Some updates, and a challenge question

Phew! It's only been like three weeks since I last posted but in Internet time that's forever. Sorry about that. I've got about two weeks left in my Master's program, and I've been completely inundated by papers and presentations. My thesis is basically eating me alive. I seriously don't know how I'm going to get it all done. But that's school, right? Yay, school!

But I'm not posting to complain! I'm posting because an awesome challenge question popped into my head last night, and because I wanted to tell you what I'm planning to do as soon as school is over and I have more than a few minutes a day to devote to SAT pwning.

Coming soon

• A print version of the Essay Guide
• 7 Deadly Math Problems, Volume 2 (click here for Volume 1)
• Some other cool things I've been dreaming up that I don't even know what to call yet
• Some videos maybe? I dunno.

The point is that things are going to pick back up around here. Soon.

And now that challenge question

As always, first correct response in the comments wins a Math Guide. Full contest rules here, but the important ones are:

• You can't be anonymous because I need to be able to contact you
• You will have to pay for shipping if you live outside the US
• You can't win more than once, so please refrain from answering if you've won a contest in the past. 
• If your comment doesn't appear on the site right away, don't panic—you're just getting stuck in my spam filter because you're not registered. I receive comments in my email in the order they're posted. 

The sum of n consecutive integers is 1111. What is the greatest possible value of n?

Good luck!

Do some good, get some PWN (part 2)

Back when Hurricane Sandy pummeled the east coast, I offered to give people who donated to the Red Cross a free copy of my Math Guide. The offer generated a few bucks for charity, and I got to give away some books to people with generous hearts. I was pretty happy with how it worked.

I don't spend a great deal of time talking about why I started this site because I don't want to bore you, but a large part of my motivation was that I believe in a level playing field. I don't think access to good test prep—a proxy for access to education in general—should be limited by geography or wealth. The Internet provides an opportunity to mitigate both barriers, and 2+ years and thousands of posts later, here we are.

But by the time students want or need access to good SAT prep, they've already completed the lion's share of their primary education. I do SAT prep because that's what I'm good at, but inequality in education spans all grade levels, and I've been thinking a lot about ways I can leverage this site to make a difference for students long before they start thinking about the SAT.

Last month, I decided that I'd start donating 10% of the royalties I receive from Math Guide sales to education-related charities. One charity I really like is DonorsChoose.org, because it lets me decide which classrooms to donate to. (I favor projects involving early math education in low-income public schools.) Today, I've decided to take things a step further.

If you (or your parents) donate $50 or more to a project on the PWN the SAT DonorsChoose.org Giving Page, I'll give you a free copy of the Math Guide (US residents only). To participate, simply make your donation at the link above, then forward your receipt and your shipping address to mike@pwnthesat.com.

Challenge question: Triangles, parabolas, and earth tones

You know, It's funny. When I sit down to write one of these, I usually have a rough sketch in my mind of how the question's going to go. But half the time, the problem completely evolves into something else by the time I'm done with it. I'm not sure why that is. It could just be that I'm capricious, but it's probably because math comes into much sharper focus once it's written down. Which I only mention to gently encourage you to write everything down when you take tests.

Anyway, you know the drill by now. First correct response in the comments wins a copy of my Math Guide. Full contest rules here.

Note: If you're new to the site, your comment might not appear right away. Don't panic—I receive comments in my email in the order they're submitted. If you're the first to get it right, you'll win even if your comment doesn't immediately appear.

In the figure above, A is the vertex of f(x), B and C are the x-intercepts of f(x), F is the origin, D is the midpoint of AF, and E is the midpoint of DF. If f(x) = px² + 8 and the area of concave quadrilateral BDCE is equal to 8, what is p?

Good luck!

UPDATE: Peter got it first. Solution below the cut.

Free vocab prep via email

I recently heard from a student who programmed a neat little application to help himself study vocabulary, and then decided to share it with everybody else for free. My kinda guy.

I signed up to try it out, and after receiving a few words a day in my email over the last few mornings, I figured this might be something y'all might like to check out, too. So I'm giving him a little signal boost.

SAT Hot Words has a nice, clean presentation—no frills. It's just words, parts of speech, and definitions. You get 5 in your email every morning. That's it.

Note that I don't think this is all you should do to learn vocabulary, especially if your test is coming up soon. I just think it's a nice daily reminder to pay attention to the words around you, and to make at least a little progress every day. Enjoy!

Read a bit more, and sign up, at SATHotWords.com.

The problem with kitchen sink test prep

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.

Know when you're beat, and try something else.

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 being 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:

19. 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 the cut.

Challenge question: Functions are awesome

It's been a while since I've posted a challenge question. There are two main reasons: my own schoolwork, and the fact that most of my PWN time in the past month or so was dedicated to the Essay Guide. Now that I'm done with that, it's time to get back in the swing of things with some pull-your-hair-out, way-harder-than-the-SAT-but-work-the-same-muscles math questions!

As always, the first non-anonymous commenter to get this right will win a copy of my Math Guide. Note that I'll only count your first answer, which is emailed to me when you submit your comment—editing later doesn't work. Don't submit until you're sure! (Full contest rules.)

Let's do this.

The figure above shows the graph of f(x). If a is an integer such that
–6 < a < 6 and  f(a) = a, what is f(f(f(a – 3)) + f(f(f(f(2)))))?

MUAHAHAHAHA. Good luck!

UPDATE: It's over! This one fooled a bunch of people. Solution below the cut.