Exploring Sticky Problems in Algebra II using Algorithms

1. Teaching and doing TI-83 BASIC programming in Algebra II Exploring Sticky Problems in Algebra II using Algorithms Liz McClain and Steve Rives NCTM, October 2007, Kansas City www.NCTM.mrrives.com

2. Exploring Possibilities

3. You Can Do It! … They Need You To.

4. Programs: Simple and Powerful :Input "Real Comp: ",R :Input "Imaginary: ",I :Input "Size: ", S :ClrDraw :For(J,1,94) :For(K,1,64) :0->C :R+J*S->A :I+K*S->B :0->G :0->H :Pt-On(J,K) :Lbl TP :C+1->C :G^2 - H^2->T :2*G*H+B->H :T+A->G :G^2+H^2->V :If C>=25:Goto EP :If V>=4:Goto EP :Goto TP :Lbl EP :If V>4:Pt-Off(J,K) :End :End

5. Solving Problems, Exploring Ideas • Problem Solving with Algorithms • Exploring Patterns with Algorithms

6. Use an Algorithm to Find Where the Graph is Positive: f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive? I.e., where is f(x) > 0

7. Toolbox of Possibilites • How can we find the roots? Math Knowledge Thinking Chair Assign open-ended problems

8. Where is 6x4 – x3 + 4x2 – x – 2 above zero? • Algorithms Promote THINKING. Check left and right of each root

9. Newton’s Method!?

10. Rational Root Theorem! f(x) = 6x4 – x3 + 4x2 – x – 2

11. Use Rational Root Theorem in a TI-83 BASIC Program • List all factors of p and all factors of q • Loop through all p/q • Check to left and right of each root to find where function is positive or negative • Print results in interval format f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive? I.e., where is f(x) > 0

12. 1. Create a New Program f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

13. Name the Program f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

14. The Blank Program f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

15. 2. The Concept of a LIST f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

16. Two LISTs Prime Factors: L1 and L2 f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

17. 3. Loop Through Each List: FOR Loop f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

18. A FOR Loop will count for us f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

19. “I” is the variable that will count, starting at 1 f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

20. Count each element in our L1 list of factors f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

21. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

22. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

23. 4. L1(i) is the ith element. Calculate “R”, a possible root f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

24. 5. Plug “R” into f(x) and get “Y” as the result. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

25. 6. See If “Y” is zero (and therefore “R” is a root). f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

26. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

27. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

28. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

29. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

30. See If “Y” is zero (and therefore “R” is a root). f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

31. Use “End” to define the extent of the For loop f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

32. Two For Loops, therefore two “End”s f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

33. Ready to Run

34. 7. Exit Edit-mode and run the “POS” program f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

35. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

36. f(x) = 6x4 – x3 + 4x2 – x – 2 Where is f(x) positive. I.e., where is f(x) > 0

37. TO DO: CHALLANGES An Open Ended Problem

38. YOU CAN PROGRAM! Pass it along…

39. Algorithms Let Us Explore and Solve Sticky Problems in Algebra II

40. …Chasing Future Possibilities • 3D-Rotation and Rendering (Trig) • Monte Carlo Method (Calc) • Newton’s Method (Calc) • Snake Game (Algebra I and II) • Finding Roots (Algebra II) • Gaussian Elimination (College) • Fractals and Complex Numbers (Pre Calc) • Finding area under the Bell Curve for z-scores using a for loop taking small steps (Pre Calc).