How Far is Yonder Mountain? -- A Trig Problem

Len Vacher, University of South Florida
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This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

Summary

In this Spreadsheets Across the Curriculum activity, students work through Polya's problem-solving heuristic to find the distance of a peak using vertical angles sighted from a wagon train heading directly for the peak. They build a spreadsheet to do the calculation. The spreadsheet also calculates the height of the peak above the plain. After calculating the distance and height, the students "look back" (in Polya's terminology) and consider the reasonableness of their answer. They also work through another way of solving the problem to see if they get the same answer. In this second way, they use Excel to find a solution by trial and error. In the end-of-module assignment, they use their spreadsheet to examine the effect of uncertainties in measuring the angles on the calculated lengths.

Module inspired by El Capitan in Guadelupe National Park, which is discussed in one of the links. Geology of National Parks collection.

Learning Goals

Students will:
  • Gain experience in figuring out how to solve a trigonometric problem in geologic/geographic context.
  • Apply algebra to solve a problem.
  • Develop a spreadsheet to carry out a calculation.
  • Look back at the solution and consider an alternative way to solve the same problem.
  • Develop a spreadsheet to find a solution by trial and error.
  • Think about the reasonableness of the calculated answer.
  • Consider the effect on the calculated answer of small uncertainties in the measured quantities and thus the sensitivity of a result to the input.
In the process the students will:
  • Begin to recognize the power of Polya's problem-solving heuristic.
  • Relearn, or maybe learn, the tangent ratio and the concept of radian.
  • Become more skilled at manipulating equations to solve a problem.
  • Be impressed with the usefulness of Excel in solving a problem by trial and error.

Context for Use

I use this module in my Computational Geology course, GLY 4866 (Acrobat (PDF) 39kB Sep25 06). The course is aimed at geology majors who have completed or are about to complete the mathematics courses required for the geology major. The class consists of students who anticipate graduating in three or fewer semesters.

This module comes early in the semester, typically in the third week of classes after the students have had a couple of computer-lab sessions aimed at introducing Excel to beginners. The focus of Yonder Mountain is "How to Solve It" in the sense of George Polya. The module uses Polya's problem-solving heuristic explicitly. It includes links to external information about Polya and his four steps for solving mathematical problems. It asks the students in the end-of-module assignment to summarize Polya's heuristic and how it applies to the trig problem discussed in the module.

I use Yonder Mountain as a homework assignment to follow up on an in-class problem-solving session. The in-class session goes approximately as follows:

  • I start by posing the problem on the whiteboard (the problem is on Slide 2 of the PowerPoint). The students divide up into groups to brainstorm about how to solve it. After about 10 minutes, we (the class) inventory the various thoughts that have been bandied about at the tables. I remind them of Polya and his "How to Solve It", which I have introduced the previous week in the in-class session accompanying How Large is a Ton of Rocks? The students then brainstorm some more in their groups.
  • Some students have no difficulty with the problem. Many others, however, come to realize that they cannot access their trig, and a few doubt that they ever had any. A few charge forth with the Law of Sines, which is not what I had in mind for this problem. For the benefit of the students who are struggling or are not sure, we go over the fundamentals of triangles including the trigonometric ratios (sine, cosine, and tangent). I mention specifically that I do not want this problem to be solved with the Law of Sines, although it is certainly a good way to solve the problem; we will be doing the Law of Sines later, in a different context. So the students return to brainstorming in their groups.
  • When I notice that one or more groups are combining the equations involving the two tangents, and actually doing algebraic manipulations, I ask for progress report-outs so the idea is disseminated. With a little more work, the groups are all getting somewhere. The Law of Sines enthusiasts are checking the answers against what they got with the Law of Sines.
  • About 10 minutes before the end of the 1-1/2 hour class time, I show the class the first spreadsheet on the module (Slide 7) and introduce them to the module's second way of solving the problem. This second method is new to everyone in the class. The idea of guessing distance until you guess right (i.e., until the distance produces the tangent of one of the known angles) comes as a complete suprise. The reaction of some of the students is as if I had just shown them how to cheat.

After class, the students work through the entire module, which has "gone live" on Blackboard during the class session. Each student emails the completed end-of-module assignment to the TA within a week of the class session.

Description and Teaching Materials


PowerPoint SSAC2006.QA531.LV1.2-student version (PowerPoint 278kB Feb15 07)

The module is a PowerPoint presentation with embedded spreadsheets. If the embedded spreadsheets are not visible, save the PowerPoint file to disk and open it from there.

This PowerPoint file is the student version of the module. An instructor version is available by request. The instructor version includes the completed spreadsheet. Send your request to Len Vacher (vacher@usf.edu) by filling out and submitting the Instructor Module Request Form.

Teaching Notes and Tips

The module is constructed to be a stand-alone resource. It can be used as a homework assignment or lab activity. It can also be used as the basis of an interactive classroom activity, with just-in-time teaching of the relevant mathematics (see "Context for Use").

The module's second way of finding the distance is by trial and error. Excel has a solver that accomplishes the same thing. I prefer that the students experience using Excel to find an answer by trial and error many times before I tell them about "Solver" near the end of the semester.

Assessment

The end-of-module questions can be used for assessment.

The instructor version contains a pre-test

References and Resources

http://www.sci.uidaho.edu/polya/
about Polya's heuristic