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Technology: A Portal to Exploration and Discovery GCTM October 18th, 2012

Technology: A Portal to Exploration and Discovery GCTM October 18th, 2012. Kenn Pendleton kennmg@aol.com. Background: Student experiences prior to this exploration.

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Technology: A Portal to Exploration and Discovery GCTM October 18th, 2012

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  1. Technology: A Portal to Exploration and Discovery GCTM October 18th, 2012 Kenn Pendleton kennmg@aol.com

  2. Background: Student experiences prior to this exploration Using natural number exponents, the three laws of exponents were developed through exploration. This likely was accomplished using a scientific calculator.

  3. Background: Student experiences prior to this exploration An exponent of zero was explored. A conjecture of the definition of a zero exponent was made and confirmed by applying the second law of exponents. The fact that the laws of exponents still held when exponents of zero were used was verified.

  4. Background: Student experiences prior to this exploration Negative exponents were explored. A conjecture of the definition of a negative exponent was made and confirmed by applying the second law of exponents. The fact that the laws of exponents still held when negative exponents were used was verified.

  5. Part I: Exploring Fractional Exponents

  6. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT tan →OFF AC X,θ,T Before turning on the calculator, notice the following keys. /ON

  7. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT tan →OFF AC X,θ,T REPLAY, or cursor: repeats processes and enables movement around the screen /ON

  8. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT tan →OFF AC X,θ,T MENU: accesses the main menu screen /ON

  9. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT X,θ,T EXIT: returns to the previous menu level when nested menus are accessed. OFF AC /ON

  10. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT tan →OFF AC X,θ,T Function keys: immediate access to screen functions and, when graphing, graph options. For example, /ON

  11. F1 F2 F3 F4 F5 F6

  12. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT tan →OFF AC X,θ,T AC/ON/OFF key: clears screens, turns the calculator on (and off, after having pressed SHIFT). /ON

  13. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT tan →OFF AC X,θ,T At the bottom-right, EXE(cute): performs intended operations and stores input /ON EXE

  14. F1 F2 F3 F4 F5 F6 ▲ ◄ REPLAY► ▼ SHIFT OPTN VARS MENU ALPHAx2 ^ EXIT tan →OFF AC X,θ,T Turn on the calculator. /ON

  15. This is the Main Menu Screen. We will start with Statistics activities. Notice the “2” in the upper-right corner.

  16. Either enter “2” from the keyboard, or cursor to the Statistics Icon and press EXE(cute) to select.

  17. F1 F2 F3 F4 F5 F6

  18. LIST 1 contains the exponents in the following expressions; LIST2 contains the values of the expressions. 42 41 40 4-1 4-2

  19. Create a graph. F1 F2 F3 F4 F5 F6

  20. Your calculator is set to use GRAPH1. F1 F2 F3 F4 F5 F6

  21. Return to the main MENU . F1 F2 F3 F4 F5 F6

  22. Graph a function. Enter “5,” or cursor to Graph and EXE.

  23. A number of functions have been entered. F1 F2 F3 F4 F5 F6 Try to draw a graph.

  24. EXIT back to the previous screen. F1 F2 F3 F4 F5 F6 EXIT is below MENU .

  25. With Y1 highlighted, SELECT this function. F1 F2 F3 F4 F5 F6

  26. Notice the verb is highlighted. F1 F2 F3 F4 F5 F6 DRAW the graph..

  27. This the same view window used earlier. F1 F2 F3 F4 F5 F6

  28. What is the value of 4½ ?

  29. Trace the function to find y when x = ½. Trace F1 F2 F3 F4 F5 F6

  30. y = 4x; 4½ = 2 What connection might students make between the expression and the result?

  31. 4½ = 4 × ½ = 2 If the above conjecture were correct, what would be the value of the following: 8½ = ? 8 × ½ = 4

  32. To test this conjecture, graph the function y = 8x and trace the graph to find the value of the function when x = ½ . EXIT back to the Graph window.

  33. Un-SELECT Y1; cursor to and SELECT Y2. F1 F2 F3 F4 F5 F6

  34. DRAW the graph of Y2. F1 F2 F3 F4 F5 F6

  35. Trace the graph to see whether 8½ = 4 as expected. Trace F1 F2 F3 F4 F5 F6

  36. The conjecture was incorrect. 4½ = 2 How else are 4 and 2 related? How is that relation connected to the fractional exponent?

  37. If the above conjecture were correct, what would be the value of 8½ ?

  38. Return to the main MENU . F1 F2 F3 F4 F5 F6

  39. Arithmetic is done in Run-Matrix. Enter “1,” or cursor to the icon and EXE.

  40. The square root symbol is in gold above the x2 key (SHIFT: x2). Find the square root of 8. To change from simplest radical (F)orm to (D)ecimal form: F↔D (above “8”). Return to the Graph window (MENU; 5).

  41. This new conjecture proved to be true.

  42. EXIT back to the Graph window. F1 F2 F3 F4 F5 F6 Un-SELECT Y2; SELECT Y3.

  43. DRAW the graph of Y3. F1 F2 F3 F4 F5 F6

  44. What is 9½ expected to be? Trace the graph. Trace F1 F2 F3 F4 F5 F6

  45. 2

  46. EXIT back to the Graph window. F1 F2 F3 F4 F5 F6 Un-SELECT Y3; SELECT Y2; DRAW.

  47. Trace the graph to x = 1/3 (0.3333333333). Trace F1 F2 F3 F4 F5 F6

  48. Change the viewing window (V-Window). V-Window F1 F2 F3 F4 F5 F6

  49. V-MEM; RECALL; 2; EXE . F1 F2 F3 F4 F5 F6

  50. EXIT back to Graph window.

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