Smokey the Bear Takes Algebra

Transcription

1 Smokey the Bear Takes Algebra Grade levels: 7 12 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 Materials Activity Sheet 1 Activity Sheet 2 Activity Sheet 3 Activity Sheet 4 Graphing calculator Computer spreadsheet program for one of the extensions Instructional Plan In the late spring and early summer of 1998, fires in Florida made headlines across the country when they raged uncontrolled over a vast area, threatening people, livestock, and property, not to mention the timber, valued at millions of dollars, that was destroyed. Controlling fires has become a significant environmental issue not just in the United States but in Central and South America as well. In particular, Brazil suffered tremendous damage during the 1998 "fire season". Areas under the jurisdiction of the United States Forest Service often have huge billboards of Smokey the Bear uttering his famous reminder, "Only you can prevent forest fires." Next to Smokey is a color-coded, fan-shaped meter indicating forest-fire danger. If the arrow on the meter points to the green end of the spectrum, the danger of fire is minimal; as the colors change from green to yellow to orange to red, the danger increases to very high. Permission to light back-country campfires depends on where that arrow rests on the scale. How does Smokey know the fire danger? What determines whether the arrow points only to yellow-orange instead of orange-red? What is the difference between a high danger of forest fire and a very high danger? This scale quantifies an abstraction: forest fire danger. What mathematics is at work here? The United States Forest Service uses a sophisticated computer program whose prototype was developed in the 1930 s to predict forest-fire danger. The Forest Service is linked to a computer network called Affirms, which enables individual rangers to enter weather and other data at their own work-stations and instantly obtain the necessary data concerning the fire danger in their locales. Because of its multi-layered nature and many subprograms, studying the program itself is not a topic for a high school mathematics course. But a great deal of interesting mathematics is involved in even a simplified look at the National Fire Danger Rating System or NFDRS. The remainder of this article gives a brief synopsis of the development of the NFDRS, a brief explanation of how it works, and possible ways to use it in a modeling or problem-solving unit in a secondary school classroom. The NFDRS: A Brief History

2 Earlier in the twentieth century, forestry agencies in Sweden, what was then the Soviet Union, Australia, Canada, and the United States independently began working on systems to manage fires. This development occurred because of rapid growth in the lumber industry. Between 1920 and 1950, foresters developed more than nine "meters" that link weather and forest-fire danger. These meters were specific to certain climates; geographic areas; different kinds of fuels, for example, grass as opposed to brush as opposed to timber; or some combination of these factors. The goal of developing a single measuring device that would consider all the variables was partially realized by the Forest Service in the mid-1960 s when it established a rating system for the rate of spread of a fire. The rating system depended on indexes that related surface spread to fuel type and moisture, that is, relative humidity. The NFDRS, put into effect in 1972 and updated in 1978, is a refinement of that model. Used today by all federal agencies and thirtyfive state agencies, it is accessible by more than two thousand stations across the United States. What Does The NFDRS Do? In essence, the NFDRS is a tool for predicting and containing fires. Input variables include weather conditions and data on the fuel, climate, and slope of the area. Weather data are consistently collected at the hottest time of the day, 1 P.M., on southern exposures with open surroundings, so that the program will in general overestimate the probability of fire. The output from the NFDRS takes the form of several indexes, some dependent on weather conditions and others related to local fire history. The occurrence index measures the risk of natural or human-caused fires and predicts the probability of the occurrence of a fire on the basis of similar conditions in the past. The burn index depends on the type of fuels available, the wind speed, and the slope or hilliness of the terrain. The fireload index is determined by examining both the occurrence index and the burn index. It can be calculated by using tables, with a handheld calculator, or by the Affirms network. Rangers patrolling remote areas can also calculate this index by using a kit of weather instruments that they can carry on a belt. Each of the indexes mentioned depends on subindexes. For example, the burn index combines a spread component with an energy-release component. These indexes in turn enable wind speed, slope, and fuel combustibility to be integrated with reaction intensity. Since each set of weather data leads to an approximation, uncertainty piles on uncertainty, so the usefulness of the final results is often more general than desired. However, the NFDRS gives individual outputs as well as a fire-load index, so that experienced fire managers can use these less general results to make decisions for an area. After a burn index and a fire-load index have been obtained, the indexes are compared with recent fire history for a given area, and one of five ratings is given: low, medium, high, very high or extreme. Fire-management personnel use these ratings in making decisions about deploying personnel and other resources that will aid in fire suppression. Teacher s Guide Goals: Students learn an important application of mathematics to a real-world situation, practice using linear functions and the meaning of slope and y-intercept, and work with fitting lines to data. Students also work with open-ended questions and give written explanation for their answers. Prerequisites: These activities presume knowledge of linear equations and their graphs, as well as slope-intercept and point-slope equations of straight lines. Students facility with a graphing calculator is also assumed. The ability to work with a computer spreadsheet program is optional but will enable students to take advantage of one of the extensions. Students also need to be able to work with sigma notation to understand the activities relating to the Nesterov index and need to be able to determine a regression line.

3 Sheet 1 Before distributing copies of sheet 1, discuss with students the Smokey the Bear billboards described in the opening paragraphs of this article. This discussion might include why the billboards are located where they are and why they are important to park visitors. Students might be interested in learning Smokey s history. A badly burned young bear cub was found by Forest Services personnel in 1950 after a destructive fire in the Lincoln National Forest in New Mexico. The cub, originally named "Hotfoot," was healed and rechristened Smokey. The resultant publicity made the bear an ideal icon for the continuation of an advertising campaign, begun during World War II, that was intended to dissuade campers from carelessly destroying the war effort s timber supply. Smokey was moved to the National Zoo in Washington D.C., where he became so popular that he was given his own zip code. He died in 1976 and was returned to Capitan, New Mexico, for burial. To complete the first activity sheet, have students work in small groups; fifteen minutes is a generous amount of time to allow for this activity. Some factors that students should list in response to the first question include relative humidity; wind speed; the number of days since the last rainfall; the terrain, addressed more explicitly in question 3; and the nature of the available fuel, for example, 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. Students may need help interpreting the word fuel here. Later, on sheet 4, the terms fine fuel and large fuel may need clarification. Answers to the second question will vary, but students should justify their hypotheses. The fact that hilly terrain causes fires to spread more quickly will be made obvious later in this activity, but students may be able to explain the phenomenon before completing the activity by thinking about the greater surface area of the ground exposed to flame. Flames will preheat the fuel that is upslope of the fire, thus making ignition easier. The fourth question on this sheet presents the justification for the development of the NFDRS. To combat fires effectively, firefighters need to deploy personnel and physical resources in the optimal way. Questions to be answered include the following: 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 does fire danger affect public use of the threatened area? Students can see that the model is useful in planning presuppression activities, allocating firefighting resources, controlling or constraining a fire, and limiting or denying public access to areas, as well as in planning the possible evacuation of local residents. Sheet 2 Because the actual NFDRS is far too complicated to examine in a secondary school classroom, we look here at some simpler precursors. One of the simplest fire-danger rating systems devised in the first half of the twentieth century is the Angstrom index, conceived in Sweden and still used in some parts of Scandinavia. The index is deliberately simple, designed to be calculated mentally. In fact, it is the only fire-danger meter in use that can be so computed. The effects of precipitation, wind, fuel type and fuel moisture, and relative humidity are not factors in this index. Students need to see that the value of I and the chance of fire danger are inversely related: the greater the value for I, the smaller the chance of fire danger. 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. For the given information in question 1, students should obtain a line with negative slope in the first quadrant. The slope of the line represents how much the fire-danger rating changes for each degree of change in temperature for the given humidity, whereas the y-intercept represents the fire danger when the temperature is 0 degrees. Students should make the connection that for the given relative humidity, the fire danger increases as the temperature increases, and they should see why this connection makes sense. The x-intercept, 44.5 in this example, represents the temperature at which the fire index becomes 0. Any value of x greater than 44.5 will result in negative values for the fire index. 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

4 scale, and with the horizontal line y = 4 at x = 4.5 or 40.1 degrees Fahrenheit. 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 To obtain these last answers algebraically, students need to solve the linear inequality with the appropriate values substituted for R. 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 danger s 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. 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. Sheet 3 To use the Nesterov index, students will need a month of weather data, preferable for the area in which they live. Perhaps the best source for the data is a local newspaper, and the data can be collected before beginning the activity. As an alternative, a table included in the appendix gives the dew point, temperature and relative humidity for a typical thirty days in the summer in Asheville, North Carolina. A Web site maintained by Pennsylvania State University gives the mean temperature and dew point for the United States as a whole but not for individual localities. The address is Also, the Web site of the National Climate Data Center, maintains weather data that are updated daily, but the data sometimes must be ordered in advance. Students may not know what the dew point means. The Weather Channel Web site, has an excellent glossary that students with Internet access can use. The glossary defines dew point as "The temperature to which air must be cooled at a constant pressure to become saturated." 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, terrain, or fuel moisture. In a possible extension of this sheet, students could compare the results of the Angstrom and Nesterov indexes when applied to a given set of local data. Sheet 4 In the final worksheet, students learn some rules of thumb that forest-fire-fighting professionals have incorporated into their set of tools. The moisture content of the potential fuel is a very important aspect of fire danger. This issue was addressed obliquely with the Nesterov and Angstrom indexes, a major failing of each. The first question on this sheet deals with fuel moisture. Students should indicate the fuel-moisture level on the independent axis, and the domain is

5 generally [0, 20]. The dependent axis will be rate of spread. 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. The question relating wind speed to rate of spread gives students a practical application of geometric or exponential growth while emphasizing the correct use of units. Students must also decide what is reasonable for wind speed: 28 meters a second is roughly equivalent to 60 miles and hour, surely an unreasonable wind speed in fire season. Students must interpret the word slope in the last question. Here, the word takes on a literal rather than a mathematical meaning, although students should be able to see the connection between meanings. The first rule is another example of a geometric or exponential relationship. The second rule can be modeled by the piecewise-defined function y = { where y is the rate of spread, Y 0 is the initial rate of spread, and d is the slope in degrees. The data in the chart for question 3 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 x x 4.49 Quadratic.06x 2 1.5x x x x x 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. would like to thank Judd Edeburn, Duke Forest resource manager, Durham, North Carolina, who was a tremendous help to me in researching and preparing this article. Appendix Temperature, Dew Point, and Relative Humidity Data For One Summer Month in a Southeastern U.S. Locale Day 1:00 P.M. Temperature ( o C) Dew Point ( o C) % Relative Humidity

6 * * * * * (Asterisks indicate that more than 3 mm of precipitation fell on that day.)

7 Standards and Expectations Algebra 9-12 This lesson covers the following Algebra Standard Expectations: analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior. generalize patterns using explicitly defined and recursively defined functions. use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts. MEET SMOKEY THE BEAR SHEET 1 1. Working in your group, list all the factors that you think would affect the spread of a forest fire. 2. Which of these factors do you think is most important? Least important?

8 3. Do you think that a fire will spread more quickly on a steep hillside or on level ground? Explain your reasoning. 4. Why must rangers know the likelihood that a fire will start? Why must rangers know the likelihood that a fire will spread? From the Mathematics Teacher, October 1999 THE ANGSTROM INDEX SHEET 2 A simple fire-danger rating system, the Angstrom index, was devised in Sweden and has been used all over Scandinavia. The index, I, is given by where R is the percent of relative humidity and T is the air temperature in degrees Celsius. The values for I translate into fire danger as follows: Use another sheet of paper to answer questions 1-8. I > 4.0: fire occurrence unlikely 4.0 < I < 2.5: fire conditions unfavorable 2.5 < I < 2.0: fire conditions favorable I < 2.0: fire occurrence very likely 1. Hold R constant at 35 percent, and graph I versus T on your calculator. Describe your results and the window that you used. 2. With humidity at 35 percent, how hot would it have to be for fire occurrence to be considered very likely? Unlikely? Which portion of the graph do you think is most reasonable for the geographic area in which you live, that is, what is the normal temperature range for your geographic area? 3. What is the real-world meaning of the slope of the line that you have graphed? The y-intercept? The x- intercept? Find the value of T that makes I negative.,

9 4. How does the graph obtained in question 1 change when the relative humidity is 40 percent? Use your calculator to determine how the 5 percent change in humidity affects the temperature at which fire occurrence becomes very likely. Repeat your work to determine how the humidity change affects the temperature at which fires become unlikely. 5. The work that you did in problem 4 can be done algebraically, with using the calculator. Explain how you might do this work. 6. Next hold the temperature constant, and vary the relative humidity. Fix the temperature at 30 degrees Celsius, or 86 degrees Fahrenheit, and graph I versus R on your calculator. Describe your results and the window that you used. With a temperature of 30 degrees Celsius, what humidity would make fire occurrence very likely? Unlikely? What is the real-world meaning of the slope of the line that you have graphed? 7. What happens to the graph in question 6 when you change the temperature to 35 degrees Celsius? Use your calculator to determine how much the 5 degree change in temperature affects the relative humidity at which fire occurrence becomes very likely. Repeat your work to determine how much the temperature change affects the humidity at which fires become unlikely. 8. On the basis of your answers to the previous questions, is this fire index more sensitive to temperature or to humidity? Explain your analysis. From the Mathematics Teacher, October 1999 THE NESTEROV INDEX SHEET 3 Another simple fire-rating system, the Nesterov ignition index, was devised in the Soviet Union in 1949 and is as follows: Where P represents the ignition index, W is the number of days since the last rainfall greater than 3 mm, t is the temperature in degrees Celsius, and D is the dew-point temperature in degrees Celsius. The computations begin on the first spring day when the high temperature is above freezing after snow melt and continue until a rainfall of 3 mm, whereupon the process starts anew. When P is between 0 and 300, the fire danger is minimal; between 301 and 1000, the fire danger is moderate; and between 1001 and 4000, the danger is high. Any value of P more than 4000 means extreme fire danger. 1. Consult your local newspaper or the Internet to obtain data for the most recent August in your area or state. Compute the Nesterov index for each of the thirty-one days in August. For simplicity, assume that it rained on 31 July.

10 2. Compare the Nesterov and Angstrom indexes. Which seems easier to use, and why? Which seems more comprehensive, and why? Extension. Construct a spreadsheet that reflects the Angstrom index for relative humidities ranging from 25 percent to 95 percent and for Celsius temperatures from 20 degrees to 45 degrees at intervals of 5 degrees. Summarize in a paragraph what you learn from the spreadsheet output. Use a local newspaper forecast, a radio or television forecast, or weather information form the National Weather Service s Web site to find the average temperature and relative humidity for a day in mid- August in your community. On the basis of these data, what does the Angstrom index predict as the likelihood of a fire in your area? From the Mathematics Teacher, October 1999 SOME RULES OF THUMB SHEET 4 Use another sheet of paper or the reverse side of this sheet to record your work. 1. When fuel moisture is below 5 percent, fires in both fine fuels and large fuels tend to spread equally quickly. When moisture levels are between 5 and 10 percent, fine-fuels fires spread more rapidly than large-fuel fires, whereas at levels above 10 percent, the rates of spread are about the same again. When fuel moisture is above 15 percent, the fine-fuel fires will tend to extinguish themselves, whereas large-fuel fires will continue to spread. On a single set of axes, draw possible graphs for the rate of spread of fine-fuel and large-fuel fires. Share your graphs with another student. Discuss any differences. 2. A general rule of thumb states that rate of spread, a dimensionless index that measures how quickly a fire will grow, will double for each increase of 4 meters a second in wind speed. The finest fuels cause fires to spread more quickly; the densest fuels, less quickly. a. What is the nature of the relationship between rate of spread and wind speed? Draw a possible graph that relates rate of spread to wind speed. b. Assume that the rate of spread is 6 when wind speed is 0 meters per second. Make a table of values for rate of spread as wind speed increases to 28 meters per second. Plot the graph of spread versus wind speed. How many miles per hour is 4 meters per second? What do you think is the reasonable part of the graph that you just drew? What is the nature of the relationship between spread and wind speed? 3. Several rules of thumb concern spread rate and the slope of the terrain on which the fire is spreading. One suggests that the rate of spread will double for every increase of 10 degrees in

11 slope; a second, that the rate of spread doubles for every 15 degree increase in slope up to 30 degrees and then doubles for every 10 degrees thereafter. These differences occur because other factors affect the rate of spread, including how packed the fuelbed is. a. What is the nature of the relationship in the first rule? b. What piecewise-defined function best models the second rule? c. Use the data in the following chart to obtain scatterplots for each of the different kinds of fuel. Superimpose on the scatterplots the graphs of the functions previously obtained. How closely do these equations model the given rules of thumb? Relative Rate of Spread Slope in Degrees Grass Loose Litter Tightly Packed Litter From the Mathematics Teacher, October 1999 (Activity taken from