Detailed Description

The purpose of this module is to familiarize mathematics teachers at the high school and community college levels with discrete dynamical systems. These systems, which can be analyzed with the tools of high school algebra, are discrete analogs to differential equation systems, which are based on calculus. As such, discrete dynamical systems can be used to introduce advanced algebra students to some of the most important ideas of calculus and differential equations. Also, discrete dynamical systems can be used to model many interesting problems on such diverse topics as: loans and annuities, population growth, radioactive decay, medication dosages, water and air pollution, and heating/cooling problems.

Teachers will learn how to set up and solve such problems and will also learn how to integrate content on discrete dynamical systems into their advanced algebra, precalculus and calculus courses.

A discrete dynamical system consists of a recurrence relation (or difference equation) describing some relationship, pure or applied, with a given set of initial conditions. The course examines a variety of discrete dynamical systems using an iterative and graphical approach facilitated by the use of calculators such as the TI-82, 83.. Algebraic methods for solving difference equations are also developed.

Particular emphasis is given to seeing the relationship between such discrete dynamical systems and real-world applications. Such applications are used to motivate the mathematical ideas, and both the illustrative examples and the homework exercises encourage those participating in the module to apply their knowledge to interesting and realistic settings. Through the experiences gained via these activities, the concepts "limit" and "solution" take on new meaning.

This course draws on the approach to Discrete Dynamical Systems employed in the Department of Mathematical Sciences at the U.S. Military Academy at West Point. The faculty there have been using this approach as an introduction to the calculus strand since the late 1980s. The material parallels the approach taken in the text Discrete Dynamical Modeling by James Sandefur (New York, NY: Oxford University Press, 1993. ISBN:0-19-508438-1). This approach, which bridges the gap between precalculus and the calculus, analyzes the sequences of values that result from iterating difference equations under varying initial values. These equations, by their very nature describe the shifts from one, or more, state(s) to another. As such, they are the discrete analog to differential equations.

This module is self-contained. You do not have to buy any additional textual materials. You do need to set aside ample time to wrestle with the problems and the content presented. Your TI-82 or TI-83 calculator will become a good friend before the module is over, in new ways that you may have never imagined possible before. If you are not familiar with the operation of the sequence mode giving dot output, the web format for the window, or the use of tables, you can acquire these skills through the Interactive TI-82 and TI-83 Tutorials.

The course is especially suited for Educators seeking ways to enhance their precalculus and calculus curriculum.