My son, who is 2 1/2, often gets upset when a big truck that he has been watching abruptly drives away.  I’ve learned to eliminate the tears and disappointment by reminding him to, “Keep your ears open!” and frequently he moves onto the hunt for the next big vehicle.

Today in geometry, I was reminded about what can happen when you simply listen.

We were beginning the chapter on congruent triangles and I wanted to hear their intuition on what it means for two polygons to be congruent.  Their warm up was the following problem:

Imagine that this hexagon is made from 12 toothpicks.

1) How could 6 more toothpicks of the same size be used to divide the hexagon into 3 congruent regions?

2) Keeping two of the toothpicks from part 1 in the same place and moving four, use six toothpicks to divide the original hexagon into 2 congruent regions.

Although using manipulatives or GeoGebra was a natural approach to solving this problem, I didn’t have toothpicks on hand, nor the mobile computer lab.  I  figured that students would sketch out different diagrams to find a solution. After a few minutes, I looked over and saw that a student had taken out 12 Sharpies of equal length and had constructed a solution to the first question.A student creating a scale model of the problem that was over 2 feet wide felt like the proper time to have the rest of the class drop everything and come huddle around this student’s desk.  We gathered up, some on stools, some on desks so that all could have a clear areal view.  There was unanimous agreement that this was a correct solution to the first problem.

“How do you know that these 3 regions are congruent?” I asked.

While answers ranged from, “It just is.” to “You can measure it and see.” the most interesting one to me was the student who claimed that the 3 regions were congruent because each was equilateral and congruent figures had to be equilateral.  About a dozen students joined in the debate, I mostly just listened, so I could learn more about their intuition. The idea that congruent figures must be equilateral was a misconception that I hadn’t come up in previous years.  I’m glad my ears were open.