# The hierarchy of hexagons

True confessions: I find a great deal of the school geometry canon tedious.

Does a trapezoid have exactly one or at least one set of opposite parallel sides? Circumcenters and orthocenters. Dull, dull, dull. Boring, boring, boring.

School geometry seems to me one of the most lifeless topics in all of mathematics.

And the worst of all? The hierarchy of quadrilaterals.

This representation of relationships among the special quadrilaterals bored me in fourth grade and I cannot muster energy for it as an adult. But I gotta teach it with my future elementary teachers.

So a year ago, I had an insight; an idea about breathing some life into this dead horse. What if we classified hexagons instead?

We began with these:

We cut these out. I had students choose one that seemed special to them for some reason, and to identify what property or properties make the hexagon special.

Students identified this one as being special because it has all right angles:

We clarified, defined interior angle and right angle, and agreed that this hexagon is special because it has exactly five right angles. The shape needed a name and we chose Bob. So a Bob is a hexagon with five interior right angles.

We also agreed that we would only specify interior in the future if there was likely to be confusion; we gave ourselves permission to refer to the interior angles of a polygon simply as angles in most cases.

Students identified the next figure as being special because it has three congruent acute angles.

Again, we needed a name and it became a Stacy. So a Stacy is a hexagon with three congruent acute angles.

We identified several of our hexagons as being concave, so we defined a concave hexagon as one that has at least one interior angle greater than 180°. (Side note: It turns out that this is the standard definition; I had remembered something about diagonals staying in the interior. In any case, we had to do some work to get from the visual shape with a dent definition to this one.)

I threw a couple of useful terms into the mix: equilateral and equiangular, and pretty soon we had enough to work with.

We took these properties two at a time and made Venn diagrams. Is there such a thing as a concave hexagon that is not a Bob? (Yes) Is there such a thing as a Bob that is not concave? (No) Is there such a thing as a concave Bob? (Yes) Etc.

Having polished off all of the pairwise possibilities, we took to the whiteboard to categorize and to argue.

Concave hexagons, Stacys, equilateral hexagons and equiangular hexagons are all special hexagons that don’t necessarily have anything to do with each other. But you can have an equilateral hexagon that is also equiangular. We named that a “Norm”. And a Stacy can be equilateral. That’s a Mercedes. The Stacy above is a Mercedes. We weren’t sure whether a Mercedes must be concave.

My students proved that no Bob is equilateral.

I would like to repeat that.

My students proved that a Bob cannot be equilateral.

I have never before been able to say that my future elementary teachers proved something. I could say before that they followed a proof I presented. Or that they produced a proof that closely mirrored one they had seen. But never that they proved something. This group did.

Their argument was based on parallel sides in a Bob-how there are two sets of three sides, and that the lengths of two sides in a set add to the length of the third. If a Bob were equilateral, one of these sides would be of length zero, which means it’s not a hexagon and so not a Bob. QED. I have spared the reader some details.

Behold the hierarchy of hexagons:

After this, the hierarchy of quadrilaterals was a mostly trivial exercise. We built it in like 20 minutes and used it as practice for the skills we developed with the hexagons.

We marveled at the bizarre relationship between the definitions of quadrilaterals and their relationships. Why is a rhombus defined in  terms of its side lengths, while a parallelogram is not? This makes it hard to see why a rhombus is a special parallelogram.

The question of the concavity of Mercedes was an open one for a couple of weeks. Then yesterday we got out the polystrips. Boom!

Not all Stacys are concave.

If we had more time, we would revise our hierarchy to incorporate this fact. But we have to move on to measurement.

Our work here is done.

Total time? Five weeks.

### 23 responses to “The hierarchy of hexagons”

1. Don

Love this…I feel like I’m boring my class every time I start into quadrilaterals because the first thing I encounter is my students’ belief that every 4-sided figure is a square. Then I pull out the old hierarchy…I think I might just modify your activity for my classes…somehow…time to ponder…Thanks!

2. I likely will be stealing this. (with attribution, of course).

3. I’ve had some fun classifying the quadrilaterals based on what symmetries they have. This gets you everything including isosceles trapezoids very nicely, but doesn’t get the more general trapezoid.

This is a great idea, too — there’s so much more room to explore with different kinds of hexagons! I would have worried about leaving so many types of hexagons uncategorized or about not recognizing some of the important types but seeing your results here changed my mind about that. Thanks!

4. Jenn

We only proved that a Bob cannot be equilateral because we were too stubborn to let it go!

But in reality, you do make this class quite interesting and it makes us want to succeed. Your enthusiasm is contagious!

5. This is excellent! I’m almost certain I’ve seen this before. Have you written about this somewhere else? I will definitely file this idea away for future use.

I would add that the study of the hierarchy of quadrilaterals is quite likely an example of over-applying the van Hiele model. The Van Hieles noted that students understand such class inclusions when they reach the third van Hiele level, where children understand how properties relate to each other. This means that class inclusion became one easy way to test children’s van Hiele level. In the 1980s, well-meaning educators decided that, if only children at a relatively high van Hiele level understand class inclusion, then we must start teaching class inclusion to the children! But the van Hieles never suggested that class inclusion was something to be taught for its own sake, only that it was a good indicator of children’s understanding of property implications. Too late now. Now everyone thinks class inclusion must be terribly important for its own sake and it is enshrined in national standards. Enter rote lessons on quadrilateral hierarchies and endless confusion.

6. Duncan

I enjoyed reading this. I will admit it is something I am did because I want to earn 10 extra credit points on a hereto ungraded geometry exam. Many in our class found the test difficult. And by difficult I mean we do not understand it well enough to answer questions beyond defining things we have seen. I admit I am jealous that you spent so much time having your students investigate and make discoveries rather than just give them facts to memorize. And while I did not fully understand the lesson you just presented upon first reading I understood enough to know that your approach and probably your results are not typical.

Thanks very much for sharing!

7. In defining Bob’s, Stacey’s, and the like, did you run into situations where your definition admitted shapes the students didn’t actually want. For example, once you defined a Stacey as a hexagon with three congruent acute angles, did you draw some other Staceys and have students blurt out “wait, that’s not a Stacey — that’s not what I meant!” If not, how did you privilege the *definition* over some sense of “I know a Stacey when I see one.” Is it because their definitions were based on some property they liked about one example, rather than trying to say what was the defining quality of some *group* of hexagons?

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