Abstract

This lesson is designed to get students to think about several of the concepts from fractals, including recursion and self similarity. The mathematical concepts of line segments, perimeter, area and infinity are used, and skill at pattern recognition is practiced.

The fractals generated here all start with simple curves made from line segments. They display the curiosities that intrigued the mathematicians looking at infinity at the turn of the century. The Hilbert curves demonstrate that a seemingly 1 dimensional curve can fill a 2-d space, and the Koch snowflake demonstrates that a 1-d curve can be infinitely long and surround a finite area.

Objectives

Upon completion of this lesson, students will:

• have seen a variety of line deformation fractals
• have developed a 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 Paper
• Copies of supplemental materials for the activities:
  • tortoise and hare race worksheet
  • Cantor's comb worksheet
  • Hilbert curve worksheet
  • another Hilbert curve worksheet
  • Koch snowflake 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.

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:

   • Can anyone explain what infinity means?

2. Objectives

   Let the students know what it is they will be doing and learning today. Say something like this:

   • Today, class, we will be talking about fractals while looking at Tortoise and Hare Race, Cantor's Comb, Hilbert Curve, Another Hilbert Curve, and Koch's Curve.

3. Teacher Input
   • Introduce the terminology:

     Initiator:

     The starting curve or shape

     Generator:

     The rule used to build a new curve or shape from the old one

     Iteration:

     The process of repeating the same step over and over

4. Guidied Practice
   • Describe the Tortoise and Hare Race to the students and ask them to speculate on who will win. Then have them run though several steps of the race, stopping when they think they see what is happening.

   • Have students run several steps of the Cantor's Comb. The students should look at the patterns made by the lengths of the segments and the total length. It may take drawing two or three iterations before the number pattern becomes obvious.

   • Repeat the previous exercise for the Hilbert Curve.

   • Lead a class discussion to clarify what "infinitely many times" means.

   • Repeat the previous exercise for Another Hilbert Curve, this time also asking students to discuss how a small change in the generator can lead to a large change in the final object.

   • Repeat the previous exercise for the Koch Curve, this time also asking about patterns in the area enclosed as well as the length of the curve.

   • Lead a class discussion to introduce the formal idea of recursion.

   • Lead a class discussion to introduce the formal idea of self similarity.

5. Independent Practice
   • Allow the student's time to complete any or all of the worksheets that are provided with the applets used during this lesson.
   • Have the students try to draw a couple iterations of any of the fractals used during this lesson and discuss why computers are useful when studying fractals.
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.

• Choose fewer of the activities to cover; for example, covering Cantor's comb, the Hilbert curve and the Koch snowflake still allows for discussion of infinity, self-similarity and recursion.
• Have different groups of students do different activities and give group presentations.
• Leave out one or more of the concept discussions and focus on pattern recognition and fractions.
• Have the students draw several steps of each of the activities by hand before trying the computerized version. Graph paper and rulers would be needed for this. Plan on an additional 10-15 minutes per activity.
• Combine this lesson with the Geometric Fractals lesson, to give the students a well rounded picture of regular fractals, including a formal definition.
• 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 a few of the classic line deformation fractals. The next lesson, Geometric Fractals, continues the student's initial exploration of fractals with those formed by repeatedly removing portions from plain figures such as squares and triangles.