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

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

Lesson Outline

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?

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.

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

Guided Practice

Have the class choose a fractal they have worked with previously. Have the students figure out
the fractal dimension of it by hand using the log function on a scientific calculator.

Guide the students through the first fractal on the computer version of the
Fractal Dimension activity explaining how the activity works.

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.

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:

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 student will have a basic definition of regular
fractals and will have seen the method for calculating fractal dimensions for fractals such as
those explored in the
Infinity, 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.