### GRE Geometry: The Impossible Task

In one of my recent classes, I told the students “You’ll never know how to answer a geometry question.” The reaction was fairly predictable: “Why would you say that?!? That’s so discouraging!!”

Of course, I certainly was **NOT** trying to discourage them. I used that statement to illustrate that geometry questions are often a type of quantitative question that can feel *immensely* frustrating! You know what shape you have, you know what quantity the question wants, but you have no idea how to solve for that quantity.

This is what I meant when I said you’ll never know how to answer these questions. That “leap” to the correct answer is impossible. You can’t get to the answer in one step, but that’s all right: you’re not supposed to!

(An important aside: if you’ve read my post regarding calculation v. principle on the GRE, you should be aware that I am discussing the calculation heavy geometry questions in this post.)

The efficient, effective approach to a calculation-based geometry question is NOT to try and jump to the final answer, but instead to simply move to the next “piece”. For example, let’s say a geometry question gives me an isosceles triangle with two angles equaling *x*. I don’t know what *x* is, and I don’t know how to use it to find the answer to the question. But I **DO** know that the third angle is 180-2*x*.

That’s the game. Find the next little piece. And the piece after that. And the piece after that. Let’s see an example.

The correct response to this problem is “Bu-whah??? I know nothing about the large circle!”

But you do know the area of the smaller circle. What piece will that give you? Ok, you say, area gives me the radius. A = pi*r^2, so pi = pi*r^2, so r^2 = 1, so r = 1. Done, and let’s put that in the diagram.

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### What Does the GRE Test? Calculation versus Principle

Some of you may have already read an excellent post discussing how you should study for the GRE, differentiating the application of skill as opposed to the application of knowledge. (Hint: you need both, but many people struggle to progress past pure knowledge!) If you have not read that post, you can find it here.

Today (or whenever you may be reading this) I would like to “riff” on that concept inside the quantitative section. Many, many students that I work with want to treat the GRE quantitative section as a math test: there’s an equation I should use, and a number I should solve for.

And sometimes, yes, that’s exactly what the test wants you to do. But there are other questions. Questions that don’t feel quite so … “math-y”. If you’ve taken a practice test, you probably know what I’m talking about, even if you can’t put your finger on an exact definition. You saw some questions that didn’t have an equation, or questions that had an equation but no definitive “x = 243” final answer. If you had a gut reaction of “This doesn’t feel like math?!?” to these questions, congratulations! You are well on your way to a more nuanced understanding of what the GRE quantitative section wants from you!

This is what I mean in the title “Calculation versus Principle”. Some GRE quant questions are best approached through the application of various math principles; running calculations on these questions is often too time-consuming.

(As an aside, when I use the term “calculation” I am **not** referring to questions you would plug into a calculator. Any questions that require mathematic manipulations to find a definitive numerical result are calculation questions.)

If I were teaching a class, this is about the point where I would get tired of talking. I’m tired of talking, let’s see an example!

Ah, yes, a lovely quant comparison question. What follows is a transcription of a hypothetical test-taker’s calculation approach. Feel free to skim the next two paragraphs; the purpose here is NOT for you to know the calculation approach, but instead to compare this approach to a principle-based approach.

***Begin hypothetical calculation test-taker.***

“I need to compare the area of a triangle to the area of a square. Well that’s easy! Area = ½ b*h , and Area = side*side. Ok, what’s the …. Uh-oh. What am I supposed to do with this? They haven’t given me numbers. No wait, when they don’t give me numbers, I’m allowed to choose numbers that fit the problem. Ok, a triangle and a square have the same perimeter. Let’s make the perimeter 12, so I can easily make a 3-sided and 4-sided figure. Ok, square with sides = 3, area is 3*3 = 9. All right, quantity B is 9. Let’s get quantity A.”

“What triangle should I make? Right triangles are easy, could I make a right triangle? Hey, a 3-4-5 right triangle has a perimeter of 12! Ok, so it’s ½ b*h, and that’s ½ (3)(4) so the triangle has an area of 6 – that’s definitely less than 9. But the problem didn’t tell me it was a right triangle; am I allowed to assume that? No, I should probably try another triangle. Well, I could make an equilateral triangle – 4-4-4. What would the area of this triangle equal? The base is 4, but what’s the height? Ok, I’ll have to draw the height. Ah, I have a 30-60-90 triangle inside here, and the 60 side is going to be . This will have an area of ½ (4)(3.7) – that will be 2*3.7, which is 7.4. Still less than 9. Ok, the answer is B.”

***End hypothetical calculation test-taker.***

Well, this person is correct. The answer is B, quantity B is always larger. But wow, that was a **lot** of work, and in all honesty, I tried to make this hypothetical test-taker an extremely accomplished GRE quant test-taker. The immediate jump to number testing, the recognition that we need to actively try to find the maximum area triangle to correctly compare that to the square area, the immediate recognition of easy right triangles and the immediate ability to calculate the area of the equilateral triangle, the quick estimation of … these are all possible, but to do them **all** in the same problem, and do them correctly? I would prefer an easier way.

So let’s see what happens when we apply a general principle to this problem.

***Begin hypothetical principle approach test-taker***

I’m comparing the area of a square to the area of a triangle. The perimeters have to be the same. Ok, I know that all else being equal, if I want to maximize the area of a shape, I want it to be symmetrical. A square has more area than a rectangle with the same perimeter. What’s the most symmetrical shape? A circle. So the closer my shape gets to a circle – the more sides I put in it – the more I’m maximizing my area. Ok, the square has more sides, and therefore the larger area. B.

***End hypothetical principle approach test-taker***

Hopefully you agree that the principle-based approach is far simpler, just as accurate, and requires much less time.

So now comes the fun part – how do we learn the principles, and how do we know when to apply them?

**Learning the Principles**

There is no easy answer to this, but I can provide some guidelines. Look through your GRE study sources. If they look anything like mine (which are, of course the Manhattan Strategy Guides), there are certain concepts that are in boldface. Compare the following options, all of which at least partly appear in bold in my strategy guides:

1) “Sides correspond to their opposite angles…. The longest side is opposite the largest angle, and the smallest side is opposite the smallest angle.”

2) “The internal angles of a triangle must add up to 180 degrees.”

3) “Rate x Time = Distance”

4) “For some grouping problems, you may want to think about the most or least evenly distributed arrangements of the items.”

Items 1 and 4 are what I would call principle statements. They give relationships or strategies, but don’t readily lend themselves to equations. Items 2 and 3 are calculation statements. They either state clearly defined numerical quantities (and therefore easily lend themselves to equation creation, a la “a+b+c = 180”) or literally state an equation.

Look through your study materials. **The more the content seems to address relationships or ideas that don’t correspond to exact numbers or exact equations, the more you should consider applying these ideas as large principles.**

There is one particular area that I feel deserves special mention: number properties. GRE questions that revolve around positive vs. negative, even vs. odd, prime vs. composite numbers are more often than not principle based. There are broad principles that define specific relationships across these types of numbers. Similarly, the GRE often asks questions that either revolve around or take advantage of what I call “trick” numbers: -1, 0, and 1; and proper fractions, either positive or negative. These numbers have special properties; learning these properties, as opposed to needing to do exact calculations, can save you much heartache on the test.

**Applying the Principles**

When should we apply the principles? **This question relies on you closely reviewing your work**. Whenever a question asks for a relationship between items without providing solid numbers, perhaps you could apply a broad principle. Whenever a question seems to rely less on solving for a specific quantity, and more on identifying what kind of quantity will result – “which of the following must be odd” – perhaps you could apply a general principle. And finally, if a question permits trick numbers, there may be a principle you could apply.

As you review your work, ask yourself the following question: **“Is there a way I could have answered this question without doing any actual math?”** If the answer is yes, you have found a principle question.

Good luck, and happy studying!!

### 3 Tips to Stay Engaged on Long Passages

We all know the feeling. You come to the end of a sentence, or a paragraph, or a page—and suddenly realize you have no idea what just happened.

From a psychological perspective, this is a fascinating phenomenon. Somehow, while a part of us thought we were reading happily along, another part was off somewhere else—ruminating on some joke we recently heard, fretting about an upcoming assignment, or planning dinner.

But whatever it is in the mind that allows us to basically be just wrong about the contents of our own thoughts (to believe we’re learning about mating practices of chimpanzees but really be hankering for spaghetti carbonara), one thing that’s certain is we can’t have this happen while we’re tackling a long reading passage in the Verbal section of the GRE on Test Day. Spacing out may be fine to varying degrees in the course of everyday life, but it can’t happen during the GRE.

Fortunately, there are some tricks and strategies you can learn now to help prevent this type of thing from happening, and to improve your overall comprehension of reading passages. The main goal, remember, is not to know the entire passage by heart—but rather to have a solid grasp of two basic things: first, the purpose and structure of the passage; second, where to find certain details in the passage should you encounter a question about them. Here are three tips to help you accomplish this:

**Tip 1: Put yourself in the author’s shoes**. GRE passages are often culled from, or imitations of, genuine texts from scientific, literary, or historical publications. What that means is that someone spent time and energy crafting the argument you see before you. Someone had a real-live thought, opinion, or belief that he or she wanted other people to know, and sat down in front of a keyboard and carefully deliberated about how best to convey this idea to a non-expert reader. By imagining this person’s motivations, you can often end up with a much more vivid picture of the content and purpose of the passage. What is the principal idea the author is trying to get across? If you were the author, how would you express these ideas? Visualizing a real person typing real ideas onto a real computer screen is a great way of plucking abstract notions from the ether and dragging them down to earth.

**Tip 2: Engage emotionally.** If someone asked you comprehension questions about what happened in the gripping last season of Breaking Bad, you would have no problem picking out the right answer. Why? Because human beings remember better things they actually care about. When something matters to us, our brain is more active, forming neural pathways that you can draw from in subsequent memory tasks. The more you can bring yourself to care about the content of the passage, the stronger your activation signal will be, and the clearer your mental picture. Many people find science passages particularly daunting, and immediately zone out at the sight of words like “electrochemical” and “tectonic.” If you imagine, though, some epic drama taking place between, say, the earth’s molten core and the hardened outer crust above it, you may find that previously yawn-worthy topics take on a certain pizzazz.

**Tip 3. Know what NOT to read.** The trickiest part of the GRE is timing. Many people feel like they’d have no trouble getting all the answers if they only had enough time. Unfortunately, given these temporal limitations, our job is rather to read as efficiently and effectively as possible—so get good at knowing what not to read. When you see a list of complicated terms, make a note of where it is, but just say No to laboring over each of its tiny details. See an in-depth description of some tangential topic? Just say No—and make a note of where it is. Come across a lengthy aside that seems unrelated the main idea? Again: say No. You don’t have time to get bogged down in these details. Sure—if a question comes up about them, you’ll know where to look. But for now, you’re reading Big Picture.

Overall, then, the key to your success is going to be about striking the perfect balance. Engage deeply the text, but don’t get too sucked in. The more you can cultivate these strategies as you practice, the better off you’re going to be when facing those initially unnerving—but ultimately conquerable—passages on Test Day.

*Manhattan GRE*

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