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College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 1 through 5 in the “calculator permitted” section. See questions 6 through 11 here, 12 through 15 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here.

Question 1 (link)

On the one hand, this is as easy as it gets. On the other hand, it already feels like a departure from what you see on the current SAT. This kind of question, where a scenario is explained in English and then the tester has to identify the right equation (or, in this case, inequality) in the answer choices, feels much more ACT-ish. But whatever. Stuff like this is what you can expect from “Heart of Algebra” questions.

The main thing this question is testing is whether you, the tester, can read that the “recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg),” and recognize that you want your answer to say, “≥ 1000,” not just “> 1000.” That’s because if a 20-year-old has exactly 1,000 mg of calcium in a day, he meets the recommended daily intake. He doesn’t need any more.

I doubt many folks reading this would be fooled by C or D, but it’s not hard to see why they’re in there as traps.

Question 2 (link)

Oh for crying out loud. There are 107 words in that question. 107! Ugh.

This is a question that doesn’t require you to be able to calculate anything. It just requires you to know, on a very superficial level, how a margin of error works.

Basically, you need to know that if you want to figure out the mean hours spent reading for the population of psych students at this university, you can ask a small sample of that population and arrive at a pretty good estimate of the actual mean of the whole population, within some margin of error. If you want to be more sure, you ask more people from the same population. The more people you ask, the smaller your margin of error. This makes sense if you think about it: say I wanted to know the average height of 100 people. I could feel pretty OK about my estimate if I only measure, say, 20 of them. I’d feel even better if I measured 60 of them. If I measured 99 of them, I’d be super confident that my mean is close tot he actual mean—even if the 100th person is a 7-foot giant, he can’t bring the average up very much! The larger the sample, the more sure you can be that the sample mean is close to the actual population mean.

So C is the answer because it increases the number of people in the experiment without changing the population from psych students to students from all across the university.

Question 3 (link)

More superficial statistics knowledge being tested here. We’re three questions into the “calculator allowed” section and we haven’t had to calculate anything yet. Given that this graph will be the basis of the next three questions, that means we’ll be through 5 out of 30 questions in this exercise without so much as needing to turn our calculators on. Make a mental note of that.

What the question is saying is that there’s a correlation between the length of a person’s metacarpal bone and that person’s height. It’s not a perfect correlation, which is why all the dots aren’t right on the line of best fit, but it’s a correlation. All the question’s asking you to do is count the dots that are more than 3 inches from the line in either direction. There are 4 dots that are at least 3 cm off the line of best fit, so the answer is B.

Things to watch out for on a question like this:

  • Axis scale. Every horizontal line represents 1 cm on this graph, like you’d expect it would, and that’s the axis you need to pay attention to for this question since it asks about outlier heights. The vertical lines represent 0.1 cm, though, and I bet a future question will test that somehow…
  • Dots they hope you miss. In this case, there’s that one all the way on the bottom left. Did you see that one? Not everyone will. The answer choice you’d expect to be there to catch the people who miss that dot, 3, isn’t there, but I don’t think we can take that as a sign that the new test won’t do that kind of thing with regularity.

Question 4 (link)

More on the same graph. Here, they’re asking about slope. Ah, slope! Our old friend! I hope we’ll be seeing more of you!

Slope, as you know, is a measure of rise over run. Here, given the axes, it’s a measure of height increase over metacarpal length increase. That’s precisely what choice A describes. If you’re having trouble distinguishing between choices A and B (choices C and D aren’t very tempting) pay attention to units. The metacarpal length axis only increases by 1 cm, total. From this graph, it’s easy enough to say that height would be expected to increase about 18cm with a 1cm increase in metacarpal bone length. It would not be easy to predict from this data how much longer someone’s metacarpal bone would be if they were 1 cm taller.

Question 5 (link)

Still on this silly graph. I was expecting these questions to get harder as they go, but this might be the easiest of the bunch! All you need to do is trace the graph. 4.45 cm metacarpal is going to be in the middle of 4.4 and 4.5—trace that up to the line of best fit, then look left to the height axis and see where you are. You’re right on 170!

The danger here, of course, is that you misread the scale on the bottom axis, and there is the 169 there in the answer choices to make you feel good about your mistake if you trace the 4.4 line up by mistake. Still, if you can read a graph, you’ll have no problem nailing this question.


Here we go, folks—another graphs quiz to entertain/torture you. Good luck! Note: This quiz will be available to everyone until 11/25/14. After that, it will be available only to registered Math Guide Owners. If you own a Math Guide, make sure you forward your receipt or other proof of purchase to so that you can…

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A few updates you might be interested in.

First, if you like discounted digital SAT prep books, then you’re in luck! For the next week—11/17 through 11/24—there will be crazy promotional prices on the digital versions of the Math Guide and Essay Guide. (Why? No particular reason—it’s just something I figured out how to do, so I’m doing it.) Anyway, for the entire week, the Math Guide will cost $1.99, down from $13.99. The Essay Guide will be a “Kindle Countdown Deal,” which means it will start at $0.99 and gradually get more expensive until it gets back up to its normal price of $8.99.

I think the Math Guide deal works for the whole world, although I can only guarantee that it’s going to work in the US. The Essay Guide deal is only for US customers. This is all a function of the stores the books are in (Google Play and Kindle, respectively). If you’re outside the US and the discounts don’t work, well, I’m sorry. I can’t help. :/

Second, just for fun, I’m going to print a limited sticker run featuring the image at the top of this post, which an old friend made for me. If you’d be interested in a sticker, fill out the form below. The first 100 responses will get a 3.5″ square sticker in the mail. Sorry, international friends—this is only for folks in the US, (unless you want a sticker bad enough to pay for international postage.)

(Update 11/25/14: If you requested a sticker, you should get it in a few days.)

It’s been a long time since I’ve done one of these, but I’ve been itching to give away a Math Guide for a while, and I woke up with this question in my head, so I figured I’d let ‘er rip.

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Full contest rules are here, but the important rules are below:

  • You must live in the US to get the book for free. If you live outside the US, you must pay shipping. If you are not willing to do that, please don’t enter.
  • You cannot have won a Math Guide in a contest in the past.

Assuming there’s a winner tonight (11/12/14), I will update this post tomorrow. Good luck!

UPDATE 11/13/14: Congratulations to Ashish, who made me worry that this question wasn’t hard enough when he got it so quickly. Then a bunch more people got it wrong (about 15% of people got it right during the contest) and I was relieved.

Now that there’s a winner, you can answer the question again (if you already attempted it). This time, when you submit, there’ll be a solution. So if you want to see how it’s done, just try the question again!

How’s everyone else doing on this challenge?

14% were graded 'PWND'
86% were graded 'Ouch'

Note: This quiz will be available to everyone until 10/31/14. After that, it will be available only to registered Math Guide Owners. If you own a Math Guide, make sure you forward your receipt or other proof of purchase to so that you can access all the exclusive Math Guide Owners stuff on this…

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How’s everyone else doing on this quiz?  …

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Math Guide 3rd editionThis is just a quick post to let you know that finally, at long last, the 3rd edition of the Math Guide is available now at It’s pretty cool—it contains two all new chapters, some new drill questions, and a bunch of other small updates that I think are cool but that no one else is likely to even notice. (I changed some fonts!)

For more information on the 3rd edition, click here.

Oh, one more important thing! Those of you that own previous editions of the book are entitled to view all the updates in this new edition online for free—just click here! (You need to be registered as a Math Guide Owner and logged in to view that page. If you own a Math Guide but aren’t registered, forward your receipt or some other proof of ownership, like a pic of the book with your name written on it, to and I’ll send you a code to get you all set up.)

It almost never fails when I’m working with a new student that the first time a question involving remainders rears its ugly head, we need to spend some time talking about what remainders are and how to find them. This happens fairly early in the process with me, intentionally. I always start with new students by working through the plug in and backsolve chapters in my math book, and the plug in chapter prominently features a remainder question. That’s what’s discussed in the video above.

A remainder, just so we’re clear, is what’s left over when one positive integer is divided by another. When 17 is divided by 5, for example, the remainder is 2: 5 goes into 12 three times—making 15—but there’s still 2 left over to get to 17. Remember long division?

long division

Remainders are always whole numbers, never decimals. However, there’s a handy shortcut that’ll allow you to convert decimals to remainders:

  1. Do the division on your calculator.
  2. If there’s no decimal, then there’s a remainder of 0. If there is a decimal, then there’s a remainder.
  3. Convert the decimal to a remainder by subtracting the part before the decimal point from the quotient you have on your calculator.
  4. You’ll be left with only a decimal. Multiply that decimal by the original divisor.
  5. Boom! You’ve got a remainder.

Here, I’ll show you. Let’s do 17/5 again. When you put 17/5 into your calculator, you get 3.4. Subtract 3 from that, and you’re left with 0.4. Multiply that by 5, and you’re left with 2—that’s your remainder! Note that if there’s a repeating decimal, you shouldn’t round it or you won’t get an integer remainder.

Here’s another example with the exact keystrokes I enter into my TI-83, and what the calculator displays.

What is the remainder when 52,343 is divided by 92?

52343/92 [ENTER]
Ans–568 [ENTER]
Ans*92 [ENTER]

Therefore, 52,343/92 = 568 R 87.

Cool right? Any remainder operations you’ll be doing on the SAT will be far less involved and easily done with long division, so you don’t need to memorize this trick, but it’s there for you if you want it.

Is that all?


If you end up having to deal with remainders on your SAT, you’ll almost definitely have to do more than just divide two integers and find the remainder. You’ll probably be asked (as you are in the problem featured in the video above) to figure out something about unknown constants given some information about remainders. When that happens to you, here are the things it’s important to know:

If n is divided by k and leaves a remainder of r, then n must be r greater than a multiple of k. For example, if a number divided by 8 leaves a remainder of 3, then that number must be 3 greater than a multiple of 8. You’ll do well to plug in a nice, low number that fits that description, like 8 + 3 = 11.

The greatest possible remainder when dividing by k is k – 1. For example, if you’re dividing by 10, then the greatest possible remainder you can get is 9.

When you divide a bunch of consecutive integers by the same divisor k, the remainders will form a repeating pattern of consecutive integers from 0 to k – 1. For example, when you divide a bunch of consecutive integers by 3, you’ll get a repeating pattern like: 0, 1, 2, 0, 1, 2, 0, 1, 2, … The pattern might begin with any of those numbers, depending on which consecutive integer you begin dividing by 3, but the pattern will be the same.

Think you’ve got this? Try a short quiz!

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How are people doing on this quiz?

45% got 3 right
40% got 2 right
12% got 1 right
3% got 0 right


I made a video today that I hope will help you understand something I get asked about an awful lot. The SAT loves asking a particular kind of Venn diagram question—not so common that it appears on every test, but common enough that high scorers need to know it. The test writers keep coming back to this kind of question because it’s so easy to trick students with it. Here’s an example:

  1. In a certain neighborhood, 11 children play baseball and 13 children play soccer. If 16 of the children play only one of the two sports, how many children play both sports?(A) 1
    (B) 3
    (C) 4
    (D) 8
    (E) 10

I solve this two ways in the video above (beginning at 4:05)—first I backsolve, then I do the algebra. My general preference is to backsolve a question like this because once you’ve practiced doing so it’s really fast and easy, but it’s good to know the algebra in case you ever come across such a question as a grid-in.

Think you’ve got this question type mastered? Try this short, 3-question quiz:

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How are people doing on this quiz?

56% got 3 right
23% got 2 right
7% got 1 right
14% got 0 right

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How’s everyone else doing on this quiz?

10% got 5 right
17% got 4 right
15% got 3 right
26% got 2 right
31% got 0 or 1 right


How’s everyone else doing on this quiz?  …

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A quick post today to focus your attention on something I think is really cool about this new iteration of—your stats page.


You can reach it from any page on the site as long as you’re logged in—just to the right of your username in the top right corner. Once you’re in, there’s a whole lot going on. I encourage you to explore every tab, but the two I really want you to pay attention to are the first two tabs.


The “Skills by Category” tab will tell you how well you’ve done by question category on all the quizzes you’ve taken. So, for example, if you wanted to see how well you’ve performed on right triangle questions, you can do that here.

The “Progress” tab (EDIT: now renamed the “My Quizzes” tab for clarity) links you to all the quizzes you’ve taken, and all the ones you haven’t. This is really useful, since I’m adding new quizzes all the time and you won’t find all of them listed on the home page. If you’re looking to take a quiz you haven’t taken before, this is where you should look.

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How’s everyone else doing on this quiz?

15% got 5 right
31% got 4 right
23% got 3 right
12% got 2 right
18% got 0 or 1 right


How’s everyone else doing on this quiz?  …

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You need to be registered and logged in to take this quiz. Log in or Register

How’s everyone else doing on this quiz?

12% got 5 right
20% got 4 right
19% got 3 right
16% got 2 right
33% got 0 or 1 right