1. The Numeracy Professional Development Project in Secondary Schools Kevin Hannah National Coordinator, Secondary Numeracy Project Addition and Subtraction Strategies

2. Addition and Subtraction strategies The Number Strategy Framework A Model for a Teaching Progression Some Questions Using materials Encouraging Imaging Towards number properties Subtraction strategies Subtraction and addition problems From Number to Algebra - solving equations

3. Strategy Framework 0Emergent 1One-to-one counting 2Counting from one with materials 3Counting from one by imaging 4Counting on 5Early Part-Whole 6Advanced Part-Whole 7Advanced Multiplicative 8Advanced Proportional

4. Objective To explore links between numeracy and algebra In particular: To show that images used to help solve number problems also develop understanding for solving complicated linear equations.

5. Answers Only Please 1. 46 + 19 2. 61 – 27 3. 14 + ? = 101 4. 78 + 124 5. 9000 – 8985 6. 403 - 98 7. 47 + y = 83 8. 53 - m = 27 9. 2x + 1 = x + 7 10. 2x - 1 = 8 - x 11. 26 + 7 = ? + 12 12. 88 + x = 120 + ?

6. A Teaching Progression Start by: Using materials, diagrams to illustrate and solve the problem Progress to: Developing mental images to help solve the problem Extend to: Working abstractly with the number property

7. Using Materials 46 + = 83 4683 4 10 3 37

8. Using Materials 48 + = 8133 48 2 10 1 81 29 + = 7546 29 1 40 5 75

9. Encouraging Imaging 39 + = 63 3963 1 10 3 24 3040506070 10

10. Encouraging Imaging 28 + = 5426 204060 28 2 10 4 3050 10 54 16 + = 7357 16 4 50 3 2070 73

11. Using Number Properties 18 + = 6244 From 18:add 2 to get to 20 add 40 to get to 60 add 2 to get to 62 Total:add 44

12. Using Number Properties 39 + = 93 27 + = 52 46 + = 82 55 + = 72 17 + = 64 54 25 36 17 47

13. Where to from here? A subtraction strategy Other strategies on number line Link to Algebra - solving equations

14. Using Materials- subtraction 81 - 47 = 4781 3 10 1 34

15. Imaging - subtraction 3764 3 10 4 3040506070 10 64 - 37 =27

16. Imaging - subtraction 83 - 26 =57 26 4 50 3 3080 83

17. Using Number Properties From 17:add 3 to get to 20 add 60 to get to 80 add 2 to get to 82 Total:add 65 82 - 17 =65

18. Other strategies & problems The number line is a versatile tool and image. It can be used to support and explain a variety of strategies. It can be used to solve a wide range of problems. It can prepare students for algebra.

19. Other subtraction strategies 81 - 47 = 4181 6 10 1 34 10 34

20. Other subtraction strategies 81 - 47 = 3181 3 10 34 10 34 10

21. Other subtraction problems 92 - = 65 2792 10 27 510 92 - 65 =

22. Other subtraction problems 92 - = 65 6592 10 27 5 2 65+ = 92

23. Other subtraction problems - 18 = 43 4361 8 10 = 43 + 1861

24. Other subtraction problems - 18 = 43 1861 3 10 18 + 43 = 61 10

25. Other addition problems + 27 = 72 2772 10 45 510 + 27 = 27 +

26. Other addition problems + 27 = 72 4572 10 45 7 = 72 - 2745

27. Other addition problems 57 + 26 = 5783 10 6

28. What is Numeracy? It’s about making sense of numbers. It’s about problem solving with numbers. It’s about understanding base 10. It’s about learning with meaning. It’s about preparing for algebra

29. Solving Equations 47 + = 83 47 83 4783

30. Solving Equations 53 - x = 27 53 - 53 - x = 27 - 53 - x = -26 x = -26 ÷ -1 x = 26

31. Solving Equations 53 - x = 27 53 - x +x = 27 +x 53 = 27 +x 53 - 27 = 27 -27+x 26 = x

32. Solving Equations 53 - = 27 27 53 2753

33. Solving Equations 2X + 1 = X + 7 7X XX1

34. Solving Equations 2X + 1 = 7 7 X X 1

35. A Teaching Progression Start by: Using materials, diagrams to illustrate and solve the problem Progress to: Developing mental images to help solve the problem Extend to: Working abstractly with the property

36. Solving Equations 2X - 1 = X + 7 7 X 1 X X

37. Solving Equations X - 1 = 2X - 7 7 1 X X X

38. Solving Equations X - 1 = 2X - 7 7 1 X X X 7 = X + 1 X = 6

39. Solving Equations 2X - 1 = 8 - X X 1 8 X X

40. Solving Equations 2(X + 1) = 18 X 18 X 1 1

41. Solving Equations 2(X + 1) = 18 X X 1 1 9 9

42. Solving Equations 2(X + 1) = 18 X 18 X 1 1

43. Solving Equations X + 3 = 2 X 2 3

44. 10 X 4 Solving Equations

45. X4 10 Solving Equations

46. What is Numeracy? It’s about making sense of numbers. It’s about problem solving with numbers. It’s about understanding base 10. It’s about learning with meaning. It’s about preparing for algebra