use times tables to multiply multiples of the multiplier: 4 x 6 = 24 so 40 x 6 = 240 and 400 x 6 = 2400 Know square & cube numbers and prime numbers to 100 Identify factor pairs for numbers Doubling Use doubling and halving to multiply by 5 e.g. 58 x 5 is ½ of 58 (29) x 10 = 290 Use practical resources and jottings to explore equivalent statements (e.g. 4 x 35 = 2 x 2 x 35) Grouping Multiply decimals by 10, 100, 1000: 3.4 x 100 = 340 Use partitioning to multiply where helpful : 402 x 6 (400 x 6) + (2 x 6) Multiply near multiples: 32 x 29 as (32 x 30) - 32 Written methods (up to 4 digit x 2 digit) Explore how the grid method supports an understanding of long multiplication. Long multiplication of 2,3,4 digit number by teen numbers Short multiplication by one digit numbers Grid multiplication is a default method for all children Multiplying fractions by single digit numbers: ¾ x 6 = 18/4 which is 4 2/4 = 4 ½ Mental methods Identifying common factors and multiples of given numbers. Use times tables facts with decimal place e.g. 6 x 4 = 24 and 0.06 x 4 = 0.24 Doubling Double decimals numbers: Use doubling and halving for mental multiplication e.g x 5 as half of 3.42 x 10 Grouping Partitioning e.g x 4 (3000 x 4) + (60 x 4) Use factors e.g. 421 x 6 as 421 x 3 doubled. Use near multiples with decimals: 4.3 x 19 as (4.3 x 20) – 4.3 Written methods Long multiplication Short multiplication by 2 digit numbers by one digit numbers Grid method as default Multiplying proper and improper fractions

Division Year 1Division Year 2Division Year 3 Count in 2s, 5s and 10s. Talk about what they notice in number patterns. Group AND share small quantities- understanding the difference between the two concepts. Sharing Develops importance of one-to-one correspondence. Using objects Grouping Children should apply their counting skills to develop some understanding of grouping. Use of arrays as a pictorial representation for division. 15 ÷ 3 = 5 There are 5 groups of ÷ 5 = 3 There are 3 groups of 5. Halving: Find half of even numbers to 20. Fractions: Find ½ and ¼ and simple fractions of objects, numbers and quantities. Counting in 2s, 5s and 10s Learn the 2, 5, 10 division times tables Know and understand sharing and grouping- introducing children to the ÷ sign. Halving Find half of a number to 40 and discover that half of and odd number leaves a remainder of one or half of a number. Know half of multiples of 10 to 100: Half of 70 is 35 Grouping Relate division to multiplication - Group from zero in jumps of the divisor to find our ‘how many groups of 3 are there in 15?’. 15 ÷ 3 = 5 Relate grouping to repeated addition and multiplication How many 5s do I count to get to 20. Fractions: Sharing Find ½, ¼, or ¾ of quantities by sharing into two or four piles. Inverse Multiplication and division are inverse. Look at an array – what do you see? Counting in 3s, 4s, 6s, 8s and 50s Learn the 3, 4, 6, 8 division times tables Halving Find half of even numbers to 100 using partitioning Use halving to divide by 2 and by 4 Know halves of multiples of 10: Half of 170 = 85 Grouping How many 6’s are in 30? 30 ÷ 6 can be modelled as: Relate division to multiplication with missing numbers. 5 x ____ = 35 is the same as 30 divided by 5. Relate to division tables e.g. 32 divided by 8 is 4 so 320 divided by 8 is 40 Partition the dividend in different ways. 48 ÷ 4 = groups 2 groups Remainders 49 ÷ 4 = 12 r groups 2 groups Use place value to divide by 10 and ÷ 10 = How many groups of 10 in 60? 600 ÷ 100 = How many groups of 100 in 600? Fractions: Using division facts: ¾ of 48 is 3 lots of (48 divided by 4)

Division Year 4Division Year 5Division Year 6 Mental Methods Count in 6s, 7s, 9s, 11s, 12s, 25s, 1000s Learn the 6, 7, 9, 11, 12 division facts Halving Find halves of numbers to 200 and beyond with partitioning Find half of money : £9 halved is £4.50 Use halving as a strategy in dividing by 2,4,8: 164 ÷ 4 is half of 164 (82) halved again (41) Grouping Using division facts s facts : 77 ÷ 7 Use multiples of 10 times the divisor : 91 ÷ 7 (7 x 10 is 70, 3 x 7 is 21) Divide multiples of 100 by single digits using division facts: 3200 ÷ 8 = 400 Remainders Remainders are interpreted according to the context. (i.e. rounded up or down to relate to the answer to the problem) Fractions Use division facts to find fractions of amounts within the times tables (7/8 of 56 is 7 x (56 ÷ 8) Formal Written Methods Written version of a mental method Mental methods Use division facts to divide multiples of powers of 10: 3600 ÷ 9 using 36 ÷ 9 Halving Halve decimals using partitioning : £14.84 Use doubling and halving as a £7 42p strategy in dividing by 2,4,8,5,20 £8.42 e.g. 115 ÷ 5 as double 115 divided by 10 Grouping Divide numbers by 10, 100, 1000: 370 ÷100 = 3.7 Use multiples of 10 of the divisor: 186 ÷ 6 as 30 x 6 (180) and 1 x 6 (6) Fractions Find 3/5 of 265 as 3 x (265 ÷ 5) Formal Written Methods Bus stop division method for dividing by 1 digit numbers. Long division for dividing by 2 digit numbers Using 10 x multiples of the divisor Children begin to practically develop their understanding of how express any remainder as a decimal or a fraction. Mental methods Use of division facts to divide decimal numbers by single digit numbers: 1.17 ÷ 3is 1/100 of 117 ÷ 3 (0.39) Continue using a range of equations but with appropriate numbers Halving Halve decimals using partitioning : Use doubling and halving as a strategy in dividing by 2,4,8,5, e.g.5200 ÷20 as half of 5200 divided by 10 Grouping Use multiples to 10 or 100 of the divisor : 378 ÷ 9 as 40 x 9 = 360 and 2 x 9 = 18 Fractions Divide fractions by whole numbers e.g. ¼ ÷ 3 = 1/12 Formal Written Methods Use multiples of the divisor (listed for accuracy) May extend bus stop method, leaving more space for laying out. 0 Jottings 7 x 10 = 70 7 x 3 = 21 Division facts

With Thanks to National Centre for Excellence in the Teaching of Mathematics Affinity Teaching Schools Alliance Hampshire Mathematics Advisory Team Leicestershire School Improvement Service Hamilton Trust Calculation Methods and Strategies University of Northampton / Nottingham Trent University