Abstract
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 threedimensional 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
Student Prerequisites
Teacher Preparation
Teachers will need:
Students will need:
 access to a web browser.
 scissors.
 tape or glue.
Lesson Outline
 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.
 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.
 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:
 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.
 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.
 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.

