The 11-10-11 Triangle

Today’s date, 11-10-11, reminded me to re-visit a recent post that posed the question “Which triangle is more equilateral:  the 10-10-11 triangle, or the 10-11-11 triangle?”

The original post elicited lots of great comments from readers, who weighed in on what they thought the question meant and how they might go about trying to answer it.  I offered one approach, and an answer to the question, in this follow up post.

As a math teacher, there are many reasons I try to create problems like this.  Here are a few that I think are important.

First and foremost, in order to address this question, a significant amount of thought must be put into deciding what the question means.  This process involves analysis, synthesis, reflection, and ideally discussion, all of which will be substantially mathematical in nature.

A second, related, virtue is that there is no obviously correct interpretation of what this question means.  Mathematics is often viewed in stark terms:  answers are either right or wrong.  But the certitude of mathematics comes only after we agree on mathematical models for our given problem.  There is often great debate about what those models should look like; the history of mathematics is full of such debate.

Problems like this one invite students into the modelling process, where they can discuss and debate the validity of various approaches.  Moreover, the problem allows solvers to create multiple different models to explore, compare, and contrast.  And in the end, we can pose and explore meta-mathematical question like “Which model most closely aligns with our intuitions?” and “Which model is the most useful?”

Lastly, this problem demonstrates one role creativity plays in mathematics.  A simple response to the question, one I heard many times, is “Neither of these triangles are equilateral.  They are both equally unequilateral.”  Given our rigid definition of what equilateralmeans, this response is technically correct.  But by relaxing our ideas about equilateral, by allowing ourselves to ponder what the phrase “more equilateral” might mean, by thinking creatively about what kinds of questions we can ask, we create an opportunity to explore, and possibly uncover, some new mathematical ideas.