Abstract

This lesson is designed to continue the discussion of fractals started in the Infinity, Self-Similarity and Recursion, Geometric Fractals and Fractals and the Chaos Game lessons. Students are introduced to the notion of irregular fractals and given an idea of the difficulty involved in calculating the fractal dimension as outlined in the Properties of Fractals lesson.

Objectives

Upon completion of this lesson, students will:

• have learned about irregular fractals and built a few
• have practiced their pattern recognition skills
• have practiced their plane geometry 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

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:
  • manipulate basic geometric shapes
  • work with angles
• Arithmetic: Students must be able to:
  • work with integers as scale factors and in ratios
• 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
• Copies of supplemental materials for the activities:
  • fractured pictures worksheet
  • flake maker worksheet

Key Terms

• fractal
• irregular fractals
• iteration
• recursion
• self-similarity

Lesson Outline

This lesson is best implemented with students working individually. Allow the students at least 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:

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 fractals.
   • We are going to use the computers to learn about fractals, 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

   • tead a class
   • Lead a class discussion on calculating the fractal dimensions for the fractals in the Fractured Pictures activity
4. Guided Practice

   • Practice calculating the frctal dimensions using the Fractured Pictures activity

5. Independent Practice
   • Have the students try the computer version of the Flake Maker activity to investigate what sorts of interesting patterns and fractals can be generated.
   • Also have stundents practice calculating the fractal dimension of the fractals they generate.

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.

• Do only one activity -- the flake maker makes more interesting fractal pictures.
• Add the additional task of trying to build an image that looks like an actual object with flake maker. Some suggestions: mountain ranges, ocean waves, flowers, animals.
• Have a contest in which the students are asked to find the most interesting image, with a panel of teachers or the entire class being the judge. (Have the students print out their images so that a display can be set up.)
• If connected to the internet, use the enhanced version of the software, Snowflake, to explore line deformation fractals more fully.

Suggested Follow-Up

After these discussions and activities, the students will have seen how complex fractals can be generated by generalizing the ideas for making regular fractals, introduced in the Geometric Fractals and Fractals and the Chaos Game lessons. The next lesson, The Mandelbrot Set, is a cap-stone activity, designed to introduce the student to the most celebrated modern fractal object, the Mandelbrot set.