1. From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy Project

2. Algebraic Thinking When students demonstrate they can use principles that are generally true and do not relate only to particular numbers or patterns they may be said to be using algebraic thinking.

3. 7 x 99 Recognising Structure 25 x 99 97 + 56

4. Algebraic Thinking A student who is using smart computational strategies implicitly understands the structure of our number system - place value, base 10, associative, commutative and distributive properties. They are thinking algebraically.

5. Extending from Computation Step 1: Build the computational strategies of students using visual images. Step 2: Exploit what the students are using computationally to develop algebra.

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

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

10. Extending from Computation Step 1: Build the computational strategies of students using visual images. Step 2: Exploit what the students are using computationally to develop algebra.

11. Extending from Computation Video Extending the picture used to build the computational skills Adjusting from a number line to a strip diagram

12. Solving Equations 19 + = 43 19 43 1943

13. Solving Equations 2X = 28 3n + 1 = 28 2p + 1 = p + 9

14. Solving Equations 2X = 28 X 28 X

15. Solving Equations 3n + 1 = 28 Does the 1 have to be that small?

16. Solving Equations 3n + 1 = 28 28 n 1 n n

17. Solving Equations: what students did 2p + 1 = p + 9 9p pp1

18. Solving Equations: what students did 2p + 1 = p + 9 9p pp1

19. Solving Equations: what students did 2p + 1 = p + 9 9p pp1

20. Solving Equations: what students did 2p + 1 = p + 9 5p pp1 4

21. Solving Equations: what students did 2p + 1 = p + 9 1p pp1 p

22. Solving Equations: what students did 2p + 1 = p + 9 1p pp1 8

23. Solving Equations: some more 4x = 28 6x + 2 = 44 5x + 4 = 2x + 25

24. The students’ equations 7x + 3 = 8x 9x + 7 = 10x + 3 4x + 3 = 3x + 13 3x + 5 = 29 5x + 5 = 2x + 38 5y + 7 = y + 23 11x + 2 = 10x + 5

25. Next: Abstract to... 13p + 6 = 10p + 42 3p = 36 p = 12 m + 41 = 7m + 5 6m = 36 m = 6

26. Solving Equations:brackets 2(X + 1) = 18 X X 1 1 9 9

27. Solving Equations: brackets 2(X + 1) = 18 X 18 X 1 1

28. Solving Equations: subtraction 19 + = 43 19 43 1943

29. Solving Equations: subtraction n - 17 = 64 n 6417 64? 17

30. Solving Equations:subtraction 2X - 1 = X + 7 7 X 1 X X

31. Reminder: A 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

32. Solving Equations:subtraction The goal is for students to work abstractly with algebraic equations. The strips are a prop on the way to doing this. There will come a point when the mental load of trying to construct a diagram is higher than the load required to manipulate an algebraic expression. The strips will have been dispensed with by this time. The following two examples probably come into this category.

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

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

35. Solving Equations: Integers Remember the goal is for students to work abstractly with algebraic equations. The strips are a prop on the way to doing this. And when it comes to integers, this visual representation doesn’t support them. So students need to have explored the structure of the equations and abstracted the principles before integers are introduced.

36. Solving Equations: Integers X + 3 = 2 X 2 3

37. Extending from Computation Step 1: Build the computational strategies of students using visual images. Step 2: Exploit what the students are using computationally to develop algebra.

38. 97 + 56 Recognising Structure

39. 97 + 56 = 100 +

40. Recognising Structure 97 + 78 = 100 +

41. Recognising Structure 97 + = 100 +

42. Recognising Structure 97 + x = 100 +

43. Recognising Structure 97 + = 100 + y

44. Recognising Structure 97 + = 100 +

45. Recognising Structure 88 + = 100 +

46. Recognising Structure 88 + = 120 +

47. Recognising Structure 88 + x = 120 + y x is 32 more than y y is 32 less than x