Matt L.'s Blog at WyzAnt.com

Why Test Prep Books Aren't Worth Your Money: The Case of Princeton Review

posted by WyzAnt tutor: Matt L.

All the major test prep books for the SAT, ACT, and GRE -- published by companies like Kaplan, Princeton Review, Barron's, and Manhattan Test Prep -- are poorly written, conceptually deficient, and, worst of all, riddled with serious errors. Students can't be expected to learn from books that aren't even right! And I don't mean the books are riddled simply with typos, which unfortunately is also true, because they are so poorly edited; I mean they really are riddled with serious conceptual errors.

Here's a simple example from the very beginning -- the diagnostic test, of all things! -- of Princeton Review's "1,014 GRE Practice Questions." The problem is on page 24, and the answer key and explanation is on page 38. Not only is their answer wrong; what's worse, their *explanation* is wrong, too! I'll set off the problem by dashes (----) and then add more commentary after.

NOTE: The question is a classic GRE "quantitative comparison," so it's hard to represent in HTML. The goal is this: for the two quantities, A and B, below, you must select which of the following is true: (A) quantity A is larger than quantity B, (B) quantity B is larger than quantity A, (C) the two quantities are equal, or (D) the relationship between the two quantities cannot be determined.

----

x and y are positive numbers

Quantity A: sqrt(x) - sqrt(y)

Quantity B: sqrt[x-2*sqrt(x*y)+y]

Princeton Review's Explanation: "The answer is (C): the two quantities are equal. Quantity B contains a common quadratic pattern. Factor the right-hand side sqrt[x-2*sqrt(x*y)+y]=sqrt[(sqrt(x)-sqrt(y))^2]=sqrt(x)-sqrt(y). Both quantities are equal, so the answer is choice (C)."

----

Princeton is right in saying that

x-2*sqrt(x*y)+y=(sqrt(x)-sqrt(y))^2

but they are absolutely wrong in saying that

sqrt[(sqrt(x)-sqrt(y))^2]=sqrt(x)-sqrt(y)

The worst part about this error isn't even that it is serious, but rather that it is commonly tested! It's not just that the Princeton Review writers made a conceptual mistake: they made the very sort of mistake that standardized tests like the GRE and GMAT are *designed* to test! Do yourself a favor -- don't waste your money on test prep books written by people who don't even know how to avoid the most commonly tested concepts. Hire an expert tutor instead!

Why Test Prep Books Aren't Worth Your Money: The Case of Manhattan GRE

posted by WyzAnt tutor: Matt L.

Here's a simple example from the Introduction (page 23) to Manhattan's Strategy Guides for the Revised GRE. This passage appears in all eight of Manhattan's strategy guides, so it somehow went unnoticed after at least eight rounds of editing by allegedly "expert" readers and test-takers. See if you can spot the error!

----

"If ab=|a|x|b| which of the following must be true?

I. a=b

II. a>0 and b>0

III. ab>0

A. II only

B. III only

C. I and III only

D. II and III only

E. I, II, and III

Solution: If ab=|a|x|b|, then we know ab is positive, since the right hand side of the equation must be positive. If ab is positive, however, that doesn't necessarily mean that a and b are each positive; it simply means that they have the same sign.

I. It is not true that a must equal b. For instance, a could be 2 and b could be 3.

II. It is not true that a and b must each be positive. For instance, a could be -3 and b could be -4.

III. True. Since |a|x|b| must be positive, ab must be positive as well.

The answer is B (III only)."

----

In fact, none of these statements must be true, so the question, as written, doesn't even have an answer! (You can easily see that III need not be true if you choose a or b equal to zero.) The serious conceptual error is to assume that "the right hand side of the equation must be positive." See if you can't add something to the question to make the correct answer be B -- or, alternatively, try tweaking the options labeled by roman numerals so that one of them becomes correct.

This is just one particularly glaring example of an unfortunate trend. Don't waste your money on prep books that are poorly edited and often seriously confused. Hire an expert tutor, instead!

Resources for Improving Your Vocabulary

posted by WyzAnt tutor: Matt L.

The most lasting way to improve your vocabulary is to learn new words (1) in context (by looking up unknown words when you read and keeping a journal of their definitions) and (2) in thematic groups -- NOT by memorizing huge lists of unrelated words. These are some of the resources I use with my students; feel free to comment to add your own favorite vocabulary book!

BOOKS

-- English Words from Latin and Greek Elements

An excellent etymological resource that helps students learn how to recognize Latin and Greek roots in modern English words and use them to predict the meaning of a word. Useful for students of all levels, from high school to college.

-- Roget's Thesaurus of Words for Intellectuals

This is the best book of advanced thematic word lists I know; I assign it to all my GRE students. The vocabulary is a bit too advanced for most high school students prepping for the SAT, but it's still an excellent resource for anyone -- in high school or not -- seriously dedicated to improving their vocabulary.

-- Merriam Webster's Vocabulary Builder

Another excellent resources, for both high school and college students. Highly recommended alongside the book above.

-- Word Smart

FREE ONLINE RESOURCES

-- Word Dynamo: http://dynamo.dictionary.com/

-- Tyrannosaurus Prep: http://www.tyrannosaurusprep.com/

Sample Worksheet: Algebraic Shortcuts for the SAT, GRE, GMAT

posted by WyzAnt tutor: Matt L.

Most students taking the SAT, GRE, or GMAT know their algebra fairly well, but many find they can't complete all the problems in the allowed time. Why? It's NOT because those students are just naturally slow: it's because they're doing more work than they need to! It's not their speed but their very approach --- the very way they conceive of the process of problem-solving --- that's flawed. To ace the math sections of standardized tests, you have to learn how to attack problems in new ways so that you get the right answers by doing as little work as possible! (Part of the reason so many students don’t already know how to do this is that it’s not taught well throughout middle and high school math classes. Learning how to think quickly and deeply often requires UNLEARNING habits your math teachers instilled in you in school!)

To see if you’re up to par, try the following problems, which test your ability to make deep algebraic connections that will save you time. If your algebraic skills are what they really should be, you should be able to do all the problems in TWENTY SECONDS OR LESS! If you can’t, send me an email and start working with me today!

EXERCISE SET: ALGEBRAIC SHORTCUTS

Suppose 2x+7=19. Find the value of each of the following expressions WITHOUT solving for x!

a) 4x+14
b) 6x+21
c) 20x+70
d) 2x+6
e) 2x+3
f) 2x
g) 2x-7
h) 2x-20
i) x+3.5
j) 0.5x+1.75
k) -2x-7
l) 10x+7
m) 8x+10

Sample Worksheet: Critical Points and Local Extrema -- Functions, Precalculus, Calculus

posted by WyzAnt tutor: Matt L.

Many first-year calculus students fall into a common trap: they tend to make bad assumptions about how functions behave. In particular, they tend to think all functions are "nice," in the sense of easy to draw and understand -- because most of the pictures their teachers draw in school to illustrate examples tend to be of nice, familiar functions they are comfortable working with, like polynomials. But functions, in general, are extremely unwieldy, and to truly master differential calculus, you have to learn to be on guard against making simplifying assumptions: what we often imagine to be the case turns out to be false on closer inspection.

Here's a simple example in the context of critical points and local extrema. A classic application of derivatives involves finding the local minima and maxima of a function. You may recall that the first step to finding these values is to find the function's critical points. Here's the common trap: most students mistakenly confuse "critical points of a function" with "points where the derivative of the function is 0." But an arbitrary function -- even an arbitrary continuous function -- need not be differentiable at every point in its domain! Critical points are NOT ONLY points where the derivative of a function is zero; they ALSO include points where the derivative is undefined. If you were looking for the local extrema of the absolute value function, for example -- the function f(x)=|x| -- you would not find its local (in fact, global) minimum at the origin if you looked only for points where f'(x) is zero, because there is no such point!

Who cares? Well, it's important to get this fact straight in any situation where you might be tested on the relationship between critical points and local extrema. To see if you're up to speed, check out the following worksheet I created below. If you work with me, you'll learn to solve all these problems and have access to many more specialized materials I've developed over the years, including problem sets and suggestions you won't find anywhere else.

PROBLEM SET: CRITICAL POINTS AND LOCAL EXTREMA

Problem 1. For each of the following claims, first decide whether it is true or false. If a claim is false, give a counterexample; if a claim is true, explain why.

a) If a function f(x) has a local extremum at x = c, then c is a critical point of f.

b) If a function f(x) has a local extremum at x = c, then f is differentiable at c.

c) If a function f(x) has a local extremum at x = c, then f is continuous at c.

d) If f'(c) = 0, then c is a critical point of f.

e) If f'(c) = 0, then f has a local extremum at x=c.

f) If f''(c) = 0, then c is a point of inflection of f.

g) The critical points of a function f(x) are those x values for which f'(x) = 0.

h) If f'(c)=0, then f(c) is either a local maximum or a local minimum.

i) If f''(c) = 0, then f'(c) is not zero.

j) If f''(c) = 0, then f'(c) = 0.

Problem 2. Fill in the blanks to make each sentence true.

a) In order for f to have a local extremum at c, c must be a ___________, but this requirement is only necessary and NOT sufficient.

b) Similarly, in order for f to have a point of inflection at c, c must satisfy ___________, but again this requirement is only necessary and NOT sufficient.

CHALLENGE PROBLEM, FOLLOW-UP: Can you find an example of a differentiable function that has a point of inflection (i.e., changes concavity) at the origin but whose second derivative is not defined there? (Hint: One way you might start is to think about what you might want the FIRST derivative of such a function to be.)

Problem 3. Which is the stronger condition: being a critical point, or being a local extremum? Which is the weaker condition: being a point of inflection, or being a point that satisfies f ''(c) = 0?

Sample Problems: Combinatorics -- SAT, GRE, GMAT, Statistics, Discrete Math

posted by WyzAnt tutor: Matt L.

Of the vast amount of math taught in high school, combinatorics is usually the most baffling for students. In my ten years of teaching, I've never had a student who felt totally confident about counting problems -- I myself didn't feel I really understood them until I went to college! -- and the most typical reaction to them is immediate fear or frustration: students often give up as soon as they see one, before they even attempt a solution. Why? Probably because many high school math teachers don't do a good job of explaining the basic concepts with concrete examples; instead, they often present a bunch of formulas to be memorized, without conveying any intuition about when to use the formulas or where they come from.

But you won't earn a stellar score on the SAT, GRE, or GMAT if you can't master the basics of combinatorics. To see if you're up to speed, take a look at the following challenging problems, all based on the following scenario:

PROBLEM

Luigi's Pizza Shop offers three types of crust, five types of toppings, and four types of cheese. Suppose a pizza at Luigi's consists of a crust and at least one type of cheese; it may have no toppings, one topping, or more than one topping, as well as more than one cheese. How many different types of pizza can be made:

0. with exactly 5 toppings and 4 cheeses? (Hint: I called this #0 because it's very easy!)

1. with exactly 3 toppings and 2 cheeses?

2. with exactly 2 toppings and 3 cheeses?

3. with exactly 2 toppings and 4 cheeses?

4. with exactly 3 toppings?

5. with exactly 4 toppings?

6. with exactly 4 cheeses?

7. with at least two cheeses?

8. with at most four toppings?

9. with at least two cheeses and at most four toppings?

10. in total?

Extra credit: Can you say which of the above problems were easier than others, and explain what made the harder problems more difficult?

---

If you work with me, you'll learn how to attack these problems and countless others like them so you feel absolutely confident your answer is correct. We'll start out by drawing very concrete pictures so you understand exactly what's involved in answering a counting problem and where the general solution comes from.

To get the answers and start learning how to ace that next exam, send me an email and let's schedule a meeting today!

Sample Review: The Algebra of Inequalities -- SAT, GRE, GMAT

posted by WyzAnt tutor: Matt L.

Many of my students preparing for the SAT, GRE, and GMAT have decent algebraic intuition when it comes to EQUATIONS, but most are much weaker when it comes to INEQUALITIES.

On the one hand, this is entirely natural: inequalities capture less information than equations -- they establish merely a relation between two quantities, rather than their equivalence -- so they are inherently trickier to think about. But on the other hand, it's crucial to have a very solid grasp of how inequalities work to do well on the SAT, GRE, and especially the GMAT (which tends to love data sufficiency questions that deal with tricky inequalities).

To test yourself to see how up-to-speed you are, try to decide whether the following statements are true or false. (I have intentionally made the problems very abstract and seemingly confusing to see if you really know what's going on, so DON'T WORRY IF YOU'RE TOTALLY LOST OR INTIMIDATED!)

1. If a+b=c+d and e+f=g+h, then a+b+e+f=c+d+g+h.

2. If a+b=c+d and e+f=g+h, then a+b-e-f=c+d-g-h.

3. If a is less than b and c is less than d, then a+c is less than b+d. (WyzAnt won't let me use the greater and less than symbols on this blog!)

4. If a is less than b and c is less than d, then a-c is less than b-d.

Here's a hint -- one and only one of these statements is false! Do you know which one it is? The amazing thing is that although these problems LOOK scary, they are extremely simple if you know the basic rules for handling equations and inequalities.

To find out which one is false, send me an email & start preparing to ace the SAT, GRE, or GMAT today!

Sample Worksheet: Normal Distributions -- GRE, GMAT, Statistics

posted by WyzAnt tutor: Matt L.

Many of my students preparing for the GRE or GMAT have decent algebraic skills, but most have trouble with statistical reasoning --- for a variety of reasons. Some have never had statistics; others have been away from it for years. In either case, it's crucial to get up to speed on the basics!

To get a sense of how prepared you are for some of the more challenging statistics questions on the GRE and GMAT, check out the following worksheet I've developed. When you work with me, you'll gain exactly the skills you need to ace these and similar problems --- you'll learn to complete this entire sheet in fewer than five minutes, in fact! --- and have access to a wide range of specialized materials I've developed over the years, materials full of strategies and problem sets you won't find in any published prep book. I guarantee you won't find a more helpful or expert tutor, so send me an email today!

PROBLEM SET: NORMAL DISTRIBUTIONS

Exercises: Percentiles & The Empirical Rule

Assume the variable X is normally distributed with mean m and standard deviation s. Fill in the blanks to make the statements true:

1. _____ percent of the data for X lies within one standard deviation of m.

2. _____ percent of the data for X lies between m and m+2s.

3. _____ percent of the data for X lies further than one standard deviation from m.

4. _____ percent of the data for X is less than m and greater than m+s.

5. _____ percent of the data for X lies between m-2s and m-s.

6. _____ percent of the data for X lies within three standard deviations of m.

7. _____ percent of the data for X lies further than two standard deviations from m.

8. _____ percent of the data for X lies above m+3s.

9. _____ percent of the data for X lies below m-s.

10. ____ percent of the data for X lies between m-2s and m+s.

11. If x is at the 2.5th percentile, then x is ____ standard deviations from the mean.

12. If x is at the 99th percentile, then x is more than ____ standard deviations from the mean. (Choose the greatest integer that makes this statement true.)

13. If x is at the 20th percentile, then x is between ____ & ____ standard deviations from the mean. (Choose the tightest bounds.)

14. If x is at the 51st percentile, then x is between ____ & ____ standard deviations from the mean. (Choose the tightest bounds.)

15. If x is at the 38th percentile, then x is between ____ & ____ standard deviations from the mean. (Choose the tightest bounds.)

16. If x is two standard deviations above the mean, then x is at the ______ percentile.

17. If x is one standard deviation below the mean, then x is at the ______ percentile.

18. If x is two standard deviations below the mean, then x is at the ______ percentile.

19. If x is one standard deviation above the mean, then x is at the ______ percentile.

20. If x lies between m-2s and m+3s, then x lies between the ______ and ______ percentiles. (Choose the tightest bounds.)

Problems

These problems are harder than the routine exercises above; you'll have to be more creative to answer them.

1. (Quantitative Comparison) Suppose that the variable X is normally distributed with a mean of 250 and that 70% of the data lies between 240 and 270.

Column A: the standard deviation of the variable X

Column B: 10

2. (Quantitative Comparison) Suppose that the variable X is normally distributed with mean 50 and standard deviation 10.

Column A: the percentage of data in X that lies between 30 and 35

Column B: the percentage of data in X that lies between 35 and 40

HINT: Think about the shape of the normal distribution!

3. (Quantitative Comparison) Suppose that the variable X is normally distributed with mean 50 and standard deviation 10.

Column A: the percentage of data in X that lies between 35 and 40

Column B: 7

HINT: Think about what you learned from Problem 2!

4. (Multiple Answer Multiple Choice) Researchers collected 400 observations of GRE verbal scores and found them to be approximately normally distributed. Suppose Jane scored exactly two standard deviations above the mean, while John, sadly, scored only at the 5th percentile with a verbal score of 350.

Based on this information only, which of the following must be true? (Select all that apply.)

A. Fewer than 380 people scored higher than John but lower than Jane.

B. More than 8 people earned a score higher than Jane’s.

C. Jane scored closer to the mean than John did.

D. More than one person earned the same score as John.

E. Jane scored higher than a 600.

5. Suppose that X and Y both measure IQ, but in distinct populations, and that X and Y are both approximately normally distributed, each with a mean of 100. The standard deviation of X is 15, however, while the standard deviation of Y is 5.

Part I (Quantitative Comparison)

Column A: the probability a person randomly chosen from population X has an IQ between 70 and 115

Column B: the probability a person randomly chosen from population Y has an IQ between 75 and 100

Part II (Quantitative Comparison) A person is randomly chosen from one of the two populations and found to have an IQ of 140.

Column A: the probability that the person was randomly chosen from population X

Column B: the probability that the person was randomly chosen from population Y