Interactivate: Emerging Patterns

Abstract

In this lesson plan, students examine the patterns that emerge from repetitive behavior. First, students will consider fractals, exploring the concept of recursion. Then, students will explore an agent model, where an overall pattern emerges from repetitive individual behaviors.

Objectives

Upon completion of this lesson, students will:

understand the concept of recursion
understand how patterns emerge from repetition
understand the basic principles of computer modeling

Standards Addressed:

Student Prerequisites

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

Teachers will need:

access to a java-enabled browser
pencil for each student
a copy of the worksheet for each student

Key Terms

fractal	Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration
pattern	Characteristic(s) observed in one item that may be repeated in similar or identical manners in other items
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

Introduce patterns by asking students to give examples of patterns they've seen recently.

If students are having trouble naming things, you might point out clothing with patterns, for example: "Will's shirt has a plaid pattern on it".
Ask students how they know these things are patterns.
Lead the students in a discussion about patterns.

Introduce the concept of recursion by leading the students in a discussion.

Ask students if they can think of recursive patterns.
- For example: 1, 2, 4, 8, 16; where each step doubles to create the next step
Ask students how these patterns are different from the patterns discussed earlier.

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 recursion, modeling, and patterns.
We are going to use the computers to learn about this, but please do not turn your computers on until I ask you to. I want to show you a little about these activities first.

Teacher Input

Draw an equilateral triangle on the board. Tell the students that you're going to make a recursive pattern with this triangle by replacing each line segment with the design shown below. Ask students what they expect the pattern to look like. Now draw the new shape on the board and ask the students to imagine what the shape would look like if you repeated the rule again.

Have student volunteers come up to draw each step. When the process starts to take too long for each step (after the third or fourth iterate), pull up the Koch's Snowflake applet. Click through the steps to watch the snowflake start to form. Discuss the results as a class:

Was this the pattern you were expecting?
How is this pattern different from the patterns we talked about at the beginning of our discussion?
Do you see the recursion in this pattern?

Take a moment to ask the students how they think the computer helped them in creating this pattern. Help them see that when doing lots of repetition, the computer can do the repetitive calculations over and over so that the thinking human beings can analyze the results.

Guided Practice

Lead a discussion on agent modeling. Then introduce the Rabbits and Wolves applet. Discuss how to use the applet, specifically showing students the population graph. Do not discuss the individual agents or their behaviors yet.

Independent Practice

Have the students work in pairs to run the simulation and answer the questions on the worksheet.

Closure

As a class, discuss the findings of the activity. Ask the students the following guided questions:

What patterns did you notice?
Would you have noticed those if you had only looked at an individual agent?
Would you have noticed those if you had only looked at the populations at one given moment?

Alternate Outline

If only one computer is available for the classroom, this lesson can be rearranged in the following way:

Project the Rabbits and Wolves applet for the whole class to see.
Have students work in pairs to complete the worksheet as you run the simulation for the whole class to observe.
Be sure to have the students observe multiple trials of the simulation in order to draw accurate conclusions.

Suggested Follow-Up

Students can explore more about modeling in the Fire: Modeling Probability lesson plan.

Interactivate

Emerging Patterns