Newton and Descartes channel Dan Meyer

There's a definite pause the first time you show parametric equations to  students well conditioned to Cartesian representations.  I like to imagine Descartes himself staring at the equations pondering : "Why would you do that????"

We're not in Kansas any more!
Descartes: "Why would you do that? It's the same end result!"

Here are three teaching ideas I used this year with my senior mathematics class which may supplement the traditional approach of showing the different representations are functionally equivalent.

1. Extend the function machine idea to show a weird new parametric function machine. Now we have two outputs!  Here are the two function machine images I use for my resources:

Based on a function machine diagram at http://raider.mountunion.edu/ma/MA125/Fall2011/Chapter7/IntroToFunctions.html
I removed text from the original image, then adjusted it to make the parametric machine.

2. Explore the reasons why we might want to use parametric expressions to describe a relationship. 

The best I answer I came up was this (click on the image for a larger view):

Newton and Descartes ponder Dan Meyer's "Will it hit the hoop" lesson.
My students did this activity in a previous lesson, so they got the joke.

In other words, a parametric description of this scenario lends itself to a deeper understanding of the physics of the situation. 

Another reason for using parametric equations is that the maths can be much more interesting - and possibly a lot easier to work with.  Parametrics also give us another way to get a feel for the constraints at work in a locus.  I love this wonderful "move the robot" explanation from James Tanton - and it speaks to my IT background where parameter go in, and things move accordingly!

3. Get a feel for parametrics by controlling the parameter using dynamic geometry software.  I found it really helped my students to build a parametric representation, then adjust the parameter by moving sliders and then seeing points move under their control. Actually touching and moving and parameter reinforces the idea of a point travelling along a path under a constraint.  Here is a resources for students to explore parametric representation of the parabola using GeoGebra:

HowTo Guide: Exploring the Parametric Representation of the Parabola

This guide is part of my collection at GeoGebra HowTo