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.
Data and Models Activity Sheet
Data and Models Answer Key
Rainforest Deforestation Activity Sheet
Rainforest Deforestation Answer Key
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.
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
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.
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
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.]
- How well were
students able to reason with data that did not exactly fit a particular
- 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?