## The Area of a Square

13 Mar

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.

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 ….

### 6 responses to “The Area of a Square”

1. March 14, 2013 at 2:11 am

4 diagrams would be helpful in one split screen with shaded regions :^)…make the largest figure first to ensure proper shading

• March 26, 2013 at 5:08 pm

Great idea, Travis. I will add that in for next year when we are summarizing. Thanks!

2. March 14, 2013 at 2:48 pm

That’s pretty interesting Mrs. Wilson! I didn’t really think about doing it that way.

• March 26, 2013 at 5:07 pm

I missed having you in geometry, Annie 🙂

3. March 26, 2013 at 4:22 pm

You don’t specify the exact grade of this class, but your original problem can be solved by 4th graders far more easily than by using rhombus area formulas, Pythagorean theorem, and the like.

If you begin with a square of side 10 and connect the 4 midpoints, you form another square that is the square you describe. Or proceeding in the opposite direction, if you start with your square and draw the second square, you can see the first square is comprised of 4 right isosceles triangles, the outer square, 8. Thus its area is 10 * 10 ÷ 2.

This type of problem solving, drawing things not in the original problem, is fairly standard practice in A+ countries. Technology would only be a hindrance.

Some related problems:
http://fivetriangles.blogspot.com/2012/08/34-area-ratios.html

• March 26, 2013 at 5:06 pm

Thank you for the suggestion…another great approach to “look for and make use of structure”.

I also should have made it clear that we weren’t using technology to solve the problem. I absolutely agree that it would be a hindrance to getting a solution. I was only using it to collect the student responses. It was the time stamp that alerted me to a student’s quick solution!