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Interactive Algebra Using TI-89 Calculators and TI-InterActive! Software NCTM Central Regional Conference Minneapolis, M

Interactive Algebra Using TI-89 Calculators and TI-InterActive! Software NCTM Central Regional Conference Minneapolis, Minnesota November 12, 2004. Lynda Plymate, Ph.D. lsm953f@smsu.edu ; math.smsu.edu/~lynda and David Ashley, Ph.D. dia059f@smsu.edu ; math.smsu.edu/faculty/ashley.html

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Interactive Algebra Using TI-89 Calculators and TI-InterActive! Software NCTM Central Regional Conference Minneapolis, M

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  1. Interactive AlgebraUsing TI-89 Calculatorsand TI-InterActive! SoftwareNCTM Central Regional ConferenceMinneapolis, MinnesotaNovember 12, 2004 Lynda Plymate, Ph.D. lsm953f@smsu.edu; math.smsu.edu/~lynda and David Ashley, Ph.D. dia059f@smsu.edu; math.smsu.edu/faculty/ashley.html Southwest Missouri State University Department of Mathematics Springfield, Missouri 65804-0094

  2. Can Calculators and Computers Increase Mathematical Reasoning? "At home and at work, calculators and computers are "power tools" that remove human impediments to mathematical performance – they extend the power of the mind as well as substitute for it – by performing countless calculations without error or effort. Calculators and computers are responsible for a "rebirth of experimental mathematics" (Mandelbrot 1994). They provide educators with wonderful tools for generating and validating patterns that can help children learn to reason mathematically and master basic skills. Calculators and computers hold tremendous potential for mathematics. Depending on how they are used, they can either enhance mathematical reasoning or substitute for it, either develop mathematical reasoning or limit it.” Steen, L.A. (1999). Twenty questions about mathematical reasoning. In L.V. Stiff & F.R. Curcio (Eds), Developing mathematical reasoning in grades K-12: 1999 Yearbook (pp. 270-285). Reston, VA : NCTM.

  3. Technology Used $49.95 Student Edition $64.95 Teacher Edition with Activity Book $135 School Pricing $150 Viewscreen Calculator $375 Viewscreen Package

  4. Computer Algebra System (CAS) • Recursion • Drugs and alcohol • Tim and Tom • Exact Arithmetic • Fractions • Radicals • 2D & 3D Grapher • Linear Functions • Level Curves • Parametric Graph • Numeric Solver • Area of Trapezoids • Symbolic Algebra • Solve equations and formulas • Function operations • Units and Conversions • Simulations • Spaghetti triangle simulation • Matrices • Polio Pioneers

  5. Algebra Standard Instructional programs from prekindergarten through grade 12 should enable all students to— • Understand patterns, relations, and functions • Represent and analyze mathematical situations and structures using algebraic symbols • Use mathematical models to represent and understand quantitative relationships • Analyze change in various contexts

  6. Number of Balloons Cost of Balloons (in cents) 20 40 60 80 ? ? ? 1 2 3 4 5 6 7 Number of Balloons 1 2 3 4 5 n Cost of Balloons (in cents) 20 40 60 80 ? ? Cost of Balloons Looking for PatternsGrade LevelExpectations in MissouriGrade 3 (left) Grade 6 (below) Cost of Balloons

  7. Looking for PatternsGrade-8 LevelExpectation in Missouri Write an equation which names the relationship between the two variables x and y for each of the following two graphs.

  8. Looking for PatternsGrade-10 LevelExpectation in Missouri Different views and explanations of the rational function

  9. Match the following scenarios with the graphs. Name & label (with appropriate numbers) the selected axes. • A 24 inch string when tied together can be made to form a infinite number of rectangles, with the area of the rectangle changing as the width of the same rectangle is made to get larger and larger. • The temperature of the filling in a frozen cherry pie increased dramatically when it is placed inside a preheated oven, then tapered off to a relatively steady hot temperature. • The population of frogs decreased as the pond became more polluted. • The diameter of the cocoon increased rapidly at first, then increased more slowly as the caterpillar prepared to change into a butterfly. • The temperature inside an oven increases when it is turned on, and then fluctuates a bit as the oven turns off and on briefly, trying to maintain a preset temperature. 6. The length of time it takes to paint the gymnasium changed as the number of people painting increased.

  10. The box plots shown below represent the ratings given to the 257 episodes, in the seven seasons, of Star Trek: The Next Generation (top plot is season 1 and bottom plot is season 7). These ratings, with 1 as the best and 257 as the worst, were determined by Entertainment Weekly magazine personnel. Use the center and spread in these plots to defend your choice of the “best season” for this program.

  11. Martha made the pattern shown below on her TI-89 calculator. One line in this pattern has equation y = x - 1. Determine the equations of the other 7 lines in this pattern. Use the numerical limits onthe x-axis and y-axis as references for your lines.

  12. Equal Pay for Equal Education?

  13. Name:_________________________ STUDENT-GENERATED EXPONENT RULESDr. Lynda Plymate Confidence Level • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________

  14. Power Versus Exponential FunctionsLynda Plymate f(x) := x a g(x) := a x f(x) = x 2 g(x) = 2 x Power vs. Exponential Functions

  15. Interactive Explorations Using TI-89 Calculators and TI-InterActive! sliders • Use recursion for interest Story of Tim and Tom • Explore parameters of functions Function Parameters • Explore area and perimeter of polygons Area and Perimeter • Use random numbers for simulation Spaghetti Triangle • Explore equations of circles Circles

  16. The Better Raise Imagine that you have been offered the position of Assistant Manager for Kinkos. You have been told that your beginning salary is to be $30,000 per year, but you must choose between the following two salary increase plans:Plan 1: $2000 raise per year Plan 2: $500 raise every 6 months Which plan is better? _____________________________________________ Are you sure about that? To verify your answer, prepare a spreadsheet similar to what is shown below and complete 20 years accumulated pay. (OR use Better Raise slider)

  17. PROBLEM: An offshore oil well is located in the ocean at point W, which is 8 miles from the closest shore point A on a straight shoreline. The oil is to be piped from W to a shore point B, which is 10 miles from A, by piping it on a straight line underwater to some shore point P, between A and B, and then on to B through an underground pipe along the straight shoreline. If the cost of laying the pipe is $50,000 per mile under water and $20,000 per mile under land, where should point P be located to minimize the cost of laying the pipe? (Note: Figure not drawn to scale) • ASSIGNMENT: • 1. Explore this problem numerically on a spreadsheet by choosing different values for the length of AP, PB, and WP, to compare and contrast the cost of laying the pipeline for locations of P. • The exact solution to the problem can be found by assigning variables to quantities that vary, finding an appropriate cost function and domain, and applying tools of differential calculus. Complete the problem again in this manner. • 1Bremigan, E. G. (2004). Note: Figures not drawn to scale. Mathematics Teacher, 98(2), 74-78.

  18. What Does Research Tell Us About Using Technology for Mathematics Instruction? In her 1997 meta-analysis of all U.S. research studies involving technology1, and her later 2001 meta-analysis of research involving CAS environments specifically2, M. Kathleen Heid reports findings from multiple researchers about using technology for mathematics instruction. She reflects on issues about the nature of technology use, learning issues, curriculum issues, and teacher preparation issues. Some of her discussion and findings are listed below: • The use of calculators does not lead to an atrophy of basic skills. • Symbolic manipulation skills may be learned more quickly in areas such as introductory algebra and calculus after students have developed conceptual understanding through the use of cognitive technologies. • The concepts-before-skills approach using CAS in algebra and calculus courses, and the inductive-before-deductive investigatory approach in geometry have been tested. • Graphics-oriented technology may level the playing field for males and females. • CAS students were more flexible with problem solving approaches and more able to perceive a problem structure. • CAS students were able to move between representations and make connections. • CAS students outperformed others on tests, with and without use of CAS during test. • 5 of 7 researching experts (on a panel) reported better understanding by CAS students 1 Heid, M. K. (1997). The technological revolution and the reform of school mathematics, American Journal of Education, 106(1)5-61. 2 Heid, M. K. (2001). Research on mathematics learning in CAS environments. Presented at the 11th annual ICTCM Conference, New Orleans.

  19. Discussion Questions • Questions of Access: • Use if all students don’t have the technology • Use technology in and out of class • Convince others of it’s value • Questions of Instruction: • Focus on mathematics not how to use technology • Concepts before skills (problem solving first) • Balance CAS usage with mental and paper work • Questions of Assessment: • New types of test questions and test format • Match testing, instruction & learning strategies

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