This lesson is designed to continue the discussion of fractals started in the , and 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 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 Addressed:

Grade 10

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 9

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Number and Quantity

Quantities

Reason quantitatively and use units 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.

Integrated Mathematics III

Geometry and Measurement

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

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.

Student Prerequisites

Arithmetic: Student 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 for experimenting with the activities

Geometric: Students must be able to:

manipulate basic geometric shapes

work with angles

Teacher Preparation

Access to a browser

Copies of supplemental materials for the activities:

Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration

irregular fractals

Complex fractals whose dimension is often difficult to determine and in some cases is unknown

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:

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.

Teacher Input

Lead a class
discussion on calculating the fractal dimensions for the fractals in the
Fractured Pictures activity

Guided Practice

Practice calculating the frctal dimensions using the
Fractured Pictures activity

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.

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.

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 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.