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 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.

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.

Functions

Linear, Quadratic, and Exponential Models

Interpret expressions for functions in terms of the situation they model

Third Grade

Geometry

Reason with shapes and their attributes.

Grades 6-8

Geometry

Use visualization, spatial reasoning, and geometric modeling 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.

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.

Integrated Mathematics III

Geometry and Measurement

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

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.

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.

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

Access to a browser

Pencil and calculator

Copies of supplemental materials for the activity:

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

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?

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.

Lead a class
discussion on exponents and logarithms to prepare students for calculating "fractal dimensions."

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.

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.

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.