1. Progression in Written Calculations North Hinksey CE Primary School

2. Understanding and Using Calculations For written calculations it is essential that there is a progression which culminates in one method. The individual steps within the progression are important in scaffolding children’s understanding and should not be rushed through. Practical equipment, models and images are crucial in supporting children’s understanding. For all calculations, children need to: Understand the = sign as is the same as, as well as makes. See calculations where the equals sign is in different positions and what these number sentences represent, e.g. 3 + 2 = 5 and 5 = 7 - 2. Decide on the most appropriate method i.e. mental, mental with jottings, written method or calculator Approximate before calculating and check whether their answer is reasonable.

3. Addition Children need to understand the concept of addition, that it is: Combining two or more groups to give a total or sum Increasing an amount They also need to understand and work with certain principles: Inverse of subtraction Commutative i.e. 5 + 3 = 3 + 5 Associative i.e. 5 + 3 + 7 = 5 + (3 + 7)

4. Models and images

5. Counting All Using practical equipment to count out the correct amount for each number in the calculation and then combine them to find the total, e.g. 4 + 2 Next, counting on, on a number track, e.g. 4 + 2

6. From Counting All to Counting On To support children in moving from counting all to counting on, have two groups of objects but cover one so that it can not be counted, e.g. 4 + 2 4

7. From using objects to numerals Relate counting on the objects to counting on the amount on a number track, e.g. 4 + 2 Then to counting on, on a number line, e.g. 5 + 4

8. The empty number line helps to record the steps on the way to calculating the total. The steps often bridge through a multiple of 10. 8 + 7 = 15 48 + 36 = 84 or: Year 2 and 3 onwards

9. Adding Two Digit Numbers Children can use base 10 (hundred, tens and units) equipment to support their addition strategies by basing them on counting, e.g. 34 + 29: Children need to be able to count on in 1s and 10s from any number and be confident when crossing tens boundaries.

10. The next stage is to record mental methods using partitioning. Partitioning both numbers into tens and ones mirrors the column method where ones are placed under ones and tens under tens. This also links to mental methods. Eg: 47 + 76 = 47 + 70 + 6 = 117 + 6 = 123 or 47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123 Next, partitioned numbers will be written under one another: Adding Two Digit Numbers - partitioning

11. Children can support their own calculations by using HTU stuff or jottings, e.g. 34 + 29 Beginning to understand ‘column addition’ – around Year 3 and 4

12. Adding Three Digit Numbers Children can support their own calculations by using jottings, e.g. 122 + 217

13. Understanding more about Column Addition – around Year 4 T U 6 7 2 4 1 8 0 9 1 + Note: Children should be taught that using a number line is sometimes more appropriate, e.g. in the context of time.

14. Continuing to understand Column Addition with larger numbers – around Year 5 e.g. 164 + 257 HT U

15. Efficient Column Addition – around Year 6 H T U + 4 1 2 1 1 164 2 5 7 T U. + 4 1 2 1.1 16.4 2 5.7

16. Subtraction Children need to understand the concept of subtraction, that it is: Removal of an amount from a larger group (take away) Comparison of two amounts (difference) They also need to understand and work with certain principles: Inverse of addition Not commutative i.e. 5 - 3 ≠ 3 - 5 Not associative i.e. (9 – 3) – 2 ≠ 9 – (3-2)

17. Models and images 5 - 2 10 - 3

18. Taking Away Using practical equipment to count out the first number and removing or taking away the second number to find the solution, e.g. 9 - 4

19. From using objects to numerals Relate taking away the objects to counting back the amount on a number track, e.g. 6 - 2 Then to counting back, on a number line, e.g. 94 - 7

20. Note: there are lots of ways to use numberlines and jottings to aid subtraction… but remember this is the progression into written subtraction.

21. Taking Away Two Digit Numbers Children can use base 10 equipment to support their subtraction strategies by basing them on counting, e.g. 54 - 23 31

22. Taking Away Two Digit Numbers Children can support their own calculations by using jottings, e.g. 54 - 23 31

23. Taking Away Two Digit Numbers (Exchange) Children can use base 10 equipment to support their subtraction strategies by basing them on counting, e.g. 54 - 28 26

24. Taking Away Two Digit Numbers (Exchange) Children can support their own calculations by using jottings, e.g. 54 - 28 26

25. Beginning Column Subtraction – around Year 4

27. Beginning Column Subtraction (Exchange)

29. Continuing Column Subtraction – around Year 5 e.g. 321 - 157 HT U 300201 100507 - 10 1 4 200 1 60100= 164

30. 1 Efficient Decomposition – around Year 6 H T U - 1 6 4 321 1 5 7 21 1

31. And we probably shouldn’t forget…Finding the Difference (Counting Back) Children need to understand how counting back links to subtraction, e.g. 7 – 4 Make the large tower the same size as the small tower.

32. Finding the Difference (Counting On) Children need to understand how counting on links to subtraction, e.g. 7 – 4 Make the small tower the same size as the large tower.

33. Finding the Difference (Counting On) To begin linking to number lines, this can be looked at horizontally instead of vertically.

34. Moving on to Number lines 61 - 52 52 61

35. Consolidating Number Lines

36. Multiplication Children need to understand the concept of multiplication, that it is: Repeated addition Can be represented as an array They also need to understand and work with certain principles: Inverse of division Is commutative i.e. 3 x 5 = 5 x 3 Is associative i.e. 2 x (3 x 5) = (2 x 3) x 5

37. Models and images

38. Use of Arrays Children need to understand how arrays link to multiplication through repeated addition and be able to create their own arrays.

39. Counting in multiples on a numberline Note: this is the progression for written multiplication, so goes alongside the practise of partitioning to multiply TUxU mentally. This is the step between repeated addition and arrays:

40. Continuation of Arrays Creating arrays on squared paper (this also links to understanding area). This also helps children to understand Fact Families. 2 x 4 = 4 x 2

41. Arrays to the Grid Method – around Year 4 7 10 6 70 42

42. Grid Method – around Year 5 7 10 6 70 42 70 42 112 + x200204 6120012024 1200 120 + 24 1344

43. Grid Method – around Year 6 600 100 90 15 805 + x203 3060090 510015 Children have to develop their understanding of related facts. e.g. 23 x 35 x20.3 481.2 8 + 1.2 = 9.2 The same process can be used for any size of number!

44. Division Children need to understand the concept of division, that it is: Repeated subtraction They also need to understand and work with certain principles: Inverse of multiplication Is not commutative i.e. 15 ÷ 3 ≠ 3 ÷ 15 Is not associative i.e. 30 ÷ (5 ÷ 2) ≠ (30 ÷ 5) ÷ 2

45. Models and images 

46. Division as Sharing Children naturally start their learning of division as division by sharing, e.g. 6 ÷2.

47. Division as Grouping To become more efficient, children need to develop the understanding of division as grouping, e.g. 6 ÷2.

48. Division as Grouping To continue their learning, children need to understand that division calculations sometimes have remainders, e.g. 13 ÷ 4. They also need to develop their understanding of whether the remainder needs to be rounded up or down depending on the context.

49. 93 ÷ 4 = ? 10 1 1 1 23 Add these together to find your answer. remainder 1 Don’t forget the remainder! Division as Grouping, counting more efficiently

50. -4 10 groups 2 groups 48 ÷ 4 = 12 Division as Grouping, counting on a Numberline

51. Division by using jottings – around Year 4 Use of multiplication tables helps the children to understand the link between multiplication and division, leading to a better understanding of chunking, and more confident use of chunking as a written method, e.g. 72 ÷ 3. 1x3=3 2x3=6 5x3=15 10x3=30 and perhaps even 20x3=60 And: 60 + 6 + 6 = 72 So: 20 groups of three + 2 groups of three + 2 groups of three = 24 groups

52. Division by using jottings – around Year 5 Children then write out the ‘chunks’ with subtotals as they go, leading to an even better understanding of chunking, and more confident use of chunking as a written method, e.g. 74 ÷ 3 10x3= 30 10x3= 30 2x3= 6 2x3= 6 72 [remainder 2]

53. Division by Chunking – around Year 6 Tracking back to using practical equipment can help this method, so the children see the link between division and repeated subtraction, e.g. 72 ÷ 3.

54. Division by Chunking e.g. 196 ÷ 6