## Investigating Fire Environments

- Lesson

Students consider the meaning of the term *variable*, both in a
mathematical and an everyday sense, by considering a text‑based
"equation." Data regarding flame length, vegetation and fire speed will
be organized in a table, and students will investigate correlations
between variables using scatterplots and lines of best fit.

Have students read the "Why We’re Worried About Wildfire" equation found on the Wildfire Equation handout. Note that the equation on the second page of the handout was written specifically for the western Nevada area. A more general version of this equation is used on the first page.

Wildfire Equation Activity Sheet |

After reading the equation, have students answer the following questions:

- How do you know that this is an equation? [The behaviors on one side of the equation are
**equivalent**to an unsafe fire environment on the other.] - Why is this information important to you? [Answers will vary.]
- How is this equation like others that you use in mathematical contexts? [An equals sign represents equivalence, which is essential for understanding algebraic expressions. Students often perceive the equal sign to indicate an answer rather than equivalence.]

Distribute the Fire Behavior activity sheet. Have students complete the chart at the top of the first page, using the information contained in "Examples of Local Fire Behavior," found on the last page of the activity sheet. You may wish to allow students to work in pairs to complete the chart.

Fire Behavior Activity Sheet |

Introduce students to scatterplots by graphing flame length versus fire speed. Work with the class to complete this scatterplot. Call on five different students; each of them should be asked to graph one point on the scatterplot. All students should complete the graph on the activity sheet as the points are plotted on the chalkboard or overhead projector. The completed scatterplot should look like this:

Ask students if they notice a relationship between flame length and fire speed. That is, as flame length increases, does fire speed change in a predictable way? [Generally, as flame length increases, the fire speed also increases, so there is a correlation. However, if the correlation were stronger, the plotted points would be closer to lying along a straight line.]

Explain to students that a "line of best fit," which is formally known as a *regression line*,
can be used to approximate a relationship between two variables if
there is a strong correlation. A line of best fit for this data will
approximate the points slightly but not perfectly. To demonstrate this,
ask students to draw a straight line that lies close to most of the
points. Students should notice that, while they are able to draw a line
that is close to some of the points, it will be far away from others.

You might also ask students to approximate how well their line of best fit would approximate the data if the point furthest to the right—the point (55, 8.5), which represents the data for big sage and bitterbrush—were removed. To some extent this point is an outlier, because the flame length greatly exceeds the flame length of the other four points. When this point is removed, the remaining four points do not approximate a line very well. On the other hand, it may be that there is a strong correlation between flame length and fire speed, and the relationship might be more obvious if additional data were considered. For instance, some sources estimate the following:

The burn rate doubles for every 2 mph increase in fire speed, while flame length increases 50%.

Do students’ estimated lines of best fit seem to agree with this statement?

On the second page of the handout, allow students to plot flame length versus burn rate on the top graph. Ask students to discuss the relationship between these two variables. [The scatterplot will reveal that there is a moderate correlation between flame length and burn rate. As with the previous graph, however, it may be that the point (55, 5900) is an outlier, but with only five data points, it is difficult to tell. This is a good opportunity to discuss the sample size needed to draw a reasonable conclusion.]

As with the previous graph, you may want to have students do additional research to find other data points. The general statement above relating burn rate, fire speed, and flame length can again be used by students to assess the validity of their estimated line of fit.

On the bottom graph of the second page, students will plot fire speed versus burn rate. Ask students to discuss the relationship between these two variables. [The scatterplot will reveal that there is a strong correlation between fire speed and burn rate. The points are very close to forming a straight line.]

To estimate a line of best fit, students should use a straight edge to draw a line that lies close to most of the points. There should be roughly the same number of points above and below the line. After students draw a line of best fit, have them use the line to make predictions. With the third graph, for example, you might ask the following questions:

- If a fire moves at a speed of 5½ mph, approximately how many acres would burn in one hour? [A reasonable guess would be about 3200 acres.]
- If a fire burns 4500 acres in one hour, what is the approximate speed of the fire? [Approximately 7 miles per hour.]
- Write an equation that relates fire speed to burn rate. That
is, write the equation for your line of best fit. [When a line of best
fit is drawn by hand, the equation must be approximated. If
*r*is the burn rate and*s*is the fire speed, a reasonable approximation is*r*= 750*s*– 500. A more exact equation is*r*= 802*s*– 1196, which can be obtained using the linear regression feature of a graphing calculator.]

- Computer and Internet connection
- Wildfire Equation Handout
- Fire Behavior Activity Sheet

**Assessments**

- Ask students to write a reflection that compares the benefits of analyzing data in a table as opposed to analyzing data with a scatterplot.
- Collect the student activity sheets and evaluate their scatterplots that compare fire speed to burn rate. In addition, require students to submit the equation of the line of best fit for this data.
- Have students ask their classmates for two pieces of information—for instance, shoe size and height; age of mother and number of siblings; or, number of hours playing video games per week and number of hours doing homework per week. Then, have students organize the data in a table, represent it in a scatterplot, and find the equation for an estimated line of best fit. (Note that a line of best fit will fit some sets of data better than others. A discussion about when it is and is not appropriate to draw a line of best fit could occur.)

**Extensions**

- Have students use the On Fire applet. For each probability, have them run five trials and determine the average of the five trials. Then, have them plot probability along the vertical axis and average percent of the forest that burned on the vertical axis. Is there a correlation? Is it linear? Can the correlation be represented by a line of best fit?
- Have students use information from the chart in the Defensible Space handout to create a scatterplot that compares steepness of slope to the recommended distance for defensible space. (This handout is the basis for the next lesson, How Steep Can You Be?) Students will need to determine how to deal with the slope; specifically, because a range is given in the chart, such as 0 to 20%, students will need to decide if they should use one of the extreme values (0 or 20) or the average value (10). Then have students determine if there is a correlation between these variables and, if there is, write an equation that relates them.

**Questions for Students**

1. How did organizing the data in a table help you see and understand the relationships among fire speed, flame length, and burn rate? How did translating this data to a scatterplot show relationships? Describe the differences between these two representations.

[A scatterplot shows a visual representation of the numbers that appear in a table. Scatterplots often show patterns that are more difficult to detect using a table of data.]

2. For what purposes would you consider using a scatterplot to show data? Are there times that using a scatterplot would be useful in providing a convincing argument to your parents or a friend?

[As shown in this lesson, a scatterplot will often make patterns obvious.]

**Teacher Reflection**

- What relationships did students articulate and use when transferring information from the table to the graph?
- What level of expertise did students demonstrate in preparing and using scatterplots? What additional experiences do they need with this type of graphical representation?
- Did students have a clear idea of what pieces to include in the graph, such as labels on the axes, a title, and so forth? What additional knowledge and skills do they need that might be presented in a mini-lesson on creating graphs?

### Heating Up

### Creating A Firewise Defensible Space

### How Steep Can You Be?

### Burning Questions

### Learning Objectives

By the end of this lesson, students will be able to:

- Recognize the impact that variables have on results.
- Draw conclusions based on numerical information organized in a table.
- Make predictions using a graph of data and a line of best fit.

### Common Core State Standards – Mathematics

Grade 8, Stats & Probability

- CCSS.Math.Content.8.SP.A.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Grade 8, Stats & Probability

- CCSS.Math.Content.8.SP.A.2

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

Grade 8, Stats & Probability

- CCSS.Math.Content.8.SP.A.3

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.