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.
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.
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 = 750s – 500. A more exact equation is r = 802s – 1196, which can be obtained using the linear regression feature of a graphing calculator.]
