Developing the standard rule of an ellipse

This activity was inspired by two people: Teresa Ryan, a fabulous math teacher tweep, and Amanda R., one of my students. A few days ago, Theresa tweeted this

Anyone have a good introductory lesson to the standard form of the equation of an ellipse? TIA #MTBoS #mathchat
— Teresa Ryan (@geometrywiz) May 7, 2014

and that question started the cogs turning. Around the same time, I had my students playing around with circles on desmos. Amanda happened to type in the equation 2x² + 2y² = 1, and notice that it had a smaller radius than our unit circle. That lead to a nice discussion as to why the radius was less than one, and then why it was equal to the square root of 1/2.

So today, again, all of this kind of gel-ed on my way into class. Here are my guiding questions, and their collective answers:

Open a desmos or ggb, and get the unit circle to show up.

Now type in 2x² + 2y² = 1. Tell me what you get, (smaller circle), what's the approximate radius? (0.7)

Type in another equation like this, which = 1, but make an even smaller circle appear, and write your equation on the eboard, plus the approximate radius.

Find pattern: as coeffs get bigger, circle gets smaller.

And what's the relation between the coefficient and the radius? Radius is the square root of one over the coefficient.

Okay, if bigger coefficients make smaller circles, what coefficients will make bigger circles? (1/2 or 1/3)

Type those in, measure the approximate radius, and write on eboard:

Is it okay if we write these equations this way instead? Are they equivalent?

Now how can we calculate the radius from the rule? It's always the square root of the number in the denominator.

Which denominator? Well, it doesn't matter. Doh. They're the same.

Oh right. I didn't notice that. Well type one in that doesn't have the same number under each term, what do you get? An ellipse!

Tell me your equations and the dimensions of your ellipses:

Unfortunately I didn't take a snip of this, but the variety was wonderful, some ellipses were horizontal, some vertical, it was absolutely no big deal for them to see that the number under the x always governed the width and the one under the y governed the height, plus that a square rooting was involved.

From there it was a piece of cake to generalize to the standard form of the ellipse! We did a bit of practice where I gave them the rule and they graphed, and vice-versa. It felt like I'd covered 2-3 days' worth of concepts just today.

Thanks Teresa and Amanda!