Polygonal Crystals

In Geometry, we’ve been trying to turn the course on its head. Recently, we’ve been working on reasoning and proof. One thing students weren’t able to prove was that the sum of the interior angles of a triangle always add up to 180 degrees. (That’s because we haven’t gone over anything involving parallel lines.)

They even blew up balloons and drew triangles with two right angles on them (using protractors and rulers).



[Note to self: students cannot tie balloons. Also, they will be found scattered around the student center later in the day if you let kids keep their balloons.]

But we said: assuming that you know that on a plane the sum of the interior angles of a triangle add up to 180 degrees, can you prove that quadrilaterals have a sum of 360 degrees for their interior angles.

And each group was able to latch onto the idea of dissection without me saying anything… breaking the quadrilateral into two triangles.

But then… then… they started to say something that scared me. They said “there are two triangles, and since each triangle is 180 degrees, the quadrilateral is 360 degrees.”

To you non-geometry teachers, this might not seem problematic. But I immediately thought: “oh gosh, these kids think of triangles and quadrilaterals and the like as having some inherent property that can be added to others. They aren’t saying the sum of the interior angles of the triangle is 180 degrees… they are saying the triangle has 180 degrees.

So I gave them a follow-up question (which I’m proud of):

triangle“The Blue Triangle is 180 degrees. The Pink Triangle has 180 degrees. So the Giant Triangle (the blue and pink triangles combined) must have 360 degrees. How is this possible? Did we just break math?”

One group had someone who figured it out right away, but the others took a good amount of time trying to figure out where this argument failed. I loved it because it really showed them a misconception they had.

It was the perfect question, because over the summer the other geometry teacher and I came up with the following (which we are in love with) involving triangle dissection:


Finally, to check if each group understood where these came from, we had them write a “triangle dissection expression” for the sum of the interior angles of this pentagon:







  1. Oh man, Sam – this is BEAUTIFUL! I want to BE in your Geometry class, and then go teach it to my kiddies and make them crazy! I love it – “did we break math?”. Did any of the kids try putting the angles together (you know, the old tear-the-angles-off-and-align-them-to-a-straight-line proof for the Triangle Sum Theorem)?

    (BTW – good think it was cold out this week, or you know what else would have happened with those balloons….) Keep writing!

    1. Oh Wendy, thanks. I pretty much think the same of you.

      As for the putting the angles together trick, we did that together and they taped the straight angle they created in their packet.

  2. Interior angles of a triangle: take a pencil, lay it along one edge, middle at the vertex. push it along the edge til the middle is on the next vertex. rotate the pencil through the interior angle. repeat for each of the next two sides. where is the pencil pointing now? what angle did it turn through? done!

  3. I’m a tad out of the SAT loop the last decade, as Michigan went to requiring the ACT and fewer and fewer students asked for SAT tutoring. Only the evil geniuses at ETS/CEEB know what the future holds for the SAT, but back when I was regularly looking at such tests, there would be problems at the high end of difficulty that involved finding sums of angles where it was impossible to know the measure of any individual angle. Many of the drawings you’ve offered up (funny, but I always thought of that as triangulating, not dissecting; maybe I had a bad experience with pithed frogs at some point) remind me of those problems: add a bunch of these things (e.g., triangles) and subtract a bunch of angles that together form one or more of these things (e.g., straight lines, circles), and voila! you’ve got the sum you’re looking for.

    A major difference here is that at least in theory, we already *know* the sum we’re looking for, and the issue is seeing that if you just “add” triangles to create new figures, you can’t just keep summing the angles. If you do, you’re “double counting” some of the angles or something like that, hence the need to subtract some of the angles.

    It’s easy to imagine how confusing this can be for kids (or adults) just starting to wrap their heads around some of the ideas. What exactly is an “interior angle” in a polygon? Without thinking about it too much or attending closely to definitions, there’s no reason not to think that some or all of those angles formed at interior intersections in those figures are to be included in the sum of the interior angles for the enclosing figure, rather than only summing the measures of the angles at the vertices of that figure. This exercise strikes me as excellent for pushing that issue to a kind of crisis point for students. Bravo!

      1. You’re welcome. Wouldn’t it be lovely if we could design and collect bunches of crisis-point problems that were known to be effective for pushing student thinking to cognitive breakthroughs?

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