## Posts tagged with: strategy

Hello there,

I’m having trouble with time on the SAT because I’m like pretty slow.

Do you have any tips on increasing my speed gradually or something like that.

Thanks for helping me

This is a really tough one—I get asked it a lot and I struggle each time with how to respond. A few things make it tricky: 1) different people struggle with speed for different reasons, and 2) for many people, speed and accuracy are inversely related: the faster you get, the less accurate you get. Frustration and diminishing returns are common results for people who just focus on speed without being thoughtful about it.

I can best address the first reason by saying that speed issues are often really content issues. If you don’t know how to approach a question because you’ve never seen one like it before, you’re obviously going to be slow on that question. The solution there isn’t to rush, it’s to become more familiar with the kinds of questions the SAT throws at you over and over. It’s a standardized test and as such, it follows certain patterns over and over again. Speed generally increases with content mastery and test familiarity.

Assuming that’s NOT your issue, your mission is to find a way to get faster while not sacrificing accuracy. You’ll have to achieve this with a careful analysis of your current approach, followed by some trial-and-error for how to change that approach. A quick example of what I mean that may or may not apply to you:

I hope that helps a bit!

There are three different kinds of SAT math sections, and it’s important to know which kind you’re working on. Lucky for you, it’s super easy to check without even having to flip through the section. Just look to the top of the first page of the section to see how many questions there are.

If the first page tells you there are 20 questions, then they’ll all be multiple choice, and go from easy to hard. By number 16 or 17, you’ll probably be into difficulty 4 or 5 questions.

If the first page tells you there are 18 questions, it’s a grid-in section. Questions 1–8 will be multiple choice and get increasingly difficult—numbers 7 and 8 will probably be difficulty 4 or 5. Questions 9–18 will be grid-in questions; they will start easy and become increasingly difficult again—numbers 16–18 are likely to be difficulty 4 or 5.

Grid-in sections are the important ones to identify—they’re the reason I wrote this post. If you’re doing any strategic skipping, recognizing grid-in sections is of paramount importance. Even if you’re not planning to do any strategic skipping, you should be conscious of how much time you’re spending on hard questions 7 and 8 while easy grid-in questions remain unanswered. Remember, easy questions are more important than hard ones.

The last math section will be section 8, 9, or 10, and will always be 16 multiple choice questions, going from easy to hard once. This section will be 20 minutes instead of 25 (for regular time students). Questions 14–16 will probably be difficulty 4 or 5.

As I’m wont to do once in a while, I’m going to expand on an idea here that might otherwise disappear forever into the murky depths of Tumblrdom. This post is inspired by (and borrows heavily from) this Q&A response. I’m revisiting it here because I think it’s a nice contrast between me and many other tutors and prep book authors. Pro-tip: When you see an opportunity to set yourself apart, take it.

The original question I received went something like this:

If an object travels the same distance at two different rates, can you use [rate 1 * rate 2 * 2] /[rate 1 + rate 2]? I got this from a “cheat sheet” from a friend, but have no idea how he derived this. Can you show me how he got this “shortcut”?

Have you ever heard the phrase “everything but the kitchen sink”? As in, “He put everything but the kitchen sink into his suitcase when he packed for his vacation”? The “cheat sheet” this person refers to is represents what I call kitchen sink SAT prep (Bad Idea #4 from this post). It’s characterized by a bunch of extraneous information that, while true, is probably not useful for the SAT, and is therefore only a distraction from the things you actually need to know.

The formula above is true—I’ll derive it below—but it’s also very unlikely to be useful on test day.

Here’s what I think you should memorize about average speed questions:

That’s a simple formula to remember, and the things that need to be plugged into it are simple to find.

Let’s test it with an example.

While stuck in traffic, Kevin traveled 30 miles at an average speed of 10 miles per hour.  Once traffic cleared up, he traveled the remaining 30 miles to his destination at an average speed of 60 miles per hour. What was Kevin’s average speed for his entire jouney?

To use the average speed formula, we just need total distance traveled and total time spent traveling. The first part’s easy: he traveled 30 + 30 = 60 miles.

As for time, you can probably intuitively calculate each one in your head, but if you’re struggling, just remember that you can divide distance by speed to get travel time. 30 miles / 10 mph = 3, so at a speed of 10 mph, it would take Kevin 3 hours to go the first 30 miles. Same deal for the second part of the journey. 30 miles / 60 mph = 0.5, so at a speed of 60 mph, it would only take him 0.5 hours to go the next 30 miles. Total time: 3.5 hours.

Here’s the thing: not only is that general formula easier to memorize than the more complicated special case one from the “cheat sheet,” but it’s also more versatile! Using my preferred average speed formula works whether the two distances are the same or not. Using the “shortcut” formula above only works when the two distances are the same. Oh, and one more thing: the simple formula is how the more complicated and less useful “shortcut” is derived! That nerdery below the cut.

Let’s use the general average speed formula above (and the example we used to illustrate it) to derive the “shortcut.”

Remember, we’re starting with:

In our example about Kevin, we found the total distance traveled by simply adding up the distances he traveled. In the case where two distances traveled are the same (call them both d), then the general “total distance” numerator is going to be:

We found the two different times above (3 hours and 0.5 hours) by dividing the distances by the rates (which we’ll call r1 and r2). So the “total time” denominator is going to be:

Put it all together:

Which, with a little algebra, simplifies to the “shortcut.” First, fix that ugly mixed fraction. When you divide by a fraction, that’s the same as multiplying by its inverse. So you can rewrite:

From there, you just need to cancel out the d, and you’re there:

Make sure the “shortcut” gives us the same answer we already know is right:

$\frac{2\times10\times60}{10+60}=17.14...$

So…yeah. It works. But this “shortcut” only works on a special case question that’s rarely asked, and it’s much harder to memorize than the general form. If the distances aren’t the same, the shortcut is useless. The general formula still works. Kitchen sink math or PWN math? THE CHOICE IS YOURS.

Since I’m all fired up now, here’s one more example of kitchen sink math vs. my kind of math. The volume of a right prism—one of the few types of 3-D figures you might see on the SAT—is the area of its base times its height. That’s the general rule. Now, one could create a “cheat sheet” with complicated formulas for each kind of prism in that class of solids—I’ve shown a few below.

Right cylinder (base radius r, height h)—actually provided in the beginning of SAT sections:

Right prism with rectangular base (base sides l and w, height h):

Right prism with equilateral triangle base (triangle sides s, height h):

But here’s the thing: if you set about memorizing those, you’re jamming a bunch of redundant formulas into your head. If you’re comfortable with triangles, and you know the general volume rule for a right prism, then you don’t need to memorize the formula for the volume of a right prism with equilateral triangle bases—you basically already know it. All you need to remember about right prism volumes is this:

The issue with the kitchen sink approach, if I’ve not beaten this completely to death yet, is this: if you memorize a bunch of special case formulas, but you don’t have a sense of where they come from, then you’re going to be in trouble on test day if you get thrown a curve ball. Kitchen sink prep means you’re over-preparing by memorizing something that’s very unlikely to be useful—both of my examples, prism volume questions and average speed questions are very rare—AND you’re still not prepared for the full breadth of prism volume or average speed questions you might see. Seems silly to me when you could memorize a simpler fact and apply it more broadly.

I don’t know if you followed the kerfuffle between Jonathan Coulton and Glee a few weeks back. It’s old news now, but I watched it unfold at the time with great interest, and I’ve been thinking about it again the last few days. The incredibly short version: Jonathan Coulton is a fairly popular musician (on the Internet, anyway) who recorded a cover of Sir Mix-a-Lot’s “Baby Got Back” in 2005 (above). Glee did a note-for-note recreation of his cover without crediting him. Then Fox’s lawyers told him he should be thankful for the exposure he didn’t get because nobody credited him.

This is a case of morality and legality not completely overlapping, and that’s all very interesting if you’re into intellectual property law (which I know is very popular among high school students these days) but that’s not where I want to go with this. The reason I bring it up is that Mr. Coulton ended up announcing that rather than pursue recourse through the courts, he’d completely change direction and try to turn this into something positive for him, and for some great charities. And there’s an SAT lesson there: know when you’re beat, and do something about it.

Coulton’s indignation was justified, but he recognized early on that he’s not going to beat an army of Fox’s lawyers, so he shifted tactics. If what you’re doing isn’t working, try something else. This is what I’m talking about when I implore you to be nimble. It’s pretty good advice for life in general, and it’s particularly germane to the SAT, on which many of the most difficult questions are vulnerable to techniques that will allow you to sidestep the math solution, if you let them. Like this one, for example:

1. Yesterday, a group of y friends went to the mall and each purchased p pairs of gym socks. If y > x > 1 and p is a positive multiple of 3, how many fewer pairs of gym socks would they have purchased if x of the members of the group had purchased only a third as many socks as they actually did?
(A)
(B)
(C)
(D)
(E)

If you’re looking for a top score on SAT math, you should be able to solve this with algebra, and you should also be able to solve it by plugging in. Being nimble in this way is how you work around the fact that you’re likely to see at least one problem on test day that thwarts your first attempt to solve it. Being comfortable solving a question like this two ways is also the best way to avoid careless errors—check your work by solving the way that you didn’t solve it the first time. If you get the same answer both ways, you’re almost certainly right. Both solutions below.

Say 10 friends go to the mall (y = 10) and each buy 3 pairs of gym socks (p = 3). So what actually happened yesterday is that the group purchased 10 × 3 = 30 pairs of socks. Now say 2 of the friends (x = 2) purchased a third as many socks as they really did. So 2 friends bought only 1 pair of gym socks each. 8 friends buy 3 pairs: 8 × 3 = 24, and 2 friends buy 1 pair each: 2 × 1 = 2. Total pairs of socks purchased: 24 + 2 = 26, or 4 fewer pairs of socks than were actually purchased. Look to your answer choices, and see which one gives you 4 when you plug in y = 10, p = 3, and x = 2.

(A) Does:
(B) doesn’t:
(C) doesn’t:
(D) doesn’t:
(E) doesn’t:

So (A) is clearly the answer.

##### Now let’s do the algebra

Note that, since you’ve already spent some time working through the problem logically with plug in, the algebra should be a bit more intuitive now than it might have seemed at first. First, create an expression for what was actually purchased. That’s easy: y people purchased p pairs of socks each. Total pairs purchased: yp. Now, figure out how many would be purchased in the alternate scenario where x of the friends purchased a third as many socks as they really did. y – x purchased p pairs of socks, and x purchased p/3 pairs of socks. Total pairs purchased in alternate scenario:

Now just do some subtraction to find out how many fewer pairs of socks would have been purchased:

Unsurprisingly, we arrive at the same answer either way. Again, if you’re shooting for an 800, you really should be able to breeze through this question (and ones like it) both ways.

##### Bonus solution

As you might have deduced from the fact that the correct answer doesn’t contain y at all, y totally doesn’t matter. All that really matters is how many fewer socks the x friends would have purchased. If they were to purchase 1/3 of what they did, they would purchase 2/3 less than they did. Since the x friends purchased xp socks in real life, they would purchase 2/3 of that is . [See also: “Is there a math way?“]

The January SAT marks the beginning of the year’s most frenzied test prep season. Seriously, between now and May, it gets real. Because so many will be ramping up their efforts in the coming weeks, I thought it’d be useful to put together a few thoughts on what not to do.

##### Bad Idea #1: Rapid-fire practice tests

This is one of the biggest mistakes kids make, and it can be a costly one, both in time and in study resources. It’s important to take practice tests in the course of your prep in the same way that it’s important to weigh yourself once in a while if you’re trying to lose weight—You need to see where you stand, but you’re not actually losing weight by weighing yourself. All the important stuff happens between weigh-ins.

If you spend too much time taking tests and not enough time reviewing those tests and learning new techniques and concepts to help you avoid making the same mistakes again, then you’re spinning your wheels. You’re also using up a lot of precious time, and if you really go overboard, you run the risk of running out of official College Board tests to take*.

Here’s an excerpt from a recent post of mine about how to take a practice test:

Do not simply grumble about your score and then take another test. Taking the test helps you build stamina, but reviewing the test is how you actually learn. A good rule of thumb is that you should take at least as long to review the test as it took you to take it in the first place. Go back and look at all your mistakes, and think them through until you’d be able to explain them to a total SAT neophyte.

##### Bad Idea #2: Heavy reliance on cool calculator tricks

Some of the more expensive calculators out there can solve algebraic equations for x. This is, admittedly, a pretty cool trick, but I’ve found that students with calculators like this tend to think it gives them a bigger advantage than it really does. And sometimes, that turns the calculator into a disadvantage.

If your calculator is on the College Board’s acceptable calculator list, that means the SAT folks don’t think it’s got too much firepower. This should tell you something.

The “solve” command is cool, but really, the SAT doesn’t ask you to simply solve algebraic equations for one variable all that often. Rather, it’ll ask you to solve for one variable in terms of another, or figure out which two algebraic expressions are equivalent to each other using some simple set of rules, like exponent rules, or factoring the difference of two squares.

Students with these high-octane calculators spend an inordinate amount of time trying to wrestle SAT algebra into a form that they can feed into their “solve” functions. If you find yourself doing that, then you might be using your calculator to your detriment.

SAT algebra is not generally time-consuming—do it by hand. Limit your calculator use to graphing the occasional function, and speeding up your arithmetic.

##### Bad Idea #3: Gimmicky testing strategies

I’ve heard them all. Start at the end of math sections section to give yourself more time on the hard questions.  Don’t read the reading passages. Always make up essay examples. Wait until you’re done with a section to bubble your answers. These are gimmicks, and whether or not you know someone who knows someone who did them and got a 2400, they’re bad ideas and they shouldn’t be taken seriously. I’ll address them in turn.

Starting at the end of a math section is probably the worst of them all. Each question is worth the same amount of points. It follows from this that the hardest questions are the least important. If you start at the end and then have to rush through the easy questions (or don’t finish the easy questions) then you have cost yourself dearly.

It’s true that there are people who can answer reading comp questions without reading the passages and score really well, but here’s the part of the story you never hear: those people are preternaturally good standardized test takers and they’d do just as well or better if they did read the passage. They didn’t go from a 550 reading the passages to a 750 skipping them. They started at close to 800, and then found they could stay at 750 without reading the passages. If you’re trying to improve your reading score, don’t give up on reading the passages. That’s where all the answers are.

The same is true of people who get their jollies by making up essay examples and getting high scores. They’re great writers already! It’s not like they were writing crappy essays until they began making up examples. Fabrication is not the path to success—it’s a parlor trick for show-offs. You should only invent evidence to support your argument if you can come up with nothing else.

And to the last point about batch-bubbling. There is actually a major test prep provider that advocates it. In real life! So this might not just be something your bonehead friend came up with. Your bonehead friend might have actually been advised to do this. Anyway, here’s why it’s a terrible idea: not every proctor will give you regular time warnings, and you don’t want a surprise section end to result in an incomplete bubbling job. Proctors will not give you time at the end to go back and bubble things you didn’t have a chance to bubble during the section’s official time.

##### Bad Idea #4: Kitchen sink SAT prep

Hopefully you’re familiar with the phrase “everything but the kitchen sink.” I’m planning a longer post about this, but for now let’s just define “kitchen sink prep” as discursive, panicked prep that forgets how circumscribed the content of the SAT really is, and therefore involves a lot of studying of things that will never be (or are incredibly unlikely to be) useful on Test Day. The SAT does not test you on how many formulas you can memorize, or how many special cases you know. There are very few things you should try to remember that aren’t given to you in the beginning of every math section. (Things to memorize include Pythagorean triples, slope-intercept form of a line, the average table.) Don’t study for a test that you’re not actually going to take.

A quick example: I was asked a question recently about a very special case of a very rare form of question: an average speed question. There is a special formula that one could employ for a very particular kind of average speed question in which an object makes two trips of the same distance at different speeds. But it’s complicated and not intuitive, and it won’t help you solve the more general average speed question where the object travels different distances. Please note that I’m not saying this formula is never useful in life, and that it doesn’t have important implications for math outside the bounds of the SAT. I’m just saying its SAT prep value is dubious at best.

Average speed questions appear incredibly rarely on the SAT. (Despite this fact, most SAT prep books I’ve read really emphasize them, stressing their readers out for no good reason.) All you need to remember is this: [average speed] = [total distance traveled]/[total travel time]. Simple to remember, and easy to deploy. To memorize anything else is to misallocate your energies.

##### Bad Idea #4a: Vocabulary obsession

This is a common enough manifestation of kitchen sink prep that it deserves its own heading. Vocabulary is important, and if you want a high Critical Reading score and you don’t already have a prodigious vocabulary you’ll need to study some. But don’t go overboard. You don’t need to learn thousands of words. Here’s the most important excerpt from a longer post I’ve written on this topic.

I’ve spoken with too many kids lately who seem hell-bent on memorizing all the words in Barron’s 3500 Word Black Hole From Which No Light Escapes, and it breaks my heart. If you do that, you will memorize literally thousands of words you won’t need for the SAT, and you’ll use valuable time doing so that would be better spent practicing your reading comprehension, writing practice essays, drilling circle questions, or trying to count the number of hairs on your head.

Quality is far more important than quantity where SAT vocab is concerned. Rather than go nuts on vocab, learn a reasonable number of words from a well-curated list. The Direct Hits books (Volume 1, Volume 2) are great for that.

Finally, don’t forget that the SAT is only one aspect of the college admissions process. A great score won’t guarantee you admission anywhere, and a score 20 points lower than a school’s middle 50% won’t necessarily keep you out. I obviously think SAT prep is important enough to have created this site and written an absurd amount about it, but sometimes I see people take it too far, at the expense of other important things.

If you ask me, this is probably because SAT scores are numbers, and other important things are less easy to quantify. It’s the same reason people chase money when they really crave happiness—money can be counted.

Don’t quit your varsity sport to study for the SAT. Don’t quit the school musical. Stay well-rounded. Being well-rounded matters.

* It’s hard to run out of College Board tests if you do prep the right way. There are 10 in the Blue Book (11 if you get the DVD version), 9 in the Online Course, and 4 more available for free download.

If you’ve ever chalked up a math error to “carelessness”(and let’s be honest—you have) then this post is for you. So often do I see students blame their mistakes on “carelessness,” in fact, that the poor word has lost its meaning in an SAT context. This post is an effort to restore its dignity.

Carelessness, in my experience, can mean one (or more than one) of the following:

• Arithmetic or simple algebra errors
• Self-delusion

To avoid misbubbling, bubble sedulously, and then double check your bubbling. You should check once as you’re bubbling, and then if you have time at the end of a section, go back through and make sure your answer sheet reflects the answers you’ve circled in your test booklet. This, by the way, is one great reason to actually use bubble sheets when you take practice tests. Because everyone misbubbles once in a great while, and if it happens to you during a practice test, you’ll be that much more likely to be careful on the real thing.

As anyone who’s ever taken a math test knows, it’s very difficult to avoid the occasional arithmetic or algebra error. Sometimes, the mind meanders. It’s also, unfortunately, very difficult to catch an error when you go back and look over your work. You can be staring right at 10 ÷ 5 = 5, but if you just wrote it 2 minutes earlier, you might not see what’s wrong with it. So to avoid bungling your simple calculations, do the following:

• Check simple arithmetic on your calculator. I know you’ve done 6 × 7 in your head a million times correctly. Just make sure.
• Try to do as many problems as you can multiple ways. If you arrive at the same answer with algebra as you did by plugging in, you can be doubly sure you’re right. Of course, you’ll need to find a healthy, comfortable balance here between caution and speed. Pro-tip: it’s usually not a good idea to sacrifice accuracy for speed. Favor caution.

As for self-delusion, this is the toughest one to fix, because it’s denial of a problem. Many students brush off every question they miss: “Oh, that was easy. Careless mistake!” This is a natural reaction—a good solution can look really obvious once it’s laid out in front of you, especially by a good teacher.

But if this sounds like you, know that you do yourself a disservice when you assume that, since a solution is obvious once someone else shows you how to do it, a similar solution will be obvious to you next time you see a similar question. By characterizing a mistake as “careless,” you tell yourself you don’t need to learn anything new about questions like the one you missed. And if that’s not true, then that question type will keep forcing you to make “careless” errors until you address your underlying knowledge gap. It’s OK not to know how to do something, and admitting that you need to learn is the first step towards doing so.

To be safe, try never to characterize a mistake as “careless” unless you’ve got a demonstrable history of cutting through similar questions like butter. Don’t even let the word “careless” enter your vocabulary until you’ve done 4 or 5 practice tests.

Early on a weekend morning?
All in one sitting?
Okay.

Practice tests are a necessary element of any SAT prep plan. The test itself is a harrowing and protracted experience, and if you haven’t put yourself through rigorous simulations a few times before you sit down for the real thing, you’ll be at a real disadvantage.

It’s important to note, though, that although practice tests are an important part of the prep experience, if you only take practice tests and do little else, your scores aren’t likely to improve much. Practice tests are, as students of philosophy are wont to say, necessary but not sufficient. But you knew that already.

Anyway, here’s how you take one. First, drag your lazy bones out of bed early on a weekend morning. Set your alarm to go off early enough that you’ll have time to eat breakfast, take a shower, and be fully alert by about 8:30, when you should start testing.

Your bedroom isn’t the worst place to practice, but if possible, get yourself to a public place that you can expect to be fairly quiet, but that will have some ambient noise—a public library is perfect. Part of the SAT experience is the fact that someone next to you might have the sniffles, or the hiccups, or…worse. A few minor distractions during your practice tests will help you to be better prepared when something noisy or smelly happens on test day.

Take the whole test in one sitting*. Yes, even the essay. And for Pete’s sake, actually bubble your answers on the bubble sheet, rather than just circling them in your book. Bubbling actually takes time, and if you’re shooting for accurate simulation, you should account for that time. As you work, make sure to circle any question you’re uncertain about on your answer sheet. That way, even if you get it right, you’ll remember that it’s something you should revisit.

No finishing early and moving on to the next section. If the section’s supposed to take 25 minutes, you work on it for 25 minutes Give yourself a 5-minute break after the 2nd section, the 4th section, and the 6th section. No finishing early and moving on to the next section.

Score that bad boy up. This might seem simple, but it’s actually a pretty important part of prepping for the SAT—to do as well as possible you need intimate knowledge of how the test is scored. Blue Book tests have a worksheet at the end to teach you how to do this.

Keep a record of all the questions you miss, or guess on and get right. Do your best to categorize them, so you can keep a tally of how many verb agreement mistakes you made, or how many right triangle questions stumped you. This way, each time you take a practice test you’ll be building a database of your weak areas, which you can then use to focus your prep.

Do not simply grumble about your score and then take another test. Taking the test helps you build stamina, but reviewing the test is how you actually learn. A good rule of thumb is that you should take at least as long to review the test as it took you to take it in the first place. Go back and look at all your mistakes, and think them through until you’d be able to explain them to a total SAT neophyte. If there are any questions that, despite your best efforts at review, you still don’t understand, ask someone for help.

* If you really want to go H.A.M., find an extra 25-minute section from another test (maybe your old PSAT practice book or something) to use to simulate the experimental section that all Blue Book tests are missing. So if, like Blue Book Test 1, your test is missing Section 4, give yourself an extra section to do between the Section 3 and Section 5.

When a student asks me how to solve a math problem, my default response is to show, if possible, how to solve it by plugging in, backsolving, or guesstimating. I do this because I figure if the “math way” was obvious, the student wouldn’t be asking me for help in the first place. Besides, problem solving—in life, or on the SAT—isn’t about following a circumscribed set of procedures. It’s about creativity and flexibility. I’ve written before about the importance of being nimble. Consider this post a sequel.

It’s fun to be good at math, and it’s nice to understand how the underlying algebra on a tough word problem works. But if you’re aiming for top scores, it’s imperative that you cast a critical eye on your own ability to tease the “math way” of solving a problem out of the problem during the fairly tight time constraints imposed by the SAT.

If x + y = p and x – y = q, what is p2 + q2 in terms of x and y?

(A) 2(x + y)2
(B) 4xy
(C) 2x2 – 2y2
(D) 2(x2 + y2)
(E) 2(x2 – 4xy + y2)

Like all questions, there’s a “math way” to do this, but unlike all questions, this one is a prime candidate for plugging in. There will be some students who can breeze through the algebra in their head and identify the correct answer almost instantly. If that’s you, then great. You needn’t plug in. But if that’s not you, or if you only kinda think that’s you, then you should probably just plug in. It’s fast, it’ll get you the right answer, and then, later on you can go home, make an awesome couch fort, and figure out the algebra when you’re not pressed for time.

##### The plugging in solution

Say x = 3 and y = 2. Then 3 + 2 = p = 5, and 3 – 2 = q = 1. 52 + 12 = 26, so you’re looking for an answer choice to give you 26. Type the answer choices into your calculator carefully, substituting 3 for x and 2 for y, and you’ll be done in a hot second:

(A) 2(x + y)2 = 2(3 + 2)2 = 50
(B) 4xy = 4(3)(2) = 24
(C) 2x2 – 2y= 2(3)2 – 2(2)2 = 10
(D) 2(x2 + y2) = 2(32 + 22) = 26
(E) 2(x2 – 4xy + y2) = 2(32 – 4(2)(3) + 22) = –22

##### The algebra
Not impossible, right? Totally doable. But arguably more involved than the plug-in solution above.
##### The bottom line

Look, I really just want you to be happy. If you want the algebra, I’ll give you the algebra. But I really think it’s a good idea for you to know how to plug in, too. Because if you have to ask me for the algebra on a question like this, that means it wasn’t obvious to you right away when you encountered it on the test. And that means there’s a good chance that when you sit down for the real thing, the algebra isn’t going to be obvious to you for every single question. And if, when the algebra isn’t obvious, you don’t have a backup plan, then you’re doing yourself a disservice.

Try the algebra first, if that’s your bent. But you should have a few other tricks up your sleeve for the questions where the “math way” isn’t jumping off the page onto your lap.

There are many ways to learn words, which is good, because there are many different learning styles. Some people like vocab books like the fantastic Direct Hits series. Some people just make it a habit to write down and learn every word they encounter that they don’t know. Still others, like myself, try to grow their personal lexicons in the long term by using a thesaurus to avoid the repetition of words in their written work.

But one of the most common methods of vocabulary augmentation is the flashcard. It’s tried and true, just like mom and dad used to use when they walked to school uphill both ways in the snow chased by sabre-toothed tigers.

I see a lot of people struggle with flashcards. They know that flashcards are supposed to be great, but although they recognize the words they’re supposed to know when they see them, they just can’t seem to make their definitions stick. If this sounds like you, read on.

## The three-pile flashcard system

For the purposes of this post, I’m assuming you’ve already either created your own cards, or purchased a set.

1. Go through the entire pile of cards. Any word you know (and I mean know—like you can recite the dictionary definition quickly and accurately) put in a different pile. That’s your KNOW pile. Any words that don’t go into the KNOW pile go into the DON’T KNOW pile. Easy so far, right?
2. Set a daily goal for yourself based on the size of your DON’T KNOW pile and the amount of time you have to learn it. You should give yourself plenty of wiggle-room in this goal. So if you have 8 weeks until your test, and 250 words you don’t know, set a goal to learn 40 words per week. That way, you’ll finish early (or won’t fall too far behind if you miss a day). And if you add words to your list as you go, you won’t have to stray from your plan to absorb them.
3. Here’s the part most people don’t do: On Monday through Saturday, try to learn your 40 words for that week. Once you’ve convinced yourself you know a word, put it in a third pile: the PENDING pile.
4. On Sunday, test yourself on the words in your PENDING pile. If you can recite the definition of a word without hesitation, you can put it in your KNOW pile. If you can’t, you leave it in your PENDING pile or, if you really don’t know it, put it back in your DON’T KNOW pile.
5. Repeat steps 3 and 4 until you’ve moved all your cards into the KNOW pile.
6. Test yourself on every word in your KNOW pile. If you can’t recite a word’s definition in 5 seconds, put its card back in the DON’T KNOW pile, and begin again.

This system is only going to work for you if you’re honest with yourself. If you don’t know a word, admit you don’t know it. Otherwise, when it appears on test day and you don’t know it as well as you should because you let it languish in your KNOW pile when it didn’t belong there, you’re going to be sorry.If, however, you stick to the plan assiduously and concede when you don’t know words, you’ll be in good shape come test day. Good luck!

This is a bit of a cliche, but you really should hold your reader’s hand and guide him through your essay. Avoid reader whiplash at all costs—your grader should never have to pause to wonder how he got to where he is, because you should be there at every juncture, reminding him exactly how he got there. Each sentence should flow neatly from the sentence before, and into the sentence after. Each paragraph should have a topic sentence, and stick to points relevant to that topic sentence.

You accomplish this by being sedulous about organizing your essay. Outline carefully before you write, jotting down the topic (and maybe the topic sentence) of every body paragraph. And then, as best you can, stick to that outline. If you come up with a brilliant idea for your second paragraph while you’re still working on your first one, it’s OK to deviate from the plan, but if you’re making up your argument as you go, your grader will be able to tell, and you’ll pay for it with a less than stellar score.

If you want your essay to have good organization and focus, you need to tell your reader what you’re going to say, say it, and once in a while remind her that you said what you told her you’d say.

##### Examples

Below you’ll find body paragraphs from the same essays whose introductory paragraphs were in this post. Note how the first writer fails to remain laser-focused on furthering his argument and inserts details that don’t really help his cause. This gives the reader the impression that, at best, the writer is a bit confused, and at worst, the writer is desperately trying to fill space. Contrast that scattershot prose with the output of the second writer, who diligently reminds his reader at the beginning and end of each paragraph that the reason he is writing about lying brothers and dead presidents is that they are germane to the topic of the value of truthfulness, and then inserts enough details to give his examples context, but not so many that his point is muddled.

Bad: After leading the Americans to victory in the Revolutionary War, George Washington became the first President of the United States. Most people have heard the story of when he was a boy and chopped down an important cherry tree. His father was very angry about the tree’s demise, and asked young George who did it. George told his father that he could not tell a lie, and his father forgave him. This story is famous because it shows how it is never OK to lie. George Washington, America’s first and best president, always told the truth.

My brother got in a lot of trouble with my Mom last weekend when he lied about where he was going on Saturday night. He told her he was going to be sleeping at a friend’s house, but really he went to a concert 45 minutes away. She caught him because she opened his duffel bag the next morning and the clothes he was wearing smelled like smoke and there was a ticket stub in his pocket. She does not like rock music (she calls it devil music) and was really mad. She took away his phone, and he had to come right home after school all week and can not leave the house this weekend.

Good: One indicator of the value people place on honesty is that of George Washington. Legend has it that long before his heroism in the Revolutionary War or his inauguration as the first President of the United States, the young George Washington was honored for his adherence to the truth. He had chopped down a cherry tree for sport, the story goes, not realizing that his actions would anger his father. When the elder Washington discovered the downed tree and demanded to know who had perpetrated the crime, young George stepped forward and said “Father, I cannot tell a lie. It was I who chopped down the cherry tree.” That this myth persists, even though these events almost surely did not transpire, speaks to the value our society places on honesty. Teachers and parents, in recounting this story, are making an effort to encourage children to be truthful, even when lying would be easier. Therefore, the child who learns this lesson well and lives by a code of honesty will be more likely to earn society’s respect.

My brother Gerald, unfortunately, could use a refresher on the fable of George Washington and the cherry tree. He recently found himself in hot water as a result of his dishonesty. Although our mother forbade him from attending a rock concert 45 miles from our home, he decided to attend the concert anyway and simply tell our mother that he was sleeping at his friend’s house that night. When my mother discovered his deception, she told him that she was more disappointed than angry (and she was pretty angry). She had trusted him implicitly, and he had betrayed that trust. His immediate punishment was a temporarily restricted social calendar, but my mother made it clear that the lasting impact of his actions would be that he would have to earn back her trust. As it is in the society in which we live, truthfulness is valued in my family. My mother’s disappointment at Gerald’s dishonesty and his appointed task of earning back her trust are further evidence that those who are honest will be better respected.

I’ve been asked a few times lately about the PSAT/NMSQT, and I figured it might be helpful to put up a brief FAQ about it. So…here that is. 🙂

How are the PSAT and the SAT different?
For one, the PSAT, at 5 sections, is much shorter than the grueling 10 section SAT. The PSAT contains no essay section (although you still get a writing score, based solely on your performance on multiple choice grammar questions). PSAT scores are a little different, too. While each SAT section’s score range is 200-800, in intervals of 10, each PSAT section’s score range is 20-80, in intervals of 1. Although the SAT is given 7 times per year, the PSAT is only given once, in October. Schools get to decide whether to administer it on Saturday or Wednesday of the appointed week.
PSAT sections break down like this:
• 2 reading sections. Each contains 24 questions and and is 25 minutes long. One section will contain 8 sentence completions and 16 critical reading questions, and the other will contain 5 sentence completions and 29 critical reading questions.
• 2 math sections.  One will contain 20 multiple choice questions, and the other will contain 8 multiple choice questions and 10 grid-in questions. Each is 25 minutes long.
• 1 writing section. It’s 30 minutes long, and contains 20 sentence improvement questions, 14 error identification questions, and 5 paragraph improvement questions. Total: 44 questions (slightly longer than the long SAT writing section).

Who takes the PSAT?
Most schools sign all juniors up for it automatically. Many schools allow (or force) sophomores to take it as well. It’s less common for freshmen to take the PSAT, but it’s by no means unheard of.

Why is the PSAT scored differently than the SAT?
Well, because the tests are different and the scores mean different things. It’s OK to make a rough estimate of your SAT score by multiplying your PSAT score by 10—so, say, a 182 PSAT score corresponds to an SAT score of 1820—but remember that it’s a rough estimate of where you were on that particular morning in October. Chances are decent that, by the time you get your scores back in December, you’re already in a different place. I’ve seen SAT scores swing much higher than PSAT scores without any additional prep, and I’ve seen them swing much lower.
What does NMSQT stand for?
The PSAT doubles as the qualifying test for the National Merit Scholarship, and that’s what the NMSQT stands for: National Merit Scholarship Qualifying Test. Only juniors are entered into the National Merit Scholarship competition. Freshmen and sophomores who take the PSAT, even if they score perfect 240s, will have to repeat that performance during their junior years if they want to enter the NMS competition.
Will colleges see my PSAT scores?
Nope.
What if I want them to?
There’s no score reporting mechanism for the PSAT. If you receive any National Merit Scholarship recommendation, obviously you can announce that as an accomplishment and colleges will know roughly the range in which your score fell, but colleges don’t generally care about the PSAT, whether you want them to or not.
Should I prepare for the PSAT?
You should take a practice PSAT and see how you do. If you score 180 or higher, you should consider doing a bit of serious prep to see if you can hit the cutoff for National Merit recognition. This number varies slightly from year to year, but is usually about 200.
Many people who aren’t shooting for National Merit still prepare assiduously for the PSAT, and that’s fine. Any prep for the PSAT is also prep for the SAT, so it’s not wasted time.
How should I prepare for the PSAT?
Pretty much the same way you should prepare for the SAT. You can read the hundreds of pages on this site for detailed advice, but the basic gist of it is this:
1. Learn some test-specific techniques and strategies
2. Take a practice test
3. Review the practice test like crazy until you understand every mistake you’ve made and could explain how to answer the question correctly to your little brother
4. Identify your weaknesses based on practice test results
5. Drill those weaknesses until they’re strengths
6. Take another practice test
7. Repeat
College Board-made PSATs aren’t so easy to come by, unfortunately. You can get one for free from your guidance office (it’s in a booklet called the Official Student Guide to the PSAT/NMSQT, which is available online except for the practice test part). After you take that one, you might have to start taking SATs instead, unless you’ve got older siblings or friends who will erase their old test books and let you use them. Using SATs isn’t the worst thing in the world. If you get yourself into the kind of shape where you’re PWNing the PSAT by October, you can sign up to take the SAT in October, November, or December, and maybe get all your testing out of the way before you’re halfway done with junior year.
Am I missing anything? Let me know in the comments.

SAT essay assignments require you to take a position and support it using “reasoning and examples taken from your reading, studies, experience, or observations.” In other words, you don’t just have to say what you believe, you need to try to persuade your reader to agree with you, or at the very least convince him or her that you have good reasons for believing what you do.

You accomplish this by crafting a well-formed argument, and not trying to overstep the scope of the assignment. That means you establish a number of premises, and tie them together logically to build towards your conclusion. Remember that, because of the murkiness of the topics essay assignments usually cover, you’re not usually going to be able to craft ironclad proofs of universal truths. Be conscious of the strength of your premises, and make sure you draw an appropriately qualified conclusion.

Resist the temptation to make your argument seem stronger than it is with words like “always,” “definitely,” “completely,” “never,” etc. Far from bolstering the strength of your essay, these extremes have a deleterious effect. They make it seem as though you don’t have enough faith in your argument to let it stand on its own without these crutches—they call attention to your argument’s weaknesses.
Instead, own up to the fact that you are limited by the time you have to write the essay and the scope of the prompt, and make a solid case for your position based on a few pieces of evidence. A nuanced, narrow argument will trump a failed attempt at incontrovertible truth every time.

Please note that I am not encouraging you to take both sides of an argument here. I’ve seen a few people do so effectively, but I don’t really think there’s time or space for that on the SAT essay, so I don’t advise it. You need to take a position. I’m just saying it should be a thoughtful one.

##### Examples

Below you’ll find two example paragraphs. Note how easily a contrarian could refute the first thesis, and how much more difficult it would be to do so to the second thesis. Note also that historical accuracy isn’t all that important on the SAT, so while the second essay might turn heads with its use of apocryphal, the first essay would not suffer from failing to mention that the story about the cherry tree is of dubious veracity.

Bad: Lying is always wrong, and examples that prove this can easily be found in history, literature, and my own life. When George Washington was a boy and cut down a cherry tree, he told the truth about it and later became the first President of the United States. When my older brother lied to our mother about where he was going last Saturday night, and then got caught, he got grounded for a week.

Good: Although there are occasions in which it might be advisable to lie, those who make the truth a priority are more likely to earn the admiration of their peers. George Washington, the first President of the United States, was so admired for his honesty that the apocryphal story of the cherry tree is one of America’s most well-known folktales. In contrast, my older brother disappointed my mother last weekend when he failed to tell the truth about his plans.

 Special nerd note: #20 in Section 4 is pretty much my favorite Guesstimate question of all time.

The College Board’s latest The Official SAT Study Guide with DVD is exactly the same as the 2nd edition, except that it comes with a DVD containing a bit of extra content. I can see the utility in the SAT timer it includes, but otherwise the only content of value on the DVD is one additional test—the January 2008 QAS. I went through that test yesterday so I could update the Blue Book Breakdown in my Math Guide (I’m calling it Test 11). A PDF of that page (with page number references, etc.) is available here for book owners. For everyone else, below is an HTML version of that table with links to some important strategies.

If you’re taking the SAT next weekend and you haven’t really started studying yet, you should know right now that you’ve not set yourself up for overwhelming success (or even regular-whelming success). Still, you’re not alone in your predicament, and Goonies never say “die.” I’m not going to say anything profound here, but I figured I’d write up a last-minute study (cram) plan to try to maximize your score in as short a time as possible. If it works, great. If it doesn’t, then use this as a starting off point for your more assiduous preparation schedule with the May or June in mind.

• MONDAY: When you get home from school, take a full SAT, strictly timed. If you have the Blue Book, use one in there (preferably one of the first 3). If you don’t, you can get a free test from the College Board, but you’re going to have to print it (I don’t recommend taking it online—the real test won’t be on a computer screen). Assuming you don’t have extra time accommodations, this should take you just under 4 hours. Correct it and score it. Scoring instructions are included at the end of the test—make sure you’re scoring it correctly. Go to sleep.
• TUESDAY: Set aside 2 hours (or more) to review all your errors in the READING section (Reading comes first because I don’t want you to forget what the passages were about). Review means understanding why every single wrong answer was wrong, and why each right answer was right. Disabuse yourself of the notion that questions on the SAT are subjective. Each right answer is right, and each wrong answer is wrong. Note line references that reveal or discredit answers. You should be able to explain any question to a complete stranger. You should also take note of any vocabulary words you don’t know, although honestly you’re unlikely to increase your vocabulary much in a week.
• WEDNESDAY: Again, set aside 2 hours (or more) to review the MATH sections of that test. If you took one of the first 3 Blue Book tests, use my technique guides (Test 1 | Test 2 | Test 3) to help you understand your mistakes and refine your approach to those question types. See if I (or one of my friends) has posted longer solutions to any questions that stump you. If we haven’t, ask me! Again, you should be able to explain each question on the test to a stranger before you call it a night.
• It’s also of paramount importance that, if you don’t already know how the math section is set up and how that should inform your test taking strategy. Read the following:
• THURSDAY: You guessed it: today is WRITING day. You know the drill by now. Set aside a couple hours, and review your mistakes. Pay special attention to the SAT’s favorite errors to test—the ones that sound simple but can be very tricky to spot: Verb errors, Pronoun Errors, Run-on Sentences. Watch out for Dangling Modifiers, too. Don’t worry too much about the essay—it won’t affect your score as much as the multiple choice grammar questions—but read this Dos and Don’ts post to avoid some of the most common errors.
• FRIDAY: Set your stuff out for Saturday: Calculator (with new batteries if possible), pencils, admission ticket. And then chill. Seriously. Just chill. The best thing you can do now is get some rest, so you can wake up on Saturday ready to go. Eat a good meal, watch a movie, and go to bed early. In the morning, you’ll have work to do.
• SATURDAY: Don’t break your usual routine. Eat breakfast if you usually do. Have coffee if you usually do. Try to get to the test center early so you aren’t stressing on the ride over about being late. Breathe.
As I said, there’s nothing sagacious in this advice. It’s a brute force solution, not an elegant one. But if you can carve out time to do this, you’ll be in a much better position on test day than you otherwise would have been. Make the best of this week, take the test with as much swagger as you can muster, and if Saturday doesn’t go as you’d hoped, use your work this week as the baseline for your prep moving forward.

This breakdown is meant to help you analyze and categorize your mistakes after you’ve taken Practice Test 3 in the Blue Book. The whole idea is that the best thing you can do to improve your score is to understand your weaknesses, and then drill the hell out of them to make them strengths. Click to see similar breakdowns for Blue Book Tests 1 and 2.

 § p # Techniques and concepts Diff. 2 514 1 I suppose you could backsolve if you wanted to. 1 2 514 2 1 2 515 3 Spatial reasoning is fun! 1 2 515 4 25% is a quarter. A quarter of a circle is a right angle. Which angles are acute? 2 2 515 5 2 2 516 6 2 2 516 7 2 2 516 8 If you’re not immediately sure what (–0.5)2 is, just put it in your calculator.. 2 2 516 9 3 2 517 10 3 2 517 11 3 2 517 12 Read the Venn diagram carefully. It might help you to darken the boundaries of A and B. 3 2 517 13 Percents, Backsolve 3 2 518 14 3 2 518 15 Percents 3 2 518 16 4 2 518 17 4 2 519 18 Draw them and count. 4 2 519 19 4 2 519 20 5 5 525 1 1 5 525 2 1 5 526 3 1 5 526 4 Read the question carefully. 1 5 526 5 Special right triangles 4 5 527 6 3 5 527 7 Draw it carefully. 4 5 527 8 FOIL it. Corresponding coefficients will be equal, so (–8 – k) = –5k and m = 8k. 5 5 528 9 1 5 528 10 2 5 529 11 2 5 529 12 3 5 529 13 This is a tricky graph. Read it carefully. 2 5 529 14 Write the equation: 5n = n + 5. Solve for n. 3 5 530 15 4 5 530 16 4 5 530 17 For every 4 inches of strip, there are 5 inches of edge. Use ratios. 4 5 530 18 Square’s side = 8, so R is (4, 8). Plug into the equation to solve for a. Parabolas. 4 8 543 1 Do the algebra: (3/4)n = 18. 1 8 543 2 2 8 544 3 Read the graph carefully. 1 8 544 4 Read the question carefully. 2 8 544 5 2 8 545 6 2 8 545 7 2 8 545 8 2 8 545 9 3 8 546 10 3 8 546 11 Do the algebra. 3 8 547 12 Reasoning: Can they all be negative? No. All but one? Yes, if the one is big enough. 3 8 547 13 List the combinations methodically. 4 8 547 14 Graph amplification 3 8 548 15 Follow the pattern remembering that all negative values are less than 100. 4 8 548 16 5