SAT 1600

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October 24, 2020

Calculus: TI-Nspire CAS Instructional Videos


Update: The CX II CAS version of the calculator is now available; I've updated the Amazon link accordingly.

The TI-Nspire CX CAS calculator is notoriously difficult to learn. I've posted a list of videos below to help you learn the calculator and will add to the list over time.

Keep the following points in mind if you haven't bought the calculator yet:
  • The Nspire has a regular (CX) version and a more powerful (CX CAS) version. Make sure to get the CAS version, as the regular one doesn't have the algebra-solving features the Nspire is known for.
  • The Nspire will make it easy for you to check your work, as it will solve most math problems if you set them up correctly.
  • Most calculus teachers allow the use of the Nspire, but their tests may include non-calculator sections. In any case, if you need to show your work to receive credit, you'll have to know how to do problems by hand.
  • It's allowed on the SAT and on any SAT Subject Tests and AP tests that will let you use a calculator.
  • It's not allowed on the ACT.
  • The Nspire has a steep learning curve, so don't buy one unless you can devote several weeks to learning it. If you're going to use it on a standardized test, where speed really matters, plan to spend a couple of months getting used to your calculator.
  • You can switch quickly between the main calculator and graphing screens using the button with the picture of a calculator on it. (It's to the left of the left arrow key.)
  • Press the book key (above the division symbol) to get all of the calculator's functions listed in alphabetical order. If you want to learn how to use a function, type the first letter of its name, then scroll down and highlight the function you're interested in. The bottom of the screen will show you what you have to put in between the parentheses to get a function to work.
    • For example, the polyRoots function needs (Poly, Var) inside the parentheses. That means that you have to type a polynomial in, then a comma, and then the name of the variable you want to solve for. The Nspire will automatically set the function to zero and solve for that variable.
  • Menu > Analyze Graph > dy/dx is a quick way to get the derivative at various points of a function you've graphed. Mouse over various points on the graph, and the derivative will show up as a light gray number.
  • The numerical solve feature (Menu > Algebra > Numerical Solve) doesn't work correctly if the equation you're solving has more than one answer. For example, nSolve(x2=81,x) gives you 9 instead of {-9,9}. 
  • Texas Instruments publishes an Nspire reference guide that you can download and print out. Open the file and hit control-F on your computer keyboard to search.
  • If you're not planning to take AP Calculus, I suggest getting a different calculator. Check out my calculator review page for more information.

Commonly Used Keys
  • The Catalog key (right above the multiplication key) is an easy way to access the calculator's advanced functions quickly.


The Blue Keys


The Pi Key



Storing and Deleting Variables
  • Storing a value into a variable and then typing (or pasting in) an expression is a very fast way to solve problems.
  • If you store a value into x and then take a derivative with respect to x, your calculator will display the answer as a number instead of as an expression in terms of x. This can be a curse rather than a blessing if you actually wanted the value in terms of x.
    • For example, 7→x (7, ctrl, var, x) stores the value 7 into the variable x. Your calculator will think of x as "7" instead of thinking about it as a letter.
      Typing 100x will then return 700.
      d/dx (x2) will return 14.
    • If you then erase the variable (Menu > Actions > Clear a-z), your calculator will think of x as a letter again.
      Typing 100x will give you 100•x.
      d/dx (x2) will return 2•x.
  • If your calculator "breaks" and refuses to display answers in terms of x (showing you only numbers instead), you probably stored a number in x and forgot to delete the variable.
    • "If you plan to do symbolic computations using undefined variables, avoid storing anything into commonly used, one-letter variables such as a, b, c, x, y, z, and so on." (Nspire reference guide, page 234 / PDF page 238)
  • Sometimes it's hard to tell whether the answer your calculator spits out is the same as what you got solving by hand. If that happens, plug in 2 for x for both the calculator's answer and your own answer and make sure that the results match.
    • You can do this by storing 2→x (2, ctrl, var, x), then scrolling to highlight the answer your calculator gave you, then hitting Enter twice to evaluate that expression with the assumption "x=2". Notice that your calculator will spit out an actual number this time instead of an answer in terms of x; you can compare that number to what you'd get by plugging 2 into the answer you derived by hand.
    • Remember to erase the variable (Menu > Actions > Clear a-z) when you're finished! If you forget, your calculator will assume that x=2 for all future calculations, not necessarily an assumption you want to make.


Converting Between Fractions and Decimals



Using Copy/Paste Features to Avoid Re-Typing Functions
  • The easiest way to copy and paste is to use the "up" arrow key to scroll up and highlight some text. Hit the Enter key in order to automatically copy and paste that text into a new line that you can then edit.
  • Control (blue key)-C copies text, and control-V pastes it. Your calculator's keyboard isn't labeled with copy and paste functions, but they function like they would for a computer. It's a useful way to copy text and paste it inside an integral or derivative symbol.
  • Watch the video below if you're interested in using Copy/Paste features to store notes on your calculator.


Defining a Function and Finding Its Derivative



Finding the Equation of a Function's Tangent Line



Implicit Differentiation


  • Note: You can also do implicit differentiate without using the impDif function. Use the Catalog key (located above the multiplication key), choose d/dx, and type your equation in.
    • For example, you can implicitly differentiate the volume of a cylinder with respect to time this way:
      d/dx ( v(t) = πr(t)2h(t) )

      You must remember to explicitly state which letters are functions. For example, v(t) tells your calculator that v is a function of t. The example below will not work, since we've forgotten to tell the calculator that v, r, and h are functions of t:
      d/dx ( v = πr2h )
    • If you do implicit differentiation this way, you must remember to clear your variables using Menu > Actions > Delete Variable or Menu > Actions > Clear a-z. If you have a number stored in the letter v, for example, and haven't cleared variables from your calculator's memory, your calculator will think of v as a number and not a letter even if you explicitly type v(t).
    • Sometimes your calculator will give you a huge mess as the answer. You can either algebraically rearrange the mess to confirm that it's the same as the answer you got when you solved by hand, or you can plug a number like 2.5 into both your calculator's answer and your own answer to make sure you get the same answer.
      • It's relatively easy to plug numbers in once your calculator has given you an answer. Store 2.5→x using cntrl-var, then use the arrow keys to highlight the expression you want to copy, and hit the enter key to paste it. Your calculator will re-evaluate the expression assuming that x=2.5.
      • You must remember to delete the variable when you're finished (Menu > Actions > Clear a-z). If you don't, your calculator will assume that you mean 2.5 every time you type x, a result you almost certainly don't want.

Differential Equation Solver




August 26, 2020

Free Sessions for September 2020 (Quantitative Finance)

Recent large losses in standard stock/bond portfolios emphasize the need for greater diversification and risk control. At the same time, students are quarantined at home without access to the educational resources they're accustomed to.

As a result, I'm offering free weekly quant finance sessions for the month of September. Please contact me using the form at the bottom of this page for more information.


Students who love math and have a deep-seated interest the stock market will want to check out the curriculum for my Quantitative Finance course, developed from thirteen years of experience reading finance research and four years working as a quant.

I've recently started trading a combination of strategies from the course, including risk parity, long/short futures factors, long/short equity factors, and long/short volatility. Here's the live performance (March 2019 through September 2020):

Return (CAGR): 16.6%/year  
CAPM Alpha: 18.5%/year  
AQR alpha: 8.6%/year (controlling for SMB, HML-Dev, Mom Large, QMJ, and BAB)
Sharpe ratio: 1.00  
Sortino ratio: 3.13 = 2.21√2  
CAPM Information Ratio: 4.22

Take a look at what others in the finance community are saying about my work:
"Just noticed that @ReformedTrader is bizarrely under-followed. He's curated, quoted, pasted, summarized, analyzed, organized and synthesized well over a hundred papers and articles on academic finance. Honestly, wtf are y'all reading if you aren't following his threads?" Adam ButlerCIO at ReSolve Asset Management (here as well)

"@ReformedTrader has put together an awesome series of Twitter “moments” that highlight research on risk premia, style premia, seasonality, and craftsmanship. Dig in." Corey HoffsteinCIO at Newfound Research and a member of Investopedia's Top 100 in finance (here as well)

"Gotta hand it to @ReformedTrader for his consistency in posting quality quant finance research links. Everyone who is interested in quant finance should follow him. Hidden gem." Pravit Chintawongvanich, Wells Fargo equity derivatives strategist

"Wow, good stuff. I didn't even know the Moments could be used like this. Really great reference and shows the power of info sharing and knowledge building on Twitter." Justin Carbonneaumanaging partner at Validea Capital Management

"Read this thread and become an expert on the quality factor. Fantastic work by @ReformedTrader." Chris Cain, Quantitative Researcher at Connors Research and author of The Alpha Formula (here as well) 
"I enjoy these long threads Darren does on some great financial books." Jim O'Shaughnessy, founder of O'Shaughnessy Asset Management and author of What Works on Wall Street

"Wow, Darren knows the paper way better than I do now." Cliff Asness, billionaire co-founder of AQR Capital Management, which makes hedge fund strategies available to investors at mutual-fund-level fees 
Some of my social media followers include

The charts below, taken from a paper written by my friend Pim van Vliet, describe one of the strategies covered in the course.

Note the strong performance in each decade, including the dot-com bust (2000-2002) and 2008 crisis periods. The Conservative leg of the strategy owns low volatility, high momentum, and high buyback yield stocks, leading to more reliable returns than those of the market as a whole.

 

This chart is from The Conservative Formula: Quantitative Investing Made Easy, one of the papers covered in my Quant Finance course.

Please contact me about tutoring and mention the words "Finance Offer."

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Why are you interested in quantitative finance, and what are your eventual goals? (required)


August 15, 2020

SAT Practice Tests

Update: I've removed Practice Tests 2 and 4 from the list, since the College Board no longer recommends that students take those two tests.

Table of Contents

College Board SAT practice tests
Ivy Global SAT practice tests, including answer explanations
PSAT practice tests

Suggestions

The PSAT practice tests at the end of the list provide useful diagnostic scores until you hit 700 in either Verbal or Math. If you're worried that you'll run out of practice tests, start with the PSAT.

The College Board no longer recommends that students take Practice Tests 2 and 4, so I've removed those tests from the list below and replaced them with newly available Practice Tests 9 and 10.

If you find yourself making small mistakes or running out of time on one or more sections of the test, consider following an objective set of timing rules so that you won't feel rushed.

I strongly suggest printing practice tests out onto real paper. It's almost impossible to take notes, cross off answer choices, or double-check your bubbling unless you're working on paper!


College Board SAT Practice Test 1

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart (This document has the scoring instructions!)
Answer Explanations (These are the College Board's explanations. The SAT Black Book has better ones.)

I've written detailed answer explanations for some of the supporting evidence questions in SAT Practice Test 1.

I've also written a detailed answer explanation for #29 in section 4 (calculator-based math).

College Board SAT Practice Test 3

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart (This document has the scoring instructions!)
Answer Explanations (These are the College Board's explanations. The SAT Black Book has better ones.)

College Board SAT Practice Test 5

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 6

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 7

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 8

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 9

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 10

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations


IvyGlobal online SAT Practice Test 1

Test, Answer Sheet, and Scoring Chart (Print this out onto real paper!)
Online Scoring

Answer Explanations
Note: This practice test has an answer key error. The answer to Critical Reading question #18 is A, not D.

IvyGlobal's SAT questions are accurate, and their answer explanations are very well-written. Their practice tests are an excellent way to learn by doing, especially if you don't have a tutor.

IvyGlobal online SAT Practice Test 2

Test, Answer Sheet, and Scoring Chart (Print this out onto real paper!)
Online Scoring

Answer Explanations


College Board PSAT/NMSQT Practice Test 1

Test (Print this out onto real paper!)
Answer Sheet
Answers and Scoring Chart
Answer Explanations

College Board PSAT/NMSQT Practice Test 2

Test (Print this out onto real paper!)
Answer Sheet
Answers and Scoring Chart
Answer Explanations

IvyGlobal online PSAT Practice Test

Test, Answer Sheet, and Scoring Chart (Print this out onto real paper!)

Additional Practice

IvyGlobal's practice tests are almost as good as College Board tests, and the IvyGlobal answer explanations are detailed and complete. You'll find links to IvyGlobal's materials (11 practice tests in print and 2 online) in my list of recommended SAT books.

Use Khan Academy to practice for SAT Critical Reading. The answer explanations aren't great, and the practice questions haven't been tested as extensively as real SAT questions have, but it's still College Board material. If you really want to challenge yourself, you can prep for SAT Literature Subject Test and AP English Language, which make SAT Reading questions seem like child's play.

ACT English and SAT Grammar/Writing are nearly identical. The SAT gives you more time per question, but it also gives you a slightly higher proportion of passage interpretation and adding/deleting sentence questions. Khan Academy is also good here.

You can find several SAT Math practice resources in my list of recommended SAT booksKhan Academy also provides decent practice, but its answer explanations may leave you a bit frustrated.



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.

Pros
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.

Cons
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.

Pros
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.


Cons
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.

Pros
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.

Cons
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.

Pros
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.

Cons
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.

Errata
#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.

Pros
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.

Cons
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.

Pros
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.

Cons
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.

July 7, 2020

Timing Rules for the SAT and ACT

Update: I've added additional comments to the Math section.

The easiest way to avoid making "stupid mistakes" on the SAT and ACT is to slow down. Skipping steps and not reading questions carefully tend to lead to small mistakes. You can slow down if you have a way to keep track of time.

For example, on SAT Critical Reading, you have 13 minutes per passage (65 minutes for a total of 5 passages).

That means that if you know the multiples of 13, you can check the clock and make sure that every passage is finished by its appropriate multiple of 13 minutes. That means you need to have read the passage, done the questions, double-checked, and bubbled within the 13 minutes that you’re allowed.

Passage 1: 13 minutes
Passage 2: 26 minutes
Passage 3: 39 minutes
Passage 4: 52 minutes
Passage 5: 65 minutes

If you find that you can't finish the Reading section even with the timing rule given above, you can choose to skip the passage type that's hardest for you. For example, it's okay to skip the 1800's-era passage and to spend 65/4=16 minutes per passage instead of 13 minutes. If you do this, choose "D" for all of the questions in the passage you're skipping and don't even bother to read that passage.

Whenever you do a problem, whether it's math, grammar, and reading, read the question carefully, but don’t stare at it for more than 10 seconds. If a method for solving the problem doesn’t become apparent within 10 seconds, look at the answer choices and try to eliminate some. For about a quarter of the questions, you can eliminate three of the four choices immediately.

Math

Show all your work on math problems. Speed increases come from using fewer steps, not from skipping steps. (For example, solving a 3-4-5 triangle using ratios is faster than using the Pythogorean theorem. We don’t skip any steps, but we do solve the problem in one step instead of five.) Small mistakes come from either mis-reading the problem or by skipping steps. If you show all your work, you’ll know immediately if you re-read everything that you solved for the right variable and plugged the numbers in correctly.

When you’re done, read the problem carefully a second time. Doing this catches about 90% of small mistakes, including ones that involve setting up equations incorrectly, solving for the wrong variable, or forgetting to convert units (for example, hours to minutes). Double-check to make sure the problem was set up correctly, that you plugged the right variables in the right places, and (importantly) that you solved for the correct variable.

If you have a lot of time, do each question two or three different ways if it looks like there’s an easy opportunity to do so. Doing this will eliminate almost all small mistakes and nearly guarantee an 800 on the Math section. (One way to teach yourself creative ways to solve problems is to work through a practice book without using any paper: this will force you to look for simple solutions. I started trying this in May 2017 and have been doing it during actual tutoring since September 2018.)

I personally like to circle the variable I'm supposed to be solving for when I first read the question, solve the problem, and put a vertical bar next to the question when I've read it a second time. This allows me to mentally be finished so that I can move on to the next question. (If I've solved a problem two different ways, I also place a check mark next to the question to remind myself not to double-check it if I have time at the end of the test.)


Doing the Math section very quickly and going back to double-check very quickly doesn’t work. That’s because the test is purposefully designed to be difficult to interpret, and if you try to go through the problems too quickly, you’ll mis-read them. Even though you’re double-checking, you’re likely to mis-read twice if you’re in a rush. Again, it makes more sense to slow down and be careful than to rush through and double-check later.

Bubbling

Work on two pages at a time and don’t bubble until both pages are finished. This will (1) minimize the distraction of moving from the test page to bubble sheet and back again and (2) reduce errors from accidentally bubbling in a question that was skipped and then being off by one.

If you want, you can print each practice test out onto double-sided pages and put the pages in a binder. Your test will then have two pages that face each other, just like the booklets you'll be using on test day.

Timing Rules

The table below has timing rules you can use for each section of the SAT and ACT. If necessary, you can modify the rules if you plan to skip a passage or to go faster in order to spend more time on difficult material.

Section Total Time Number of Questions Time for each set of 10 questions Comments
SAT Critical Reading 65 minutes 52 questions 13 minutes Instead of checking the clock every ten questions, give yourself 13 minutes per passage. If there's a passage type that's particularly hard for you, try finishing the others in 12 minutes so you can spend 17 minutes on the difficult one.
SAT Grammar 35 minutes 44 questions 8 minutes Since this section of the test has four passages (11 questions each), you can give yourself 8 minutes per passage with 3 minutes left at the end of the test to check your bubbling.
SAT Math (no calculator) 25 minutes 20 questions 12 minutes

SAT Math (calculator) 55 minutes 38 questions 14 minutes

ACT English 45 minutes 75 questions 6 minutes

ACT Math 60 minutes 60 questions 10 minutes You have one minute per question on this section.

ACT Reading 35 minutes 40 questions 8 minutes Give yourself 8 minutes per passage. The science-related passage at the end is hardest for most people, but note that 8*4=32 and that you have 35 minutes to finish the test, so you actually have 11 minutes, not eight, to finish the final passage.
ACT Science 35 minutes 40 questions 8 minutes


In order to use these rules effectively, you need to have your own timing device. Most test centers don't have digital clocks, and even if you're good at reading analog clocks, there's no guarantee that the clock on the wall is going to to be in sync with the proctor's timing device, which will probably be his phone.

Finally, remember that these timing rules are designed to make the test less stressful for you. If you forget your watch or find it inconvenient to check the time, you can still do very well if you've practiced a lot and know that you can work quickly enough.

April 29, 2020

GRE: The Best Prep Books

Update: I've updated a few of the links to GRE practice materials.

If you're taking the GRE to get into grad school, take heart: the test is very similar to the pre-2016 "old" SAT. My list of recommend prep books, below, even includes manuals for the old SAT.

You survived the old SAT and did well enough in college to apply to grad school. With some practice, you'll do fine on the GRE, too.




The GRE is a computer-adaptive test, and PowerPrep software is the only place you can find computer-based official practice tests.

Pros
GRE questions are tricky and constantly force you to double-check your assumptions. Only official test questions allow you to safely make the assumption, "The problem is with me and not with the way the question is written or with the answer key."

Since there are only two practice tests, I recommend taking the first one before you start studying and the second one once you think you might be ready to take the GRE.

Cons
The pre-July 2017 version of the software has a user interface that's laughably 1990's. I had to look multiple times to find the button that allowed me to review my answers. If you're still a fan of Windows ME and Mac System 7, you might be okay with this.

The post-July 2017 version is Web-based. I made an ETS account just so I could access the Web-based version of Practice Test #1, but as of June 14, 2017, my attempts to log in to my ETS account have failed. (There's no error message; I just get returned to the login page, where I have to type in my user name and password over and over again.)



This is the only place you can get paper-based official practice questions. The old 2015 edition is nearly identical to the 2017 edition I've linked to above, so check the prices on both before making your purchase.

Pros
This resource has hundreds of practice questions, far more than the practice tests in the PowerPrep software do.

Cons
The practice material is great, but you may find the strategies and answer explanations unhelpful.



Manhattan Prep is the gold standard when it comes to GRE prep. This is their very long, very heavy book of practice questions and complete answer explanations.

Pros
If you've taken a practice test and just want hundreds of questions to practice with, this is the book for you. They're not official GRE questions, but they're close.

Cons
This book is so thick that you might have trouble keeping it open when you practice. The actual GRE only gives you scratch paper to write on instead of a printed test booklet, so you have to manage your practice scratch paper, calculator, and Manhattan Prep book at the same time.

Errata (2nd edition)
Ch. 11, #18 (p. 453): The answer is (B). The book has the correct answer explanation but says the answer is (C) instead of (B).


Manhattan Prep GRE Strategy Guides

You can buy all eight of Manhattan Prep's strategy guides as a package, but you only need to choose one of the eight to get the most important benefit: one-year access to Manhattan Prep's six computer-adaptive practice tests.

Pros
It's a lot cheaper to buy a $15 book than to pay $39 for the practice tests on Manhattan Prep's Web site. You can also try one of their practice tests for free if you create an account on their Web site.

Cons
Don't buy all eight of their strategy guides unless you have a lot of time to kill. Spend the majority of your time studying for the areas in which you need the greatest score gains.



This bank of GRE Quantitative practice questions contains accurate material and detailed answer explanations.

Pros
This book has the same format as Dr. Warner's SAT and ACT Math books. You can jump right in and start working without having to wade through preliminary reading.

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.

Cons
Most of the content review is in the answer explanations, so you can't treat this book like a textbook. You really have to engage with the material to receive the maximum benefit.

Errata
On page 77, problem 79 has a graph drawn in such a way that it's unclear whether you should include (G) as a possible answer. The book's answer, AEFG, is still correct, however.



Nova's GRE Math Prep Course

This is the closest thing I've seen to a GRE Quantitative textbook. It's very heavy on practice problems, though, and the problems for each topic are sorted into categories (Easy, Medium, Hard, and Very Hard).

Pros
You can get content review here without feeling like you're skipping the hardest questions on the GRE. In fact, the questions labeled "Very Hard" are more difficult than the questions you're likely to see on the GRE itself.

Cons
This book, like any math textbook, is pretty dense. Expect hard work without any entertainment!


Cracking the GRE Premium

This is a decent all-around study guide. It focuses on strategy, not content review, which is advantageous if you have less than two weeks to study.

Its strategies for Text Completion and Sentence Equivalence are excellent. Unless your vocabulary is already post-graduate level, you're going need those strategies to help you guess on sentence completion questions.

Pros
The book's strategies are well-written and clearly explained.

Cons
The practice tests may not feel 100% like GRE questions to you. Some of them are worded in confusing ways, and the answer explanations are cryptic enough to leave you wondering whether the bewilderment is truly your fault.

The paper practice tests aren't computer-adaptive, so the book itself won't give you a diagnostic GRE score. You have to take an online test or use PowerPrep software for that.


The Official SAT Study Guide (2009 edition)

SAT Prep Black Book (2015 edition, Mike Barrett)

When the old SAT died in 2016, ETS didn't throw all of those SAT words away. It's using them on the GRE!

The old SAT's Critical Reading section is an excellent source of semi-official GRE Verbal questions. The old SAT's question types don't match up precisely with the GRE's, but since both tests were written by ETS, you can still use the former to practice for the latter.

Mike Barrett's SAT Prep Black Book provides excellent strategies and answer explanations for official old SAT Critical Reading questions. I recommend going through pages 39-118 in conjunction with the 2009 edition of the Official SAT Study Guide.


Flocabulary: The Hip-Hop Approach to SAT-Level Vocabulary Building

If you want to ace Text Completion and Sentence Equivalence questions, you need to build your vocabulary.

I've said before that reading is the best way to build your vocabulary. That's definitely true for most standardized tests, including the post-2016 SAT, which test your knowledge of vocabulary in the context of longer passages.

The GRE, however, tests vocabulary using only one sentence at a time. You can get away with memorizing definitions and even with using flash cards.

That's why I'm recommending Flocabulary, which embeds the definitions for advanced vocabulary words into rap songs. Rhythm and rhyme have always made large amounts of text easier to memorize, and vocabulary definitions are no exception.

You can rip Flocabulary songs from the included CD onto your phone and then play them in your car. The Flocabulary book contains the lyrics to all the songs, definitions for the songs' vocabulary words, and multiple-choice practice questions.

Pros
This is the easiest way I've seen to memorize the definitions of words like myopic and recapitulate.

Cons
The stories in some of the songs don't feel quite finished. Phobia, for example, is a song about a guy who's scared to go outside because of a dilapidated house in his neighborhood. We get to the end of the song and never find out what happens to the house or whether the protagonist eventually rallies the courage to face his fear.

You're only going to get the superficial dictionary definitions of words, which may or may not help you in real life. You're doing this to get a vocabulary fix for the GRE, but you'll learn the words properly later by reading them in normal contexts, right?


The Yo Momma Vocabulary Builder

Instead of using music like Flocabulary does, this book relies on humor and silly imagery to help you learn word definitions.

Pros
Any learning tool that connects with preexisting knowledge in your mind is going to work better than flash cards. If funny works for you, go for it!

Cons
Like Flocabulary, the Yo Momma method divorces meanings of words from the ways those words are used in real contexts. You'll get a distorted, oversimplified understanding of vocabulary, but it might be enough to get you through the GRE.

If you're going to pick up a book, though, why not read something you're actually interested in and look up the vocabulary words in the process?

Essay Topics

ETS publishes lists of all the possible Issue and Argument topics.

Going for a Perfect Score

In theory, you can get a perfect score on the GRE if you study hard enough. A perfect 170 in Quant is only 97 percentile, and a 169/170 in Verbal is 99 percentile.

In practice, it's not just about studying and strategy. You have to have a strong enough vocabulary to successfully guess on all of the Verbal questions, no matter what words the test throws at you. Crutches like Flocabulary can help, but there's no way you can memorize all of the words you need in a couple of months.

Improving your vocabulary is a lifelong process, so the GRE may be less vulnerable to tips and tricks than we'd like.

April 4, 2020

ACT: The Best Prep Books

Update: I've added an Errata section to my review of For the Love of ACT Science.

The Official ACT Prep Guide, 2019-2020 Edition

This is a much-needed update addressing the weaknesses of the 2018 and 2016-17 editions. The ACT essay changed in fall 2015 and again (slightly) in fall 2016, so all of the practice essay questions in the older editions were out-of-date.

The ACT has gradually been getting harder over time, and the 2019-2020 edition addresses that by making its own practice tests pretty tough and then making the scoring curves pretty generous to compensate. The first three tests are identical to the ones in the 2018 edition. The fourth one appears to be ACT Form 74C, which was administered June 2017. The fifth test is ACT Form A10 (December 2017).

Pros
This is one of the best and most convenient sources of practice material you'll find. The book is heavy and unwieldy, but you can tear out the practice tests and bubble sheets and take them on a flat surface.

Cons
The scoring instructions are hard to find. You'll find answer explanations immediately after each practice test, but the actual answer keys, scoring instructions, and score conversion tables are at the back of the book. It's important to use the correct score conversion table for each practice test you take, as harder tests also have more generous curves. You can miss some questions and still get 36 if you get a difficult section; this tends to happen most often for Math and English.


ACT Prep Black Book, Second Edition (Mike Barrett)

If your score goal is 30-33 and you don't have a tutor, this is the book to start with. Barrett's answer explanations are very detailed, about five to ten paragraphs per question, and he uses only official ACT practice tests.

This new second edition addresses the weaknesses of the first edition: it uses the updated (2018 and 2019-2020) Official ACT Prep guide as its companion, it explains every question on every practice test instead of skipping questions, and it presents solutions in a much more organized way.

You have to get either the Official ACT Prep Guide or the Official ACT Prep Pack, since Barrett's book has answer explanations for that edition but doesn't include the questions themselves.

Pros
A top tutor can explain any officially released ACT question to you in as much detail as you want. Barrett's answer explanations are almost as good. At $28, it's a lot more affordable than hiring a real tutor.

His book really shines in its strategy suggestions for the Reading and Science sections, where the right approach to the test is more important than reviewing content. If you apply Barrett's advice with enough practice tests, you can eventually reach a Reading score of 36 and a Science score of 34.

His treatment of ACT Math focuses 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 41-60, which tend to be much harder.

The Math content review is limited. If you're shooting for a perfect score on the Math section, you really need to know guessing strategies and the "correct" methods in order to decide which approach is the fastest. Unless your math skills are already very strong, you might have trouble breaking above 32 without additional content review.

Barrett's English grammar strategies are very easy to learn: for example, he uses the term comma sandwich instead of the more common but technical term non-essential clause. This is a strength if you want to learn the material quickly but a drawback if you want a deeper understanding of the ACT's grammar rules.

Cons
Since Barrett chooses to use only official ACT practice questions, he doesn't include any practice questions in the content review chapters. You have to read the entire book and then take an official practice test. For this reason, I suggest treating your first test as un-timed practice.


Cracking the ACT with 8 Practice Tests

This is a good all-around study guide with content review, basic test-taking strategies, and six practice tests. It's a good choice if your final target score is 30 or lower.

Pros
For unofficial practice questions, these are pretty good. My students almost always buy this book and do some of the practice tests before deciding to hire me to review the material with them. I haven't seen the kinds of confusing, badly written questions and answer key errors that plague other unofficial study books.

The strategies in this book are basic and easy to apply.

Cons
The Princeton Review is all about giving you what you need and not one iota more. The ACT is a difficult test that requires students to think critically and pay attention to detail, and the basic strategies in this book may not be enough to reliably raise your ACT score above 30.

Its advice for ACT Reading, to skip directly to the questions without reading passages first, makes it hard for some students to grasp each passage's main point.


1,511 ACT Practice Questions

This is the best bank of unofficial practice questions I've seen. It's useful if your can't get access to official ACT practice tests.

Pros
For unofficial practice questions, these are pretty good. I haven't seen the kinds of confusing, badly written questions and answer key errors that plague other unofficial study books.

Cons
It's not a full prep guide. It has answer explanations but doesn't contain content review or organized strategy suggestions.



The Complete Guide to ACT English, Fourth Edition (Erica Meltzer)

This book is a winner: it has grammar rules, practice questions, and answer explanations. It's a great choice if you're shooting for a 30-36 in ACT English.

The new Fourth Edition adds a Parts of Speech Preliminary Exercise (p. 14), includes answer explanations for the practice problems (p. 247), and fixes some typos, including a major problem with the practice tests (incorrectly labeled problems).

Pros
If you understand every grammar rule Erica teaches, ACT English questions become objective, and you can tackle them like math questions. This doesn't mean they're easy, but it does mean that you can figure out why the wrong choices are actually wrong.

Erica also has some grammar quizzes on her Web site.

Cons
Erica's strategies are great for all of the question types except those that involve adding and deleting sentences. For those, read the Relevance and Purpose article on the College Panda's blog.

Her book is like a textbook (long and possibly boring). Decide now that you're going to be dedicated enough to read the whole thing, including the answer explanations at the end of the book.

Minor annoyances include
  • the way that practice questions are numbered (passage number, followed by question number)
  • the fact that you have to remember to flip to page 247 to grade the practice problems
  • the lack of any guidance regarding the use of the score conversion chart (p. 304).

The Complete Guide to ACT Reading (Erica Meltzer)

Another winner from Erica, this book has great strategies for managing time, skimming, taking notes, and answering tricky questions. It's a good choice if you want a 30-36 in ACT Reading.

Pros
Erica includes a lot of practice questions, including two full practice tests with answer explanations. Her book is more convenient than the ACT Prep Black Book, which requires you to buy an out-of-date edition of the Real ACT Prep guide.

Erica's Web site also offers several reading quizzes.

Cons
Erica's practice questions are good, but they're not as good as real ACT questions. You should supplement her book with real ACT practice tests or consider getting the ACT Prep Black Book instead, which has excellent answer explanations for official ACT Reading questions.



This unusual book addresses vocabulary skills that can help with multiple sections of the SAT and ACT.

Pros
Instead of drilling vocab words using flash cards, Erica groups words by their function in the English language and provides practice questions for those functions. This is a practical approach that will pay dividends in college later: words like hypothesis, tentative, and analogous often occur in science-related passages, so it makes sense to group them together.

Because Erica's focus is on practical reading, her book is equally helpful for ACT English and Reading passages. The section on passage-based vocabulary for science passages (pages 31-35) will even help you on the ACT's Science section.

Erica's Web site has a complete list of grammar rules and and reading/grammar quizzes.

Cons
If you already score higher than 700 on SAT Verbal and 30 on ACT English/Reading/Science, this book might be too easy. You'd benefit more from taking practice tests and reading 10th-to-12th grade level books to build an advanced vocabulary.


Ultimate Guide to ACT Math (Richard Corn)

This is the closest thing to an ACT Math textbook I've seen. It organizes content review and practice drills by topic. If you don't feel comfortable with high school math, start your prep with this book.

Students who get A's in school math often struggle with the ACT. A school test focuses on one chapter of your book at a time, and a good teacher tells you exactly what's going to be on that test. The ACT, on the other hand, tests knowledge that ranges from 7th grade to precalculus and includes Common Core material that not all students have seen yet.

Pros
Richard Corn's book is enough like a textbook to get you comfortable with the topics that are tested on the ACT, but it's not long enough to be truly intimidating.

Cons
The main strength of Corn's book, its textbook-style organization, is also its weakness. The real ACT won't tell you whether a particular problem is testing the standard-form equation of a circle, the area of a circle, or right triangles within the unit circle. It could potentially test all three topics at the same time!

To truly be ready for the ACT, you have to learn how to think on your feet. Corn's book is great if you need content review as a primer, but you'll want to graduate quickly to more advanced practice materials.


28 ACT Math Lessons to Improve Your Score in One Month: ADVANCED Course (Steve Warner)

28 ACT Math Lessons to Improve Your Score in One Month: INTERMEDIATE Course (Steve Warner)

28 ACT Math Lessons to Improve Your Score in One Month: BEGINNER Course (Steve Warner)

320 ACT Math Problems arranged by Topic and Difficulty Level (Steve Warner)

These excellent practice books can get your ACT Math score into the 30-36 range. They're banks of practice problems with detailed answer explanations.

Pros
The practice material is very similar to real ACT Math tests.

The problems 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.

Cons
Content review is minimal. 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.

If you need content review, start with Richard Corn's Ultimate Guide to ACT Math and come back to Warner's book later.

If you feel rushed on ACT Math practice tests - a common problem - you need to make answer choice elimination and guessing your primary strategy on the easiest 80% of the test. If you correctly eliminate four choices, the fifth one has to be right, even if you haven't solved the problem traditionally. Eliminating answer choices is usually fast and less error-prone than traditional solutions, but (unfortunately) isn't taught directly in Richard Corn's and Steve Warner's books.

Errata
#1 on page 86 of the Advanced book has three possible correct answers: A, D, and E.

#4 on page 121 of the Advanced book is unsolvable unless you assume that the marbles need to be close to evenly distributed between the boxes.

#9 on page 205 of the Advanced book has a confusing answer explanation. The answer key is correct, though.

#2 on page 242 of the Advanced book is unsolvable unless you assume that shape at the upper left of the picture is a semicircle. (You should not make assumptions unless they're explicitly stated on the ACT, so the book shouldn't expect you to, either.)

#8 on page 244 of the Advanced book is unsolvable unless you assume that the two triangles shown are right triangles. (Again, you should not be expected to make assumptions that are not explicitly stated in the problem.)

#11 on page 328 of the Advanced book should state that the pyramid has five isosceles triangular faces (not four) and needs to point out that the pyramid is a right pyramid (in which the height is perpendicular to the base).


For the Love of ACT Science (Michael Cerro)

Cerro's book is an excellent strategy guide that can get your ACT Science score into the 30-34 range. He goes over each question type in detail and provides drills, practice tests, and answer explanations.

Pros
It's hard to write good practice questions for ACT Science. I suspect this is because most teachers don't know how to read scientific literature. You have to read journals regularly to understand concepts like correlation and causation, experiment design, and hypothesis evaluation, and even then, most scientific articles read like alien writing.

Given the difficulties involved, Cerro does a great job putting realistic-looking questions together. If you want a dedicated book for ACT Science that's written by a tutor and not a big test prep company, Cerro's book is the only option.

Mike Barrett's ACT Prep Black Book also has excellent strategies for ACT Science, but it requires you to buy The Official ACT Prep Guide.

Cons
The pro is also a con: Cerro's attempt at the impossible, writing accurate ACT Science questions, results in a book that's very good but has some weaknesses. You might feel that a few of his questions and answer explanations are written in a confusing way. If that bothers you, get Barrett's book instead and stick to official ACT questions.

If you want a 34-36 in ACT Science but struggle with finishing the test on time, Cerro's book probably won't be enough. You'll need to work on using your background knowledge to identify the answer that's probably correct before looking at the passage and then use the passage to verify that answer you chose. I'm not aware of any book that teaches this strategy, but I use it in my own tutoring.

Errata
#34 of the Chapter 5 Test (page 89): There is not enough information to answer the question. Based on background knowledge, the answer should be G (methane), which is the answer in the book's answer key, but Student 3 (whom we are being asked about) might argue differently.


The Master Key to ACT Science (Hugh Hung Q. Vo)

There is a noticeable lack of advanced prep guides for ACT Science on the market. My own preference is that students work on building their background knowledge so that they can reduce their reliance on reading the (often long and confusing) passages.

I don't know of any prep book that will teach you how to do this, so if you can't sign up for tutoring, the next best thing is probably this Hugh Hung Q. Vo's book.

It's very dense and will require a lot of self-discipline to get through, as the author's purpose is to help you look at science passages and break down experiments as an actual scientist would. There's also less of an emphasis on background knowledge than I would prefer.

Pros
The table of contents clearly lists the types of questions that students struggle with. If you've taken and reviewed your own practice tests and know exactly what you need help with, this book can be a useful reference manual.

Cons
There's a lot of information packed into this book's pages, and you could easily get cross-eyed trying to figure out what's going on. Unless you're a very serious student, it's probably better to go with Michael Cerro's book.

ACT Essay Sample Responses

Because the ACT essay has changed over the past few years, old editions (pre-2019) of the Official ACT Prep Guide have the wrong essay instructions. If you have one of those older editions, you'll have to go to the ACT's essay page to see the most recent version. That page also has sample essays written by students along with comments on how the essays were graded. There are a total of six sample essays; click on the links in the horizontal, purple Sample Essays bar in order to see all six.

Here's a copy of the updated instructions as of June 2018. Note that instead of having to write about all three perspectives, you can pick only one, giving you the ability to write with more clarity and focus.

The test describes an issue and provides three different perspectives on the issue. You are asked to read and consider the issue and perspectives, state your own perspective on the issue, and analyze the relationship between your perspective and at least one other perspective on the issue. Your score will not be affected by the perspective you take on the issue.

Books to Avoid

Kaplan's series of ACT prep books generally focuses on material that's too easy. The initial diagnostic practice test may give you an inflated score that is unlikely to repeat itself in a real ACT sitting.

Working with Official Practice Tests

If you find unofficial questions to be inaccurate or confusing, you can still prep for the ACT using only official practice tests.
  1. Take a practice test under timed conditions.
  2. Score your test and clearly mark the questions you missed. Use a different symbol to mark the questions that you got right by guessing.
  3. Review the questions you missed or had to guesss on. Spend at least ten minutes on each one. You have to prove to yourself that each question objectively has one right answer and three incorrect ones.
  4. Make a list of the questions you're not able to figure out on your own. Use Mike Barrett's answer explanations, the Internet, or a tutor's help to get to the point where you can clearly identify one correct answer and three incorrect ones for every single question. Your job isn't done until you can do this.
  5. Repeat the process using a new practice test. Keep the cycle going until you're happy with your scores.
This method can work really well if you have a tutor. Do a practice test and hire a tutor to go over some of the answers with you, then spend time at home going over the remaining answers on your own. In your next session, you can go over any questions that are still confusing. When you and your tutor are satisfied that you fully understand the first practice test, repeat the process with a new test.

Going for a Perfect Score

Since the ACT is an established test, you have plenty of resources to draw on if you want to practice for a perfect score.

If you get two 35's and two 36's, the four scores will round to a composite of 36. In addition, difficult test sections give you free points, so you might potentially miss one or even two questions on a section and still get a 36 on it.

Take an official test to get a baseline score and then go through all the books in the list above. After you're done, do as many official practice tests as you can. Since the ACT gives you some leeway to get questions wrong, you should work on improving your speed and accuracy on official tests rather than on unofficial books with questions that are intentionally harder than the real thing.

Most of all, go easy on yourself if you don't make it. Colleges want students who will bring them glory, and the difference between a 34 and a 36 doesn't say much about that potential in the long run.