Christopher Danielson’s great post on The Hierarchy of Hexagons begins with the observation that “School geometry seems to me one of the most lifeless topics in all of mathematics. And the worst of all? The hierarchy of quadrilaterals.” Then he describes his attempt at “breathing some life into this dead horse” by having students classify hexagons instead. Seems to me he was wildly successful: I never heard of future elementary teachers proving anything, but his class “proved that a Bob cannot be equilateral.” !! But what is a Bob, you say? Read his post — no, wait: read the rest of this post, then read his.
The chart above simply shows the relationship among all types of triangles in a Venn-diagram-like way that I think is exceptionally clear. I’ve never seen it done like this before, but if someone else has, please tell me.
All I’ve done with this chart myself is to show it to my geometry classes and ask them to draw examples of some of the types. But there other ways to use this idea, and one that seems much better to me now is: ask students to come up with their own graphical ways of showing the relationships among all types of triangle; then show some or all of them (and perhaps mine) to the class, and lead a “compare and contrast” discussion. Much better because much more likely to result in students really understanding! Comments, anyone?
For that matter, a classification like this begs the question of what types of figures there are of four or more sides, and of course the number of possibilities goes up rapidly with the number of sides. But how many are there with four sides, and how rapidly does the number go up? Well, what if a shape — say, a hexagon with five interior right angles — doesn’t have a name? (Actually, according to Christopher Danielson’s students, that’s a “Bob”.) Is a pentagon with two long sides and three short ones where the long sides are adjacent different from one where they’re not? What if the two long sides are parallel? What if the angles are all equal, or is that even possible? Etc. etc.
So a discussion of classifying triangles could lead to a discussion of partitions, of classification in general, of hierarchies, of (abstract) trees, and, no doubt, other topics.