Jack, be nimble.

Test Preparation SAT Preparation
1

<p>I've been talking in 140 character bursts to a bunch of other NYC area tutors on twitter lately about critical thinking, problem solving, test taking, etc. We've been trying to put our collective metaphorical finger on that certain something that makes great test takers great. This is what I'm coming up with after a few days of thinking it over. (I posted this at my own site too, where it's a bit more formatted, but I wrote this post with CC in mind):</p>

<p>Here's a question I love to throw at students early on in the process of working with them (let's call this a grid-in for now, to keep things simple):</p>

<p>(4^999 + 4^998)/5 = 4^x. What is x?</p>

<p>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:
[ol]
[<em>]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.
[</em>]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.
[<em>]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.
[</em>]And if my student starts wrestling with other, more manageable numbers (for example: (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.
[/ol]</p>

<p>Let's be clear here: it's best to know how to do this question the right way (full solution is at the bottom of this post), like the third student. She has a strong base of math knowledge and has seen enough similar problems not to succumb to the pitfall of misusing 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.</p>

<p>The fourth student, though, is one who, when faced with an intimidating problem that his tools seem at first not to be able to solve, finds a way to claw out the correct answer anyway. 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.</p>

<p>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 does it on game day. I'm going to worry less about student 4 on game day.</p>

<p>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 struggle with tough problems sometimes. You're going to have to learn from your mistakes and try not to repeat them. 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.</p>

<p>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.</p>

<p>Solution to the sample problem:</p>

<p>(4^999+4^998)/5=4^x</p>

<p>(4^998(4+1))/5=4^x</p>

<p>(4^998(5)/5=4^x</p>

<p>4^998=4^x</p>

<p>998=x</p>

2

<p>You could also do it like this:
4^999 = 1
4^998 = 1/4 (since it’s one fourth of one)
(1+1/4)/5
(5/4)/5
1/4
1/4 = 4^998
998.</p>

<p>It’s a bit more abstract, I guess, but it gets the job done.
Good sample problem!</p>

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<p>I just graphed it on my calculator to get the answer, haha</p>

4

<p>Clever solutions, both. That’s the kind of nimbleness I’m talking about.</p>

<p>@Stevenf: that’s a bit abstract, yes, but you’re really just plugging in. It’s not much more abstract than the solution student #4 had in my post. Instead of choosing 4^3 and 4^2, like he did, you chose 4^0 and 4^(-1)!</p>

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<p>Buddy, you want to see nimbleness? </p>

<p>(4^999+4^998)/5=4^x</p>

<p>MAGIC MAGIC MAGIC</p>

<p>x = 998</p>

<p>GOML</p>

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<p>The above solutions are all fine. But if those manipulations are daunting and MAGIC seems unavailable, there is another alternative:</p>

<p>Just plug it into a TI89.</p>

<p>Solve((4^999 + 4^998)/5=4^x,x)</p>

<p>…it takes the calculator about 5 seconds to crunch the numbers, but it works!</p>

<p>There are many students who will not think of any of the algebraic methods. It is good to remind them that their calculator can do this kind of thing. I know that the strongest math students can get 800s without any calculator at all, much less a TI89. But I would say that for the rest, the 89 can be a boost. Nimble is good. So is opportunistic.</p>

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<p>Yeah, a TI89 can handle these numbers. Punch this into an 83 or an 84 (which more people have, in my experience), though, and you get an overflow error. What do you do when your calculator can’t help you?</p>

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<p>Then you go back to “read, think and play”. And no doubt, the majority of the times, that’s the better plan. The TI89 should just be one of the methods available to you. Your experience taking practice tests (and following up the practice with thorough analysis of what went well and what didn’t) will enable you to choose the approach that’s best.</p>

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<p>The 89 is a great tool, don’t get me wrong. I just see too many kids place all their eggs in that basket. The SAT writers know very well the capabilities of the 89, so they mostly avoid questions that can only be cracked by sheer calculating power.</p>

<p>As you said, experience in testing situations will help you to decide when it’ll work and when it won’t.</p>