UnGoogleable Question - Oct 19, 2015 - Approximating Circle Area

I started the "Circles and Trigonometry" unit on my Applied Math class this week, so to get them triggering on some prior circles knowledge form Geometry, I threw up some basic circle area questions for the Do Now. I expected they would be able to do them, but my goal for the activity was mostly just to get them thinking about radii, diameters, and angles.

This first photo was as difficult as those Do Now questions got. Where the conversation got interesting was when a student called me over for a question.

"Hey, can just make a triangle in here and use the Pythagorean Theorem? Would that be close enough?"

What a great question! What's so good about it?

1. We'd done Pythagorean Theorem last week, so she was attempting to use a mathematical tool she was familiar and comfortable with in solving the problem.
2. She made a connection that triangles quite comfortably co-exist with circles.
3. She was trying to make a judgement on the reasonableness of her potential answer.

She didn't know it, but I think that's THREE of the eight Standards for Mathematical Practice she's exhibiting or about to exhibit!

HOW DOES THIS BECOME AN UNGOOGLE-ABLE QUESTION?
One of the most important things you can do to present your students with UnGoogle-able questions is to be open to opportunities for them!

Instead of reacting negatively to this student and telling her to just use the area formula, I made sure to celebrate that question.

"Hey, that's a great idea, but I don't think that's going to be close enough. I like that you're thinking, though!"

What made this question worthy of UnGoogle-able status, however, was that her question stuck with me through class and into lunch that day. I knew that one triangle wouldn't cut it, but how many might it take? I spent lunch doing some calculations on my chalkboard.

This question got better the longer I thought about, and the more I talked about it with my colleagues.

As I was playing with the triangles myself, I started getting annoyed with all the radicals in my calculations. I wondered, "Does it even to be RIGHT triangles? Does it even have to be TRIANGLES?"

Once I landed on that thought, I knew I was ready to enshrine this question because it had several of my "UnGoogle-able" characteristics.

1. It was a question I myself was interested in. (and did not know the answer.)
2. It came from a student, so I was more  confident the class would buy-in to the question.
3. It was approachable from several levels, from draw on a graph paper and literally count boxes to using angle and circle properties expertly and keeping your numbers in radical form.
4. Students would be able to have conversations about it.
5. Students would be able to write about their solution and then analyze the reasoning of others. (Another Standard for Mathematical Practice!)
6. There are multiple solutions depending on your priorities. (I suspect that triangles would ultimately get you the closest, but area calculations for quadrilaterals would be much easier. "Best method" could be either of those in different settings.)

I can't wait to see what my students do with this question!

What about you? Where else would you take this idea?