How Far is Yonder Mountain? -- A Trig Problem

Len Vacher, University of South Florida
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This material was originally developed by Spreadsheets Across the Curriculum as part of its collaboration with the SERC Pedagogic Service.

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: In the process the students will:

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:

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

about Polya
about Polya's heuristic