**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."