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# Trapezium City

### By Carissa Tobin

#### FRI MAY 15, 2020

"It pays to have a good partner," Axel said to me as I walked into his office/our bedroom the other day.

"My research partner for this class," he said.

Oh. Well, I'm sure he likes me too.

He and his research partner were working on a project that was currently causing some strife in our own partnership.

The great debate began when Axel sent me, along with many others, a set of online math problems to do for a class he is taking.

Some of them were okay to do, and some were hard. Really hard. And then one of them… just didn’t make any sense.

I was supposed to find the area of this “trapezium” (I would have called it a trapezoid, but I’m used to teaching math to kindergartners, not adults…). And it just didn’t seem right.

So if the top is 6 and the bottom is 8, that means that if you imagine a little triangle on each end, its base would be 1, its height 4. And the diagonal is 5. So using the Pythagorean Theorem, 4 squared plus 1 squared should equal 5 squared. Is 16 +1 = 25? No! No, it doesn't'! So the dimensions on this trapezoid or trapezium or whatever you want to call it are incorrect. So how am I supposed to solve its area?

Apparently there was a trick that the problem set taught you for solving the area of trapeziums. Axel and his partner report that some people got the problem "right."

"So they took the formula and memorized it like a robot? That is exactly what you shouldn’t do!" The kindergarten teacher in me was outraged.

Axel said, "I'm getting a kick out of this."

Axel texted his partner, who we'll call Sam, to explain my complaints. Apparently the two of them had discussed this before. "She's very passionate about math."

"Yeah, no she's right." Honestly, that was all I wanted to hear. Thank you, Sam. He said something about it being called an "impossible shape." But I stopped at the "she's right" acknowledgment. He was going to look into it further.

Axel added a footnote for his professor: "One participant submitted a lengthy critique to the authors about calculability issues related to the figures in the pilot study. The participant requested that a particular question be excluded from the study because it was invalid. This participant may be related to one of the authors and may also be a former elementary school math teacher who is very passionate about math education. This participant is likely to publish a humorous blog post on her website about the participant’s outrage at such questions and would like to know if the professor agrees with the calculability / validity issue re: the green trapezoid in Fig. 1."

This wasn't just about me getting the question "wrong." What about all those other people that got wronged? Either by not being able to solve it, or, worse, by having thought they solved it correctly when really they were using a memorized algorithm and not even thinking?

"You've got this indignant righteousness about you," Axel told me, slapping his leg. "This project was worth it just for this."

And then Axel ran back into the room laughing, holding his laptop. The professor had emailed back almost immediately, having jumped directly to the footnote with curiosity.

The verdict?

"I mean, she's not wrong. In fact she's quite right."

Small wins doesn't really suffice for this one.

Vindication!