Circles present a dilemma for me. I think it's because the way I typically introduce the standard equation of a circle is so abstract for something that is rather straight-forward.

The way I see most textbooks present circles, and the way I've presented circles in the past is very similar to this:

Aye aye aye.

Every semester I think, "Maybe I can make this derivation make sense to them this time around!"

And every semester I fail. Miserably. I mean, you can practically taste the glazed-over looks. It doesn't matter how many times I say, "It's just the Pythagorean Theorem, guys!" I've lost them.

So, I tried something different this time around. It's still far from perfect, but the equation of a circle came much faster and from many more mouths than usual. I did what most of us probably do when trying to verify a general concept: apply it to a specific example.

I showed them this picture and asked them to relate the lengths of

Then I showed them the same picture as before and asked for the same information, but this time--no grid, no numbers. I told them to call the center (

They came to the result pretty quickly.

Some concerns I have with this strategy: Do they understand the significance of

Like I said, I'm happier with how things went this time around. But not yet satisfied (hopefully I never fully will be). Any suggestions are gladly welcomed!

So, I tried something different this time around. It's still far from perfect, but the equation of a circle came much faster and from many more mouths than usual. I did what most of us probably do when trying to verify a general concept: apply it to a specific example.

I showed them this picture and asked them to relate the lengths of

*a, b,*and*r*using the Pythagorean Theorem, and then find the exact lengths of*a*and*b*. And, most importantly,*how*did they arrive at the lengths for*a*and*b*(without just counting units).Then I showed them the same picture as before and asked for the same information, but this time--no grid, no numbers. I told them to call the center (

*h,k*) and the other point, which represents any point on the circle, (*x,y*).They came to the result pretty quickly.

Some concerns I have with this strategy: Do they understand the significance of

*x*and*y*? Did they just regurgitate the example from before without thinking about the fact that they're**deriving a general equation**? I desperately want to teach my students how to think abstractly and generally because, well, that's what pure mathematics is about. But is it more important for some ideas than others?Like I said, I'm happier with how things went this time around. But not yet satisfied (hopefully I never fully will be). Any suggestions are gladly welcomed!

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