## Rainforest Deforestation- Problem or Myth?

• Lesson
9-12
1

This lesson allows students to explore the idea that rainforest deforestation is occurring at an exponential rate. Students will use provided research about Amazon deforestation and conduct their own research to determine whether deforestation is occurring exponentially.

In this lesson, students use tables to determine which model, linear, quadratic, or exponential, best fits a set of data. Students use the common difference and common ratio to determine the appropriate model for data. The common difference is used to determine a linear model, and the second difference can be used to determine a quadratic model. The common ratio can be used to determine an exponential model.

Then students use their knowledge of analyzing data to find a data model to see if deforestation is occurring at an exponential rate.

Before the lesson, be sure to make each student a copy of the Data and Models Activity Sheet and the Rainforest Deforestation Activity Sheet.

Students use the Data and Models Activity Sheet to get acquainted with the idea of using constant differences and ratios to determine an appropriate model for given data. Instruct students to calculate the common difference by finding the difference between the next y-value and the current y-value. Be sure to point out that when using real world data, these differences may not be exact, but they should be relatively close. If the common differences are close to being equal to the same value, then the data may be represented by a linear model.

The second difference method can be used to determine if the data may be represented by a quadratic model. To calculate the second difference, select 3 consecutive y-values, labeling them y1,y2, and y3 and perform the following calculations:

y2-y1=a and y3-y2=b, and then b-a=c. c is called the second difference. If all second differences are close to being equal to the same value, the data may be represented by a quadratic function.

The common ratio is found by dividing the next y-value by the current y-value. If all the ratios are equal, or close to being equal, to the same value, then the data may be represented by an exponential model. Examples are shown below.

After students have determined which model fits each set of data, have them graph the data in each table separately using a graphing calculator or graphing software. Students should work in groups of two or three, so they can discuss the work they are doing. When students have finished the activity, they share their answers with the whole class. An example of what students' work should look like is shown below.

Students then use the Rainforest Deforestation Activity Sheet to decide whether the rainforest is suffering deforestation at an exponential rate. In order to introduce this activity sheet, you may want to find a short video on deforestation online to share with the class.

As a closing activity, students should share their ideas about whether they think rainforest deforestation is occurring exponentially, based on the research data. They will also critique research conducted on the subject.

### Ideas for Differentiation

• For students who struggle with the idea of the data not being exactly linear, quadratic, or exponential, suggest that they generate data from a quadratic or exponential equation and then calculate constant differences to see the results when the model is a perfect fit.
• Additionally, students can be given the option to choose between linear and quadratic models only, to enable them to focus on the idea of constant difference.

Assessment Options

1. Have students come up with their own sets of data that represent a linear, quadratic, or exponential function. Students should trade data sets with a partner, and the partner should use constant differences and ratios to determine the type of function.
2. Ask students to compare and contrast the three different function types, using real-world examples.

Extensions

1. Students could use graphing technology to graph tangent lines at the data points on the graph. Students should understand that linear models have tangent lines with the same slope (thus the same first difference) while quadratic and exponential functions' lines continue to have a different slope. The slope of a tangent line to a parabola does change at a constant rate (hence the same second differences). This is a great introduction to the idea of calculating the slope at a point.
2. Students could determine rules for 3rd and 4th degree polynomial equations, expanding their pattern to nth degree polynomial functions.
3. Depending on the student audience, this is a golden opportunity to teach how to generate the equations once you establish what kind they are. This would require some previous experience with combinatorics.

Questions for Students

1. Why is it necessary to have at least 5 data points when trying to determine whether a linear, exponential, or quadratic model might best fit the data?

[You must have at least 5 data points to find three second differences. Having two second differences from 4 data points doesn't make a pattern.]

2. How can the idea of a constant difference or ratio be related to the rate of change of a linear, quadratic, or exponential model?

[Lines have a constant rate of change, as shown by the first difference.  Parabolas and exponential curves do not have a constant rate of change, as shown by the different first and second differences. An exponential curve will approach a horizontal asymptote but not cross it. A quadratic curve will keep increasing at a faster rate until the rate becomes infinitely large.]

3. When compared to looking at a graph, how do using constant differences and ratios better indicate the type of function? Can an exponential function ever appear linear?

[Simply looking at a graph of the data may make a slowly growing exponential curve or parabola look linear, especially if one "zooms in" on the data by looking at a relatively short time period. Using constant differences and ratios will make the relationship between the data points more clear.]

Teacher Reflection

• How well were students able to reason with data that did not exactly fit a particular model?
• What observations did you record regarding whether students were able to understand problems with predicting which model might be used to represent a certain set of data?
• How would you rate your students' level of enthusiasm and involvement (high or low) for this lesson? Explain why.
• How did you challenge the high achievers?
• How did you appropriately adapt the lesson for the diverse learner?

### Learning Objectives

Students will:

• Use first and second differences as well as constant ratios to determine whether data models linear, quadratic, or exponential growth.
• Use graphing technology to fit a model to data.
• Use their own models to make predictions about data.

### NCTM Standards and Expectations

• Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.
• Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.
• Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.
• Approximate and interpret rates of change from graphical and numerical data.
• Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.