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 

Toys and tools for exploring the Parabola

Following on from three ideas to introduce locus, here are three ideas I used to help make the locus of the parabola come alive for my students. Regular readers of this blog will know how much I believe in the benefits of hands on exploration of mathematical objects - and these are very hands on!

The three ideas are:

• Use a MIRA mirror to construct a parabola. My senior students loved this activity- a chance to revert to back to childhood, while still being challenging.
• Use GeoGebra to construct a parabola given any arbitrary focus point and directrix. Try this with non-standard orientations.
• Be entranced by a wonderful 3D optical illusion toy that exploits the properties of the parabola.

1. Using a MIRA mirror (MIRA math tool) to construct a parabola
I'm extremely fortunate to have a box of these in my faculty storeroom:

Source:  http://www.enasco.com/product/TB14953T 

While they look like tools for the junior math room (and they are wonderful to use in this context!), there's no reason our senior students should be locked out from using them! Here is a worksheet that give instructions on constructing a parabola with the MIRA mirror.  It's a really fun activity - a chance for senior students to play a little - and a great opportunity to ask the "why" question - reinforcing the idea of locus and the locus definition of a parabola.

Locus and Parabola MIRA Parabola GHT0501

2. Constructing a parabola using GeoGebra.
Why do I need special help to construct a parabola in GeoGebra you may say? Of course GeoGebra can construct a parabola with zero effort. But this guide explores how to construct a parabola using the locus approach.

Can you find the parabola given any arbitrary focus
point F and directrix AB?

Locus and Parabola Make a Parabola GHT0301

3. Discover something special about the parabola
A terrific toy worthy of being in your mathematics (and science) classroom is the Optigone Mirage®.

The Optigone  Mirage® is a pair of twin parabolic mirrors, arranged to project
a 3D image floating above the top of the kit. In this image from a paper by Christian Ucke,
the pig is actually inside the mirrors.

As always, encourage students to play with the toy (not that they will need encouragement - my students had their smartphones out takings photographs within seconds!), then ask the Why? question. Even though I purchased* one of these for my senior mathematics class, students across all my classes were entranced by it - and it gave me great pleasure to say to the juniors "you will learn how it works in your maths and science classes in a few years".

* Sadly I could not find a convenient way to purchase one of these in Australia - so I ended up buying a  clone from Australian Geographic.

How do you do that in GeoGebra?

Looking back on my own high school mathematics education, I realise I never really knew what a parallelogram was. I never knew how it 'worked', how its angles and diagonals operated, how they changed when the slope of the parallel lines was changed. The rhombus? All I could really say - if I remembered it at all - was it was a kind of squashed up square. If only I had  been given a dynamic geometry tool to play with! As a teacher now, I strive to have my students actually touch mathematical objects - to move them, push them, pull them, to watch what happens. I'm convinced that if students do that, so long as they are reflecting on what is happening to the objects (and why), they will remember them for life.  And the ideal tool for hands-on interaction: GeoGebra.  Free software, runs on Windows, Apple and Linux (anything that runs Java), backed by a community of hundreds of thousands of teachers using and sharing GeoGebra resources.

A resource I haven't found yet though is a set of simple, one page instructions I can give to students showing how to construct a certain mathematical objects in GeoGebra, so I have begun building some.

Here's the first installment:

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For Junior and Middle School

How to construct a parallelogram.

How to construct a rhombus. This one appears simple but can be confusing - practice it first before giving to students.

For Senior School

How to construct a parabola using a focus point and a directrix.
How to build a polynomial factor exploration tool.

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I have put these links on a new GeoGebra HowTo page in this blog. These files are also available at the Maths Faculty sharing repository.

Art photography in the maths classroom - thanks to flickr

So much of the mathematics content we teach appears to many students to be fairly dry on the surface - we hope we bring them on the journey to see its wonder, but there is nothing like a great visual to create interest and start a quality discussion. Recently I tapped into an amazing resource: flickr photos provided under Creative Commons licenses.  There are thousands of high quality inspirational photographs available just waiting to go into a mathematics lesson.

For the topic "The Quadratic Function", water was my visual theme. I use just one of these images for each lesson - we start the lesson with a full screen, high definition version and then I use smaller versions to create a visual link for transitions between lesson activities.

All images under Creative Commons. Click on the photograph for details.

For the topic "Locus and the Parabola" I blended an astronomy theme (think: parabolic reflectors) and more abstract designs:

All images under Creative Commons. Click on the photograph for details.

These images look spectacular in high definition projected onto a screen in class - and give an opportunity to engage in some broader discussion about the application of mathematics, and the relationships between art, science and mathematics. Interesting to see students also noticing the use of Creative Commons - a chance to model good practice and spread the CC message.

Because the creators of these images generously put their work 'into the commons',  I can use them in my own works and then in turn, share those works with other teachers without any copyright constraints - hopefully helping students in many other classes.

Finding Creative Commons licensed content on flickr
Use the Advanced Search feature:

scroll down and select these boxes:

I also select "Interesting" which tends to return richer images. When using the image, make a CC attribution and provide a link back to flickr page. I also like to leave a thank you comment to author. And thank you flickr for coming to the CC party. Now if only Google would make CC searching available on its main image search page! I believe it is there - just hidden - and life is too short to look for hidden options.