Abstract

This activity is designed to further the work of the Infinity, Self-Similarity and Recursion, Geometric Fractals, and Fractals and the Chaos Game lessons by leading the students to build a working definition of fractal.

Objectives

Upon completion of this lesson, students will:

• have built a working definition of regular fractal
• have looked carefully at the concepts of dimension and scale
• have been introduced to the concept of logarithms
• solved simple exponential equations for the exponent both by trial and error and using logs

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 use ratios and proportions to represent quantitative relationships
• develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation

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
  • understand the basic notion of Euclidean dimension
  • measure figures to find the scale factor in similar objects
• Algebraic: Students must be able to:
  • understand formulas involving exponents
• 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 activity:
  • the fractal dimension worksheet

Key Terms

• infinity
• iteration
• recursion
• self-similarity

Lesson Outline

This lesson is best implemented with each student working individually. Plan on 1-2 hours for the initial discussions if logarithms are introduced. Then allow the students 20-30 minutes to explore the 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 remember what a fractal is?
   • What are some fractals that we have looked at thus far?
   • Does anyone know what dimensions are?

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 dimensions and how to calculate fractal dimensions.
   • We are going to use the computers to learn about fractal dimensions, 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 what students see as the common features of the fractals they have encountered in the Infinity, Self-Similarity and Recursion, Geometric Fractals, and/or Fractals and the Chaos Game lessons.
   • Lead a class discussion on dimension and scale to prepare them for the idea of "fractional dimension."
   • Lead a class discussion on exponents and logarithms to prepare students for calculating "fractal dimensions."

4. Guided Practice
   • Have the class choose a fractal they have worked with previously. Have the students figure out the fractal dimension of by hand using the log function on a scientific calculator.
   • Guide the students throughthe first fractal on the computer version of the fractal dimension activity explaining how the activity works.

5. Independent Practice
   • Once the students have begun to grasp how to calculate fractal dimensions have them work independently with the remaining fractals.
   • If you choose to pass out the accompanying worksheet you may choose to have the students complete it 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.

• Leave out all references to logarithms, using only trial and error for finding the fractal dimensions. This reduces the required time significantly.
• Add an additional discussion session: Build a class list of all the fractals whose dimensions have been calculated in order by size of dimension, and have students use the pictures as evidence for why this ordering makes sense visually.

Suggested Follow-Up

After these discussions and activities, the students will have a basic definition of regular fractal and have seen the method for calculating fractal dimension for fractals such as those explored in the Self-Similarity and Recursion, Geometric Fractals, and Fractals and the Chaos Game lessons. The next lesson, Chaos, delves deeper into the notion of Chaos introduced in the Fractals and the Chaos Game lesson. An alternate follow-up lesson would be the Irregular Fractals lesson, in which the students learn how the notion of calculating fractal dimension is much more difficult with irregular fractals.