I first remember asking this question seven years ago, which was pre-TI-Nspire Navigator; at the time, I used Promethean Activotes (a form of “clickers”) with my students. The clickers only worked with multiple choice questions, but now that I use TI-Nspire Technology, I am not limited to collecting results electronically for only multiple choice questions.
This seems like an ordinary geometry problem. But seven years ago when I gave the problem, I realized by using the time stamp on the electronic results that Emmalee had solved the problem correctly in just a few seconds. Emmalee was one of my quiet students about whom I am speaking when I say that the TI-Nspire Navigator system gives every student in my classroom a voice. Without using a response system, I would have never realized how quickly Emmalee had solved the problem and known to ask her how she worked the problem. I’ve been asking students for years whether anyone solved a problem differently than the others, but Emmalee would have never volunteered that information to us.
I actually didn’t start our discussion by asking Emmalee how she had solved it. I started with some of the students who successfully solved the problem in the longest amount of time. You see, just like you, I knew something about Effective Classroom Discourse, even though I didn’t know how to name what I was doing. After some discussion, we realized that most of the students recognized that the square could be decomposed into two 45°-45°-90° triangles. They went from the given hypotenuse back to the leg using the generalizations that they had made about the side lengths of Special Right Triangles. Once they found the leg, some rationalized it and some didn’t, but they all squared it to get the area of the square. At that point, I asked Emmalee how she had solved the problem. Emmalee looked up and said, “We were just talking about how to calculate the area of a rhombus. Since a square is a rhombus, I used that area formula instead.” The area of a rhombus is half the product of the diagonals, and it had taken Emmalee less than five seconds to calculate ½*10*10. Emmalee had entered the practice of look for and make use of structure long before we even had the Standards for Mathematical Practice. Emmalee’s explanation was so compelling that I realized I needed to provide my students the opportunity every year to think about this problem the way Emmalee had. I’ve probably asked this problem for most of the years that I taught geometry, but it was seven years ago that the problem was memorable – it was then that I realized the problem could be solved quickly because of the technology that we were using in the classroom. Many students before Emmalee might have worked it the same way, but that discussion had never come up in my classroom – maybe other students were quiet, too, and I hadn’t the technology to give them a voice at the time.
This year I had to find a different way to include the problem. We don’t really have a unit on area anymore. Unfortunately, we haven’t had a day to make sense of area formulas like we have in the past. Every once in a while, in passing, we will talk about an area formula. So most of my students didn’t explicitly know the area formula for a rhombus. I put a sequence of questions on a recent bellringer to see what would happen.
So as we were talking about these problems, we had to spend some time developing the formula for the area of a rhombus. The students first recognized that one diagonal of a rhombus decomposes the rhombus into two isosceles triangles – we generalized the formula for the area of the rhombus that way. (I was surprised that they didn’t first notice that we could decompose the rhombus into four right triangles.) Then we talked about the area of the square – and I had to lead them through the practice of look for and make use of structure. Now that you know how to calculate the area of a rhombus, would you have calculated the area of the square differently? They thought and talked and made sense of the structure of the square. And they figured out that they could calculate ½*6*6 for the area of the square.
These are small steps in getting them to extend their reasoning to shaded area problems:
- A circle is inscribed in a square with a side of 12 ft. What is the area of the region between the square and the circle?
- A square with a side of 12 ft is inscribed in a circle. What is the area of the region between the square and the circle?
- A square with a diagonal of 10 ft is inscribed in a circle. What is the area of the region between the square and the circle?
- A circle is inscribed in a square with a diagonal of 12 ft. What is the area of the region between the square and the circle?
And so the journey continues ….