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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?


  • 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.

  • ShefCris

    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.”

  • Jasper

    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.