Patterns in Fractals

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.

• 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:
  • Open the applet Another Hilbert Curve.
  • 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 will use these additional sites:
  • Sierpinski's Triangle
  • Koch's Snowflake
  • Sierpinski's Carpet
• 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.