Before I go any further, I want to make it clear that the audience for this is people fairly comfortable for mathematics, particularly teachers. I’ll be heading into woods that will likely disorient many students and others who struggle with mathematics. That said, if you want to challenge your thinking, please read on.
Earlier today, I saw this video by the inimitable polymathematic:
He is talking specifically about question 15, which is “Find \(\frac{13}{16}\div 2\).”
My immediate thought: This is 13 divided by 16, divided by 2. He proceeded to evaluate this using KCF, converting the 2 to 1/2.
To be clear, I don’t have an immediate problem with KCF, or Keep Change Flip, or “division is multiplication of the reciprocal” (the way to say it if you want to sound all technical and stuff), and yet the video made me realize (and grumble about) what I do have a problem with.
I think we (hopefully inadvertently) convince students there’s a huge, important conceptual difference between fraction notation and obelus notation. And on the one hand, I can see an argument that they’re not identical, that they capture different things, but at the same time, there’s a reason why we usually abandon obelus notation some time in high school or college.
The obelus (this thing: \(\div\)) is clunky. It is front-and-center in the zombie math troll (e.g., “Simplify \(4 \div 2(3-1)\)”). I don’t really hate it; I think it’s fine for younger grades. For inline mathematics at higher levels, I do prefer the slash, but I have a warm nostalgic spot in my heart for the obelus (enough, after all, to learn its name).
Anyway, the issue for me in \(\frac{13}{16}\div 2\), and with \(\frac{a}{b}\div\frac{c}{d}\) in general, is not KCF or the poor lil obelus, it’s how it feels like students struggle with the connection that BOTH \(\frac{a}{b}\) and \(a\div b\) are division. They’re just two different ways to write division.
Here’s why “keep-change-flip” works, but be aware: It involves seeing the obelus in a way you might not have before (unless you’ve read my earlier piece on this).
I’ll rewrite \(\frac{a}{b}\div\frac{c}{d}\) entirely in obelus terms: \(({a}\div{b})\div({c}\div{d})\).
Now I’ll get rid of the parentheses, but this requires doing something not allowed in standard mathematical notation: \({a}\div{b}\div{c}\div\div{d}\).
If I were using subtraction instead of division, there’s nothing weird here: \(-(-a) = a\). The opposite (additive inverse) of the opposite of a number is the original number. It turns out that the same is true for division: Except for zero, the reciprocal (multiplicative inverse) of the reciprocal of a number is the original number.
In other words, if you’ll allow me to use a little poetic license with the symbols, \(\div(\div a)=a\).
So that gives us: \(\frac{a}{b}\div\frac{c}{d}={a}\div{b}\div{c}\div\div{d}={a}\div{b}\div{c}\cdot{d}\).
And another thing… we tend to insist that strings of multiplication and division have to be interpreted from left to right. That’s not exactly true, but for a long time I struggled with understanding where the thinking is going astray for some students.
So this is a fairly recent realization for me, certainly the first time I’ve written it down: There are two ways to interpret \(x\otimes y\), where \(\otimes\) is some operator (addition, subtraction, etc.): As “Apply the operator to both x and y” and as “Starting with x, perform the operator using y”.
For instance, \(3+4\) might be interpreted as “add three and four” or as “starting with three, add four.” If we just have the one operator, the interpretation isn’t important: Either way, it’s seven.
But if we have multiple operators, the interpretation matters: \(3-2+4\) could be either “apply the subtraction to the values on either side, and apply the addition to the values on either side” or “starting with three, subtract two and add four”.
Under the first interpretation, we have to go from left to right: Applying the addition first incorrectly gives us -3, while applying the subtraction first correctly gives us 5.
Under the second interpretation, order doesn’t matter: We can subtract two first, then add four, or vice versa. Which is to say, the commutativity that makes us love addition and grumble about subtraction returns. Huzzah! The additive layer, both addition and subtraction, are fully commutative.
Ditto multiplication, meaning that \({a}\div{b}\div{c}\cdot{d}\) is commutative as long as the division sign sticks with the value after it. That is, as long as we interpret it as “take value a, divide by b, divide by c, and multiply by d… in some order”.
So I can finally wrap this up: \(\frac{a}{b}\div\frac{c}{d}={a}\div{b}\div{c}\cdot{d}={a}\div{b}\cdot{d}\div{c}=\frac{a}{b}\cdot\frac{d}{c}\). Yay, we got KCF! With a side of potatoes and gravy.
Again, the key here is being able to move fluidly between \(\frac{a}{b}\) and \(a\div b\): It’s two different ways to write division. Ultimately, the conceptual gain we have with fraction notation (rates, ratios, and proportions) as well as the notational clarity far outweighs any nostalgic value of the obelus, so the latter gets the heave-ho.