Do you run on autopilot?

Quick brainteaser for you: If 3 bunnies can eat 3 carrots in one hour, how many carrots can 1.5 bunnies eat in one hour?

Really take a moment to think about it.

Do you have an answer?

Ok. If you answered, or even were tempted to answer, “1.5 carrots,” then I’m glad that you’re reading this article! The thing is, while the bunnies are eating carrots at the rate of one per hour, it doesn’t follow that 1.5 bunnies will eat 1.5 carrots. 1.5 bunnies will only eat one carrot, because 1.5 bunnies is really just 1 bunny. (That half a bunny isn’t feeling very well and doesn’t want any carrots.)

I don’t mean to say that this is a GMAT-style problem, but if you found yourself answering “1.5 carrots,” then you are prone to operating on autopilot. Students who operate on autopilot will often fall into traps, and they may become overwhelmed by questions that don’t fall clearly into easily recognized patterns. And let me tell you – you will likely see many problems on the GMAT that don’t fall into common patterns!

It’s imperative that you understand the role that active thinking plays in GMAT success. Sometimes I tell my students, “The GMAT is a math test the way that hockey is an ice skating contest.” They usually look at me like I’m crazy until I explain, so let me explain. Of course all hockey players can skate. They can skate backwards, forwards, sideways…. They can stop on a dime and accelerate just as easily. But they’re playing hockey while they’re doing all of that, right? They watch the puck, they shoot the puck, they defend their goal and attack the other goal. I’m not a hockey expert, but it seems to me that skating skills aren’t nearly enough. Hockey requires that you skate expertly, and yet it isn’t exactly an ice skating contest.

And that’s my entire point. The GMAT isn’t a math test, exactly. It assumes you can do math. It assumes you can solve for x in your sleep. It assumes that you can tick off the major exponent rules on your fingers and reverse-FOIL without taking a breath. But… then what? It’s not enough that you can do the math – you also have to do the thinking. A lot of students just work automatically through GMAT problems. But that’s not enough.

The GMAT often gives you questions that test your ability to think creatively. For example, consider this problem. (It’s based off of a real Official Guide question.)

A café sells cupcakes for $1.00 each and donuts for $1.30 each. If a customer buys some cupcakes and some donuts for a total of 13.10, how many cupcakes did the customer buy?

A)    4

B)     5

C)    6

D)    7

E)     8

Take a moment to see if you can solve the problem.

Really, try!

Did you give it a try? Ok.

A lot of students would begin working here on autopilot. Did you? The “automatic” response would be to write something like 1C + 1.1D = 13.10. Sure, there’s nothing particularly wrong with that. It’s a true representation of the mathematical relationship here. Unfortunately, it doesn’t help us solve the problem.

No one can actually use the equation C + 1.1D = 13.10 to solve for C or D. There are infinite solutions to that problem, if it’s looked at solely algebraically. You would need either another equation or the value of D in order to find C. Then how can this be solved?

The question makes it clear that there is only one solution, so there must be only one way that  $1 cupcakes and $1.30 donuts can add up to $13.10. The trick is to focus on the .10 part from the 13.10. How can we ensure that the sum ends in 10 cents? 

The cupcakes cost exactly $1 each, so it isn’t the cupcakes’ fault that the price includes ten cents. So, it must be that the price of the donuts ends in 10 cents. One donut would cost 1.30, two would cost 2.60, three would cost 3.90, four would cost 5.20, six would cost 6.50, seven would cost 7.80, and eight would cost 9.10. So it must be that the person bough eight donuts, so that the price would end in 10 cents. It’s the only way that it would end in 10 cents (knowing that the sum is only 13.10). Then we would just need enough cupcakes to bring the total to 13.10, so we’d need four cupcakes. The customer must have bought 4 cupcakes.

As you can see, this problem depended as much on your logic skills and focus as it did on your math skills. And that’s exactly my point. By all means, study your math skills. They’re foundational. But as you work through each problem, stay engaged and focused. Thinking – active, creative, and purposeful thinking – is just as necessary for success on the GMAT as strong math skills are. All the algebra drills in the world won’t tell you when not to use algebra, and no perfectly constructed proportion can explain why 1.5 bunnies can’t eat 1.5 carrots.

So, try this tonight. Try a practice set of quant questions, preferably from the Official Guide to GMAT review, with the goal of staying fully engaged. Begin each problem by keeping your hands behind your back for ten seconds while you think through the problem fully and make sure that you’re not operating on autopilot.

I never would have thought of that.....

Have you ever found yourself saying, when you heard or read the explanation to a problem, “I never would have thought of that?” Ultimately, this will do nothing but undermine your success.

Let’s unpack what it means when students say this self-defeating mantra, and why it’s so important to stop saying it!

Assumption #1: Because I didn’t come up with it myself, I never will be able to do it.

A lot of the time, students expect themselves to be able to invent solving methodologies themselves. “I never would have thought of that,” they sigh, and what they mean is: “What is wrong with me that I didn’t think of it myself? The fact that I didn’t think of it myself is all the proof I need that this test is beyond me.”

I was talking about this phenomenon with a friend of mine who also tutors for the GMAT, and he just laughed and said, “I never would have thought of the Pythagorean Theorem on my own! I don’t know why students often seem unhappy to learn from the past.”

Think about it though: Nearly everything you know how to do is something that someone taught you to do. Your parents taught you how to brush your teeth and tie your shoes. Your teachers taught you science and foreign languages and all sorts of other topics. From books you may have learned to cook or trade on the stock market, and friends may have taught you fashion tips or how to play poker. You didn’t invent any of these skills from scratch. You learned these things. That’s what humans do – we learn from earlier generations. (That’s why we have such a lengthy maturation process, compared to other species.)

Don’t beat yourself up because you don’t invent your own methods for solving data sufficiency or critical reasoning problems. If a teacher or a book shows you a new method, focus on how much it will help you next time, instead of focusing on the fact that you didn’t know it before.

Assumption #2: Other people can do that because they’re inherently better at the test than I am.

Sometimes when a student says, “I never would have thought of that,” they sort of smile as if that’s a compliment to me. In this case, “I never would have thought of that” sort of implies that the only reason that I, Laura, thought of it is that I have some sort of intrinsic ability that the student lacks. But that’s just not the case!

I wasn’t born on this Earth knowing how to do the GMAT. In fact, when I was younger, I got pretty bad grades in math. I just hated it! I learned how to do math as an adult and found that when I had the right mindset, I could succeed at it in a way that I had never been able to when I was a teenager.

The reason that I often have an intuitive or quick solution to tough problems is that I have learned a lot about math and verbal content. I’ve also learned how to approach word problems when I get stuck, and how to take methodical steps through strange and confusing data sufficiency problems. I’ve learned when it’s worth it to spend an extra moment or two checking my work, and I’ve learned when it’s time to cut and run. You can learn these things too.

In other words: people who are good at the GMAT don’t necessarily have some sort of inborn and unattainable talent that other people don’t have. Skill depends on practice and honest reflective assessment of your performance. Knowledge comes from study. And scores come from skill and knowledge.

Don’t stand under a flag that says, “I’m not one of those people who are good at the GMAT.” There’s no such thing as “those people,” and you can be whoever you want to be.

Assumption #3: That isn’t the way I think, so I’m going to stick with the method I’m used to.

A while back I was working with a tutoring student on a problem that required her to multiply 54 by 17 and then divide by 27. She diligently multiplied out 54 x 17, and then set up the long division to divide the result by 27. I asked her to wait for a moment, and re-wrote the problem as a fraction:

(54)(17)
    27

I asked her if she saw a shortcut, and she said that she didn’t. I suggested that something might reduce, and the second I said the word “reduce” the student immediately saw that 54 could cancel with the 27:

(54)2(17)    =    (2)(17) =     34
    27                      1  

She looked at what she had done, how quickly she had solved the problem with the help of that shortcut, and sighed and said, “I never would have thought of that.” And by that she meant: “I never will think of that in the future. That isn’t the way my mind works. So there’s no point in trying to learn.”

She absolutely knew how reduce. Her arithmetic skills were very sharp! However, she didn’t look for opportunities to reduce. I suggested to her that she make an effort in the future to remember to look for opportunities to use her excellent skills with reducing in fractions, so that she could perform her calculations more quickly and easily.

At first, though, she really didn’t believe that she ever would recognize an opportunity to reduce. “I’ve been doing it the long way forever,” she said. “It’s just how I think.” Her argument was that since she was never going to learn how to do it the faster way, she’d have to just stick with the way she was comfortable with – even though it hadn’t been getting her the scores she wanted and even though it was actually harder and slower than the fast way. I finally helped her see that saying “it’s just how I think” was really just an excuse to not try something new.

We finally decided that the student should write “reduce!” at the top of each page of her scrap paper when she worked on quant. She made it her goal of the week to find opportunities to reduce. It wasn’t easy for her at first, but after a while it got to be more natural. These days she’s reducing like a pro and her pacing has improved a great deal in quant. I’m sure she can barely remember the day that she said that she’d never learn to reduce.

You’re not so set in your ways that you can’t learn something new. If you ever catch yourself thinking, “I would never have thought of that, so that’s not the right method for me,” ask yourself: Is this method really too hard for you? Or have you decided that good enough is good enough for you? While it’s true that some methods are better than others for some people, don’t rule out a method completely just because it isn’t your initial instinct.

Assumption #4: If it doesn’t come easily to me, then it means that there is something wrong with me and I’ll never get better at this.

In this case, “I never would have thought of that” means something a lot closer to “I give up.” I’ve seen students say “I never would have thought of that” while they lean back sadly in the chair and sigh.

I know that studying for the GMAT can be exhausting and sometimes even a little depressing. But nothing worth having is easy, and the work you put into the GMAT absolutely will be worth it down the road. It doesn’t come easy to most people. The difference is that some people keep working through it until it gets easier.

There’s nothing wrong with you if you find the GMAT tough. Don’t give up just because it doesn’t come to you quickly. A friend of mine used to say, “The only difference between success and failure is one more try,” and while I think that’s almost intolerably cheesy, I also think it’s true.

So:
Next time you find yourself saying “I never would have thought of that,” stop yourself and say, “But next time, I WILL.”  

Thinking Carefully Through Data Sufficiency

 I *was* going to write a post on zero, but I wanted to write this one first. Hey, it's my blog. :)

A student recently came to me to review a question that was a lot like this:

Does 4x + 7y = 19?

(1) 8x/ 7-y = 6

(2)  x= 3

Now, whether you find this question easy, hard, or somewhere in between, stay with me while I explain how and why the student got the question wrong, and how he could have gotten it right, because I think that his mistakes were ones students frequently make in DS.

I asked him to explain his thoughts on the problem. He said,

"Well, first I see that there is an x and a y, so I know I need to know more about x and y, so see, I wrote down, x = ? and y = ?. And then looking at (1), I simplified it and got 4x + 7y =21, so that was wrong. Then, looking at (2), if you know x, then you can do it, so it's sufficient. That's why I got B."

I asked the student, "You just said that with statement 2, you can 'do it.' What can you 'do'?" 

The student said, "I can solve plug in x."

"And that makes statement two sufficient?"

"Yes."

I asked: "Sufficient to do what, exactly?" 

The student paused, started to say something, then laughed, and said, "I get it. Wow, I really lost the thread, huh?"

We still had to talk about it some more, but I was really glad that he got what went wrong in the question -- that he had lost the thread. Do you get it? 

Let's go back to the beginning. The question itself was,

"Does 4x + 7y = 19?"

The student thought: "I need to know more about x and y."

The better approach would have been to think: "This is a yes or no style question. I need to know if it's true or false that 4x + 7y =19. If I'm given x and y each separately - great. But I might not need to know their individual values, just whether 4x + 7y = 19 or not.

Statement (1):  8x / 7-y = 6

The student had a good start to this one. He multiplied both sides by 7-y, to get 8x = 6(7-y).

Distributing, he got 8x = 42 - 6y.

He added the 6y over and got 8x + 6y = 42.

He divided both sides by 2, and got 4x + 3y = 21.

So far? Faultless. But his next step is where he went wrong, and it was all in how he explained the situation to himself. He looked at the simplified equation, thought, "that's wrong," and marked the statement insufficient.

You may have heard this before, but it bears repeating, and in bold letters: The statements are never, ever, ever wrong. They just might be insufficient to answer the question. It's imprecise, and therefore not a good idea, to say that the statements are "right" or "wrong" when you mean "sufficient" or "insufficient."

Here's how the student could have been accurate in his assessment of the statement. He could have done all that good algebra he did, and then he could have asked himself the original question again and attempted to answer it. His thoughts might have sounded like this:

"Does 4x + 3y =19? No, it doesn't. I just found out that it equals 21. That means I have enough information to answer the question with a resounding confident NO. I can answer this question, so it's sufficient."

Let's look at statement (2).

(2)  x= 3

The student thought: "I have x, so I can do it." 

As I mentioned above, this was imprecise thinking. What "it" could he "do"? He wasn't even really sure! It's so easy to confuse yourself when you are not clear in your thinking. He wouldn't have gotten lost if he had thought, "The question asked, 'Does 4x + 7y = 19?' If I know x, can I figure out if that equation is true or false? I really can't, because it depends on y. Statement 2 is insufficient." 

(So, in case it wasn't clear here, the answer was choice A, statement one only is sufficient.)

So there are two morals of the story today:

Moral #1: The statements are never wrong.

Wrong the statements never are! Right they must be. This means they aren't wrong.... hey, am I making the point clear and memorable? :)

Moral #2: Always ask yourself the original question and attempt to answer yourself in Data Sufficiency. This way you'll never "lose the thread" and confuse yourself into the wrong answer.

Two incidental notes:

*I asked the student if I could talk about this on my blog, and he said sure. I wouldn't just write this without asking first.

*Incidentally, that question is one I just made up for the purposes of this blog entry, but it's based off of dozens of questions I've seen just like it in Official Guide and real GMAT questions over the years.

On Rates, Speeds, and Out of Date Test Prep Materials

Often, when I begin working with a new tutoring student who has been studying for a while, he or she will sigh in our first meeting: "It's those awful rate and speed problems that really get me down." I often reply, as I did in my tweet today:

"Oh, something like, 'If A heads towards B at x mph, and B heads towards A at y mph, when will they meet?' "

The student usually says, "exactly!"

And I say: "The answer is: 'you probably don't care.' "

Let me explain.

The GMAT has changed over the years. The content that you see in the Official Guide is not always reflective of what is currently on the exam. Many of the questions in official guide 12 were also in official guide 11 and official guide 10 and official guide 9...... How reflective do you think that question really is of what the test really is like?

Further, many test prep companies write their practice problems by mimicking particular problems from the Official Guide. Then, those companies will often write more problems by mimicking the problems they already wrote! Also, if a particular question is retired from the Official Guide, do you really think that the test prep company will dig through their materials and find the questions that they based off of that particular question, and retire those? No way! They paid someone to write that question and they're not going to waste that resource. So you go to a class at a major test prep company, and get taught by someone who might never have taken the real GMAT (some test prep companies allow their teachers to score qualify on practice tests instead of real ones), and you work your way diligently through practice problems that are clones of clones of clones of questions that were out of date four or five years ago. Your teacher, who might never have seen a real GMAT, might have score qualified 4 or 5 years ago, or might just be forced to spout company rhetoric against his/her own better judgment, will probably not set you straight.

One of the most obvious examples of this is the preponderance of awful rate and speed questions in GMAT practice problems. Could you potentially see one actual real rate/speed word problems, with cards driving around and crashing into each other or whatever? Yeah, maybe one. Maybe two on a freakish day. But if you took the GMAT four or five times, I bet that you'd see just one or two actual rate and speed questions on all the tests put together.

So, if you can do rate and speed questions, great. You might get one, and it's always worth it to tag every base. But if rate and speed questions are causing you untold misery, you can heave a sigh of relief.... and go open a number properties book and start studying that, because that is what you'll be tested on.

Further, if you are having rate and speed issues, think about why you're having rate and speed issues. Are you having trouble with word problems in general? Are you having trouble with algebra? Do you have trouble visualizing things, or organizing lots of details? Those are things to be concerned about. Take a step back and look at the root causes, not the surface expression of the issues.

The only reason I can see to learn how to do them is that they're not that bad! I'll write a post some time on how to do them. Until then, I was serious about that number properties thing. Get to it!

How to write good flashcards

A lot of people use flashcards to study for the GMAT, and I am 100% behind this. I think it's a great idea! But it seems that some people don't use their flashcards very well. Let's talk a little bit about good flashcards versus bad ones.

Essentially, good flashcards should be:

• specific
• situational
• concise
• necessary 
• question and answer

A tutoring student of mine showed me a flashcard that looked something like this:

The definition of the word "integer" is any number that is a whole number with no fraction or decimal parts. Integers include positives, negatives, and zero. An example of an integer is 7. When an integer is added to, subtracted from, or multiplied by another integer, the result will always be another integer. When one integer is divided by an integer, the result may or may not be another integer.

The back was blank.

It wasn't her fault; no one had ever taught her how to make a good flashcard! So we spent some time going over why it wasn't a helpful study tool, and how it could be improved.

Basically, this flashcard met only one of my criteria -- it was specific. However...

1. It wasn't situational. How does this long paragraph of words connect to anything you'd actually see on the test? It doesn't look like anything you've seen in a GMAT question at all, does it?
2. It wasn't concise. I mean... just look at it! A person can't just expect to memorize an entire paragraph.
3. It wasn't completely necessary. My student had been studying for months and had learned the definition of the term "integer" completely -- she had no problems with that concept. The whole first half of the flashcard was a waste of her time.
4. It wasn't question and answer. If you expect to really learn this material, you will have to interact with it. Simply looking at a piece of information will not have the impact of asking yourself a question and answering it.


 I suggested this flashcard (below) as a replacement:

If x and y are integers, which of the following is not necessarily an integer?

A) x + y

B) x - y

C) xy

D) x/y

On the back it said:

When you add, subtract, or multiply with integers, you always get more integers. If you divide with integers, you may or may not get another integer. Therefore, x/y doesn't have to be an integer.

I gave my student the instruction that when this flashcard comes up in her studies, she wasn't supposed to just look at it and say, "yeah, it's D" and keep moving. She was supposed to actually say something like "it's D because if you divide with integers you might not get an integer, but with adding, subtracting, and multiplying, you do." And say it OUT LOUD. Before she looked at the back.

I will have to write some more posts on this topic, but I hope it's enough to get you started.