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From the archives: Horizontal fraction bars and “for every”

July 21, 2011

While I’m on vacation, I’ve decided to repost some of my older posts from my early days in blogging, in the hopes that they might get a bit more discussion than before.

Here are two simple things, that if taught well in middle/elementary school, could make a huge difference in how students understand science, at least the routinized parts of dealing with rates and units.

First, is horizontal fraction bars. I don’t know why it is, but my kids still love to write out fractions as 1/2, and sometimes even 1\div 2. This isn’t such big deal when dealing with pure numbers, but if you toss in some units, things can get very confusing very quickly, as in 10\textrm{m}/\textrm{s}^2 \cdot 5\textrm{s} = 50 \textellipsis What should the unit be? It isn’t nearly as obvious as

10\frac{\textrm{m}}{\textrm{s}^2} \cdot 5\textrm{s}=50\frac{\textrm{m}}{\textrm{s}}

So that’s my first suggestion. Get rid of the \div symbol as soon as possible. In fact, what’s the pedagogical reason for teaching this symbol at all? Wouldn’t it be easier just to start introducing fractions from the get go, and saying things like \frac{4}{2} means “how many times does 2 go into 4? I’m probably missing something here, since I’ve never tried to teach division in elementary school.

Here’s my other suggestion. Replace the word “per” with “for every.” My kids mostly get that the word “per” means divide. But even this doesn’t really help them figure out how to write the unit, when it’s complicated: “meters per second per second,” and moreover, per doesn’t really give any insight into why you are dividing, or what that means. If you replace this little word with “for every” suddenly, the meaning is much, much clearer. “My car gets 50 miles for every gallon of gas” instead of “My car gets 50 miles per gallon.” This makes it so much easier to figure out how far I can go on a tank of gas. And it helps with solving the inverse problem as well. Suppose I travel 400 miles, how many gallons of gas did I use? “Well, if I go 50 miles for every gallon, all I need to think about his how many 50’s go into 400 to figure out how many gallons of gas I used.”

These two little changes make concepts like density, velocity, acceleration, pressure and so many more “ratio” concepts much easier for students to grasp.

Anyway, if you’re a middle school or elementary school teacher, I’d love some feedback here. What is the advantage of the slash fraction bar or the \div symbol? How do you introduce the ideas of rates (with units) to kids?

2 Comments leave one →
  1. July 21, 2011 12:58 pm

    I don’t know anyone, at least in Canada, who still uses the /, but I can see an argument for the division symbol.

    The idea of division starts off being separate from the idea of fractions, probably because we don’t always have equal fractions (2/5) but we do always, to start with, have equal division (27 divided by 9).

    As a student, I would possibly find it confusing to have to divide sometimes (27/9) and not others (2/5).

    On the flip side, it would certainly make mixed numbers and decimals a lot easier to teach…

    I’m not sure the best approach because I have a serious hate on for long division. When it’s taught in Grade 6 it’s barely used for the year, and then in Grade 7 we tell students to use a calculator to do the division because the bigger question is more important than the long division.

    • July 24, 2011 1:40 am

      I am certainly not an expert in elementary math ed, but I don’t see why you can’t introduce division without the division symbol $\div$

      Why not just have students start with problems like \frac{9}{3} and \frac{8}{2} and then once they’re very comfortable with that, ask them what \frac{8}{3} might mean.

      And I’m with you—learning the algorithm of long division simply for its own sake (without any basis for why it works) seems to be pretty fruitless, and one of the first places where students decide they are “not math people”—many adults I know make jokes about not knowing how to do long division.

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