1. Developmental Stages in Calculation Colehill First School 1st March 2013

2. Addition and Subtraction Part 1

3. Concrete Stage: Addition Concrete stage: with real objects putting together two sets of objects, with a number sentence: 3 + 2 = 5

4. Concrete Stage: Subtraction Concrete stage: with real objects taking a set of objects away from a larger set, with number sentence 4 – 2 = 2

5. Counting Forwards (Addition) and Backwards (Subtraction) Counting forwards along labelled number tracks or lines, with number sentences: 3 + 4 = 7

6. Counting Forwards (Addition) and Backwards (Subtraction) Counting backwards along labelled number tracks or lines, with number sentences: 5 – 2 = 3

7. Place Value Addition and Subtraction Place value addition using tens and units: 20 + 3 = 23

8. Place Value Addition and Subtraction Place value subtraction using tens and units: 15 – 5 = 10

9. Addition and Subtraction using the Hundred Square:

10. Addition By Counting On Using A Blank Number Line: 47 + 36 = 83 47 50 80 83 47 77 83

11. Subtraction By Counting Back Using A Blank Number Line: 37 – 24 = 13 13 20 30 37 13 17 37

12. Expanded Vertical Layout For addition: For subtraction: 47 +327 – 116 76 300 20 7 - 13 (7+6) 100 10 6 110 (40 + 70) 200 10 1 123 = 211

13. Subtraction Using The Expanded Vertical Layout With Decomposition: 53 – 28 = 50 3 - 40 13 20 8 20 5 = 25

14. The Compact Written Method For addition: 47+ 76 123 1

15. The Compact Written Method For subtraction: 53 – 28: 4 5 1 3 – 2 8 2 5

16. Multiplication and Division Part 2

17. Multiplication Concrete stage putting together equal sets, with counting: 1 2 3 4 / 5 6 7 8 / 9 10 11 12

18. Multiplication Drawing stage representing the concrete stage in pictures, with repeated addition: 5 + 5 = 10

19. Multiplication Counting forwards in jumps of greater than 1, both mentally and along labelled number tracks or lines, or on the hundred square, with number sentences written as repeated addition: 3 + 3 + 3 = 9

20. Multiplication Introducing the multiplication symbol as a shorthand form of recording: 2 + 2 + 2 = 6 or 3 x 2 = 6 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35 or 7 x 5 = 35

21. Multiplication Recognising equivalent multiplication, e.g. using arrays: 2 x 4 = 8 * * * * * * * * 4 x 2 = 8 * * * * * * * *

22. Multiplication Place Value: Multiplying by 10 and multiples of 10, using 0 as a place holder: 5 x 10 = 50 5 x 100 = 500 5 x 1000 = 5000

23. Multiplication Multiplication using partitioning: e.g. 38 x 7 = (30 x 7) + (8 x 7) 124 x 6 = (100 x 6) + (20 x 6) + (4 x 6)

24. Multiplication Grid Layout (expanded method): 124 x 6 X 100 20 4 6 600 + 120 + 24 =744

25. Multiplication Vertical method for multiplication: 38 x 124 x 7 6 210 (30 x 7) 600 (100 x 6) 56 (8 x 7) 120 (20 x 6) 24 ( 4 x 6) 266 744

26. Multiplication Multiplication using the compact written method: 38 X 124 X 7 6 266 744 5 1 2

27. Division Concrete stage : sharing sharing a set of objects between a group of people: 4 shared between 2 people gives 2 each.

28. Division Drawing stage : sharing representing the concrete stage in pictures, recording using the division symbol. e.g Sharing 6 sweets between 3 friends: Joe Lucy Lee * * * * * * 6 : 3 = 2

29. Division Concrete stage: grouping making groups or sets of a certain number: e.g. 15 grouped into 5s gives 3 groups: # # # # # # # # # # # # # # #

30. Division Division as repeated subtraction: Counting backwards in jumps of greater than 1 along labelled number tracks or lines, or on hundred squares, with number sentences: 8 -:- 4 = 2

31. Division Division as repeated subtraction without the number line, continuing until 0 (or a remainder) is reached, e.g. for 18 -:- 6: 18 – 6 = 12 – 6 = 6 – 6 = 0 I took 6 away three times, and so 18 -:- 6 = 3 And for 20 -:- 6: 20 – 6 = 14 – 6 = 8 – 6 = 2 I took 6 away 3 times, leaving 2 at the end, so: 20 -:- 6 = 3 remainder 2

32. Division Division using ‘chunking’ Chunking means subtracting larger groups, or chunks, of the divisor number. This saves time and reduces the length of the repeated subtraction: e.g. for 72 -:- 6 72 – 60 (10 X 6) 12 – 12 (2 X 6) 0 So, 72 -:- 6 = 12

33. Division Short formal written method for division: 12 6 72 and with remainders: 20 r 4 or 20.5 8 164

34. Division Long formal written method for division: 78 remainder 3 7 549 49 59 56 3