22Oct/114

## A great triangle exploration

Mr. Honner has a great exploration at his blog. It starts with a simple question, that has subtlety and depth to it: How do you determine the "equilateralness" of a triangle? Can you compare two triangles and determine which is more equilateral than the other?

The post introducing the investigation is here. I encourage you to do your own exploring before reading the 28 comments which are rich in ideas. Once you've played around with the ideas yourself then take a look at what Mr. Honner came up with in Part II.

I love this kind of exploration for a number of reasons:

- The question is simple to understand.
- Just like in the real world there are multiple approaches.
- It's not clear that there is a right solution but some are better than others.
- Students get to think about properties of triangles in new and different ways.
- Students get to think deeply about the notion of "metric."
- This problem is more interesting than many other geometry problems I've seen.

Nice!

Patrick HonnerOctober 22nd, 2011 - 09:24

Thanks, Sol! Glad you like this problem. And thanks for so succinctly summarizing its virtues. I’ve had lots of great conversations with students about this, precisely because of the reasons you’ve articulated here.

Most importantly, to me, is that by thinking about this problem, students are forced to examine the concept of “right answer”. As you say, it’s not clear there is a “right” approach, but some approaches are better than others. And how do we measure

that?David BrooksOctober 23rd, 2011 - 17:48

By definition, neither is equilateral. They are equally “not equilateral”.

PJ OrtizNovember 7th, 2011 - 08:59

Well, Mr. Brooks is right by saying that by definition, neither is equilateral. If we use the ” eyemeter” and determine it by looking at it, I would say that the triangle on the left looks more equilateral the the one on the right. Too bad the “eyemeter” does not work in mathematics. It would help everyone solve problems way more easier! Lol.

Stuart PJanuary 21st, 2012 - 04:35

This reminds me of an activity I explored with students a few years ago, looking at the ‘squareness’ of a natural number. We created a function from N to the interval (0,1]. It was a good exercise in definitions, and we extended it to limits of sequences, subsequences etc.