This lesson is designed to introduce students to the concept of fractals and show them some examples of fractals in the real world.

Standards (NCTM)

Geometry Standards:

• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties
• describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling
• examine the congruence, similarity, and line or rotational symmetry of objects using transformations
• draw geometric objects with specified properties, such as side lengths or angle measures

• Educational

Students should know:

• basic measurement concepts (e.g. area, perimeter).
• basic geometrical concepts (e.g. line segments, angles, degrees).
• Technological

Students must be able to:

• use a web browser such as Netscape.

Teachers will need:

• a large sheet of paper to tape the students' fractals to in order to create a giant fractal.
• Fractal & Chaos Fact Sheet.
• Class Project Documents.
• Transformations Handout.
• Genetic Code Handout.
• Blank Fractal Grid Pages.
• Tortoise and Hare Applet, Hilbert Curve Generator, Another Hilbert Curve Generator, Koch's Snowflake, Sierpinski's Triangle, and Sierpinski's Carpet.

• access to a web browser.
• scissors.
• tape or glue.

1. Guided Practice
• Divide the students into small groups so that each group has one computer
• Allow the students to play with the Tortoise and Hare Applet.

2. Objectives
• Introduce students to the concept of fractals.
• Introduce student to some basic calculus concepts.
• Make the students proficient in recognizing and generating image transformations.

3. Teacher Input
• Explain how the applet illustrates Zeno's Paradox
• Use Zeno's Paradox to introduce and explain the concept of limits
• Brielfy explain that calculus is the branch of math that deals with limits

Guided Practice
• Allow the students to explore the following applets:
• Hilbert Curve Generator
• Another Hilbert Curve Generator
• Koch's Snowflake
• Sierpinski's Triangle
• Sierpinski's Carpet
• Encourage the groups to try to figure out how each applet works.

Teacher Input
• Discuss how each applet works, inviting groups to explain what they discovered.
• Compare the applets to one another.
• Use the applets to help students visualize the concept of limits.

Teacher Input
• Pose the question: How many orientations can a piece of paper have on a flat surface? Explain the concept of transformations.
• Using the transformations handout, ask students to assign a label to the transformations at the bottom of the page based on their degree of rotation/reflection with respect to the identity figure.
• Hand out the Geometric Genetic Code worksheet and have the students identify the codes that correspond to the fractals.

Guided Practice
• Hand out two copies of the fractal grid worksheet to each student (or one copy with the grid set printed on both sides). On the first sheet, give them a starting image and the code they need to create a full fractal with it.
• Once they are confident with creating a fractal based on a given code, let them create their own fractal using the other grid set. Once they complete the fractal, have them determine it's code.

Teacher Input
• Compare the algorithm used to generate fractals in the Interactivate applets with the process the students used to make their own fractals.

5. Independent Practice
• Divide the class into groups and give each group a "Making a Stage 8 Fractal" handout. Follow the instructions to create a giant fractal, combining each group's stage 7 fractal by taping them all onto the large sheet of paper.

6. Closure
• Present some examples of fractals in real life, referring to the "Uses of Fractals and Chaos" worksheet. Other possible examples include the art of Jackson Pollock and the pattern of the veins in a leaf.