Finding Patterns Using Fractals

This lesson will introduce students to patterns in fractals using resources of Shodor Education Foundation, Inc. Permission has been granted for the use of the materials as part of the workshop Interactivate Your Bored Math Students.

A lesson plan for grade 6 Mathematics

By Wendy Korbusieski

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• Learn more about mathematics.

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Learning outcomes

Upon completion of this lesson, students will:

• have been introduced to patterns.
• have practiced finding patterns in the observable process of fractal generation.

Goals taken from http://www.shodor.com

Teacher planning

Time required for lesson

1 hour

Materials/resources

• Pencil and paper
• Computer lab with access to the Internet for all students
• Chalk board or overhead projector
• Transparency copy of data chart

Technology resources

• Color monitor with access to the Internet for each student
• Overhead Projector
• Calculators(optional)
• Projection capability for teacher’s computer

Pre-activities

The teacher should familiarize him or herself with Hilbert Curve, Another Hilbert Curve, Sierpinski’s Carpet, Sierpinski’s Triangle and Koch’s Snowflake activities on the Shodor website.

Activities

Begin by introducing the lesson with something like:

Hand out the attached worksheet on patterns in fractals. Demonstrate the process for the students, especially if they are unfamiliar with computer applets.

Open your browser to The Hilbert Curve http://www.shodor.org/interactivate/activities/hilbert1/index.html 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 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.” (shodor.com)

Have students record their results in their worksheet. The teacher will need to record the results on the overhead transparency of the worksheet, open another window in Excel and simultaneously show the curve and spreadsheet with recorded results on screen or make a copy of the chart on the board. Continue to fill in the chart for all six levels of the curve. The applet will stop computing at the sixth level.

Ask students, “How can we generate the next three stages of this pattern?” Taking note of the pattern needed, finish the next three stages for the Hilbert Curve. Ask for questions, then give directions for continuing with the rest of the applets.

Assessment

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 will use these additional sites:
Another Hilbert Curve: http://www.shodor.org/interactivate/activities/hilbert2/index.html

Sierpenski’s Carpet: http://www.shodor.org/interactivate/activities/carpet/index.html

Siepenski’s Triangle: http://www.shodor.org/interactivate/activities/gasket/index.html

Koch’s Snowflake:
http://www.shodor.org/interactivate/activities/koch/index.html

Students should complete this worksheet with a mastery level of 80% or higher.

Supplemental information

Attachments:

• Patterns with Fractals Worksheet
• Fractals Answer Key

Related websites

The Hilbert Curve http://www.shodor.org/interactivate/activities/hilbert1/index.html
Another Hilbert Curve http://www.shodor.org/interactivate/activities/hilbert2/index.html
Sierpenski’s Carpet http://www.shodor.org/interactivate/activities/carpet/index.html
Siepenski’s Triangle http://www.shodor.org/interactivate/activities/gasket/index.html
Koch’s Snowflake http://www.shodor.org/interactivate/activities/koch/index.html

Comments

An extension to this lesson can include writing rules using variables for each of theses patterns.

As another extension or review, have students practice reading and writing large numbers and/or place value.

Students may need to use Converter http://www.shodor.org/interactivate/activities/conversions/index.html while in Sierpinski’s Carpet and Sierpinski’s Triangle in order to change decimal data into fraction data or the teacher may prefer for students to try and determine the pattern while in decimal form.