Students who are reasonably comfortable with math itself often complain, “I could solve most GRE quantitative problems if I had enough time, but I always seem to run out of time,” or “I just wish I could do GRE math faster.” Does this sound like something you’ve said? Then this section is for you.
Solve this question. Allot yourself a strict one-and-a-half-minute time limit.
The brain
First, let’s quickly review a few things about the brain. Before we do, though, please know that we’re not neurologists. Instead, our goal here is to introduce you to another way of thinking about the problem above. So below is just a little background to set the scene. The brain, of course, is much more complex than the amount of space we have in this book!
The cerebrum, the “intelligence” part of our brain, is divided into two halves— the left and right hemispheres. Each half controls all the sensation and muscular movements on the opposite side of the body. So when you move your left leg, it’s the right half of your brain that controls it. The two hemispheres also process information very differently.
The left hemisphere is all about logic, organization, precision, and detail management. It specializes in differentiation, that is, telling the exact difference between closely related things. It’s very good at following clear rules, recipes, formulas, and procedures in a step-by-step, logical way. The left brain controls the grammar and syntax of language. If someone said, “I are happy,” it would be your left brain that recognized that as incorrect.
In the figure, ABCD is a square, and all the dots are evenly spaced. Each vertical or horizontal distance between two adjacent dots is 3 units. Find the area of the shaded region.
60 72 81 96
120 (Answer and explanation below.)
B A C D A B C D E
The right hemisphere controls intuition and pattern-matching. It specializes in integration, that is, seeing the underlying similarity or unity behind things that seem pretty different. It’s often called the “artistic” side of the brain, and it’s good at interpreting information presented as symbols or images. It’s also good at facial recognition and voice recognition.
Math and the brain
Okay. So which hemisphere is better for math? Well, the detail-management,
organization, and precision of the left brain are a huge help in arithmetic and algebra. Meanwhile, the right brain's pattern-matching skills can play a role in some branches of math, such as geometry. Now, we're going to make an oversimplification in order to illustrate two ways of approaching GRE math. Imagine that we’re all split up into “left-brain people” and “right-brain people,” each with a “dominant” hemisphere that has a larger influence over the way we think. In general, left-brain people usually feel reasonably comfortable with math and can certainly follow the methodical procedures with ease. Typically, right-brain people tend to have a more difficult time keeping all the details straight, although many times they tune into the “big picture” ideas faster.
The magic of right-brain thinking
Left-brain people usually know the rules reasonably well. They’re often the “good- at-math” kind of students. Faced with most GRE quantitative questions, they could figure out the right answer, given enough time. The catch, of course, is that they don’t have unlimited time on the GRE—just thirty-five minutes for twenty questions, or 1:45 per question.
An overwhelming number of GRE math questions are designed specifically to punish someone who is overly left-brained. In other words, the questions are written to be looked at in different ways. Yes, the methodical, step-by-step, plodding approaches can work, but they take too much time. The best test takers can reframe a question or see it in a different way—one that can be answered more quickly.
Example: the practice problem
Let’s take a look back at the practice problem. Yes, you could figure out separately the area of each triangle, each square, each trapezoid, and then add all of those shapes together. That would take some time. So let’s look at it a different way.
Here’s a right-brain solution to this question: rearrange the pieces! First, slide the little triangle up into the corner.
B A C D B A C D
GRE Quantitative Reasoning
Now, notice that the left-most trapezoid would fit nicely into that blank space on the upper right.
Now, flip that remaining long trapezoid over the diagonal BD:
Would you look at that! The shaded region accounts for exactly half the area of the big square. The big square is 12 × 12 = 144, so the shaded region is 144 ÷ 2 = 72. Answer = (B). Once you see the trick, the pattern, there’s only the most minimal of calculations needed.
Another reasonably quick right-brain approach would be to simply count all of the little triangles.
The original shaded figure can be broken into 16 equal triangles. The full square is 16 little squares or 32 triangles. Therefore, the area of the shaded figure is exactly half the area of the square.
Learning to see
Some right-brain readers might celebrate such a process, while some left-brain people might be frustrated or annoyed at this point. They may be thinking, Great! Now that it’s pointed out, yes, that’s an efficient way to solve, but how am I supposed to see that on my own? B A C D B A C D B A C D B A C D B A C D
Mastering the strengths and skills of your non-dominant hemisphere is never an easy task, but it can be done.
There are a wild variety of things you can do to enhance right-brain function. Read poetry. Look at art. Make art. Read about patterns in comparative mythology. Free-associate. Imagine. Follow chains of word associations. Slow down and really look at things.
o o
1. If the problem asks for the value of an expression involving variables, chances are good there will be some way to solve for the value of the expression directly, without solving for the individual variables.
2. For a “find the area of the shaded region” question in which the region is
particularly complicated (as in the practice question), look for a way to rearrange and simplify.
3. If the problem is a geometry one stated in words, always sketch a rough diagram, unless you can visualize the diagram easily.
4. If the problem is purely numerical or algebraic, consider whether there would be a way to visualize the problem (number line, xy-plane, etc.)
From this point forward, whenever you practice GRE quantitative questions, first of all, always practice against a strict time limit. Second, the criterion is no longer whether you got the answer right or not. Even if you got the correct answer, compare your solution to the official solution. If your solution was a slow methodical approach and the official solution shows a shortcut, then for your purposes, consider this a question you got wrong. For every such question, force yourself to write down the shortcut and consider what you could have done to “see” that shortcut. Force yourself to put it into words and explain it—a practice that will strengthen your interhemispheric connections. As you collect more and more write-ups, go back to reread the collection. Keep doing this consistently. Learn from your mistakes, and before you know it, you’ll start “seeing” solutions to GRE quantitative problems!