Archives For GRE Strategies

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gmat square root

Have you ever gotten a GRE question wrong because you thought you were supposed to take a square root and get two different numbers but the answer key said only the positive root counted? Alternatively, have you ever gotten one wrong because you took the square root and wrote down just the positive root but the answer key said that, this time, both the positive and the negative root counted? What’s going on here?

There are a couple of rules we need to keep straight in terms of how standardized tests (including the GRE) deal with square roots. The Official Guide does detail these rules, but enough students have found the explanation confusing – and have complained to us about it – that we decided to write an article to clear everything up.

Doesn’t the OG say that we’re only supposed to take the positive root?

Sometimes this is true – but not always. This is where the confusion arises. Here’s a quote from the OG 2nd edition, page 212:

“All positive numbers have two square roots, one positive and one negative.”

Hmm. Okay, so that makes it seem like we always should take two roots, not just the positive one. Later in the same paragraph, though, the book says:

“The symbol √n is used to denote the nonnegative square root of the nonnegative number n.”

Translation: when there’s a square root symbol given with an actual number underneath it – not a variable – then we should take only the positive root. This is confusing because, although they’re not talking about variables, they use the letter n in the example. In this instance, even though they use the letter n, they define n as a “nonnegative number” – that is, they have already removed the possibility that n could be negative, so n is not really a variable.

If I ask you for the value of √9, then the answer is 3, but not -3. That leads us to our first rule.

Rule #1: √9 = 3 only, not -3

If the problem gives you an actual number below that square root symbol, then take only the positive root.

Note that there are no variables in that rule. Let’s insert one: √9 = x. What is x? In this case, x = 3, because whenever we take the square root of an actual number, we take only the positive root; the rule doesn’t change.

Okay, what if I change the problem to this: √x = 3. Now what is x? In this case, x = 9, but not -9. How do we know? Try plugging the actual number back into the problem. √9 does equal 3. What does √-9 equal? Nothing – we’re not allowed to have negative signs underneath square root signs, so √-9 doesn’t work.

Just as an aside, if the test did want us to take the negative root of some positive number under a square root sign, they’d give us this: -√9. First, we’d take the square root of 9 to get 3 and then that negative sign would still be hanging out there. Voilà! We have -3.

What else does the OG say?

Here’s the second source of confusion on this topic in the OG. On the same page of the book (212), right after the quotes that I gave up above, we have a table showing various rules and examples, and these rules seem to support the idea that we should always take the positive root and only the positive root. Note something very important though: the table is introduced with the text “where a > 0 and b > 0.” In other words, everything in the table is only true when we already know that the numbers are positive! In that case, of course we only want to take the positive values!

What if we don’t already know that the numbers in question are positive? That brings us to our second and third rules.

Rule #2: x2 = 9 means x = 3, x = -3

How are things different in this example? We no longer have a square root sign – here, we’re dealing with an exponent. If we square the number 3, we get 9. If we square the number -3, we also get 9. Therefore, both numbers are possible values for x, because both make the equation true.

Mathematically, we would say that x = 3 or x = -3. If you’re doing a Quantitative Comparison problem, think of it this way: either one is a possible value for x, so both have to be considered possible values when comparing Quantity A to Quantity B.

Rule #3: √(x)2 = 3 means x = 3, x = -3

Okay, we’re back to our square root sign, but we also have an exponent this time! Now what? Do we take only the positive root, because we have a square root sign? Or do we take both positive and negative roots, because we have an exponent?

First, solve for the value of x: square both sides of √(x)2 = 3 to get x2 = 9. Take the square root to get x = 3, x = -3 (as in our rule #2).

If you’re not sure that rule #2 (take both roots) should apply, try plugging the two numbers into the given equation, √x2 = 3, and see whether they make the equation true. If we plug 3 into the equation √x2 = 3, we get: √(3)2 = 3. Is this true? Yes: √(3)2 = √9 and that does indeed equal 3.

Now, try plugging -3 into the equation: √(-3)2= 3. We have a negative under the square root sign, but we also have parentheses with an exponent. Follow the order of operations: square the number first to get √9. No more negative number under the exponent! Finishing off the problem, we get √9 and once again that does equal 3, so -3 is also a possible value for x. The variable x could equal 3 or -3.

How am I going to remember all that?

Notice something: the first example has either a real number or a plain variable (no exponent) under the square root sign. In both circumstances, we solve only for the positive value of the root, not the negative one.

The second and third examples both include an exponent. Our second rule doesn’t include any square root symbol at all – if we have only exponents, no roots at all, then we can have both positive and negative roots. Our third rule does have a square root symbol, but it also has an exponent. In cases like this, we have to check the math just as we did in the above example. First, we solve for both solutions and then we plug both back into the original equation. Any answer that “works,” or gives us a “true” equation, is a valid possible solution.

Takeaways for Square Roots:

 

(1) If there is an actual number shown under a square root sign, then take only the positive root.

(2) If, on the other hand, there are variables and exponents involved, be careful. If you have only exponents and no square root sign, then take both roots. If you have both an exponent and a square root sign, you’ll have to do the math to see, but there’s still a good chance that both the positive and negative roots will be valid.

(3) If you’re not sure whether to include the negative root, try plugging it back into the original to see whether it produces a “true” answer (such as √(-3)2 = 3) or an “invalid” situation (such as √-9, which doesn’t equal any real number).

* The text excerpted above from The Official Guide to the GRE 2nd Edition is copyright ETS. The short excerpts are quoted under fair-use statutes for scholarly or journalistic work; use of these excerpts does not imply endorsement of this article by ETS.

knowledge-vs-skillFor me, the material you need to study for the GRE can be divided into two groups. No, not verbal and math. Knowledge and skills. Differentiating these two groups is important because they are learned in very different ways.

Learning Knowledge

So far, I would bet that most of your study time, from elementary school through college, was devoted to learning information. The skill of remembering facts is something that most of us have practiced quite a bit in the school realm. And sure, some of us are better than others at doing so, but mostly we at least have an idea where to start.

The knowledge, or information and facts, tested on the GRE would include vocabulary words, properties of numbers, mathematical definitions, and mathematical formulas.

I’ve written in the past about lots of unique ways to learn vocabulary, but ultimately I think that the techniques for learning knowledge fit into four categories:

(1)   Drill. This would include writing words and definitions, making and reviewing flashcards, listing out numbers that fit a certain property, and writing and re-writing formulae. All these methods have their place.

(2)   Explain.  It’s generally easier to remember something if you understand it. For that reason, trying to explain a fact is a good way to learn it. This category would include studying with a partner, defining a word using its roots, and proving a mathematical formula.

(3)   Link. Tying new information to information you already know is a good way to remember it. This would include finding a vocabulary word in a TV show or song, linking a word to its antonym, using one math formula to remember another, or building more specific geometry rules from the rules you already know.

(4)   Use. I find that the saying “use it or lose it” is pretty applicable to learning. This category would include using new vocabulary in conversation or emails, writing sentences with vocab words, and doing practice math exercises.

There’s probably not that much new here so far. But that’s the key: we’re only halfway done.

That’s not enough!

Many students feel frustrated with the GRE because they feel like they know and understand the underlying math or vocabulary, but still aren’t seeing their scores improve as much as they would like. If you’re in the boat, don’t panic!

If you feel like you understand the underlying material but aren’t seeing your score improve as quickly as you’d like, or even at all, it might be that you’ve only worked on the knowledge and haven’t yet worked on the skills. Or, you’ve worked on the skills, but in the wrong way.

It’s not that your time has been wasted – you need that underlying knowledge to succeed on the test. But on its own, it won’t be enough.

So, what are the skills we need, and how do we learn them?

Learning Skills

Skills are learned differently than knowledge. You didn’t make flashcards to learn to play the piano. You didn’t learn to ice skate by writing the names of ice skating moves over and over in a book.

If you want to know the capital of Maine, and you don’t know, there’s no way to figure it out on your own. You have to look it up, and once you look it up, at least for that moment, you know the answer. That’s knowledge, and it’s often learned in that way: don’t know, give up, look at the answer, know, repeat.

Skills don’t work like that. If you show up at a piano lesson, and the teacher asks you to play a song for the first time, you’ll probably try it and make a lot of mistakes. What then? Well, what you don’t do is ask the teacher to play it for you and then say, “Oh yeah, that sounds right – I got it now!” and then move on without ever looking at it again.

I hope that piano lesson scenario sounds crazy to you. And similarly, I hope you can see why doing a math problem, getting it wrong, reading the answer, understanding it, and moving on is equally crazy. Being able to solve a math problem requires some underlying knowledge, but ultimately, it’s a skill, like playing the piano or running a marathon.

Because of that, you have to practice it like a skill. The skills on the GRE would include things such as solving a multiple choice geometry problem, solving a quantitative comparison question, guessing on a quantitative comparison question, solving a sentence completion question, staying calm during a timed exam, and deciding when to move on from a question.

How do you practice skills? Generally, I would employ a 4-part process:

(1)   Try it timed. Just like the piano student in the above example, you should give the problem a try from the beginning. This lets you practice your own set of testing skills: assessing the problem, timing, guessing, and moving on.

(2)   Re-work untimed.  What do you think that piano teacher would have the student do next? Most likely, go back and try to work on the parts of the song that were hard. Similarly, you should go back and try to work on the problem on your own. See if you can get unstuck and get yourself to the right answer.

At this stage, the piano teacher might also interject some tips or reminders. You can do the same for yourself by using resources such as your strategy guides, other problems you’ve done, or definitions you don’t remember if you need them.

(3)   Use the answers (sparingly). If that piano student is really stuck, the teacher might show him or her what to do – but only until the student gets unstuck. You should do the same with your answers. If you need to, start reading the answer, but only until you come across something you did wrong and didn’t recognize. Then, stop, and go back to working on your own as far as you can. Repeat this process as needed.

(4)   Record a take-away. When you’re playing the piano, you create muscle memory that lets you reuse what you’ve learned in other contexts later. Recording a take-away has a similar effect. This is the chance to look back at the problem and say, “Hmm, what could I have seen/known from the beginning that would have let me get this problem right the first time?” Then, write down a sentence that takes the form of, “When I see _________ in a problem, ____________________,” where the first blank tells you what trigger to look for, and the second blank tells you what to remember, what rule to apply, what to think about, or what you can expect to happen in the answer.

It’s not that most of us have never learned a skill – all of us have. Even if you haven’t played a sport or a musical instrument, you probably know how to drive, use a computer, and do all kinds of unique things at your job. It’s just that we don’t often apply those skill-learning skills to academic tasks – but for the GRE, they will make a big difference.

test-anxietyI want to preface this article by saying that I’m not a psychiatrist, psychologist, therapist, or expert in test anxiety. I’m simply a tutor who has helped students prepare for standardized tests for the past 15 years.

Studying math is important. Studying verbal is important. Studying the test itself is important. But what about when you’ve done all that, and you can’t overcome the anxiety that holds you back from achieving your dream score? What about that panic that makes your brain fuzzy? What about the flustered feeling that stops you from showing what you know?

Everyone feels some pressure during the test, but test anxiety may be having a negative impact on your score if any of the following are consistently true for you:

  • Your real exam scores are significantly lower than your practice test scores.
  • When the timer is set, you feel unable to answer a question that is easy for you once the timer is off.
  • You find yourself unable to move forward through a real or practice test and resort to panicked guessing.
  • Many of your practice tests remain uncompleted because you are overwhelmed by pressure during the test and find that you need a break.

Everyone experiences anxiety in different ways. But the good news is that there are many strategies you can use to mitigate test anxiety and improve both your comfort level and your store. Here are a few strategies you can try.

1.     Work small to big. Many times students do their homework one question at a time, and then take a practice test. That’s like going from finding your golf grip to competing in a tournament. No wonder it makes you anxious!

Instead, think about increasing the amount of timed questions in small intervals. Start by timing one question at a time, then two, then four, and slowly increase the amount until the test doesn’t feel so daunting.

2.     Focus on calmness, not scores. We know that the GRE score is important, but math and verbal aren’t the only areas you need to study. So is staying calm, keeping your mental focus, and honing the ability to work quickly and effectively in mental “crisis” mode instead of hurried and frazzled in “panic” mode.

How can you practice such a thing? Try practicing a full test (or a smaller problem set) with your only goals as staying calm and staying on time. Those are both things that need active practicing, and it can help to experience the exam while calm, when the focus is off the score. You might even find that your score holds its own… or goes up!

3.     Mix topics slowly. If all your studying has been one topic at a time, it can be overwhelming to take a real exam. Not only are many topics mixed, but also it can be the first time you’ve had to actively identify what is being tested in the question.

You can help remove anxiety caused in this way, and also increase your score significantly, by mixing topics slowly. When you study one topic, force yourself to identity how you can tell what topic is being tested just by looking at the question. Once you’ve done that with two topics, try mixing them together. Then add one more topic at a time.

4.     Make a plan, take a break. This seems simple and straightforward, but it really can help. I know that every second is precious on the GRE, and many of us feel time pressure during the exam. Sometimes, that time pressure can put us in panic mode, where we feel like any second we aren’t “doing something” is a second wasted, so we rush into working without a plan.

Generally, making a plan is worthwhile. Taking a moment to figure out why type of question you’re doing and how to attack it will generally be faster than jumping right into solving, which may send you down the wrong path and not allow you to pick up key signals.

In addition, you may find that a short break, just 10 or 20 seconds where you close your eyes and take a deep breath, helps you to refocus and ends up saving you time.

5.     Study for question recognition. This suggestion applies whether you have test anxiety or not, because it’s the clearest and most direct way to improve your score (after learning the underlying basics). When you get a question wrong, you don’t just want to learn what the right answer is. At least as importantly, and I would argue more importantly, you want to answer the question, “What did I need to recognize or know to get this question right?”

Asking this question forces you to learn from each question in a way that can be applied to future questions. It pushes you to recognize patterns and to learn how to notice what’s being tested in a question, which can help you make a plan, use what you’ve studied, and avoid common traps.

6.     Meet with a test anxiety specialist. Yes, there is such a thing as a test anxiety specialist. And while most students won’t need one, if you find that your mastery of the material can’t shine because you are paralyzed in the face of the real exam, working with a specialist may help you get through the roadblock that’s holding you back.

Studying for and taking a big exam such as the GRE is an inherently stressful process. But when that stress gets in the way of your success, try taking active steps make it more manageable. After all, you want to show off all the stuff you’ve learned as best you can!

Zoom In, Zoom Out

Jane Cassie —  September 16, 2013 — Leave a comment

GRE zoomI’m a terrible photographer because I don’t zoom in a reasonable manner. I try to zoom in on things that really can’t be appreciated without context. I try to zoom out and capture a whole panorama when the scene is too busy for any viewer to appreciate it. I suppose I could practice, but instead I’ve just stopped taking pictures and let other people do it for me.

But when it comes to the quantitative section of the GRE, I know exactly how to zoom, and I try to make sure my students know how to do the same. If you took all your photos with the camera on its factory setting, they would all be okay, but none of them would be really great. You want to get closer and more into the minutiae sometimes, and take a broader view other times, skipping all the details. The same is true when studying for (and taking) the GRE.

Zooming Out

Consider the following problem:

If x is the median of all the even multiples of 7 from 15 to 100, and y is the mean of all the even multiples of 7 from 16 to 104, what is the value of xy?

With your mental math camera on the regular setting, your approach might sound something like, “Okay, I know how to find the median. I’ll list out all the terms, and choose the middle one. After I’ve done that, I can find the mean by average out the first and last terms, because this is an evenly spaced set. Once I find both those numbers, I’ll subtract to find the difference.”

This approach is okay, and it will get you to the right answer. But zooming out a little allows you to look at the problem collectively, as a whole, and think something like, “Hey, both these sets of numbers are the same, since they both start at 21 and end at 98. And in an evenly spaced set, the mean and median are the same. So their difference is zero.”

Zooming out lets you pick up patterns in the exam and take advantage of the fact that you’ve studied them and notice them. It allows you to notice trends in the exam, which helps you know quickly what issues to consider and can also help you make an educated guess. Let’s take a look at another sample problem.

What is the average (arithmetic mean) of all the multiples of ten from 10 to 290 inclusive?

  1. 140
  2. 145
  3. 150
  4. 190
  5. 200

On a regular setting, I’m looking at this and thinking, “Great, I know how to find the mean. I’ll list all the multiples of ten, add them up, and divide by the number of terms.” By zooming out, I can realize, “Hey, I know the GRE doesn’t want me to do that. This test rewards me for reasoning; is there a faster way? Yeah, this is an evenly spaced set of terms, so the middle one is the mean. And I can find that by just taking the mean of 10 and 290.

Making a Plan

The purpose of zooming out (or zooming in, as we’ll see in a second) is to make a plan. Each question should cause you to clarify what information you’re being given (“What are they telling me?”) and what you’re being asked to find from it (“What are they asking me?). Then, make a plan. Continue Reading…