Triangle Types, Quadrilateral Quotas, & Hexagon Hierarchy

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.

In classroom management, the more concrete, the better (WIL #5)

For my first semester as a student teacher, I taught 6th grade math. Many of my students had a hard time controlling themselves, and one of my standard tactics was to make a list of offenders I would make stay after class. A very common problem, and a common way to handle it. But it was obvious that the clearer the connection between the behavior and the consequences, the more effective a deterrent it was. So my partner and I started putting a timer on the smartboard to keep track of how much class time the offending students had wasted, and announcing that they’d be staying after class for the same length of time.


To make the connection even more concrete, I looked for a simple, easy-to-use program that would count down whatever length of time it had gotten up to, and preferably one that could run in a Web browser with no installation. But I couldn‘t find one: every program I ran across treated going up and down as unrelated functions, typically calling them “stopwatch” and “timer”.

Well, I know how to program a computer: I spent years working as a software engineer. So I finally (long after my semester in middle school) wrote it myself. Following a suggestion by my friend and colleague Jeremy Sebens, you can even set a time of day — the time the class ends is the obvious candidate — at which it’ll start counting down automatically!


“Don’s Up/Down Timer” is available at

There are also instructions for using (and, if you want, customizing) it there, but it requires no installation and it’s very easy to use.

Another classroom application might be the opposite of the above scenario: you want to reward students for doing something they don’t like — and the longer they do it, the longer they get to do something they do want to do.

I think Don’s Up/Down Timer would be most helpful in middle school, but it could be useful in any situation where you want to measure an unknown interval of time, then set a timer to go off when the same interval elapses again. Prof. Kathy Marrs of IUPUI tells me she’s used it with her college-freshman biology class — I’m not sure for what. Certainly there’s nothing specific to math about it.

(4 Feb. 2013: revised to add the “rewarding” application and update the URL.)