Fractals and the Chaos Game

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.

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.

Objectives

Upon completion of this lesson, students will:

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

Standards Addressed

Grade 9

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 10

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Geometry

  • Similarity, Right Triangles, and Trigonometry

    • Understand similarity in terms of similarity transformations

Grades 6-8

  • Geometry

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

Grades 9-12

  • Geometry

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

Geometry

  • Data Analysis and Probability

    • Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.

  • Geometry and Measurement

    • Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Technical Mathematics I

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

Integrated Mathematics III

  • Geometry and Measurement

    • Competency Goal 2: The learner will use properties of geometric figures to solve problems.

Student Prerequisites

  • Geometric: Students must be able to:
    • recognize and sketch objects such as lines, rectangles, triangles, and 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 for experimenting with the activities.

Teacher Preparation

Key Terms

experimental probability

The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played

infinity

Greater than any fixed counting number, or extending forever. No matter how large a number one thinks of, infinity is larger than it. Infinity has no limits

iteration

Repeating a set of rules or steps over and over. One step is called an iterate

probability

The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability

recursion

Given some starting information and a rule for how to use it to get new information, the rule is then repeated using the new information

self-similarity

Two or more objects having the same characteristics. In fractals, the shapes of lines at different iterations look like smaller versions of the earlier shapes

theoretical probability

The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a four-sided die is 1/4 or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast with experimental probability

Lesson Outline

  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 to 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.
    • 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. Place 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, 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 Outline

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 the Sierpinski's Carpet and Sierpinski's Gasket explored in the Geometric Fractals lesson also appear from playing The Chaos Game. The next lesson, Properties of Fractals, is a capstone 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).