Abstract

The following discussions and activities are designed to lead the students to explore various incarnations of chaos.

Objectives

Upon completion of this lesson, students will:

• have experimented with several chaotic simulations
• have built a working definition of chaos
• have reinforced their knowledge of basic probability and percents

Standards

The activities and discussions in this lesson address the following NCTM standards:

Number and Operations

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

• work flexibly with fractions, decimals, and percents to solve problems

Understand and apply basic concepts of probability

• use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations
• compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models

Links to other standards.

Student Prerequisites

• Geometric: Students must be able to:
  • recognize and sketch objects such as lines, rectangles, triangles, squares
• Arithmetic: Students must be able to:
  • understand and manipulate basic probabilities
  • understand and manipulate percents
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• Access to a browser
• Pencil and calculator
• Copies of supplemental materials for the activities:
  • the fire! worksheet
  • the better fire! worksheet
  • the game of life worksheet
  • the rabbits and wolves worksheet

Key Terms

• chaos
• experimental probability
• iteration
• probability
• theoretical probability

Lesson Outline

This lesson is best implemented with students working in teams of 2, alternating being the "driver" and the "recorder." Allow the students about 30 minutes to explore each computer activity.

1. Focus and Review

   Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

   • Does anyone know what predictabilty means?
   • Can anyone explain what chaos means?
   • If I lit a fire in the middle of the room can anyone predict what else would catch on fire?
   • What would influence the way the fire might spread?

2. Objectives

   Let the students know what it is they will be doing and learning today. Say something like this:

   • Today, class, we are going to learn about probability and chaos.
   • We are going to use the computers to learn about probability and chaos, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

3. Teacher Input
   • Lead a class discussion on basic probability to prepare students for working with the activities.

4. Guided Practice
   • Have the students try the computer version of the Fire! activity to investigate how large the burn probability can be and still consistently have trees left standing.
   • Lead a class discussion on chaos.
   • Ask the class to think about why the fire activity is not very realistic. Be sure the point that controlling the probability of the spread of the fire is out of a person's hands. Motivate the next activity by pointing out that if we assume that fire will spread 100% of the time, then leaving some empty space in the forest (which a person can control) may keep the entire forest from burning.
   • Have the students try the computer version of the Better Fire! activity to investigate how large the forest density can be and still consistently have trees left standing after a fire.
   • Lead a class discussion on how prevalent chaos is in science.

5. Independent Practice
   • Have the students try the computer version of the Game of Life activity to investigate this classic demonstration of chaos.
   • Have the students try the computer version of the Rabbits and Wolves activity to investigate how the effects of small changes in the initial values of things changes the outcomes.
   • If you choose to hand out the accompanying worksheets you can have the students complete them now.
6. Closure

   • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outlines

This lesson can be rearranged in several ways.

• Reduce the number of activities; for example, use only the Better Fire! and Game of Life activities to give the classic examples of simulations with chaotic behavior.
• Add the additional activity using the Flake Maker activity with the following three starting shapes:
  

  Use the results of these three generators as an analogy for how small changes in structure cause large changes in cell growth in biology.

Suggested Follow-Up

After these discussions and activities, the students will have seen more ways in which chaos, first introduced in the Fractals and the Chaos Game lesson, is used to model behavior. The next lesson, Pascal's Triangle, reintroduces Sierpinski-like Triangles, as seen in the Geometric Fractals and Fractals and the Chaos Game lessons, in yet another way, demonstrating the rich connections between seemingly different kinds of math.