The homework for 3rd Graders was to draw as many lines of symmetry as they could for a regular pentagon. I thought they knew their way around this stuff — they’d had a lot of success working with a cut-out hexagon the week before, and had been getting a bunch of practice — but I was surprised by the huge variety of responses in their homework, so I decided to figure out what was going on.
I figured we could settle this by having a whole-group conversation. Probably, the kids who found less than five lines stopped looking too early. If I shared a variety of work, I bet that kids would quickly recognize that these other lines split the pentagon into two congruent pieces. (In 3rd Grade speak, this is criterion is usually expressed as “you could fold it over perfectly.”)
It didn’t go down like that. I noticed that some of the kids who saw 2 or 3 lines switched positions during the early part of the discussion, but the 1s and 5s weren’t budging.
F and I offered (what I thought was) a pretty convincing argument against the possibility of there being only one line of symmetry. After all, they argued, if you think that the vertical line of symmetry works, just rotate the pentagon a bit and you have a new “top” of the shape to split in half.
It was around now — especially when T shared — that I started to form a theory about what the “one line” camp was thinking. Their responses to the arguments made me think that they didn’t see the regular pentagon as actually having this rotational symmetry. They saw it as irregular, in some way, so that these other lines wouldn’t actually fold over perfectly.
“Why don’t we cut them out?” R asked. Perfect, I said. I was in for a surprise!
With the cut out pentagons, you could just see that the rotations returned the pentagon to its original shape, and made room for a new “vertical” line of symmetry. And, for sure, this helped some kids.
But Bo and Annie [pseudonyms] called me over, because they were caught in argument and couldn’t make progress.
Annie: “I know that you can rotate it, but there’s still only one line.”
Annie: “I’m saying that you can draw that line if you rotate it, but it’s the same line?”
Do you mean that it wouldn’t be a new line?
Annie: “No, it would be a different line on the page, but I’m saying that it would still be the same line.”
Hold on, I think I’ve got you. Are you saying, look, you could draw a new line if you rotate it, but that’s not actually a different line, since it’s still basically that same, vertical line that you started with?
Huh. So if you have all these different lines, you wouldn’t say that there are five lines of symmetry or anything?
Annie: “No, because they’re all the same.”
At the moment that I understood her, I no longer knew what to say. I’m used to thinking of all sorts of mathematical similarity and difference — congruence, modular congruence, similarity, equivalence, symmetry — but I hadn’t ever thought of mathematical sameness quite in the way that Annie did.
She’s saying (what is she saying, exactly?) that these lines aren’t different because they’re intellectually similar. There is, for her, just one line of symmetry because there’s basically only one idea of a line of symmetry. Sure, you could repeat that process five times, but that’s not the point. The point is that you get a line of symmetry by connecting a point and its opposite side.
And maybe she’s also saying that when you rotate the shape, you’re really making a new shape. That the regular pentagon is a brand new regular pentagon when you turn it in the air, and you have a new opportunity to draw that vertical line and divide it into parts.
I wonder what she’d say if she compared her pentagon thoughts to her hexagon results. And earlier in the week she had drawn a ton of lines of symmetry in a circle.
Maybe this has something to do with the unloveliness of the pentagon? Maybe pentagons feel different to her than these other shapes?