Decimals are *hard*.

What would we even want the student to do here if he’s working in decimal? Like, how do standard multiplication algorithms handle something like a repeating digit?

That’s what I’m getting out of this mistake right now: the deviousness of decimal representation, and the way it can obscure numerical properties.

How about you? What do you make of all this?

## 4 replies on “A decimal mistake”

And if you convince the student that the answer is actually 191.9999…, there’s another subtlety of decimals revealed and more understanding to be gained.

I’m thinking more of an opportunity to bring up commutative property of multiplication. Taking 1/3 of 9 is much easier than 1/3 of 64.

Hmmm. First, let me say that I agree with Fawn’s comment concerning the use of mathematical structure to simplify calculations – students should always be encouraged to attend to structure. But I’m not seeing a mistake here, unless, somewhere in the instructions, students were asked to express their answers with appropriate scientific precision (significant figures.) In that case, the “correct” answer is 200 cm^3. This would be a good opportunity to discuss precision and significant figures but, outside of that context, there is danger here in focusing on a “textbook answer” rather than the student’s mathematical thinking and application. I would hate to think that this student would look at the back-of-the-book answer and think, “Oh, darn, I got it wrong.”

The student’s mistake was to drop the repeating decimal between 64 cm^2 / 3 = 21.3… cm^2 and 21.3 cm^2 * 9 cm. This is why intermediate calculations should be done using absurdly precise values, and rounded off to an appropriate number of significant figures at the very end.