In the previous lesson, students learned about defensible space. Specifically, they learned about preparing defensible space on a flat terrain. In this lesson, students will learn how to measure percent slope, and they will consider the impact of slope on defensible space.
Distribute the Defensible Space guide. Have students look at the chart on the bottom of the first page, the table in the lower right corner of the second page, and the chart in the lower left corner of the third page. All three of these presentations suggest that the steepness of terrain has an impact on preparing defensible space. (Note that each of these topics could lead to a rich math discussion and lesson. Therefore, depending on time limitations, you may choose to focus on just one or two of these ideas, rather than attempting to cover all three.)
Ask students, "Why do you think the steepness of the terrain matters when trying to prepare defensible space?" [As students learned in Heating Up, wildfire travels faster and more easily up a hill than across a flat surface. To discourage the progress of a fire moving up a hill, more space should be left between trees, plants, and other flammable objects.]
Explain to students that the steepness of a hill is measured as percent slope. A synonym for percent slope is grade, and students may have seen this word used on road signs like the one shown below:
Ask students what they think the percent slope number means—for instance, when a sign on the highway indicates than an upcoming hill has a "9% grade," should they be worried about how steep the hill is?
To allow students to discover the way that percent slope works, play a game of "Guess My Rule" with them. As the name implies, students are to guess the rule (in this case, the formula for percent slope) by examining examples that you share with them. Start the game by drawing some of examples of percent slope on the chalkboard or overhead projector. Some possible examples to use are shown below:
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| Percent Slope = 60% |
Percent Slope = 27.3% |
Percent Slope = 40% |
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| Percent Slope = 25% |
Percent Slope = 80% |
If students are not able to guess the rule from these examples, give them some more. Or, draw a right triangle, have students suggest the lengths of the legs, and then compute and report the percent slope to them. The formula for computing percent slope, as is probably obvious from the examples above, is:
Once a student thinks that she has figured out the rule, do not allow her to announce the rule to the class. Instead, draw a right triangle and give the lengths of the legs, and ask her to tell you the percent slope. If she gives the correct percent slope and you are satisfied that she has determined the rule, then a reward is to let her lead the class through another example. As new students figure out the rule, they can then take over as the class leader.
If some students have not discovered the rule after many examples, allow classmates to help them. (Note that this is a good means of assessment. You may need to offer extra assistance during the rest of the lesson to those students who had difficulty recognizing the pattern.)
Ask students to describe a situation in which the percent slope would be 100%. [Most kids will think that a 100% slope corresponds to a 90°angle, but that is not the case. Given the formula for percent slope above, if rise = run, then the percent slope is 1, or 100%. This occurs for a 45°.]
The next portion of the lesson focuses on the percent slope measuring tool. A version of this tool is drawn on the second page of the Defensible Space guide, in the box titled "Homeowner’s Guide to Calculating Percent Slope." However, this tool is not labeled correctly. The error is actually very subtle, but it points out a misconception that many students hold.
Allow all students to inspect the tool, and ask them what is wrong with it. Some students may be able to offer an explanation. [The percent slope is correct, but the angle measure is incorrect. That is, a 20% grade does not correspond to a 9° incline. In fact, a 20% grade corresponds to an 11.3° angle.] To help students see the error, distribute protractors and allow them to measure the angles. They will only need to measure one angle to see that the degree measurements are incorrect, though you may want to have them measure all of the angles. Students are often surprised to learn that the angle measures are not 9°, 18°, 27°, 36°, and 45°, but instead they are 11.3°, 21.8°, 31.0°, 38.7°, and 45.0°.
The reason for the error and the misconception is that percent slope is a linear function. That is, as the length of the vertical rise increases, the percent slope increases proportionally. However, that is not how angle measures work. You may need to remind students that angle measures are based on a round object (i.e., a circle), but percent slope is based on a straight object (i.e., a line). This is why percent slope increases linearly, but angle measure does not.
This idea may be difficult for kids to grasp, since it seems counterintuitive. But here’s an explanation that might help: Consider a triangle such that the base of the triangle never changes, but the height of the triangle increases. As described above, if the height is equal to the base, the angle is 45°, as shown below left. Ask students, "If the height is doubled, what will the angle be?" Most students will suggest that the angle will also double and have measure 90°. However, this is not the case, as shown below right. If θ = 90°, this triangle would have two right angles, which is impossible! If students measure θ with a protractor, they will find that it is closer to 64°.
Proceeding with the lesson, students should then construct their own percent slope tool. Distribute one copy of the Percent Slope activity sheet to each student.
To build the percent slope tool, students will need to cut out the rectangle from the activity sheet and glue it to heavy cardstock. Then, they will need to punch a hole through the circle, pass one end of the string through the hole, and tape the string to the back of the cardstock. Finally, tie a weight to the other end of the string so that it hangs taut.
The completed percent slope tool will look like this:
Students should now use the tool to measure a slope. Ideally, take the students outside to measure the slope of some nearby hills. Alternatively, you can tape several pieces of rope or string in various locations throughout the room, and students can determine the percent slope of each of them.
As homework, students should complete the How Steep Can You Be activity sheet, which will be used in the next lesson.
To help students complete this activity sheet, you may want to work with a science teacher in your school. Collaborating with a science teacher could integrate the content of both mathematics and science. The science teacher could present information about various types of vegetation and unique features that require special consideration when using them in landscaping. Students may need to refer to field guides to identify the vegetation at the location they select.
