This lesson is designed to introduce students to the idea of finding patterns in the generation of
several different types of fractals.

Objectives

Upon completion of this lesson, students will:

have been introduced to patterns.

have learned the terminology used with patterns.

have practiced finding patterns in the observable process of fractal generation.

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 6

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 7

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 8

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

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.

Fifth Grade

Operations and Algebraic Thinking

Analyze patterns and relationships.

Fourth Grade

Operations and Algebraic Thinking

Generate and analyze patterns.

Geometry

Congruence

Make geometric constructions

Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations

Prove theorems involving similarity

Grades 6-8

Algebra

Understand patterns, relations, and functions

Grades 9-12

Algebra

Understand patterns, relations, and functions

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.

Number and Operations

Competency Goal 1: The learner will apply various strategies to solve problems.

Technical Mathematics II

Geometry and Measurement

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

4th grade

Algebra

Standard 4-3: The student will demonstrate through the mathematical processes an understanding of numeric and nonnumeric patterns, the representation of simple mathematical relationships, and the application of procedures to find the value of an unknown.

7th Grade

Algebra

The student will demonstrate through the mathematical processes an understanding of proportional relationships.

Intermediate Algebra

Algebra

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

4th Grade

Patterns, Functions, and Algebra

4.21

5th Grade

Geometry

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

Patterns, Functions, and Algebra

5.20 The student will analyze the structure of numerical and geometric patterns (how they change or grow) and express the relationship, using words, tables, graphs, or a mathematical sentence. Concrete materials and calculators will be used.

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.

If the students have studied fractals previously, you may ask, "Do you remember fractals? What
can you tell me about them?" or "Can anyone tell me what fractals might have to do with
patterns?"

If students are not familiar with fractals, that is okay. They do not need knowledge of
fractals for this lesson. You can begin with questions such as, "Does anyone know what a
pattern or a sequence is?" or "Can anyone tell me a sequence that we see everyday?"

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 patterns. After this lesson you will understand them
better, be able to talk about them, and be able to pick them out of a process.

We are going to use the computers to learn about patterns, but please do not turn your
computers on or go to this page until I ask you to. I want to show you a little about patterns
first.

Teacher Input

Explain to the students how to do the assignment. You should model or demonstrate it for the
students, especially if they are not familiar with how to use our computer applets.

Open your browser to
The Hilbert Curve in order to demonstrate this activity to the students.

Ask the students what they see. They should tell you that they see a line segment. Point out
to the students that the box at the top of the applet tells you that the line segment has a
size of 1.0 units.

Explain to the students that when you press the button to go to the next stage, a process will
take place or that the applet will do something to the line segment on the screen.

Press the button to proceed to the next stage. Ask the students to describe what they see.
They should tell you that there is now a rectangle in the middle of the line segment standing
on end.

Ask the students to describe the lengths of the segments in the rectangle and the line. Help
the students to see that the new figure is made up of 9 line segments that are all the same
length. Point out to the students that the box at the top of the applet tells us that there
are 9 line segments of size 1/3.0 units.

Ask the students what 1/3.0 means. They should tell you that it means one-third. Ask them,
"One-third of what?" Help the students see that these line segments on the screen are
one-third of the length of the
original line segment.

Have the students guess what will happen when you press the button to go to the next stage.
Explain to them that the process that happened before will happen to every line segment in the
figure.

Press the button to go to the next stage. Ask the students if they are surprised. Have a
student explain why the picture looks as it does. Point out the box at the top of the applet
that tells the students how many segments there are in the figure and how long the segments
are.

Ask the students, "Does it make sense that when we divided each of the line segments in the
previous stage into three parts, that these line segments should be 1/9 in length?" Have a
student explain why this is true.

Pass out the
Patterns in Fractals Data Table. With the students, show how you would answer the questions for The Hilbert Curve.

Guided Practice

Try another example, letting the students direct your moves. Or, you may simply ask, "Can anyone
describe the steps you will take for this assignment?"

If your class seems to understand the process for doing this assignment, simply ask, "Can
anyone tell me the steps that you will need to take to fill in the rest of this chart?"

If your class seems to be having a little trouble with this process, do another example
together, but let the students direct your actions:

Ask, "What do I need to do to answer this first question on our data table?"

Let the students take the class through the steps to answer the questions for the second
applet. If they seem to be having trouble, give them a hint. If they do something
incorrectly, see if they find their own mistake, or gently suggest they try another way.

Independent Practice

Allow the students to work on their own to complete the rest of the data table worksheet.
Monitor the room for questions and to be sure that the students are on the correct web site.

Students may need help with the questions involving finding areas on the last three applets.
You may choose to let the students work in groups to determine a method for finding the area
of the requested spaces. If the question seems too difficult, allow the students to move on to
another applet or complete the challenging questions for extra credit.

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. You may make a
list of characteristics for the students to keep in their notebooks.

Alternate Outline

This lesson can be rearranged in several ways.

You may choose to only use a couple of the computer applets for this lesson.

You may assign each of the five applets to a different group that would report their finding
back to the class.

You may invent your own way of using this lesson to suit the needs of your students.

Suggested Follow-Up

This lesson can be followed by
Patterns in Pascal's Triangle, which will allow students to continue to build the skills necessary to identify patterns.
Another lesson,
An Introduction to Sequences will introduce students to sequences of numbers unrelated to geometric shapes or fractals.