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 Addressed:

Grade 10

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 9

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Eighth Grade

Geometry

Understand congruence and similarity using physical models, trans- parencies, or geometry software.

Geometry

Congruence

Make geometric constructions

Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations

Grades 6-8

Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Apply transformations and use symmetry to analyze mathematical situations

Use visualization, spatial reasoning, and geometric modeling to solve problems

Grades 9-12

Algebra

Understand patterns, relations, and functions

Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Use visualization, spatial reasoning, and geometric modeling to solve problems

Discrete Mathematics

Algebra

Competency Goal 3: The learner will describe and use recursively-defined relationships to solve problems.

Geometry

Data Analysis and Probability

Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.

Geometry and Measurement

Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Integrated Mathematics III

Geometry and Measurement

Competency Goal 2: The learner will use properties of geometric figures to solve problems.

Technical Mathematics I

Geometry and Measurement

Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

Geometry and Measurement

Competency Goal 1: The learner will use properties of geometric figures to solve problems.

Intermediate Algebra

Algebra

The student will demonstrate through the mathematical processes an understanding of sequences and series.

5th Grade

Geometry

5.15b The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will identify and explore congruent, noncongruent, and similar figures

5.15c The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will investigate and describe the results of combining and subdividing shapes

6th Grade

Geometry

6.15 The student will determine congruence of segments, angles, and polygons by direct comparison, given their attributes. Examples of noncongruent and congruent figures will be included.

Student Prerequisites

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

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

Teacher Preparation

Access to a browser

Pencil and Paper

Copies of supplemental materials for the activities:

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

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:

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

Guided 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's 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 self-simlarity.

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

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.

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.

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.