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
- 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.]
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.]
- 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?
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
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
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
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.