Math Etiquette V: Divide and Conquer

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Most students I meet tend to have a mental block about division. They see long numbers and immediately think, “I didn’t learn those!”—and then, they refuse to even try. That's the kind of fear that can sabotage scores, either on test day or during test prep.

Fortunately for YOU, you’re reading this, and I, your faithful math servant, know the churnings in your head and can make the connection for you. There are really only a few ways to divide, and they’re all pretty easy. It’s just a matter of not assuming it’s a BIG DEAL. In fact, it’s really NOT a big deal. You’ll see.


Division Trick #1

This one is really simple. Remember all those multiplication facts...the ones you already know from somewhere around first grade? Yeah, those multiplication facts.

Well, division is merely the reverse of them! So, the first step to division is knowing your multiplication facts forward and backwards. I’d start with all your multiplication tables up to 12.

For example: 9 × 12 = 108—you knew that, right? Sooo...if you see 108 ÷ 12, you should immediately know the answer is 9.

Do you really see what’s going on? If you know your multiplication facts, you start to associate three different numbers together. So if you really know them, then when you see only two of those numbers, your brain will immediately think of the third in the group.

This set of associations also happens with addition, so you just need to know that two different groups exist, and consciously pick the “multiplication” group. For example, if I think of the numbers 3 and 5, I’ll probably think of two possibilities: 8 (because 3 + 5 = 8) AND 15 (because 3 × 5 = 15). Since my brain recognizes 3-5-8 as a group AND 3-5-15 as another group, I just need to be aware of which one I want.



Division trick #2

“But Kristina! The numbers I’M trying to divide are waaaay longer than that pedestrian example you just wrote. I swear, I don’t know this!”

Yeah, yeah, yeah. But don’t you see, long numbers are just made up of small numbers…and you already know all the small numbers!

So here's my advice: start at the beginning digit or two (the digit or two on the left side of the long number) and do that small division problem. Then keep moving over another digit to the right, and keep doing the small division problems. Soon, you’ll be done with the number!

Let's look at an example, one with a really big number: 19,032 ÷ 6 = ?

Take the first digit, the 1—6 doesn’t go into it.

So now take the leftmost two digits, 19—6 fits into 19 3 times, with 1 left over. Write down the 3 and save the 1. So now your solution is looking like this:

19,032 ÷ 6 = 3__

With your “saved” 1, add on the next digit over: 0. That gives us 10. 6 fits into 10 only 1 time, with 4 left over. Save the 4. Where are we now?

19,032 ÷ 6 = 31__

So, do the same thing: take your saved 4 and tack on the next digit: 3. Thus, we’re now dealing with 43. 6 fits inside 43 7 full times, with 1 left over. Write down 7 and save the 1.

19,032 ÷ 6 = 317__

There’s only one digit left! Take your saved 1 and tack on the final digit in the long number: 2. This gives us 12. 6 fits inside 12 evenly 2 times, with nothing left over. Write that 2 down and you’re done!

19,032 ÷ 6 = 3172

Does this process sound vaguely familiar?? It should. It’s JUST LONG DIVISION, which you already learned in second or third grade...but got out of practice because you probably use your iPhone calculator for everything. (Tell me I'm wrong!)

Not to say this doesn’t take a few extra seconds. It does. Maybe you shouldn’t do long division by hand on the ACT Math section or the SAT Calculator section, but the whole point is: you can if you need to.


Division Trick #3

This one is my favorite! And the most “Math Etiquette” way of dividing. And it stems from something a lot of you hate: fractions. (Hold on! Don't close the window just yet. I swear I'm going somewhere with this.)

You know how when you divide one fraction by another, you’re really just multiplying the first number by the reciprocal of the second?

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Well, let’s just take it a step further. If you divide by a number—ANY number, not just a fraction—you’re always multiplying by its reciprocal! Like this: 

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Thus, if you want to divide by a number, just place it in the denominator! DONE.

In fact, if you are multiplying by several numbers AND dividing by several numbers, it’s MEGA-easy: the first number and all the numbers you’re multiplying by get placed in the numerator, and each of the numbers you’re dividing by get placed in the denominator. Cross out and simplify and you’re done!

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I hope these tips help give you a more instinctive sense of the way division works. That's what math etiquette is all about, after all! The more you have that grasp, the less time you'll have to expend in a high-pressure situation like the No Calculator Math section...or any standardized math test section. It's important to go beyond memorizing the formulaic "tips and tricks" that are so frequently where people stop in their math test prep—at least if you want the highest possible scores! Still need help? You know where to find me—and you know I love working with students on the kind of foundational math knowledge that saves time, effort, and anxiety while maxing out math scores. Until then, happy dividing!