2. Enhancing Algebra Instruction Through the Use of Graphing Technology Bill Gillam 10/18/02 wbgillamjr@aol.com Http://billandcimgillam.com

3. Carla’s Lemonade Stand # glasses$ Profit 0 1 2 3 4 5 6 -9.50 Input: number of glasses Output: Profit Initial costs are $9.50. Each glass costs $0.75

4. Carla’s Lemonade Stand # glasses$ Profit 0 1 2 3 4 5 6 -9.50 -8.75 Input: number of glasses Output: Profit Initial costs are $9.50. Each glass costs $0.75

5. Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit 0.000 1.000 2.000 3.000 4.000 5.000 -9.500 -8.750 -8.000 -7.250 -6.500 -5.750

6. Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit

7. P = -9.50+0.75g Carla’s Lemonade Stand Input: number of glasses Output: Profit

8. Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit Rule: P(n) =0.75n - 9.50 n = 0, 1,2,3….. \Y1= 0.75X – 9.50 \Y2= \Y3= \Y4= \Y5= \Y6=

9. Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit Rule: P(n) =0.75n - 9.50 n = 0, 1,2,3….. 12.000 13.000 14.000 15.000 -.500 0.250 1.000 1.750

10. Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit Rule: P(n) =0.75n - 9.50 n = 0, 1,2,3…..

11. Variables Related by Rules # Lawns$ Profit 0 5 10 15 20 25 30 -195 -135 -75 -15 45 105 165 Rule:

12. Variables Related by Rules # Lawns$ Profit 0 5 10 15 20 25 30 -195 -135 -75 -15 45 105 165 Rule: -195 + 12x = profit

13. Variables Related by Rules Summary 1. Tables, Graphs & Rules 2. Income, Expense & Profit Power Point by Bill Gillam Adapted from Concepts in Algebra: A Technological Approach

14. Variables Related by Data in Graphs: Part II - Interpreting Graphs X Y AGE

15. Variables Related by Data in Graphs: “The Hiker”

16. Variables Related by Data in Graphs: Hike This! Time

17. Variables Related by Data in Graphs: Hike This!

18. Time Variables Related by Data in Graphs: Hike This!

19. Time Variables Related by Data in Graphs: Hike This!

20. Time Variables Related by Data in Graphs: Hike This!

21. Time Variables Related by Data in Graphs: Hike This!

22. Time Variables Related by Data in Graphs: Hike This!

23. Time Variables Related by Data in Graphs: Hike This!

24. Time Variables Related by Data in Graphs: Hike This!

25. Time Variables Related by Data in Graphs: Hike This!

26. Time Variables Related by Data in Graphs: Hike This!

27. Time Variables Related by Data in Graphs: Hike This!

28. Variables Related by Data in Graphs: Interpreting Graphs 1. CBL Stations 2. Graph Stories

29. Variables Related by Data in Graphs: Interpreting Graphs Summary 1. Speed 2. Rate of Change

30. Variables Related by Data in Graphs: Interpreting Graphs Power point by Bill Gillam

31. Rule: Y = *x + f(0) Variables Related Linearly # Lawns$ Profit 0 5 10 15 20 25 30 -250 -200 -150 -100 -50 0 50 5 5 5 5 5 5

32. Rule: Y = 10x + - 250 Variables Related Linearly # Lawns$ Profit 0 5 10 15 20 25 30 -250 -200 -150 -100 -50 0 50 5 5 5 5 5 5

33. Variables Related Linearly # Lawns$ Profit 0 5 10 15 20 25 30 -195 -135 -75 -15 45 105 165 Rule: Y = + f(0)

34. Variables Related Linearly # Lawns$ Profit 0 5 10 15 20 25 30 -195 -135 -75 -15 45 105 165 Rule: Y = 12x -195 5 5 5 5 5 5 60

35. Variables Related Linearly Rule: Y = 12x - 195 ProfitProfit lawns 4 48 12=195/x x =16.25

36. Variables Related by Rules Fundraiser Activity

37. Variables Related Linearly Summary 1. Ratio of change between any two points is constant. 2. Rule: y = Mx + b 3. Graphs a straight line.

38. Variables Related by Rules Summary Powerpoint by Bill Gillam

39. Investigating Linear Patterns in Data 7/22/2002

40. “If he was all on the same scale as his foot, he must certainly have been a giant.” - Sherlock Holmes The Adventure of Wisteria Lodge. Investigating Linear Patterns in Data

41. Footlength and Height Question: Is the length of a person’s foot a useful predictor of his/her height?

42. Footlength and Height 1. Measure and record each student’s footlength and height in a table. 2. Graph the ordered pairs: (footlength, height). 3. Discuss

43. Strategies: 1. Studying numerical patterns (table) 2. Matching a graphical pattern (graph) 3. Using function fitting tools (symbolic) Investigating Linear Patterns in Data

44. Studying numerical patterns (table) FootHeight 21.2156 22.1159 22.2161 22.3154 22.5158 22.6162.5...

45. Matching a graphical pattern (graph)

46. Using function fitting tools (regression) y = mx + b

47. Symbolic Representation of Line of Best Fit y = mx + b

48. Strategies: 1. Studying numerical patterns (table) 2. Matching a graphical pattern (graph) 3. Using function fitting tools to obtain a modeling function (symbolic) Investigating Linear Patterns in Data

49. Power point by Bill Gillam Investigating Linear Patterns in Data

52. Exponential Growth According to legend, chess was invented by Grand Vizier Sissa Ben Dahir, and given to King Shirham of India. The king offered him a reward, and he requested the following: "Just one grain of wheat on the first square of the chessboard. Then put two on the second square, four on the next, then eight, and continue, doubling the number of grains on each successive square, until every square on the chessboard is reached."

53. According to legend, chess was invented by Grand Vizier Sissa Ben Dahir. He presented the game to King Shirham of India. The king offered him a reward, and he requested the following: Exponential Growth eight, and continue, doubling the number of grains on each successive square, until the last square is reached." ”Place one grain of wheat on the first square. Put two on the second square, four on the third, then

54. Exponential Growth You may give me the wheat or its equal value on the 64th day. This is all I require for my services. The king agreed, but he lost his entire kingdom to Sissa Ben Dahir. Why?

55. Exponential Growth square/dayriceSum 111 223 347 4815 51631 63263 764127... 64__________ How much wheat did the King owe for 64th day? How much wheat in all?

56. Exponential Growth In all, the king owed about 18,000,000,000,000,000,000 grains of wheat. This was more than the worth of his entire kingdom!

57. Exponential Growth There is a function related to this story: f(x)=2^x day rice sum rice dayrice2^(day-1) 2^day-1 112^0 = ____ 2^1 - 1 = ____ 222^1 = ____ 2^2 - 1 = ____ 342^2 = ____ 2^3 - 1 = ____ 482^3 = ____ 2^4 - 1 = ____... 64____2^63 = ____2^64-1 = ____ Copy and fill out this chart.

58. Exponential Growth

59. Moore's Law (from the intel website): http://www.intel.com/research/silicon/mooreslaw.htm

60. Exponential Growth Gordon Moore (co-founded Intel in 1968) made his famous observation in 1965, just four years after the first planar integrated circuit was discovered. The press called it "Moore's Law" and the name has stuck. In his original paper, Moore predicted that the number of transistors per integrated circuit would double every 18 months. He forecast that this trend would continue through 1975. Through Intel's technology, Moore's Law has been maintained for far longer, and still holds true as we enter the new century. The mission of Intel's technology development team is to continue to break down barriers to Moore's Law.

61. Exponential Growth chip Year Transistors 4004 1971 2,250 8008 1972 2,500 8080 1974 5,000 8086 1978 29,000 286 1982 120,000 386 processor 1985 275,000 486 DX processor1989 1,180,000 Pentium® processor 1993 3,100,000 Pentium II processor 1997 7,500,000 Pentium III processor 1999 24,000,000 Pentium 4 processor 2000 42,000,000 Produce a plot of year vs. transistors

62. Exponential Growth- Moore’s Law chip Year Transistors 4004 1971 2,250 8008 1972 2,500 8080 1974 5,000 8086 1978 29,000 286 1982 120,000 386 processor 1985 275,000 486 DX processor1989 1,180,000 Pentium® processor 1993 3,100,000 Pentium II processor 1997 7,500,000 Pentium III processor 1999 24,000,000 Pentium 4 processor 2000 42,000,000 Produce a plot of year vs. transistors (from the intel website): http://www.intel.com/research/silicon/mooreslaw.htmhttp://www.intel.com/research/silicon/mooreslaw.htm

63. Exponential Growth Review of how to do a point plot: "STAT" "Edit" enter year in L1 and transistors in L2. "2nd" "Y=" "Plotsoff" "Enter" “Enter" "2nd" "Y=" Choose Plot1 {On, Scatterplot, L1, L2, mark} "Zoom" 9

64. Exponential Growth Moore’s Law indicates that the growth should be modeled by: Y=2250*2^((12/18)x) or Y=2250*2^(x/1.5) Is this accurate? Use exponential regression to find a curve that fits.

65. Exponential Growth 1. Describe the graph: 2. How does this relate to the rice problem? 3. Can you think of other things that “grow” this way (ie. Doubling over a constant period of time?)

66. Exponential Graphs *************************************

67. Exponential Graphs 1. Which graph is most likely 4^x? 2. Which is most likely 0.25^x? 3. Which is most likely 1^x? ************************************* Y= 4^xY = 0.25^x Y = 1^x

68. Exponential Graphs Graph the following pairs of functions on your calculator: A) y = 0.5^x and y = 2^x B) y = 0.25^x and y = 4^x C) y = (3/4)^x and y = (4/3)^x ************************************* Make a conjecture about the relationship between these graphs.

69. Exponential Graphs ************************************* Make a statement about the relationship between these graphs.

70. Exponential Graphs *************************************

71. Exponential Graphs ************************************* 1. Sketch a guess for the graph of y = 4*3^x.

72. Exponential Graphs ************************************* Sketch a guess for the following graphs, then check them on your calculator to see how close you came. Y = 5^x; Y = 2*5^x; Y =0.2^x; Y = 6*(0.2^x); Y = -3^x and Y = -0.2^x. Do exercises 1 - 4