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Showing posts with label Math. Show all posts
Showing posts with label Math. Show all posts

July 17, 2020

SAT Math Level 2 Subject Test: The Best Prep Books

Update: I've updated the errata section for my review of 320 SAT Math Subject Test Problems - Level 2 (Steve Warner).

The Official SAT Subject Study Guide: Math Level 2

This book has four official College Board Math Level 2 practice tests.

Tests 3 and 4 are the same as the tests in the old edition of the College Board's Math Level 1 & 2 book. Test 3 is also the same as the test included in the Official Guide for All SAT Subject Tests.

The great news is that Tests 1 and 2 are completely new. They haven't been published before in any form, and they're even not the same as the two official but unpublished College Board Math Level 2 tests that are floating around.

Official material is a true confidence builder. Every question you get wrong contains skills you need to practice.

Most prep books have poorly written questions, answer key errors, and questions that are unrealistically easy, difficult, or off-topic. If you get questions wrong or run out of time on unofficial tests, you'll have trouble figuring out whether the fault lies with you or with the book you're using.

Most students who take practice tests for Math Level 1 and Math Level 2 find that they do better on Level 2 because of its generous curve. Based on the raw-to-scaled score conversion tables in the book, a raw score of 43/50, 40/50, 44/50 and 43/50 will get you perfect 800's on the first, second, third, and fourth Math Level 2 practice tests, respectively. To get a perfect score on Math Level 1, you usually need a raw score of 49/50 or 50/50.

The book only offers practice tests, warm-up questions, and answer explanations for the practice tests. It doesn't address strategy in any kind of detail.

SAT Math 2 Prep Black Book (Mike Barrett)

Barrett's method focuses on shortcuts and calculator tricks to get you through problems you're not sure how to do. He includes very detailed answer explanations for the problems in two College Board practice tests.

Since Barrett de-emphasizes content review, you may have trouble breaking above 750. It's great, however, if you use it in conjunction with another book that teaches you how to solve problems more traditionally.

This book's greatest strength is its focus on strategic guessing. It's faster to cross off three or four answer choices and choose from what's left than it is to solve problems traditionally. Unless you're very good, you'll need to use strategic guessing for the first forty questions in order to garner enough time for questions 41-50, which tend to be much harder.

Most of Barrett's answer explanations are one to two pages per question, so his book can really help if you don't have time to find a tutor.

Barrett's book doesn't contain any practice problems. You have to get the College Board's Math Level 2 study guide: all of his answer explanations are for the problems from that book.

Keep in mind that Barrett's answer explanations are tied to the old edition of the aforementioned study guide. That means you'll either have to use that edition or match his explanations up with tests 3 and 4 of the new edition yourself.

Cracking the SAT Math 2 Subject Test

This is a good all-around study guide. It contains content review, useful strategies, and decent practice tests.

The practice tests have no answer key errors. They're not quite the same as official practice tests (some of the problems lack elegant solutions and will take you longer than 30-60 seconds to solve), but the differences aren't large enough to keep you from getting an 800 on the real thing.

The book's helpful content review chapters can keep you from feeling lost when you're trying to re-learn your entire high school math curriculum.

The Princeton Review is all about giving you what you need and not one iota more. It's a good idea to supplement this book with the extra practice material in Steve Warner's book.

320 SAT Math Subject Test Problems - Level 2 (Steve Warner)

I use this as a textbook in my own tutoring for students who score 700 or higher on practice tests. It's a huge set of practice problems with detailed answer explanations.

The practice material is very similar to real Math Level 2 tests.

The problems in this book are arranged by topic and difficulty level, so students who don't need any content review can jump straight to the chapters that contain what they want to work on.

The answer explanations provide more than one way to do each problem, and the fastest method is marked with a star.

The book doesn't provide any content review. Dr. Warner does define terms like range and domain in his answer explanations, but his book doesn't have an index. You'll need to label important pages with Post-It notes.

In order to finish on time, you should first eliminate as many answer choices as you can and then decide whether to solve each problem in thirty seconds, guess from the remaining answer choices, or skip the problem and come back later. Dr. Warner's answer explanations don't talk about eliminating answer choices, so you may find that you run out of time if you solve problems the way he does.

If you're scoring below 700 on practice tests, start with the Princeton Review's book and come back to Dr. Warner's book later.

#108 on page 101 has two correct answers: (D) and (E).

The answer to #65 on page 164 is 55, which is not one of the answer choices.

The answer to #98 on page 173 is (A), not (B). Since the first term is k0, k4 is actually the fifth term in the sequence.

The answer to  #99 on page 173 is 13/4, which doesn't match any of the answer choices.

The answer to #156 on page 140 is 9.54, not 8.43. Dr. Warner arrived at the wrong answer because he plugged in 115° for ∠SOT instead of 145°.

#16 on page 149 doesn't give you enough information to solve the problem. Assume that the heights of the two cones are equal, and you'll get the correct answer.

The answer to #112 on page 176 is (E), not (A). The complex number z + 2 can be in either Quadrant I or Quadrant II, depending on how big its real component is, so i(z+2) can be in either Quadrant II or Quadrant III. In fact, if z = -2+i,  the answer ends up lying on y-axis, which isn't in any of the four quadrants.

The answer to #135 on page 184 should be "none of the above." Since a square root can't be negative, f(g(x)) will never be equal to -1, and the quantity a + b is an imaginary number.

#156 on page 192 should read 0 < θ < π/2, not 0 < x < π/2.

#160 on page 193 is written in an unclear way, as it's not evident whether the order of the positive integers matters when you're adding them together.

Barron's SAT Subject Test: Math Level 2

Barron's practice tests are harder than real College Board tests, and I'd only recommend them if you really want to challenge yourself.

That said, I've changed my view on this book over the past couple of years. Most of my students want a perfect 800 on the test and constantly seek out difficult practice questions.

If you really like math and think the hardest questions are the most fun, even when the answer explanations aren't perfect, this could be the right book for you.

If you decide to try the practice tests, add 50 points to your score in order to compensate for difficult (and, occasionally, poorly written) questions.

Barron's guides tend not to change much from one edition to the next. For example, except for question 18, Model Test 1 is basically the same in the 10th and 11th editions. You can buy a used 10th edition for five dollars if you want.

Because Barron's is a major publisher, you can find its books at the public library. That's not the case for self-published books like Steve Warner's and John Chung's.

The pro listed above is also a con: errors tend not to get fixed from one edition to the next.

You have to be proactive in order to check your work, as the answer explanations are short, and it may be hard to tell whether a question is badly written or if you've simply answered it incorrectly.

Ivy Global's Online Math Level 2 Practice Test and Answer Explanations
Ivy Global, which has published fairly accurate SAT practice tests, has recently released a Math Level 2 practice test.

Their material is pretty challenging and includes topics that the real Math Level 2 test doesn't cover. That said, the answer key is accurate, so go ahead and take their test if you need the extra practice.

Dr. John Chung's Mathematics Level 2

This book has problems that are much harder than the real thing and covers some obscure topics. The problems are all doable in 30 seconds or less, though, so they're hard in a way that may be helpful if you're already scoring 800 and want some extra practice.

Only two official practice tests have been released for Math Level 2, so you may need the extra practice in this book if you absolutely must get an 800 on your test.

Though the book tests some obscure concepts that rarely show up on official tests, you could see those concepts in your precalculus class. The difficult problems in this book might be fun if you're obsessed with math.

The practice tests are very difficult, so don't treat your scores from this book as accurate diagnostics.

Books to Avoid

The Arco and McGraw-Hill books contain inaccurate questions and answer key errors.

Going for a Perfect Score

A raw score of 44/50 will usually get you a perfect Math Level 2 score. Even after the test deducts a quarter of a point for every question you get wrong, you can afford to miss five of the fifty problems. That's like getting an A-minus on an advanced high school math test.

The books above contain everything you need to get an awesome score, but if you'd like personalized help, you can sign up for in-home or online tutoring.

May 18, 2019

How to Update Your TI-84 Plus CE's Software

Your TI-84 Plus CE calculator has some really cool features, including a circle/ellipse/hyperbola grapher (Conics) as well as a polynomial solver and system of equations solver (PlySmlt2):

If you hit the Apps key and don't see the menu above, you'll need to upgrade your calculator's software.

The current OS software version is 5.3. Use the instructions below to check which version you currently have installed:

To upgrade to the latest OS version, you'll need to connect your TI-84 to your computer using the mini USB cable that came with your calculator. (Old digital cameras/camcorders/GPS systems as well as other Texas Instruments calculators use the same type of cable, so you may be able to find one lying around the house.)

You'll then need to download two pieces of software from the Texas Instruments Web site:

  1. TI Connect CE software for Windows (or TI Connect CE software for Apple Macintosh computers)
  2. TI-84 Plus CE OS and apps bundle

Install the Connect CE software and open the installed application. You'll see a screen that looks like this:

Connect your calculator to your computer using the mini USB cable I mentioned earlier.

Then go to the Actions menu on your computer and choose Send OS/Bundle to Calculators.

Navitate to the TI-84 Plus CE OS and apps bundle that you installed earlier and open that file.

The update process will take a few minutes, after which you'll able to use all of the apps listed in the menu below. Enjoy!

August 24, 2017

SAT Math Level 2: Seven Key Strategies

Here's a list of strategies to get your Math Level 2 score into the 750-800 range.

  1. Skip questions as often as possible. If you can't figure out how to solve a problem within 10 seconds, skip it and come back after you've finished question 50. Remember that there's a 0.25-point guessing penalty, so you can miss 5 questions or skip 7 and still get a perfect 800.
  2. Start by eliminating choices that are clearly wrong. If you can cross of three or four choices, don't bother to solve unless you see a very quick 30-second solution. Smart guessing is faster than solving.
  3. The extra time you gain from guessing will keep you from feeling rushed near the end. Don't rush on questions. It's better to skip two and do two carefully than to rush on four.
  4. Remember to use your calculator's functionality. Most graphing calculators can do inverse trig functions, imaginary numbers, absolute value, systems of equations, and conic graphing.
  5. Use your calculator to store numbers into x and then regenerate complicated expressions so that you don't have to keep typing those expressions over and over. Storing into x also eliminates having to use parentheses when you plug in negative numbers. (In order to re-create an expression you've already typed in, click the up arrow until the expression is highlighted and then press Enter twice.)
  6. Because you skip questions on this test (but not the SAT), hold off on bubbling until either the proctor calls, "5 minutes left!" or you finish question #50. That keeps you from making bubbling mistakes after you skip questions. It also ends up saving time. You can always go back and check the skipped questions after you bubble.
  7. The week before the test, focus on re-doing the practice tests you've already taken. Using the strategies above, you should start to get scores in the 750-800 range.
If you have time to study, check out my list of the best Math Level 2 prep books!

May 30, 2017

Quadratic Sequences: How to Find Explicit and Recursive Definitions

Your math book probably doesn't explain how to get explicit and recursive definitions of quadratic sequences. Most of the solutions on the Internet involve systems of three equations. Fortunately, I've come up with something simpler.

Quadratic Sequences

A sequence is quadratic if the second difference, also known as the difference of the difference, is constant. In the picture below, the second difference is equal to 2, and it's constant, so the sequence is quadratic.
Note that the first difference is just the slope of whatever quadratic function the sequence comes from. If the first difference is the slope, that means the second difference is the slope of the slope.

Getting Explicit Definitions

To get an explicit definition, we need to make the sequence above fit a quadratic function:

At this point, you've probably been told to create a system of three equations using f(1) = 5, f(2) = 10, and f(3) = 17 in order to solve for a, b, and c.

I'm happy to tell you that there's an easier way.

Astute Calculus AB students may already have noticed that the second slope is just the second derivative of f(n). In other words,

The second difference is equal to 2a.

(If you're not in Calc AB, write down the bold-faced words in the previous sentence.)

That means we can figure out that a = 1 just by looking at the sequence!

We an also figure out c just by looking. If we plug in n = 0, we get f(0) = c. In other words,

The constant c is equal to the n = 0 term of the sequence.

We can work backward from f(1) to get to f(0). Kudos to one of my students for pointing this out!

Again, we've figured out that c = f(0) = 2 just by looking at the sequence!

Since we know that a = 1 and c = 2, our closed definition is almost complete:

Now we can get b by plugging in one of the terms from the sequence.

Let's plug in f(1) = 5:

After we simplify and solve, we'll get b = 2 along with the complete explicit form of the sequence:

You can check this definition by regenerating the original sequence starting at n = 1. It works!

Here's a quick summary of what you need to know to get the explicit form of a quadratic sequence:

  1. The second difference is equal to 2a.
  2. The constant c is equal to the n = 0 term of the sequence.
  3. Get by plugging in one of the terms from the sequence. 

Getting Recursive Definitions

Here's the sequence again in case you need it:

The recursive definition of a quadratic sequence has the form

The first part of the definition is the first term of the sequence: f(1) = 5. It's easy, but don't forget to write it down when you do problems on your test!

The second part is almost as easy. Remember that the second difference is equal to 2a, so just put the second difference in front of n. (If you're starting with the explicit form, you can just multiply a by 2 and stick the resulting number in front of n.)

Now we can just plug in terms from the sequence and solve for the constant d:
Knowing d gives us the complete explicit definition:

f(1) = 5
f(n) = f(n-1) + 2n + 1

You can check this definition by regenerating the original sequence starting at n = 2. It works!

You can also check the recursive definition intuitively. Remember that the first difference is the slope of the sequence. Our recursive definition says that the slope should be 2n + 1, which matches the first difference we got from inspection. The second difference should be the slope of the slope. The slope of the slope is just the slope of 2n + 1, which is 2, which also matches the second difference we got by inspection. If you're in Calculus AB, you'll note that we just took a second derivative.

Here's a quick summary of what you need to know to get the recursive form of a quadratic sequence:

  1. The first part of the definition is the first term of the sequence.
  2. Put the second difference in front of n.
  3. Plug in terms from the sequence and solve for the constant d.
I wish you hours of fun as you study for finals!

May 13, 2017

TI-84 Statistics Functions

The TI-84's Statistics functions are notoriously difficult to figure out. Fortunately, Dr. Claude Moore at Cape Fear Community College has published step-by-step instructions for each function.

This is what her explanation looks like for the normalcdf function.

WARNING: Some versions of the TI-84 have calculation errors if you enter extreme values for z, such as 1099. (The answer that pops out of normalcdf might be off by as much as 0.5!) It might be tempting to do this in order to capture every bit of area under the bell curve, but for all practical purposes, z = -1000 will work just as well as z = -1099 without giving you a calculation error.

January 31, 2017

Khan Academy's SAT and Math Missions

This post contains a list of links to Khan Academy's excellent set of interactive classes.


If you're really dedicated, it's possible to learn all of high school math through Khan Academy. Look, Ma: no teacher!

I personally used Khan Academy's math missions (Geometry, Algebra II, Trigonometry, Precalculus, and High School Statistics) to prep for a perfect score on the SAT Math Level 2 subject test. If you prefer to study on paper, check out Steve Warner's Math Level 2 workbook.


Algebra Basics

Algebra I


Algebra II



High School Probability and Statistics

Differential Calculus

Integral Calculus

All Math


Khan Academy's SAT program

Khan Academy provides good SAT practice, but its answer explanations aren't always helpful. SAT Math problems always have more than one solution, and the fastest solution takes 30 seconds or less. Khan Academy's explanations only include one solution per problem, and it may not be the one that is fastest or easiest for everyone.

Its answer explanations for the SAT's Critical Reading section can be particularly incomplete and confusing. To be fair, I've looked at most of the prep books for the new SAT, and all of them have this problem to some extent.

It's a good idea to supplement Khan Academy's test prep program with Erica Meltzer's Complete Guide to SAT Reading and Steve Warner's SAT Math course.


These missions are still under development. There's helpful practice here, especially for AP Statistics, but there's not enough material to learn everything on your own without using a separate textbook.

AP Statistics