1. Calculation at Coton School

2. The national curriculum for mathematics aims to ensure that all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. (The National Curriculum in England September 2013)

3. Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. Pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects. (The National Curriculum in England September 2013)

4. The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on. (The National Curriculum in England September 2013)

5. Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject. (The National Curriculum in England September 2013)

6. In the following slides, there are examples of progression through a variety of calculation strategies. Decisions about when to progress are based on the security of pupils’ understanding and their readiness to progress to the next stage. Being able to calculate fluently enables children to solve problems more successfully.

7. Addition

8. Counting Children use a wide variety of concrete objects and pictorial representations to solve simple addition problems counting all. Then they use counting on, putting the largest number first.

9. Numberlines Teachers model the use of the numberline. Bead strings are used to illustrate addition including bridging through ten by counting on 2 then counting on 3 when doing 8 + 5. Children begin to use bead strings to support their own calculations.

10. Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on irrespective of the order of calculation. First counting on in tens and ones, then children become more efficient by adding the units in one jump. (e.g. by using the known fact 4 + 3 = 7). +10 +10 +3 34+23=57 34 44 54 57

11. This is followed by adding the tens in one jump and the units in one jump. +20 +3 34+23=57 34 54 57

12. Children will continue to use empty number lines with increasingly large numbers and decimals, including compensation where appropriate. +40 38+186=224 -2 186 224 226

13. Partitioning Children also use other informal pencil and paper methods (jottings) to support, record and explain mental methods building on existing strategies. 47 + 76 = 123 47 + 76 40 + 70 = 110 110 + 13 = 1237 + 6 = 13 110 + 13 = 123 arrowCards_revised_v5

14. Formal Written or Vertical Addition Secure informal methods lead to the development of more formal written methods. These all use, and require an understanding of, partitioning and place value. 60 7 + 20 4 80 11 = 91 6 7 + 2 4 8 0 1 1 9 1 2 6 7 + 8 5 1 2 1 4 0 2 0 0 3 5 2

15. From this, children will begin to regroup above the line. 3 6 7 2.84 + 8 5 + 1.36 1 1 1 1 4 5 2 4.20 With an understanding of place value, children know that decimal points line up under each other, particularly important when adding or subtracting mixed amounts, e.g. £3.59 + 78p.

16. Adding Fractions First add fractions with the same denominator. 2 + 2 = 4 5 5 5 Then apply their understanding of addition and equivalent fractions to add fractions with different denominators. 1 + 1 = 5 2 8 8

17. Subtraction

18. Counting Children use a wide variety of concrete objects and pictorial representations to solve simple problems ‘taking away’ finding how many are left when some are removed

19. ‘finding the difference’ by using practical objects/resources to make a comparison between the numbers ‘part/whole’ situations finding how many in a set fit a criteria e.g. 5 monsters, 3 are blue and the rest are red. How many are red?

20. Numberlines Teachers model the use of the numberline. Bead strings are used to illustrate subtraction including bridging through ten by counting back 2 then counting on 3 when doing 13 - 5. Children begin to use bead strings to support their own calculations.

21. Children begin to use numberlines to support their own calculations. Taking Away 1st) COUNTING BACK in tens and ones -1 -1 -1 -1 -10 20-14=6 6 7 8 9 10 20

22. 2nd) COUNTING BACK tens in one jump and units/ones in ones -1 -1 -1 -1 -20 40-24=16 16 17 18 19 20 40 3rd) COUNTING BACK tens in one jump and units/ones in one jump -4 -20 40-24=16 16 20 40

23. Finding the Difference COUNTING ON to be used when numbers are close together. +5 +60 +3 153-85=68 85 90 150 153 Children will continue to use empty numberlines with increasingly large numbers and decimals, including compensation where appropriate.

24. Partitioning Children also use other informal pencil and paper methods (jottings) to support, record and explain mental methods building on existing strategies. 78 - 43 = 3578 - 43 70 – 40 = 30 30 + 5 = 358 – 3 = 5 30 + 5 = 35 (example that does not need the children to regroup)

25. From this the children will begin to regroup 71 – 46 60 70 11 - 40 6 20 5 20 + 5 = 25 They should continue to use an empty number line where the numbers are close together or near to multiples of 10, 100 etc to count on.

26. Partitioning and decomposition 754 - 86600 140 700 50 14 - 80 6 600 60 8 = 668 Decomposition 6 14 14 754 - 86 7 5 4 - 8 6 6 6 8

27. Children should extend the decomposition method to numbers with any number of digits, including decimals in a range of contexts including money and measurement. Number lines and mental methods should also be used, if more efficient than decomposition.

28. Subtracting Fractions First subtract fractions with the same denominator. 3 - 2 = 1 5 5 5 Then apply their understanding of subtraction and equivalent fractions to subtract fractions with different denominators. 1 - 1 = 3 2 8 8

29. Multiplication

30. Counting Children count in twos, fives and tens. They use concrete objects and arrange them into groups or sets. They record their work using pictures, repeated addition or arrays. 2 lots of 3 = 6

31. Repeated Addition Three times five is also 5 + 5 + 5 = 15 or 3 lots of 5 This is easily shown on a numberline. 5 5 5 0 5 10 15

32. Commutativity The numberline also shows how 3 x 5 = 5 x 3 5 5 5 3 3 3 3 3

33. Arrays Arrays help children to model multiplication calculations. 5 x 3 = 15, 3 x 5 = 15, 15 is 3 lots of 5, 15 is 5 lots of 3

34. Scaling Up Children will develop an understanding of scaling and use comparative language. Doubling Understanding the concept of twice as big using partitioning as numbers get bigger. Double 46 80 12 = 92

35. Multiplying by 10, 100, 1000 Understanding the concepts of ten, one hundred and one thousand times bigger using place value charts, moving digits. THHTUthth

36. Partitioning Children also use other informal pencil and paper methods to support, record and explain mental methods and develop understanding to use in more formal written methods. 38 x 5 = (30 x 5) + (8 x 5) = 150 + 40 = 190

37. Array/Grid Method This builds on the understanding of arrays and partitioning in order to multiply larger numbers and decimals. 24 x 6 120 + 24 = 144 x204 612024

38. Short Multiplication Understanding from the array/grid method leads into more formal methods, expanded first, and then compact. 54 x 7 50 4 5 4 x 8 x 8 32 4 3 400 4 3 2 432

39. Long Multiplication Children often feel most secure using the array method/grid method for larger numbers, but it does lead into understanding of formal long multiplication. = 3290 = 987 4277 x300209 10300020090 39006027

41. Multiplying Fractions Children will use practical resources and diagrams to multiply proper fractions and mixed numbers by whole numbers. 2/3 x 3 (2/3 of 3) 2½ x 3 (3 lots of 2½) This leads into multiplication of pairs of proper fractions. 1/2 x 2/8 (one half of two eighths)

42. Division

43. Sharing Children use concrete object and share them. For example, sharing 12 eggs into 3 nests by placing an egg in each nest in turn. Grouping For example, children group 12 eggs into boxes of 6 by using repeated subtraction.

44. Arrays Arrays help children to model multiplication calculations. If I have 15 sweets, how many people can have 3 each? 15 is 3 lots of 5 15÷5=3 15 is 5 lots of 3 15÷3=5

45. Numberlines Counting on or back to find how many fours there are in 12. 4 4 4 12÷4=3 0 2 3 4 5 6 7 8 9 10 11 12

46. Children will develop their use of the numberline to add or subtract multiples of the divisor for calculations (with or without a remainder). 74÷3=24r2 3 3 3 3 -60 (20x3) 0 2 5 8 11 14 74

47. Scaling Down Children will develop an understanding of scaling and use comparative language. Halving Understanding the concept of half as big using partitioning as numbers get bigger. Halve 253 100 25 1.5 = 126.5

48. Dividing by 10, 100, 1000 Understanding the concepts of ten, one hundred and one thousand times smaller using place value charts, moving digits. THHTUthth

49. Chunking This leads on from using number lines and is another name for repeated subtraction where children subtract multiples of the divisor from the dividend. 14 6 - 60 (10 x 6) 24 - 24 (4 x 6) 0 84

50. Short Division This uses understanding of regrouping and place value. 1 2 r2 6 7 1 4 This is taught using practical resources.

51. Long Division The children may use formal methods including chunking to divide numbers up to 4 digits by 2 digit numbers.

52. Dividing Fractions Children will use practical resources and diagrams to divide proper fractions by whole numbers. 1/3 ÷2 = ⅙