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!" |

**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

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