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How do you teach to teenagers or kids that if we have a square with side length $x$ m (that is, $100x$ cm), then its area is $x^2$ m$^2,$ but not $10000x^2$ c$^2$m$^2$ ?

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    $\begingroup$ Do you presume these students understand what the unit "cm" means -- that it is not "c" times "m"? $\endgroup$
    – Nick C
    Mar 26, 2023 at 17:08
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    $\begingroup$ Are you currently experiencing this, or is it just a curiosity? I ask because in my experience, it is far more likely that a student would say the answer is either $10000 x \text{cm}$ or possibly $10000 x^2 \text{cm}$ before they would consider squaring each letter in a unit. $\endgroup$
    – Nick C
    Mar 26, 2023 at 17:11
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    $\begingroup$ I mean, $\textrm{c}^2\textrm{m}^2$'s being a non-standard notation for $(\textrm{cm})^2$? Certainly, but to play devil's advocate (or to someone not knowing otherwise) formally interpreting $\textrm{c}$ as an operator that multiplies by $1/100$ -- like the percent symbol is sometimes presented -- makes it all make perfect sense. So, especially if the learner simply has their own conception of what they had meant, yet to be articulated, I would say such an answer is less wrong than reflective of missing familiarity or acculturation. $\endgroup$ Mar 26, 2023 at 23:14
  • $\begingroup$ That's the reason you usually use units like meter and not centimeter. The right way would be to expand the prefix centi to 1/100 to see what you're doing and that meter is the unit you're talking about. $\endgroup$
    – allo
    Mar 28, 2023 at 14:52
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    $\begingroup$ I like the answer below, but if I were having this issue with my students and wanted them to keep it simple I'd force them to read it out loud. They say $(cm)^2$ so they should write it. Otherwise I expect them to say centi-squared meters-squared. It stresses the difference outlined in ryang's answer without forcing them to calculate it. $\endgroup$
    – PGibbon
    Mar 28, 2023 at 15:30

3 Answers 3

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But $x^2$ m$^2$ and $10000x^2$ c$^2$m$^2$ are arguably both correct.

The prefix ‘centi’ means a hundredth, and the convention \begin{align}\text{cm}^2:={}&(\text{cm})^2\\={}&(10^{-2}\text{ m})^2\\={}&10^{-4}\text{ m}^2\end{align} does not preclude the curious-looking first intermediate step in the presentation \begin{align}(100x\text{ cm})^2&{}\color{red}{=100^2x^2\text{ c}^2\text m^2}\\&=100^2(0.01)^2 x^2\text{ m}^2\\&=x^2\text{ m}^2,\end{align} which is just as sound and solid as the presentation \begin{align} 2000\text{ cm$^2$/h}&=2000\left(\frac1{100}\text{ m}\right)^2\left(60^2\text{ s}\right)^{-1}\\ &=5.56\times10^{-5}\text{ m$^2$/s}. \end{align} In all the above, ‘centi’ and ‘$\%$’ perform the function of dividing by $100:$ the former as a prefix operator, whilst the latter as a postfix operator.

P.S. Likewise for deci, milli, micro, nano, kilo, giga, etc..

P.P.S. While obtaining the above link, I learnt that deca is the only such prefix whose symbol ‘da’ has two characters. Since milli versus mega is ‘m’ versus ‘M’, it's a shame that deci versus deca couldn't just be ‘d’ vesus ‘D’.

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    $\begingroup$ xy^2 ≠ (xy)^2, so I don't follow....Now if you said cm^2 is one hundredth of a square meter...well, I guess it would still be in violation of SI standards. Which probably should be noted. I did learn here that a centipede is so-called because it's a hundredth of a foot long. Right? :) $\endgroup$
    – user1815
    Mar 28, 2023 at 14:23
  • $\begingroup$ In particular c^2m^2 as implicitly equivalent to (cm)^2" (which steps you omitted in expanding (100x cm)^2). "c^2m^2" is not allowed in the SI/ISO standards. (Honestly, I thought your answer was tongue-in-cheek. So was my comment, at least in parts.) $\endgroup$
    – user1815
    Mar 28, 2023 at 14:34
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    $\begingroup$ $\rm (cm)^2 = c^2m^2$ is incorrect per international standards. That is what I'm disagreeing with. Mathematician can make up a system that is consistent if they want to, but as units are more important in science and engineering, they shouldn't do it in this case. I find it fun to do so, privately, or with others who also enjoy this sort of play. If taken seriously, it is just wrong per conventional definitions. (The side issue of whether one is permitted to separate c from m in $\rm(cm)^2$ but not in $\rm cm^2$ is not that important.) $\endgroup$
    – user1815
    Mar 28, 2023 at 15:24
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    $\begingroup$ @ryang We can parse $\neg\neg(\forall x Px)$ as $\neg(\neg(\forall x Px))$. The first negation operates on the statement $\neg(\forall x Px)$, not directly on the second negation. In $(\text{cm})^2\!$, we are squaring the unit $\text{cm}$, not squaring the operator $\text{c}$ and unit $\text{m}$ separately. $\endgroup$ Mar 28, 2023 at 22:14
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    $\begingroup$ I had not seen that comment, but you haven't really satisfactorily addressed my point. You are introducing a very uncommon convention ($1~\text{cm} = 1~\text{c}^2\text{m}^2$) in a context where students are already confused in a way that is likely to lead to more long term confusion. Don't do this. It is bad pedagogy / andragogy. $\endgroup$
    – Xander Henderson
    Apr 13, 2023 at 11:54
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I wanted to synthesize some of the discussion happening in the comments into an answer.

To me, a statement like $$(100x\ \text{cm})^2 = 10\,000x^2\ \text{c}^2\text{m}^2$$ indicates that the student has not associated the symbol $\text{cm}$ with the unit $\text{centimeters}$ in the context of exponentiation. You can help them make this connection by rewriting the left-hand side as $$(100x\ \text{centimeters})^2\!,$$ at which point they will hopefully see that $10\,000x^2\ \text{c}^2\text{m}^2$ is gobbledygook. In this form, it is also easier to see that \begin{align*} (100x\ \text{centimeters})^2 &= (x\ \text{meters})^2\!\\ &= x^2\ \text{square meters}\text{, so}\\ (100x\ \text{cm})^2 &= (x\ \text{m})^2\!\\ &= x^2\ \text{m}^2\!. \end{align*}

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Although being a bit late to the question...

The underlying problem I see is that we first expose our students to a math symbology where single letters are used to name things, and that simply concatenating these characters means multiplication.

Then we introduce units where this no longer holds: $mm^2$ is not the same as $m^3$ or $mmm$, and we are surprised that students struggle with this situation. Decomposing $mm$ into $m \cdot m$ needs the student to understand that $m$ as a prefix is not the same as $m$ as a unit (completely contradicting everything we taught before!), and that this "multiplication" is implicitly surrounded by a pair of parentheses. So, this will not aid in understanding the concept, but instead create more confusion.

In fact, unit prefixes need a special treatment.

  • In most situations, units with prefixes should be viewed as a multi-letter entities (just like function names like $sin$, see my answer here), and not as a multiplication of a prefix with a unit.
  • Only when it comes to unit conversions, then it might make sense to separate prefix and base unit, as in $mm = 0.001 \cdot m$ (but remind the students that they should use parentheses, so $mm = (0.001 \cdot m)$ would be the "safe" replacement).

So, it's more consistent to introduce units as multi-letter names, thus falling into a category that's needed anyway, e.g. for function names.

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    $\begingroup$ Arguably, this is part of why these kinds of mult-letter symbols are usually typeset with upright text, rather than italicized text. This may be subtle, but $mm^2 = m^3$ (the variable $m$, multiplied by itself three times), while $\text{mm}^2 = (\text{millimeters})^2$ (i.e. square millimeters). $\endgroup$
    – Xander Henderson
    Apr 13, 2023 at 0:43
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    $\begingroup$ So, we want to teach our students to look not only at the letters, but also at their typesetting. And what about handwriting? IMHO, half of the problems that students have with mathematics, are not caused by mathematics itself, but by the notations that mathematicians use. $\endgroup$ Apr 13, 2023 at 7:22
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    $\begingroup$ @RalfKleberhoff Using different typefaces to represent different kinds of objects is certainly not unique to mathematicians and in fact is more common among scientists, see this NIST document which cites ISO 31. I agree that notation can cause many problems for students, but often it is because we are using the same symbol to mean different things. This convention allows us to distinguish the symbols $mm$ and $\text{mm}$. $\endgroup$ Apr 13, 2023 at 11:42
  • $\begingroup$ @RalfKleberhoff I did not say that we should necessarily rely on the typesetting. Rather, I indicated that this is part of why we typeset these things differently. It is a subtle hint (beyond context alone, which should be clear enough) that these kinds of objects are different. Also, this isn't really a problem with mathematics---it is a problem with physics and chemistry and engineering and other fields where units are a daily part of life. Those dirty applied people are messing up our pure notation. :D $\endgroup$
    – Xander Henderson
    Apr 13, 2023 at 11:49
  • $\begingroup$ And, for the record, the only reason that I commented is that it really stood out to me that you had not properly typeset the sine function as $\sin$ (using $\sin$). Because I was trained by my instructors and mentors, over a career in mathematics, to understand that $\sin$ should be upright. I really stood out to me. Which is another reason that it might be a good idea to teach students to write mathematics using a computer at an earlier age. $\endgroup$
    – Xander Henderson
    Apr 13, 2023 at 11:57

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