Introduction to Fractals:
Infinity, SelfSimilarity and Recursion
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 2d space, and the Koch snowflake demonstrates that a 1d 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, selfsimilarity 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
Use mathematical models to represent and understand quantitative relationships
 model and solve contextualized problems using various representations, such as graphs, tables, and equations
Geometry
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
Use visualization, spatial reasoning, and geometric modeling to solve problems
 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:
Key Terms
This lesson introduces students to the following terms through the included discussions:
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 1520 minutes for each exploration.
 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:
 Objectives
Let the students know what it is they will be doing and learning today. Say something like this:
 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
 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.
 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 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,
selfsimilarity 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
1015 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 FollowUp
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.
