Interactivate: Introduction to Fractals: Geometric Fractals

Abstract

This activity is designed to further the work of the Infinity, Self-Similarity, and Recursion lesson by showing students other classical fractals, the Sierpinski Triangle and Carpet, this time involving iterating with a plane figure.

Objectives

Upon completion of this lesson, students will:

have seen the classic geometric fractals
have reinforced their sense of infinity, self-similarity and recursion
have practiced their fraction, pattern recognition, perimeter and area skills

Standards Addressed:

Student Prerequisites

Geometric: Students must be able to:
- recognize and sketch objects such as lines, rectangles, triangles, squares
- understand the concepts of and use formulas for area and perimeter
Arithmetic: Student must be able to:
- build fractions from ratios of sizes
- manipulate fractions in sums and products
Technological: Students must be able to:
- perform basic mouse manipulations such as point, click and drag
- use a browser for experimenting with the activities

Teacher Preparation

Access to a browser
Pencil and Graph Paper
Copies of supplemental materials for the activities:
- Sierpinski's triangle worksheet
- Sierpinski's carpet worksheet

Key Terms

generator	The bent line-segment or figure that replaces the initiator at each iteration of a fractal
infinity	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
initiator	A line-segment or figure that begins as the beginning geometric shape for a fractal. The initiator is then replaced by the generator for the fractal
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

Does anyone remember what infinity means?
Can someone explain to the class what an iteration is?
Who knows what self-similarity is?

Objectives

Today, class, we are going to learn more about fractals, the idea of self-similarity, and recognizing patterns within fractals.
We are going to use the computers to learn more about fractals, the idea of self-similarity, and recognizing patterns with in 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

Walk students through several steps of the Sierpinski Triangle. The students should look at the patterns made by the areas of the individual triangles and the total area. It may take drawing two or three iterations before the number pattern becomes obvious.
Discuss the number of triangles present in each iteration see if any of your students can recognize the pattern.
Have the students discuss what they believe will happen to the area of Sierpinki's Triangle as the number of iterations go beyond the computers computational capability. Will the area of the triangle ever reach zero?

Guided Practice

Have the students repeat the previous exercise with Sierpinski Carpet .
Lead a class discussion to make note of how these are similar to the line bender fractals from the Infinity, Self-Similarity, and Recursion lesson.

Independent Practice

If you choose to hand out the worksheets that accompany these applets you can have the students work on them.
An alternative is to have the students calculate the area Sierpinski's carpet and triangle at several different iterations.

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.

Combine these activities with those in the Infinity, Self-Similarity, and Recursion lesson.
Leave out the concept discussion and focus on pattern recognition and fractions.

Suggested Follow-Up

After these discussions and activities, the students will have seen a few of the classic plane figure fractals to compare with those from the Infinity, Self-Similarity, and Recursion lesson. The next lesson, Fractals and the Chaos Game, continues the student's exploration of fractals by showing how other, seemingly different, ideas can generate the same kinds of fractals.

Interactivate

Introduction to Fractals: Geometric Fractals