As technology has become more advanced and widely accessible, numerical methods have become increasingly important and powerful tools in the world of mathematics. The purpose of this project is to give students the opportunity to explore Newton's Method for finding zeros of polynomials in both the real and the complex plane, to see the link between the topic and beautiful fractal art, and to explore more general iterative processes.
This investigation has three main components. The first component introduces and explores Newton's Method in the real plane. This component is essential to the project and must be completed first. The next two components are independent of one another, and teachers can use some or all of these materials, depending on the background of their students and the amount of time they would like to commit to the project. We generally try to give at least two weeks for projects like this one. Student materials and teacher materials, most with solutions, are provided so that teachers can see what content is involved in each section.
In the first component, students will learn the theory behind Newton's Method for finding real roots of polynomial functions. They also begin to learn about the basin of attraction of a root, which is the set of initial values whose iterates converge to that particular root. Finally, there is an optional extension that gives students the opportunity to look at Newton's Method from a different perspective - using quadratic functions and maximums or minimums instead of roots.
The second component extends Newton's Method into the complex plane. It begins with examples of finding real and complex roots for simple polynomials, then extends Newton's Method to find basins of attraction of complex roots of polynomials. Also included is a short video showing how to use the free program Winfeed (created by Rick Parris at Phillips Exeter Academy), which students can use to create beautiful images displaying basins of attraction.
If your students are interested in the artistic aspect of Newton's Method and basins of attraction, you may want to look into Polynomiography, the work of Dr. Bahman Kalantarithis of Rutgers University. This article provides information about the mathematics behind his beautiful images.
The final component of this project allows students to explore other iterative methods that can be used to solve equations.
If you choose to have your students work through this investigation and find errors or have questions, please e-mail one of us at: "belledin at ncssm dot edu" or "teague at ncssm dot edu." One of us (or our colleagues) will get back to you with support.
Note: Teacher handouts are password-protected. Please feel free to email one of us in the NCSSM math dept to receive the login credentials.