```
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!

Thank you Audrey! This is wonderful! I love how it came naturally out of the discussion of circles also, just as it should. Thank you for sharing it.

ReplyDeleteThanks for reading, Teresa! I was particularly happy with how I snuck the fractions in without anyone noticing...our kids tend to not be too comfortable with them.

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