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:
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
CCBYNCSA

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, STS114 mission specialist,
anchored to a foot restraint on the International Space Station’s Canadarm2.
Photo: NASA http://spaceflight.nasa.gov/gallery/images/shuttle/sts114/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/sciencenews/scienceatnasa/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.
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 
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.
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.
Great ideas! I was hoping for more on "acting the goat" after you mentioned it before.
ReplyDeleteOne application for the locus idea of an ellipse: after learning it back in 4 unit in high school I next came across it doing a millinery course. Best way to draw a neat ellipse for a hat pattern!