# CHAOS AND COMPUTERS/Lesson Plan

Luci Ryan
Leeds Middle School

## AIM

Students will be introduced to the Chaos Theory by exploring and generatingfractals and iterative processes; Students will observe order depending on randomness relative to Sierpinski and Pascal's Triangles; Students will discover the process of celluar automata, by changing specific rules in order to observe emerging patterns/structures;Students will discuss how the chaos theory may be applied to life; Lessons will be brought to closure by playing the Chaos Game which will allow students to track orbiting after numerous iterations. Then play the Chaos Game.

## CONTENT

Fractals for the Classroom : Strategic Activities Pages 11; 23; 29; 35; 47; Mary Ann Connors, web site Fractals for the Classroom, Springer-Verlag, 1992; For more information on CHAOS, FRACTALS, etc., please refer to serendip

## PROCEDURE

Introduce new terms:(chaos,randomness,fractals,iterations,coloring look up table,cellular automata,Pascal's Triangle,Sierpinski's Triangle,cell, order, self-similarity,dimension) Display Pascal's Triangle on overhead projector. Provide background in reference to its genesis via lecture. Students will begin to fill in the missing numbers in order to create the triangle. Color the odd and even numbers in the first three rows with two distinct colors. Explain how these numbers have been used to solve various probability problems. Students continue to practice creating the Pascal Triangle through row 20 and try to discover its rule.
They will color cells by using a special rule realizing that when the rule changes, new structures are being created. By carrying out this application of cellular automata, they will discover and explore coloring look-up tables. Questions to be Answered: How many numbers are in row 9? Row 10? Row 11? How many will be in row n? What pattern emerges? Observations? Distribute handout # 1:2 Cellar Automata. Discuss how rules relate to randomness. Allow students to change rules. Make observations and solicit ways in which this process might be used to determine fluid flow, fertilization of soil,etc.
Students will construct Sierpinski's Triangle using the process of subdivision. Finally, students will play the Chaos Game in order to discover that Sierpinski's riangle arises from the chaos game. After eight to ten iterations, students will behin to track the orbit by recording.

## Materials Needed

Calculators, colored pens, pencils, dice, triangles (right,isoscles,equilateral, scalene), handouts, rulers, paper, over-head projector, transparencies, Chaos Game, graphing paper.

## HOMEWORK

Students will draw their own Sierpinski Triangles to be discussed and presented in class to show "self-similarity".