Abstract

This activity is designed to further the work of the Geometric Fractals lesson by showing students how the Sierpinski triangle can arise from seemingly totally unrelated sources. This gives the students an appreciation of the interconnections of different kinds of mathematics.

Objectives

Upon completion of this lesson, students will:

• have seen the chaos game
• have practiced their fraction, percent and basic probability skills

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 patterns, relations, and functions

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules
• relate and compare different forms of representation for a relationship

• model and solve contextualized problems using various representations, such as graphs, tables, and equations

Use visualization, spatial reasoning, and geometric modeling to solve problems

• draw geometric objects with specified properties, such as side lengths or angle measures
• use geometric models to represent and explain numerical and algebraic relationships
• recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life

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:
  • manipulate fractions in sums and products
• 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, Ruler, 6-Sided Die and Graph Paper
• Copies of supplemental materials for the activity:
  • the chaos game worksheet
  • the chaos game transparency

Key Terms

Lesson Outline

This lesson is best started on paper with each student working individually. Plan on 10-15 minutes for individual exploration. Then allow the students to work individually or in small groups to explore the computer activity. Plan on an hour for the complete set of computer activities, with additional time for discussion.

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 remember what a fractal is?
   • Can any one explain what the word pattern means?
   • Can someone explain what the word random means?
   • Who thinks that seemingly random process can result in a pattern?

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 how a seemingly random process can result in a familiar pattern.
   • We are going to use the computers to learn about patterns, 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
   • Before you say anything place three large dots on the board so that they represent the corners of an equilateral triangle. Lable the dots A,B,and C. Explain to the class that you are going to start at a random place on the board, place a dot, and call on them one by one and have them choose A,B, or C. Then Place a small dot half way between the last dot drawn and the corner called out by the student.
   • Once everyone has had a chance to call out a corner ask the class if anyone sees a pattern. (They Probably won't)
   • Ask what the class thinks will happen if this process is continued for a really long time.
   • Explain to the class that the process they were just doing is how the chaos game works.

4. Guided Practice
   • Have the students try the computer version of the basic (triangle) chaos game. Placing 20 dots at a time for as many iterations as necessary for them to recognize Sierpinski's Triangle forming.
   • Lead a discussion on basic probability to prepare the students for their independent practice.

5. Independent Practice
   • Have the students try changing the probabilities (which can be explained as movement ratios) for various starting shapes with the computer version of the chaos game making and testing conjectures about the final shapes.
   • Have the students record what they plan to change the probability to and what they expect to happen and what actually happens.
   • If you choose you may also pass out the worksheet that accompanies this applet and have the students complete it at this time.
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.

• Combine just the triangle version of these activities with those in the Geometric Fractals lesson to give a simple introduction to the Sierpinski triangle and fractals.
• Combine the triangle version of this activity with the Geometric Fractals and Pascal's Triangle lessons to give a comprehensive introduction to the Sierpinski triangle by examining three separate ways in which this figure can be generated.
• If connected to the internet, use the enhanced version of the software, Sierpinski Gasket Maker, to explore generalizations to the Sierpinski shapes in which the user can specify both a movement probability and a rotation.

Suggested Follow-Up

After these discussions and activities, the students will have seen that Sierpinski's carpet and gasket explored in the Geometric Fractals lesson also appear from playing the chaos game. The next lesson, Properties of Fractals, is a cap-stone lesson designed to summarize and formalize the notion of a fractal now that the students have seen several different kinds. An alternate follow-up lesson would be the Pascal's triangle lesson, in which the Sierpinski triangle appears again from a different source (namely Pascal's triangle).