Investigating Area With Paper Folding

Last Thursday was a half-day for students because we had an open house at the event. I decided it was a good day for my geometry students to do some paper folding. I gave each student about five sheets of hamburger patty paper and these instructions:

Task 1: construct (by folding) a square that is 1/4 the area of the original square.

Everyone got it except Daniel, who didn't follow instructions and constructed a rectangle with 1/4 the area. The kids jokingly gave him a hard time. I reminded the class that Daniel's left arm was still in a cast, so squares and rectangles were the same to him.

Task 2: construct a triangle that has 1/4 the area of the original square.

Again, no problems. Everyone folded the same way.

Task 3: construct another triangle, still 1/4 the area of the original, but this triangle may not be congruent to the one you created in Task 2.

Some time passed, but not too long before Slater got it, and Zach did. They shared it with the class.

Task 4: construct a square that is 1/2 the area of the original square.

Easy-peasy.

Task 5: Construct a square that is 1/2 the area of the original square—the same instruction as Task 4—however, it must be "oriented differently."

Then I went to sit down, put my feet up (okay, figuratively speaking only), drank some water, and peacefully watched as my students slowly but surely mangled up their papers trying to create this square. They asked for more paper, and I gave them more.

Austin: What do you mean "oriented differently"?
Me: Its orientation is different.
A: That didn't help...
M: I'm not here to help you. Not today.
Zach, waving his folded paper above his head and proceeded to walk up to me: Oh, I got it!
M: No, you don't got it. Sit down.
Josh: I got it!!
M: No you don't.
J: But you didn't even look at my paper.
M, pretending to look: There, I looked.

I watched with satisfaction as everyone struggled with Task 5.

The kids know I live for moments like that. They are stuck, and they have to think hard.

Even Slater, a self-proclaimed math prodigy (who had elevated himself to "academic prodigy" since then), was quiet and used his pencil to calculate on paper. Even Bobby—who once claimed that he and Slater should get the same IQ score if tested—sat quietly, arms to his sides, staring at the patty paper.

The bell rang. I told them to work on it for homework, and the hint was "right triangle."

Alex: Someone will look it up online, Mrs. Nguyen.
Me: That's their loss.

For the next day, we did Task 5 together. The smiles returned to their faces. I love teaching.

Updated 03/26/12