Here’s a question I love to throw at students early on in the tutoring process (let’s call this a grid-in for now, to keep things simple):

1. If $\fn_phv&space;\frac{4^{999}+4^{998}}{5}=4^x$, what is x?

It’s a beautiful question because no matter what, it’s going to show me something about the kid with whom I’m working. Almost everyone goes to the calculator first. Once it becomes clear that the calculator will be no savior I see a few divergent paths, all illuminating:

1. If my student says it can’t be done, I know one kind of question on which I’m going to have to drill her repeatedly.
2. If my student says x = 1997, then I know he just added the exponents in the numerator and completely ignored the denominator, so we’re going to need to review the exponent rules and get his vision checked.
3. If my student factors 4998 out of the numerator to see that everything else cancels out and x = 998, then I know I’m going to have to really challenge her to get her score higher than it already is (full solution explained at the bottom of the post).
4. And if my student starts wrestling with other, more manageable numbers (for example: $\fn_phv&space;\frac{4^{3}+4^{2}}{5}=16=4^2$),  to try to suss out a pattern, I know I’m dealing with a kid who knows how to struggle and who doesn’t back down from tough questions.

Let’s be clear here: it’s best to know how to do this question the right way, like the third student. She has a strong base of math knowledge, has seen enough similar problems not to misapply exponent rules, and is creative enough to try pulling out the greatest common factor to see if anything good happens (and it does). But I don’t know yet what she’s going to do when she gets to a problem that’s unlike any she’s seen before (and on the SAT, that will indubitably happen, and probably when it counts). So I’m going to keep watching her closely until I get to see how she reacts to a question that makes her squirm.

The fourth student is one who finds a way to claw out the correct answer when faced with an intimidating problem that his tools seem at first not to be able to solve. He might not be as conversant with math as the third student, but in the eyes of the SAT, she and he are exactly the same on that question. Because he’s scrappy. He’s nimble. And that will take him a very long way.

In sports, you’ll often hear a commentator say, “That’s why you play the game,” after an underdog wins a game it shouldn’t have. It doesn’t matter who looks better on paper. It matters who performs on game day. After seeing him work this question, I’m going to worry less about student #4 on game day.

If you want to take your place in the pantheon of great test takers, you’re going to have to be nimble. You’re going to have to grapple with tough problems sometimes. You’re going to have to, as poor Mario does in the video above, make mistakes, learn from them, and try not to repeat them. You’re going to have to be flexible, and willing to try more than one approach.

This is why I want you to know how to plug in, but I also want you to be able to do the math. This is why I want you to have a decent essay skeleton in mind when you walk into the test center, but I also want you to be able to handle a fakakta prompt like the one about reality TV. This is why it’s important to have a good vocabulary, but you’re wasting your time if you think the path to a higher reading score is through flash cards or word lists alone.

Kids sometimes complain that I ask too much of them. But the way I see it, I really only want one (admittedly multifaceted) thing out of my students: I want them to learn to be nimble.

##### Solution to the sample problem:
$\fn_phv&space;\frac{4^{999}+4^{998}}{5}=4^x$
$\fn_phv&space;\frac{4^{998}(4+1)}{5}=4^x$
$\fn_phv&space;\frac{4^{998}(5)}{5}=4^x$
$\fn_phv&space;4^{998}=4^x$
$\fn_phv&space;998=x$