Getting The Right Answer Right
What does it mean to get the right answer on a math question?
Some may give the quick answer and say that a right answer is one that earns you full credit. But that answer, while necessary, is not sufficient. For any given problem, there’s usually more than one way to get to the right answer, even if you’re not taking a multiple-choice test, where you know the right answer is staring back at you before you even start the question. There are textbook right answers, there are clever right answers, there are inefficient right answers, and there are lucky right answers. Of course you always want to get the answer right, but you want to do it in the best way possible.
It’s worth examining some of the different paths to the right answer:
The “Right” and “Wrong” way. Ever gotten the right answer on a test but missed points because you didn’t do the problem the way it was taught in class? Most students develop problem-solving habits through years of showing work and teachers’ expectations. Sometimes, those expectations exist for a good reason. Sometimes, they don’t. Doing a problem the way the textbook (and your teacher) has shown you may help you develop skills you’ll need in future classes or applications. Textbook methods are in the textbook because they work and because they’re often the most mathematically generalized or rigorous methods.
But consider the possibility that the textbook method may not be the best method on a standardized test, or any test where you aren’t required to show your work. Textbook methods are often cumbersome given the per-problem time limit on a standardized test.
Slow vs. Fast methods. It goes without saying that some ways of getting to the correct answer are faster than others. Especially on a standardized test, the textbook method of solving a math problem isn’t always the fastest way to the right answer. Take, for example, this algebra problem (SAT/PSAT/ACT only):
(3x2 + 4x – 2)(x – 5) – (x-1)2 =
A) 3x3 – 12x2 – 20x + 9
B) 3x3 – 10x2 – 24x + 11
C) 2x3 – 11x2 – 20x + 9
D) x3 + 18x2 – 20x + 11
E) 3x2 + 8x + 9
Yes, you could do this with straightforward FOILing, but all those cross-terms are going to take time. It’s faster to ask yourself two questions: 1) What is the constant term going to be? and 2) What is the highest-order term going to be? Once you realize that the constant has to be 9 and the leading term is 3x3, the only possible right answer is A.
Foolproof vs. Error-prone methods. Okay, so perhaps there is no such thing as a foolproof method to get the correct answer. But some methods can get pretty close. In the problem above, which method exposes you to more possibilities of mistakes: the textbook method or the first-term, last-term method? Which method are the test-makers expecting you to use, and which mistakes might they have anticipated? Answer choice B above will seem right if you forgot to distribute the negative sign in the second term. And even if the test-makers haven’t anticipated every mistake you can make, you could still spend a lot of time searching for your error if your answer isn’t even an answer choice.
And that’s if the test is multiple-choice. If it isn’t, finding and correcting sign errors in your work isn’t always easy. For open-ended questions, you may have to use the textbook method. If so, try checking your work by plugging in some numbers and making sure everything makes sense. Be wary of just doing the problem again as a means of checking your answers. Some mistakes are just as easy to make the second time around if you don’t approach the problem from a different direction.
There are often many ways to get the correct answer on a math problem, but some ways can be more “right” than others.