As per usual, Dan’s got me thinking. I’d like to think I beat him to the punch on this one, but I probably never said it quite as eloquently, nor did I manage to use skull-and-crossbones over a fancy gradient background. Either way, consider my whole blog an entry into the war on context with a lower-case “c”. That is, ways that context has been bottled, sold, neutered, and otherwise destroyed by text books and ne’er-do-well instructors world wide. (of whose ranks I unwittingly join some days)
I present unto you, the Rice Krispie Treat Cone. Ripe for the cutting and eating, this cone brings the conic sections to life! /dovesflyfromsleeves
This cone, built by yours truly in his very own kitchen, will elicit the most fundamental understanding of conic sections! Your students will be powerless against its awe-inspiring fundamentalitude! Your students will scream, “Circles, Ellipses, and Hyperbolae, Oh My!” as they nosh away on Diabetes-inducing rapture!
Multiply that by negative one. This is the kind of thing that I’ve been on about for the past few months. (this is my eighth month blogging!) The Rice Krispie Treats Cone is in no way real. It reminds of “diversity” month in grade school.
The Krispie Cone does not teach anything, unless there happened to be some fantastic reason for making a ‘baked’ good into the shape of a cone. (By the way, you’re looking at 10 cups of Krispies, 10 cups of mallows, and 3/4 cups of soy butter [I’m lactose intolerant])
Nevertheless, A Rice Krispie Treat cone has made its way into my classroom for 6 semesters, now. How do I defend myself against garnering my own philosophical indignation in the face of such hokum?
Create A Need, Then Get ‘Em With The Krispies:
Easy. We began with a simple game of rabbits vs. wolves. This year, I had enough students to actually run the game outside rather than just the computer simulation. It was basically red rover, except people literally figuratively died, man.
Rules: A replete wolf is a reproducing wolf. The wolves then eat more rabbits, the rabbits begin to show a net decrease in population. The wolves then starve, and the rabbits replenish. Repeat.
Replace “rabbits” with “kids running around trying to touch home base,” and “wolves” with “kids attempting to arm tackle other students” and you’ve pretty much got the idea. We recorded numbers at the end of each round.
This created the circle of life. We even sang the song. Laughing = 27 people trying to agree on the opening syllables to circle of life, btw. The students then had some genuine questions. How many wolves is too many? How many rabbits is too many? When are the populations growing (calculus! got ‘em) the healthiest? This was great. We needed circles, they’ve seen circles, all was well.
The conversation about circles led to ellipses. All was then not well. These kids hated ellipses. I could go on, by why not let Lockhart do it for me?
ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.
Upon substantial probing, the reason for ellipsephobia, (and hyperbolaphobia) was basically a lack of motivation. All I had to do was put up a diagram of the solar system, with a few extra players for good measure, and the situation was resolved.
What’s even better is that students began to ask why conic sections? Why is an ellipse as valid an orbit as a circle? Why is a hyperbola the highest energy orbit? When do things leave their terrestrial parabolas for orbiting circles?1 I was floored. This is all from rabbits, wolves, and a picture of the solar system. It’s the universe that happens to fit on a cone, not a cone that governs the universe. This is such a fundamental shift for students, that I have to sit down and swirl a scotch just to avoid crying.
Some really wanted to see it worked out, so I gave them this link. The cone comes from the fact that gravity pulls you towards the center of itself and has an inverse square behavior. It’s all kind of a fire hose, so be careful, if you’re going to lay any of this on them.
Other kids were just happy to have me explain that as you amp up your speed (energy) you try to leave the Earth. The harder you try, the further you get, but you usually just get pulled back, but the eccentricity of your orbit just keeps going up. Finally, though, the Earth isn’t strong enough, and bang, you’re outta here.
Here’s a game we played.
Meat and ‘Taters: Implicit Differentiation
This all leads back to implicit differentiation and the need to assess the speeds and trajectories of objects living on circles and ellipses. The mechanics of this technique are now so thoroughly motivated that perhaps students may actually want to figure them out.
Of course, the book always puts them through a meat grinder of problems that would never actually arise in real life, but this is the part of me that is still terrified of “college prep.” Or, the terrifying realization that someone else is going to force my students to memorize the derivative of secant for no other reason than that it can be done, so I can’t be the one that spent their time talking about Kepler’s laws and not teaching about emeffing secant.
So a dichotomy develops wherein we find motivation for new things (above), and then we see how far we can take it before it breaks (book problems). This has been my solution for motivating problems like this (a chain, implicit quotient, and implicit product rule, all in one shot! We sure are getting our $100′s worth, thanks book!):
When the reality is that no one on Earth ever finds the slopes of much more than a meager ellipse:
Yes, I realize the first problem is quality weight lifting, but when’s the last time you lifted for a game you didn’t even plan to dress for?
Oh yea, the Rice Krispie Cone. It’s just fun, and they’ll never forget the edible conic sections.
That is all.
Visto en THINK THANK THUNK
Universidad del País Vasco-Euskal Herriko Unibertsitatea
Facultad de Ciencia y Tecnología
Departamento de Matemáticas
Barrio Sarriena s/n
48940 Leioa (SPAIN)