## Smokey Bear Takes Algebra

- Lesson

This lesson introduces students to the many factors that play a role in
creating a forest-fire danger rating index. They will be looking at how
we use a scale to quantify the abstract idea of forest fire danger.
Using the real-world situation, students examine the meaning of the
slope and intercepts of a line. To complete the activities related to
these indexes, students should be comfortable with linear, quadratic
and exponential functions and their graphs. Students’ facility with a
graphing calculator is assumed. Students also use summation notation to
do the activities relating to the Nesterov index. This lesson plan was
adapted from the article "Smokey the Bear Takes Algebra," which
appeared in the October 1999 issue of the *Mathematics Teacher*.

To
begin the lesson, gather background knowledge from the students about
wildfires. Where are Smokey Bear billboards located? Why are these
signs important to park visitors? What is the difference between a *high* danger of forest fire and a *very high* danger? What are some of the extreme losses that occur during uncontrollable fires?

To get students thinking about the factors that are considered when rating fire-danger, distribute copies of the Meet Smokey Bear Activity Sheet. Go over the introduction with students, and have them work in small groups to answer the questions.

Meet Smokey Bear Activity Sheet

Next, explain to students how the National Fire Danger Rating System, or NFDRS, is a tool used for predicting and controlling fires. The NFDRS takes in many environmental variables, such as the factors students just discussed, to predict the rate of spread of a fire. Experts create an index using these inputs and compare it with the fire history of a given area, and one of five ratings is given: low, medium, high, very high, or extreme. Fire-management officials use these ratings in making decisions about deploying personnel and other resources that will aid in fire suppression. Since the NFDRS is a complex system, simpler indexes, the Angstrom and the Nesterov, are examined during this lesson.

Distribute the Angstrom Index Activity Sheet, and explain that this index is a simpler precursor to
the NFDRS. It was designed so that fire-danger could be computed
mentally; in fact, it is the only index in use that can be calculated
in this way. Go over the introduction with students, pointing out the
factors that are used in this index and those that are not. What does *I* represent? As *I* increases, what is happening to fire danger? Be sure they see that the value of *I* and the chance of fire are inversely related: the greater the value for *I*, the smaller the chance of fire.

Note: Teachers may wish to rewrite the formula in terms of a
Fahrenheit scale for students unfamiliar with Celsius temperatures. For
example, a window in which *x* (temperature) varies from 0 to 40
degrees Celsius will take into account all reasonable temperatures, 32
degrees to 104 degrees Fahrenheit, for the fire season in most areas of
the United States. Having students think of Celsius temperatures in
multiples of 5 makes conversion easier for them.

Next, students will examine another rating system, the Nesterov Index. To use the Nesterov index, students will need a month of weather data (dew point, temperature, and relative humidity), preferably for the area in which they live. As an alternative, data for a typical thirty days in the summer in Asheville, North Carolina, is included in the activity sheet.

Distribute the Nesterov Index Activity Sheet. Go over the introduction with students, and explain that the computations begin on the first spring day when the high temperature is above freezing and continue until a rainfall of 3 mm, whereupon the process starts anew.

As a closing, distribute the Rules of Firefighting Activity Sheet. This activity allows students to work with some rules that firefighters have incorporated into their set of tools.

Rules of Firefighting Activity Sheet

### Solutions to Meet Smokey Bear

**1:** Relative humidity, wind speed, the number of days since
the last rainfall, the terrain, the nature of the available fuel (e.g.,
grasses will burn more readily than brush, which will burn more readily
than redwood trees); in general, the finer the fuel, the greater the
fire danger.

**2:** Answers will vary. Students should justify their hypotheses.

**3:** Hilly terrain causes fires to spread more quickly. The
greater surface area of the ground that is exposed to flame, the
quicker the fire will spread. Flames will preheat the fuel that is
upslope of the fire, making ignition easier.

**4:** To combat fires effectively, firefighters need to deploy
personnel and physical resources in the optimal way. Where should
trenches be dug? Where should helicopter teams be stationed? How much
of the fire-fighting resources of personnel and material need to be
used, and when? How can people be kept safe? Students should see that
the model is useful in preparing for controlling fires.

### Solutions to the Angstrom Index

**1:** Students should obtain a line with negative slope in the first quadrant.

**2:** The intersection of the line in question 1 with the horizontal line *y* = 2 occurs at *x* = 24.5, or 76.1 degrees on a Fahrenheit scale, and with the horizontal line *y* = 4 at *x* = 4.5 or 40.1 degrees Fahrenheit.

**3:** The slope can be described as how much the fire-danger
rating changes for each degree of change in temperature for the given
humidity. Students should make the connection that for the given
relative humidity, the fire danger increases as the temperature
increases. The *x*-intercept, 44.5 in this example, represents the
temperature at which the fire index becomes 0, whereas the y-intercept
is the fire-danger when the temperature is 0 degrees. Any value of *x* greater than 44.5 will result in negative values for the fire index.

**4:** When the humidity is raised to 40 percent, students should
see that the new line is parallel to the original line with a slightly
higher *y*-intercept. If students are using the Celsius scale,
they can get a nice picture by restricting the calculator window to [0,
50] by [0, 5]. The Celsius temperatures at which fire danger becomes
likely and unlikely, respectively, are 27 and 7; those temperatures in
degrees Fahrenheit are 80.6 and 44.6.

**5:** To obtain these last answers algebraically, students need to solve the linear inequality

with the appropriate values substituted for R.

**6:** When students hold the temperature constant instead of the
humidity, they obtain a line with a positive slope that represents the
change in fire danger per increase in percent humidity. At the given
temperature, fire danger is very likely at a relative humidity at or
below 46 percent; fire danger is unlikely for relative humidity greater
than 86 percent. Again, some confusion might arise because of the
inverse nature of the meter: low numbers mean higher fire danger. But
students should be able to understand that as relative humidity
increases, the chance of fire occurring decreases. Thus their graphs
should confirm their scientific understanding. When the temperature is
raised to 35 degrees Celsius, the relative humidity at which fire
danger is likely or unlikely becomes 56 percent and 96 percent,
respectively.

**7:** Fixing the relative humidity and considering only
temperature change produces a line with a slope of magnitude .1; fixing
the temperature and considering only relative humidity produces a line
with slope .05. The index seems to be more sensitive to temperature
change.

### Solutions to Nesterov Index

**2:** The principal virtue of the Angstrom index is its ease of
computation. The fire danger can be computed mentally when the data are
available. Neither of the rating systems is particularly comprehensive,
as neither takes into account wind speed, terrain, or fuel moisture.

### Solutions to Rules of Firefighting

**1:** This rule deals with the moisture content in potential
fuel, an important aspect of fire danger that is not directly addressed
by the Nesterov and Angstrom indexes. Students should indicate the
fuel-moisture level on the independent axis, and the rate of spread on
the dependent. The nature of the information demands that the graphs be
qualitative. Graphs of fine and large fuels should overlap on the
intervals [0, 5] and [10, 15]. On the interval between 5 percent and 10
percent, the fine-fuel graph should be higher than that for large fuel,
whereas for fuel-moisture values greater than 15 percent, the fine-fuel
graph should find its way down to the horizontal axis.

**2-4:** Here, students are given a practical application of
geometric or exponential growth. Students must also decide what is
reasonable for wind speed: 28 meters a second is roughly equivalent to
60 miles an hour, an unreasonable wind speed in fire season.

**5:** The rule is another example of a geometric or exponential relationship.

**6:** The data in the chart are not articularly linear. Linear,
quadratic, and exponential-regression models obtained on a TI-83
calculator for grass, loose litter, and tightly packed litter are given
in the following chart:

Grass | Loose Litter | Tightly Packed Litter | |

Linear | 1.85x - 20.97 | .93x - 9.9 | .46x - 4.49 |

Quadratic | .06x^{2} - 1.5x+ 7.09 | .03x^{2} - .75x + 4.04 | .01x^{2} - .38x +
2.51 |

Exponential | 1.13(1.08)^{x} | .93(1.07)^{x} | .81(1.06)^{x} |

The exponential model for all three types of fuel is a much better fit than the linear models and a somewhat better fit than the quadratic; nonetheless, students with minimal experience with regression can find a line of best fit.

### Reference

Goetz, Albert, "Smokey the Bear Takes Algebra," The Mathematics Teacher, vol. 92, no. 7 (1999) pp. 596 - 605.

### Learning Objectives

Students will be able to:

- Explain the relationship between relative humidity, temperature and the likelihood of fire-danger.
- Explain the real-world meaning of the intercepts and slope in the Angstrom index.
- Use summation notation to find the Nesterov index for each of the thirty-one days in August.
- Use graphing calculators to find equations to model the relationship between the slope of the land versus rate of fire spread.

### NCTM Standards and Expectations

- Generalize patterns using explicitly defined and recursively defined functions.

- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

- Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.