The following lesson is designed to help students explore the real-world implications of functions
and independent/dependent variables. This lesson is best implemented with students working in
teams of 2, alternating between the "experimenter" and "recorder" using the computer activities.

Objectives

Upon completion of this lesson, students will:

understand the correlation between independent and dependent variables in real-world situations

understand the importance of identifying and manipulating independent and dependent variables to determine causation

Standards Addressed:

Number and Quantity

Quantities

Reason quantitatively and use units to solve problems.

Grades 6-8

Algebra

Use mathematical models to represent and understand quantitative relationships

Grades 9-12

Algebra

Use mathematical models to represent and understand quantitative relationships

Student Prerequisites

Algebraic: Students must be able to:

understand the concepts of fractions and decimals

understand and manipulate basic probabilities

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

Access to data-processing software such as Microsoft Excel

The number or value that is entered, for example, into a function machine. The number that goes into the machine is the input

output

The number or value that comes out from a process. For example, in a function machine, a number goes in, something is done to it, and the resulting number is the output

proportion

A relationship between two ratios is proportional if the two ratios are equal in value.

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:

What does it mean to say that something is independent?

What, then, do you think an
independent variable is?

What does it mean to say that something is dependent?

What, then, do you think a
dependent variable is?

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 independent and dependent variables

We are going to use the computers to learn about independent/dependent variables through a
real-world experiment, but please do not turn your computers on until I ask you to. I want to
show you a little about this activity first.

Teacher Input

Lead a discussion to prepare the students to identify independent and dependent variables in the
Advanced Fire applet. Ask questions such as the following:

In the real world, why is it important to know which variables are independent and which are
dependent?

How can we know if the dependent variable really depends on the independent variable?

Introduce the concept of inputs and outputs. Discuss the relationship between independent
variables (inputs) and dependent variables (outputs), especially in the context of
experimentation. Ask questions such as the following:

If there are multiple inputs, how do you determine a relationship between just one independent
variable and the dependent variables (outputs)?

What does it mean to "control" the inputs?

Who determines the independent variable in an experiment?

Guided Practice

Demonstrate how to use the
Advanced Fire applet. Be sure to demonstrate the following tasks:

Setting and changing the probability that fire will spread.

Setting a tree on fire and watching the fire spread.

Reviewing the result of past burns with the Burn History button.

Discuss the fact that Advanced Fire has multiple variables. Ask the following questions:

Are there any variables in this scenario?

How do you know?

What are they?

Which variables do you set and which are determined by the applet itself?

Which of these variables are the independent variables? The dependent variables?

Independent Practice

Have the students try the computer version of the
Advanced Fire activity to investigate the relationship between Burn Probability and the actual proportion of
trees left standing.

Divide students into groups of 2-3 and ask them to choose a topic related to Advanced Fire to
investigate that involves controlling independent variables and measuring their effect on
dependent variables. Students should then fill out the
worksheet for their chosen topic.

Closure

You may wish to bring the class back together for a discussion of the findings. Ask students to
identify their chosen independent and dependent variables, and then describe the relationship that
they found between the two. Ask the following questions:

What is one way to easily identify the independent variable(s) in a problem? The dependent
variable(s)?

Did every group that performed the same experiment get the same results? What does that tell
you about the randomness of the burning?

Alternate Outline

This lesson can be rearranged in several ways:

If only one computer is available, choose an experiment as a class and have each student or
group make predictions about the relationship between independent and dependent variables.
Then, conduct the experiment on an overhead projector to test your hypotheses.

Suggested Follow-Up

These discussions and experiments will have given students a greater understanding of independent
and dependent variables and their use in quasi-real-world situations. To follow up on this lesson,
consider having students conduct more complex experiments with
Directable Fire!! or
A Better Fire!!, where students have to consider variability in direction as well as proportion.