A student, Melissa, in my SAT class, shared her way of thinking about a well-known problem that appeared on this blog awhile back as part of a larger investigation. You may recall the post about the ratio of the areas of circumscribed and inscribed figures.
I told her I would celebrate her method on my blog, which would definitely be viewed by less than or equal to a million people a day! One of my students asked me why I don't simply write a book about all these ideas and I replied, "I'm thinking about it." He asked what the title would be and I replied, "What I've Learned From My Students."
Here's the problem/investigation/challenge/activity///// for you or your geometry students:
Consider the diagram above. Assume that it depicts two squares. The smaller inscribed square is formed by joining the midpoints of the larger square.
(1) Explain why the area of the inscribed square is one-half of the area of the larger.
Easy so far...
(2) Consider the circle circumscribed about the smaller square, i.e., it passes through its 4 vertices. Explain why this circle is inscribed in the larger square. This requires that you show that the 4 sides of the larger square are tangent to the circle. [By the way, most students would assume this is obvious from the diagram, but ...]
Not impressed? Deja vu all over again as Yogi would say? This is how Melissa demonstrated that, for a given circle, the area of the circumscribed square is twice the area of the inscribed square. Is this the method you have seen or used yourself? Consider that she used no variables, didn't plug in particular values for the dimensions, etc. She drew the diagram and basically said that the diagram proves itself! A proof without words, so to speak (that might even make Sidney Kung proud). I congratulated this young lady and the class applauded. Interesting how students immediately recognize someone's brilliance...