Around this time of my geometry course each year, students have learned a whole lot of geometry – parallel lines, congruent triangles, basic proof techniques. It is the perfect time for them to come up with challenge problems for me. Believe it or not, classes never disappoint in this fashion. I’ve had some great questions in the past. (One of my favorites – can you fold an 8.5 by 11 piece of paper into an isosceles triangle where the entire triangle has the same number of layers of paper. I’m still not sure about that one, but I really enjoy watching students try every year.)
I tell my students that there are two things that make a challenge problem great:
- It is relatively easy to explain to anyone.
- It is challenging to find a solution (or even figure out if a solution is possible).
This year, a student came up with a problem that fit those requirements perfectly: Is it possible to take a square, and cut it into isosceles triangles that aren’t congruent? We then restated it to require a finite number of non-isosceles triangles (though the fractal version is a pretty cool visual and a nice approach).
We spent a few minutes on this at the end of class, but didn’t get anywhere right away. I sent this out via twitter, and almost immediately got a tweet back from Henri Picciotto:
@Ethan_MidPen The answer is yes. I can do it in seven.
— Henri Picciotto (@hpicciotto) November 15, 2016
What? That brought out some resolve in me to find a solution, and later that day I did. The next morning, before school, I had a student rush in to show me her solution, which was pretty similar to mine. I suggested to both my geometry classes that there may be more solutions – how many unique solutions or approaches may there be? Can we find an even number? Fewer than seven? I ended up waking up in the middle of the night thinking about this, and came up with even more solutions. So far, I’ve got solutions for 6, 7, 8, 11, 13, and 17 triangles. But the best part of this whole story is that a week later, I have students talking about this before and after school, at lunch, outside my classroom, totally invested in a problem that they made on their own.
— Ethan Weker (@Ethan_MidPen) November 22, 2016
When you are a teacher, these are the moments that you live for. Students of varying backgrounds and supposedly different ability levels collaborating and arguing with passion about squares and triangles. Sometimes, that question of “When am I going to use this?” or “How does this apply in the real world?” never come up because the math, by itself, is just fun.
So…can anyone find a solution with fewer than 6 triangles?