Instructional Unit: Parametric Equations

by Mary Wilson Eager

Introductory Activity and Assessment

I owe many thanks to my trigonometry and pre-calculus students at Lumpkin County High, Dahlonega, GA. They participated enthusiastically in this unit on parametric equations and helped me tremendously. Every day I teach, I also learn. Thanks LCHS juniors and seniors.

The unit, designed for pre-calculus students, begins with an exploratory activity authored by Dr. Jim Wilson, UGA professor of mathematics education. Instead of beginning with definitions, theorems and worked examples of problems, this unit begins with an exploration. Students use a graphing utility to enter many parametric equations and, from their observations, make generalizations about parametric curves. The class accesses a lab with twenty Macintosh computers. The software available is Xfunctions, with its parametric graphing utility. With limited instruction the students are able to enter the equations correctly, as well as change the bounds of the parameter and minimums and maximums for the viewing window.

Objectives : Students investigate a previously unknown topic in mathematics, parametric equations. They develop their ideas on this topic by observing the behavior of many graphs of parametric equations. To conclude the introductory investigation, students prepare a summary of their findings to share with the class.

Materials: Thanks to Dr. Wilson, I give students the first eight problems from EMT 668 assignment ten. Click here to view the problems.Assignment ten

Assessment: Students present their findings in various formats. Some prepare overheads to use as they talk to their classmates. Others use the computer graphics to illustrate their information. The work of two students, Kelly Ferguson and Jon Whidby, is included in this document as a sample of the summaries the students presented. Click here to link to their file.Click here

At the conclusion of the four day exploration-presentation period, students write a journal topic on the process and their current understanding of parametric equations. I ask them to respond to these statements:

"How do you feel about learning by the discovery process"

and "Describe what you know about parametric equations".

The responses range the gamut from delight with discovery learning to total frustration with the entire process. I'm quoting select sentences from these journals.

"Its fun to learn on my own instead of having the teacher telling me what to do."
"I have a hard time staying on task. Its too easy to get sidetracked by pretty pictures"
"Investigations are very entertaining and time consuming".
"I have difficulty deciding what's important from looking at so many graphs."
"One person can't do all the investigations."
"We can see what effects the equations and learn through trial and error."
"Its good to be able to explain our thoughts to others."
"I'm frustrated because I jump to conclusions. I have no idea if my conclusions are right or wrong."