Properties of Fractals

Abstract

This activity is designed to further the work of the Infinity, Self-Similarity, and Recursion, Geometric Fractals, and Fractals and the Chaos Game lessons by leading the students to build a working definition of fractal.

Objectives

Upon completion of this lesson, students will:

  • have built a working definition of regular fractal
  • have looked carefully at the concepts of dimension and scale
  • have been introduced to the concept of logarithms
  • have solved simple exponential equations for the exponent both by trial and error and using logs

Standards Addressed

Grade 9

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 10

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Third Grade

  • Geometry

    • Reason with shapes and their attributes.

Functions

  • Linear, Quadratic, and Exponential Models

    • Interpret expressions for functions in terms of the situation they model

Grades 6-8

  • Geometry

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

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

Introductory Mathematics

  • Data Analysis and Probability

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

  • Geometry and Measurement

    • COMPETENCY GOAL 2: The learner will use properties and relationships in geometry and measurement concepts 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.

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.

Integrated Mathematics III

  • Geometry and Measurement

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

7th Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of the relationships between two populations or samples.

Secondary

  • Algebra II

    • AII.09 The student will find the domain, range, zeros, and inverse of a function; the value of a function for a given element in its domain; and the composition of multiple functions. Functions will include exponential, logarithmic, and those that have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions.
    • AII.15 The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions.
    • AII.9
    • AII.15

Student Prerequisites

  • Geometric: Students must be able to:
    • recognize and sketch objects such as lines, rectangles, triangles, and squares
    • understand the basic notion of Euclidean dimension
    • measure figures to find the scale factor in similar objects
  • Algebraic: Students must be able to:
    • understand formulas involving exponents
  • 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

Key Terms

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

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

  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 a fractal is?
    • What are some fractals that we have looked at thus far?
    • Does anyone know what dimensions are?

  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 about dimensions and how to calculate fractal dimensions.
    • We are going to use the computers to learn about fractal dimensions, 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

  4. Guided Practice

    • Have the class choose a fractal they have worked with previously. Have the students figure out the fractal dimension of it by hand using the log function on a scientific calculator.
    • Guide the students through the first fractal on the computer version of the Fractal Dimension activity explaining how the activity works.

  5. Independent Practice

    • Once the students have begun to grasp how to calculate fractal dimensions have them work independently with the remaining fractals.
    • If you choose to pass out the accompanying worksheet you may choose to have the students complete it now.

  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 Outline

This lesson can be rearranged in several ways:

  • Leave out all references to logarithms, using only trial and error for finding the fractal dimensions. This reduces the required time significantly.
  • Add an additional discussion session: Build a class list of all the fractals whose dimensions have been calculated in order by size of dimension, and have students use the pictures as evidence for why this ordering makes sense visually.

Suggested Follow-Up

After these discussions and activities, the student will have a basic definition of regular fractals and will have seen the method for calculating fractal dimensions for fractals such as those explored in the Infinity, Self-Similarity, and Recursion, Geometric Fractals, and Fractals and the Chaos Game lessons. The next lesson, Chaos, delves deeper into the notion of Chaos introduced in the Fractals and the Chaos Game lesson. An alternate follow-up lesson would be the Irregular Fractals lesson, in which the students learn how the notion of calculating fractal dimension is much more difficult with irregular fractals.