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.

Three teaching ideas for introducing locus

Here's three ideas I use for introducing locus.

1. What makes these shapes? Have you ever looked down from an airplane window and seen a sight like this? Something you will often see above large scale farming areas:

Centre Pivot Irrigation - an example of locus of the circle.
Photo: Etienne Boucher CC-BY-BY-NC-SA 

http://www.flickr.com/photos/djof/147222315/

Give the students some time to digest these images - you will hear lots of wild theories.

Then show this picture and see if they can work it out:

“Centre Pivot Irrigation” Matt Green CC-BY-NC-SA

http://www.flickr.com/photos/imjustwalkin/4875949345/

The Wikipedia page on Centre Pivot Irrigation has lots more interesting information.

2. Acting the Goat.  Tie yourself with a rope to a chair or desk and pretend to be a goat. Explain how goats will eat absolutely everything in sight. Model the behaviour. A good laugh - and your class won't forget the locus of a circle or the concept of a constraint determining the locus. Then extend the idea to different situations of a goat on a leash (tied to a fence with a sliding leash, etc).

Built the idea of a locus by "acting the goat".

3. Who cares about locus?  Show an image like this:

Astronaut Stephen K. Robinson, STS-114 mission specialist, 

anchored to a foot restraint on the International Space Station’s Canadarm2.
Photo: NASA  http://spaceflight.nasa.gov/gallery/images/shuttle/sts-114/html/s114e6647.html

This person certainly cares about locus! Also a good opportunity to raise awareness of the International Space Station. More information on the Canadarm2 at http://science.nasa.gov/science-news/science-at-nasa/2001/ast18apr_1/

I said three ideas? Sorry - I can't resist sharing three more ...

4. "Locus Pocus" : A high quality video about locus well worth showing in class is Erica Morabito's Locus Pocus.

I like this video because it creates some memorable visual representations of specific locus problems that students need to know while also being entertaining and succinct (lesson time is a precious resource!).

5. Introduce the ellipse : Most students have an idea what an ellipse is, but very few know how to make them or the locus idea behind them. This YouTube clip give a good demonstration:

and I wrap up this discussion with a picture of my favourite example of an ellipse in action:

Locus of the ellipse at work in the universe:
a stunning NASA image of Io orbiting Jupiter.
http://www.nasa.gov/mission_pages/juno/multimedia/pia02879.html

Yes - another segue into astronomy.  Locus is manifest throughout our universe - it would be remiss of us not to show such stunning images of it at work.

6. Work the Geometry 'vs' Algebra dynamic: I find it helps students to be explicit about the two different approaches to curves (geometric versus algebraic). It seems to me many students prefer the algebraic - so we need to work extra hard to show the virtue and value of the locus view. I reintroduce our fellow travelers Euclid and Descartes and show the dynamic at work - that we want to be able to switch between them with ease.

Every time I have an idea or problem to develop with the class, I invoke the Euclid/Descartes duo. It's fun to have great mathematicians in the classroom with you - and it helps develop a broader of mathematics as a dynamic, developing intellectual inquiry - something students can participate in and perhaps even extend.

Next post in this sequence: Teaching ideas for introducing the locus of the parabola.

Moves in translation with Miss Anna Parabola

Some concepts are so powerful in mathematics, they just keep popping in your course again and again - like old friends. Such an idea is the translation of a curve in the number plane. In my class, this old friend has a name: Miss Anna Parabola. Anna has been making an appearance throughout my course, starting off with an introduction to the quadratic function.

Miss Anna Parabola demonstrates $y = x^2, y = -x^2, y = x^2 + k$.
Ballerina: Alicia Alonso in 1944, photographed by Gjon Mili for Life magazine

I will admit to raising and lowering my arms in the different ballet positions in class, standing up on chairs (against OH&S regulations ....) - but I vehemently deny donning a tutu. 

I knew I was onto a good thing when I started teaching the topic "Locus and the Parabola". One of our textbooks spends an arduous 35 pages (no kidding) going through all the iterations of the different orientations and translations of the parabola - but I realised with our class understanding of Miss Anna's dance moves, we could collapse the entire thing into two lessons: one lesson to cover the different orientations, one to cover the translation.

The Four Standard Orientations of the Parabola -
as interpreted by Miss Anna Parabola (aka Alicia Alonso)
(Click image for a larger view)

Translating the vertex.

And it worked : my students can now do this effectively and efficiently. They connected our previous work on shifting curves like $x^2 + y^2 = 25$ to $(x-2)^2 + (y+4)^2 = 25$ to this work on shifting the parabolas. We cracked what would otherwise be a very arduous (and boring) part of the topic by focusing on the key idea of 'moves in translation'.   I'm a big fan of creating characters and story to build a narrative in the course, so I was thrilled to see the work from previous topics developing Anna Parabola pay off like this.

Oh - and in case you haven't heard of him, Anna has a new friend: Billy the goat. Billy helps develop the idea of locus : if you tied him to a fence, he would happily devour everything around him, following the locus constraint imposed on him.  And yes - I do admit to tying myself to a desk and 'acting the goat'.  How am I going to live this down....?

PS: I'm not sure this trick would work at a boys' school.... Might have to invent a rugby player in motion...

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.