### What is the SAT really testing?

Much hay is made about what the SAT is actually testing. Does it function as some strangely-defined "college readiness" measurement? Is it a pure reasoning test? Is the SAT a test of innate intelligence, like an IQ test? Is it a completely meaningless hoop that you just have to jump through like a circus dog because everyone else does? The College Board claims only that the SAT tests "the skills you're learning in school: reading, writing and math." Is that true?

The indefatigable Debbie Stier asked me for my take on this the other day, and I realized that I had never written about it on this site. My answer to all of the above is a qualified "no." There's a bit of truth to each claim, and you won't have to look too hard to find people who'll argue for any of them. You might even have a friend with a crazy theory of his own (Duuude, the SAT is an awesome predictor of alien abduction!). I've been working with the SAT for a while now, and I've come to my own conclusions.

The short answer: there is no short answer. Each subject tests different things, and although there are overarching themes, you're not giving the test fair treatment if you try to encompass the whole thing in just a word or two. What follows is a bit of a brain dump. Chime in in the comments if you think I'm missing something.

### Weekend Challenge - not so impossible edition

I wanted to take it easier on you guys after last week's hard-as-hell question, so here's one that could actually appear on an SAT.

This week's prize: you and your future random college roommate will have remarkably similar tastes in music. Trust me -- it matters.

1. After m days, the average (arithmetic mean) temperature for the month of June in the city of Riverside was is 87° Fahrenheit. If the temperature the next day was 99° and the average temperature for June rose to 89°, what is m?

UPDATE: Excellent job, Guilherme. Solution below the cut.

### Take a deep breath.

I've only been at this a few months, but if there's one thing that site analytics have shown me reliably over that short time it's that traffic booms in the days following an SAT score release. Consider this your official welcome, June SAT score recipients! I hope, sincerely, that you're pleased as punch with your scores, and that you've only stopped by here on a sort of strange Internet victory lap. If that's the case, congratulations!

For everyone else: this is not the time to sulk or let frustration overcome you. This is the time to redouble your efforts and start prepping efficiently, so that by the time the next SAT comes around in October, you're a lean, mean, test-stomping machine. Take today to blow off some steam -- go for a run, play video games, take a nap -- but be ready to get to work tomorrow.

Your mission this summer should be to systematically seek out particular weak areas in your SAT skills, and then drill them until they become strengths. And by "weak areas" I don't mean "I suck at math." I mean real nitty-gritty specifics. I mean "I always screw up exponent problems," or "I have a really hard time recognizing faulty comparisons."

Once you know the kinds of questions that hurt you the most, you can start working those specific areas. Remember, big score improvements can come from relatively small skill improvements. One fewer mistake per section will lead to a roughly 100 point overall score improvement.

Looking for a way to identify some weak math areas quickly? Try this drill. Or this one. Want to dive right into the deep end? I got you, boss.

Feel free to reach out to me on Twitter, Facebook, or Tumblr with questions. Seriously.

### Median and Mode on the SAT

Like the average (or, as some say, the arithmetic mean), the median and the mode are useful properties of a set of numbers and can give statisticians great at-a-glance insight into the nature of copious data. When the SAT gets its hands on them, though, they are usually stripped of any analytical utility and instead used as a framework in which to ask tricky reasoning questions. Also, the SAT doesn't really test either of these concepts very often -- doubly so for mode. So I'll leave it to your college stats class to elucidate the myriad ways median and mode are useful in real life, and just show you what you need to know to PWN the rare median and/or mode question on the SAT.

##### Median
The median is the middle value in an ordered list of numbers. If the list of numbers you're given isn't in numerical order, you can't find the median until you put it in numerical order. If there are an even number of values in your list, the median is the average of the two middle values. That's it. That's all you need to know about the median. Mouse over the following example sets to see their medians.

• {4, 6, 7, 9, 12, 16, 30}
• {9, 30, 16, 4, 7, 6, 12}
• {2, 200, 300, 700}
• {17, 22, 6, 110, 68, 52, 29, 8456}

##### Sample Median Question
1. Which of the following CANNOT change the value of a median in a set of numbers?

(A) Adding 0 to the set
(B) Multiplying each value by -1
(C) Increasing the least value only
(D) Increasing the greatest value only
(E) Squaring each value

### Direct and Inverse Proportionality (Variation)

There are two kinds of proportionality (some call these problems "variation" problems, but I'm sticking with proportionality) problems that you might see on the SAT: direct and inverse. I'm going to cover both here since I'm in the business of preparing you for any eventuality, but you should know that the the former is much more prevalent than the latter. Don't sweat inverse proportions all that much.

##### Direct Proportions
There are a few ways to represent direct proportionality mathematically. The Blue Book likes to say that when x and y are directly proportional, y = kx for some constant k. This definition is correct, of course, but I find it to be less useful since it introduces an extra value into the mix and doesn't lend itself as easily to the kinds of questions you'll usually be asked on the SAT. I much prefer to say:

When x and y are directly proportional:

Note that k is still in there, but we don't have to deal with it directly anymore. I like to streamline.

In a direct proportion, as one value gets bigger, the other gets bigger by the same factor. As one gets smaller, the other gets smaller by the same factor*. Observe:

p and q are directly proportional
p = 4, q = 10
= 8 (), q = 20 ()
= 40 (), q = 100 ()
= 2 (), q = 5 ()
= 1 (), q = 2.5 ()

For easy direct proportion questions, all you'll need to do is plug values into the proportion above, and solve. And the "hard" direct proportion questions won't actually be much harder.

##### Let's see an example
1. If y is directly proportional to x2, and y = 8 when x = 4, what is y when x = 5?

(A) 5.12
(B) 10
(C) 12.5
(D) 14
(E) 25

### Weekend Challenge - Hoarders edition

 Source: Married to the Sea.
This started out as an Old MacDonald's farm question. No wait, I thought to myself, not depressing enough.

The prize this week: You'll get the satisfaction of knowing that you probably solved this problem in less time than I spent staring at my computer screen trying to come up with a clever prize for this week. I swear I used to be more creative.

In Wendell's house, the ratio of unopened credit card offers to out-of-date phone books is 9 to 5. The ratio of magazines to crushed loose cigarettes is 25 to 7, and the ratio of McDonald's Happy Meal toys to rotting, half-eaten pizzas is 3 to 2. There are 6 used-up batteries lying around for each broken VCR. The ratio of crushed loose cigarettes to McDonald's Happy Meal toys is 5 to 8, and the ratio of used-up batteries to out-of-date phone books is 5 to 7. There are 30 magazines for each 4 broken VCRs. If there were 108 unopened credit card offers, how many rotting, half-eaten pizzas would Wendell have in his house?

Please note that, as usual for the weekend challenge, I'm taking a concept that SAT has been known to test, and extending it to an extreme to which the SAT would not go (not only in subject matter, but in scope as well). All of this is to say that these weekend challenges are meant to be fun for you, not to freak you out if you can't get them. They're usually a degree or two harder than what the SAT will throw your way.

Put your answers in the comments. I'll post a solution Monday (probably late in the day). Good luck!

UPDATE: Nice work, Chris. And special thanks to Catherine from kitchen table math, who pointed out that my original wording ("If there are...how many does...") made non-integer quantities (fractional batteries??) quite unsavory. I've changed the wording of the last sentence a bit now for the benefit of people who find this in the archives and want to torture themselves with this question later.

Solution below the cut.

### Run-on Sentences and Fragments (featuring The YUNiversity!!!)

A quick note before we begin: I'm positively elated to have teamed up with Tumblr all-star The YUNiversity for this post! Everybody knows that eye-popping visuals are a great boon to students trying to learn otherwise dry material, and nobody does them better. If you like the illustrations he provided for this post, you simply must make a habit of checking his site every day. He's amazing.

Ok, now. If you want to understand run-on sentences, first you have to understand the difference between a sentence and a fragment. Both are similar in that they contain a subject and a verb, but a sentence can stand on its own as a complete thought, and a fragment cannot. Fragments seem to end abruptly, and leave you wanting to ask something like “...and then what?” To make things super clear in this post, in the examples below complete thoughts will be in green and fragments will be in brown.#

It’s easier to show this than to try to describe it, so here are some fragments. As you look them over, ask yourself "What is it about these that prevents them from standing alone as complete sentences?"
• even though his fans booed him
• when the cows come home
• because her mother was in jail for grand theft auto
• while you were sleeping
• to whomever the taser belonged
None of the above are complete thoughts -- they're the beginnings or the ends of thoughts, but mean very little on their own. On the SAT, if you see a fragment trying to be a sentence all by itself, you have to fix it. Fragments are always wrong on the SAT.

A run-on (or "comma splice," if you like) is kinda the opposite problem. If you come across a comma that’s separating two complete thoughts, that’s a run-on. Like fragments, run-ons are always wrong and you need to fix them.

##### A run-on looks like this:
 Two complete thoughts separated by a comma? NO ME GUSTA.

### Why don't I post solutions to my drills?

A bunch of people have emailed me asking why I only post answers, and not full solutions, for my drills. That's a fair question, so I figured I'd answer it publicly. I don't post solutions because I think the best way to improve your skills is to figure the solutions out on your own. When I work with kids, I almost never give answers (although I'm almost always asked). What I do instead is help my students to find the path to the solution themselves and then keep them on it by reminding them to use all the weapons in our arsenal. To put it in old-school SAT analogy form:

Mike the tutor : SAT math :: lane bumpers : bowling

Although I'm not physically sitting beside you and acting as a bumper, I have provided on this site the tools you need to solve the problems in my drills, and I've linked to the relevant techniques for each question in the answer keys to point you in the right direction. If you want to be able to solve similar questions on test day, you'd do well to mess with the puzzle pieces now by yourself until you see how they fit together.

That said, it's completely counter to my mission to frustrate you, so I'm happy to help if you're really stuck. Ask me a for an explanation at PWN the SAT Q&A and I promise to get one to you as quickly as I can.

One last note: I don't want to be unduly dogmatic about this. I know self-study is a lot different than sitting with a tutor, and I don't want to force a round peg into a square hole. If enough people chime in, I'm happy to reevaluate my position.

### Diagnostic Math Drill #3

Here's another drill for you to use to isolate problem areas. Please remember: When you're practicing, the time you spend reviewing your mistakes is arguably more important than the time you spend doing the problems in the first place. This drill will be most useful to you if you spend some real time with the answer key and technique guide afterwards. If you just mark the drill, shrug off your mistakes as "silly" ones, and move on, you're dooming yourself to repeat them. It's important to LEARN from your mistakes.

I should also restate something I've said before: These drills of mine are not exhaustive. I'm trying to write questions that cover a broad range of possible SAT questions, but in no way can these 20 questions (or these 60 questions, if you count the first two drills too) encompass every kind of question that will be thrown your way. I'm trying to help you work out a few specific kinks, but the bulk of your work should be in the Blue Book.

The answer key, as always, is linked from the end of the drill so that you can work through the drill without peeking, but you can also access it directly here.

Feel free to print, share, etc.

--OR--

Good luck!

### Weekend Challenge - Best Band You've Never Heard Of Edition

I've spruced it up a tad, but an extremely similar question was #17 (not even #20!) on an SAT in 2006. The prize this week (pardon my proselytizing): Someday you will be as good at something as Mike Miller is at songwriting.

UPDATE: A couple folks nailed this one, and doing so represents, in my opinion, a promising level of nimbleness. Color me impressed. Solution below the cut.

### How to know whether you should guess on the SAT

Because there's a penalty of ¼ raw score point for incorrect multiple choice responses on the SAT, many students experience extreme trepidation about guessing when they aren't sure about an answer. I've stated my general advice on guessing before, but the truth is that while I almost always find that my students benefit slightly from guessing more, I'm open to adjusting that advice if it doesn't seem to serve a particular student well. If you're not comfortable just taking blanket advice from a stranger on the Internet, there's actually a very simple experiment you can perform to help you settle on a guessing strategy that works for you.*

Here it is: always, ALWAYS, ALWAYS GUESS on practice tests, and make little marks on your answer sheet to remind yourself which choices were guesses. When you're done, score your test twice: once with your guesses in there, and once with all your guesses replaced by blanks.

What you'll probably find is that there isn't much difference either way, but once you've done this on 3 or 4 tests, you'll start to get a sense of how guessing works for you. By the time the real test comes along, you'll be comfortable in your guessing strategy, knowing that it's based not on superstition or blind faith, but science.

### This is BROLTRON.

 Follow me on Tumblr if you want to help color him in.
I've got some good stuff brewing behind the scenes here (hint: it's more advice, drills, and challenge questions) but things might be a bit slower than usual this week. If you've got the itch for learning, though, and I'm not sufficiently scratching it, might I recommend TheYUNiversity? You won't be sorry you clicked.

### Weekend Challenge - June SAT edition

Good luck to all you warriors out there giving it your all one last time before the summer. May your June SAT scores be well worth all your hard work.

This weekend's challenge is a bit of a logic question. The prize if you get it: nobody in your testing room will assault your senses with unbridled body odor tomorrow morning. Awesome, right? I know.

If g, h, j, m, n, p, q, r, and s are all positive integer constants, and (g + h + j)(m + n + p + q)(r + s) is odd, what is the difference between the maximum number of even integers and the minimum number of even integers that could be in {ghjmnpqrs}?

UPDATE: We had two correct answers again this week! I'm going to have to start making these harder.

Solution below the cut.

### Review: My experience with Grockit

About a month ago I realized that, as someone who blogs about SAT prep, I really should have an informed opinion on what's out there in the online test prep space. With the June SAT prep cycle winding down, I've finally had time to sign up for a free trial and give Grockit a test spin for a few hours.

For the uninitiated, Grockit is web-based test prep built with adaptive learning technology (in other words: it assesses your skill level and tries to challenge you accordingly), garnished with a few rudimentary social gaming elements. Students are awarded ready-to-share achievements for things like "hot streaks" of questions answered correctly, and there's a "Multi-Player" practice system in which students tackle problems together, and are encouraged not only to chat with each other about answers, but to reward each other with points for being especially helpful.

I started with math, because I rightly assumed it would be the most fun, and then moved on to reading and writing. My thoughts about all three experiences follow. I should note that I didn't try every single thing Grockit has to offer; I didn't enlist the help of any of Grockit's tutors, for example (I thought that'd be weird). But I did enough poking around in the SAT sections of the site that I feel like I have a decent sense of what it's about. I should also state the obvious: I am a grown up and a test prep professional myself, so I didn't approach this exactly the same way a 16 year old might.