Mathematical Journeys: Occam's Razor and SAT Math

I mentioned this problem from one of my earliest blog posts with one of my students last week, so I thought I'd bring it back as this week's Math Journey.  Enjoy!

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The SAT messes with your head. Don't feel embarrassed, it messes with everyone's head. It's designed to. The SAT is a test of your critical reasoning skills, meaning it's actually far more about logic and figuring out the correct course of action than it is about actually knowing the material. This is nowhere more evident than on the Math section.

The SAT Math trips up so many students because they expect it to behave like a math test. The truth is, the SAT Math is about figuring out how to answer each problem using as little actual math as possible. It's all about working quickly, and the questions are structured such that they conceal the quick logic and context-based route behind the facade of a more complicated math question. They're trying to psych you out; to make you think the problem is harder than it is. In math class you're taught to be thorough, to show your work and not leave out any steps. On the SAT, it's not only possible but downright preferable to solve as many questions as you can without ever picking up your pencil.

Let's take an example that caught me on the first go-round.

If a + b = 3 and ab = 4, what is (1/a) + (1/b) ?

Since you have two equations with two variables, your first instinct might be to treat them as a system of
equations. Namely, solve one for a, plug the result in for a in the other one, get it into quadratic form and solve from there, then use the newly-found roots in place of a and b in the final equation. If you take the time to go through the problem this way, you find that your answer is almost one of the options, but the sign is wrong (your answer is negative when the answer choice is positive). This can frustrate you to no end and cause you to waste precious minutes of your testing time checking and rechecking your math. The intended course of action is much simpler.

What the SAT wanted you to do with that problem is to just look at the last expression, the one you're trying
to find. How would you normally add two fractions with different denominators? You'd find the lowest common denominator and multiply, right? So just treat the variables as you would numbers:

(1/a) + (1/b) = (b/ab) + (a/ab) = (a + b)/(ab)

By now you should have noticed that both the numerator and denominator of this final expression are
identical to the first two equations they gave you. All you have to do is plug them in.

a + b = 3
ab       4

And that's literally it. They knew you'd over-think it if they set it up to look like a system of equations rather than simply an algebraic manipulation where they happened to give you the answers in the problem. They also knew you'd likely taken advanced algebra and trig, and would probably forget that the test doesn't cover those topics. The realization that it looks like a system of equations, and that you know how to solve those, would push you into starting the (somewhat long) procedure and it wouldn't occur to you to find a simpler way. Since their goal is to trip you up, make you waste time, psych you out, and mess with your head, they'll do everything they can to camouflage the quick shortcut behind big scary problems.

It's helpful to remember Occam's Razor; the simplest solution to a problem is usually the best one. Always look for the common-sense answer, and remember not to over-think it. This is also why I think it's beneficial for a student to go into the SAT with an attitude of healthy indignation and skepticism; if you're always looking for the traps, you'll be less likely to fall into them. Rather than getting frustrated, if you can see the trick question coming and confront it with a “Seriously?!” you'll be better prepared to deal with it.

Occam's Razor relates more to hypotheses. If multiple competing hypotheses are present, the one with the least assumptions should be selected.
This is a really good example of non-trivial, algorithmic thinking applied to solving of Math problem (I would probably agree with Mark M. in regards to Occam's Razor related to hypotheses testing, and seemingly unrelated to this problem solving technique) Best regards, \$45p/h

Ellen S.

Math and Writing Geek

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