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?

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