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

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

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. 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.

2. Lead a class discussion on dimension and scale to prepare them for the idea of "fractional dimension."

3. Lead a class discussion on exponents and logarithms to prepare students for calculating "fractal dimensions."

4. Have the students choose a fractal they have explored to figure out the fractal dimension of by hand using the log function on a scientific calculator.

5. Have the students try the computer version of the fractal dimension activity to reinforce what they saw by hand.

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.