From NAEP, 1996, Grade 12 Mathematics:

This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work.

Describe a procedure for locating the point that is the center of a circular paper disk. Use geometric definitions, properties, or principles to explain why your procedure is correct. Use the disk provided to help you formulate your procedure. You may write on it or fold it in any way that you find helpful, but it will not be collected.

The NAEP results were not hugely promising in 1996.

I’ve posed this problem to my students for several years now with some success. I first give them a paper circle for playing. Then I pass out circular lids and let them play for a few more minutes. And then we talk.

As you can imagine, the first suggestion is to fold the circle in half once and then fold it in half again. Which is great. As long as the circle can be folded.

So then we restrict our investigation to circles that cannot be folded.

How can we determine the center of the circle if the circle can’t be folded?

I had some interesting answers to this question this year. We didn’t spend as long on it as usual. We had more whole class discussion earlier. But we still learned something from each other.

In one class, a student suggested that we draw a chord, and then construct a perpendicular to the chord at one of the endpoints, creating a right triangle. (We had previously explored Tangents to a Circle from Geometry Nspired.)

And then, even better, before suggesting that we could then construct the midpoint of the hypotenuse of the right triangle (which of course, would be easy at this point), he suggested that we repeat the process – and that the intersection of the hypotenuses of the two right triangles would be the center of the circle.

I did this lesson with another class last week, and they started with the usual suggestion of folding the circle in half two times. We moved to TI-Nspire. They suggested I draw in the diameter. But how would I know that it was a diameter?

They agreed that I wouldn’t. So we started with a chord. And we made the chord so that it was obviously not a diameter.

Someone suggested that we construct a tangent at one of the endpoints of the chord. Now if we had started with a tangent, it wouldn’t have been bad to construct a perpendicular to the tangent throught the point of tangency. But we hadn’t started with the tangent. We had started with the chord. I had no idea where this might go, but I was up for trying.

Then someone suggested that we construct another tangent. Another tangent where? At the other endpoint of the chord.

So now we have two tangents. Do we have a diameter? We don’t, obviously. But then they asked to move an endpoint of the chord. Make it so the tangents are parallel to each other. How do we know when they are parallel?

When two parallel lines are cut by a transversal, consecutive interior angles must be supplementary.

In both classes, we ended up discussing a lot more geometry than if I had passed out the instructions for constructing the center of a circle and asked them to follow the steps.

We still thought about the traditional construction of the center of a circle. How can we use this chord to get the center? Someone in the class suggested finding the midpoint of the chord.

How can the midpoint help us to get to the center of the circle? They decided they needed a hint. So we looked through the menu to see if anything jumped out at us.

And something did. Perpendicular bisector.

And from there, we all agreed that now that we had a diameter, the center was no trouble at all.

Of course we really didn’t need the midpoint first if we were going to construct the perpendicular bisector of the chord. But I am learning to listen to my students – and to use their suggestions even when I don’t first see where they are going to lead us. That is how I teach and learn mathematics every day with my students. And that is why I look forward to tomorrow’s classes, when the journey continues…

Fawn Nguyen

March 26, 2013 at 12:25 am

I want to pose this problem to my 8th grade geometry kids. (I love these kinds problems, so of course they should too. 🙂

Thanks, Jennifer!

jwilson828

March 26, 2013 at 5:10 pm

Of course they should! And I’m sure they do…You always give them great problems. I really enjoy reading about them on your blog.