In the past, I've never been completely satisfied just giving students the formula for a circle and having them apply it -- I always felt it was accessible enough that students could figure it out themselves.
On the other hand, whenever I lectured them and showed them how it works -- how it's based on the Pythagorean Theorem, and how to work backwards from numbers to generate the formula -- I realized 45 minutes into the lecture that I was overdoing it: in my efforts to be certain everyone understood it, I ended up just putting on a show, and my students really never had to engage very much with the content.
In response, I think I have finally found a happy compromise: I start class with a "mini-lecture" (no notes -- just the intro/opening for the day) and briefly -- 10-15 minutes -- I show them what they are going to be doing today.
I then give each student a copy of this worksheet and have them work cooperatively together in groups. This worksheet basically replaces a lecture. I circulate to make sure they are getting the point of the exercise. By the end of the block, about 45 minutes after they have been working on it, they seem to have a much better grasp of how it works and how to apply it than in years past.
The file includes spiral review problems.
This worksheet is intended to be written on directly.
Please download the pdf preview file first, so you can see exactly what's included; the product file is a word document, which you may personalize for your students.
Keywords: discovering the equation of a circle, Jonnard