Teaching Module: Iterations, Fractals and Chaos

Time spent

Brief Description

Content

Teachers’ Guide

Target

Form 4-7 students who are interested in mathematics.

Time spent

1.25 hours to 1.5 hours for each lesson.

Brief Description

In this module we aim to guide students to explore, under a unified framework, some phenomena concerning iterations, fractals and chaos. We emphasize concrete examples, and we believe that it is through working out concrete computations that students can have a better feeling of the mathematics behind these deep subjects. Therefore we have designed a series of student-oriented mathematical experiments throughout this module.

The module consists of a total of 7 lessons. In lessons 1 and 2 we introduce various examples of fractals including the Sierpinski Triangle, the Cantor Middle Thirds and the Snowflake. It will be seen how iterations of simple functions lead to complicated geometric objects like fractals. In lesson 3 the logistic map *f* (*x*) = *ax*(1-*x*) is introduced as a mathematical model of population growth. After that students are led to explore the iterations of this logistic map, and they will be asked to give an interpretation of the observed phenomenon in terms of population growth. Chaos will be observed in lesson 4 via the exploration of the logistic map, and some common misconceptions about chaos will be clarified. Lesson 5 presents a more theoretical explanation about some phenomena observed when we study the iterations of the logistic map in lesson 3. In particular the concept of stability will be emphasized. Towards the end of lesson 5 we guide students to study the iterates of the real quadratic map *f* (*x*) = *x*^{2} + *c*, which is a close relative of the logistic map. Then lesson 6 studies the iterates of the complex quadratic map, where students will meet the famous fractals: the Mandelbrot Set and Julia Sets. Finally we conclude the module with lesson 7, where students are led to explore the Newton’s method from an iteration point of view. Fractals again naturally arise when we solve equations iteratively using the complex Newton’s method. We hope finally students will appreciate how fractals, iterations and chaos come together naturally as a coherent whole.

Content

Teachers’ Guide

The teachers’ guide of this teaching module is avaliable here.