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

The activities and discussions in this lesson address the following NCTM standards:

Algebra

Understand patterns, relations, and functions

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules
• relate and compare different forms of representation for a relationship

• model and solve contextualized problems using various representations, such as graphs, tables, and equations

Apply transformations and use symmetry to analyze mathematical situations

• 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
• use geometric models to represent and explain numerical and algebraic relationships
• recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life

Links to other standards.

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: Students 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 such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• 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

Lesson Outline

Groups of 2 or 3 work best for these activities; larger groups get cumbersome. Working through one or two iterations of each curve as a class before setting the groups to work individually can cut down on the time the students need to discover the patterns. Plan on 15-20 minutes for each exploration. The discussion below assumes that the student has worked with the activities from the Infinity, Self-Similarity, and Recursion lesson.

1. 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 infinity means?
   • Can someone explain to the class what an iteration is?
   • Who knows what self-similarity is?

2. 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 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.

3. 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?

4. 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.
5. 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.
6. 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 Outlines

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.