SAT Math 1 & 2 Subject Tests

Chapter 2

Strategy

It’s easy to get the impression that the only way to do well on the Math Subject Tests is to become a master of a huge number of math topics. However, there are many effective strategies that you can use on the Math Subject Tests. From Pacing and Process of Elimination to how to use your calculator, this chapter takes you through the most important general strategies, so you can start practicing them right away.

CRACKING THE MATH SUBJECT TESTS

It’s true that you have to know some math to do well, but there’s a great deal you can do to improve your score without staring into math books until you go blind.

Several important strategies will help you increase your scoring power. There are a few characteristics of the Math Subject Tests that you can use to your advantage.

·        The questions on Math Subject Tests are arranged in order of difficulty. You can think of a test as being divided roughly into thirds, containing easy, medium, and difficult questions, in that order.

·        The Math Subject Tests are multiple-choice tests. That means that every time you look at a question on the test, the correct answer is on the paper right in front of you.

·        ETS writes incorrect answers on the Math Subject Tests by studying errors commonly made by students. These are common errors that you can learn to recognize.

The next few pages will introduce you to test-taking techniques that use these features of the Math Subject Tests to your advantage, which will increase your score. These strategies come in two basic types: Section strategies, which help you determine which questions to do and how much time to spend on them, and question strategies, which help you solve an individual question once you’ve chosen to do it.

SECTION STRATEGY

The following represents a sample scoring grid for the Math Subject Tests. The grids vary somewhat from test to test, so this is just a general guide.

Math Level 1

Math Level 2

A few points are notable:

·        While it is theoretically possible to score below a 350 on the tests, it usually requires a negative raw score (getting more than 4 times as many questions wrong as right). In practice, the tests are scored 350-800.

·        On some test dates, some scores are not possible (such as 420 on the Math Level 2 scoring given above).

·        The Math Level 2 scoring grid is very forgiving. Approximately 43 raw points scores an 800, and approximately 33 raw points (out of 50) scores a 700. The percentiles are tough, though; a 700 is only 61st percentile! The Math Level 1 has a more conventional score distribution.

Pacing

The first step to improving your performance on a Math Subject Test is slowing down. That’s right: You’ll score better if you do fewer questions. It may sound strange, but it works. That’s because the test-taking habits you’ve developed in high school are poorly suited to a Math Subject Test. It’s a different kind of test.

Think about a free-response math test. If you work a question and get the wrong answer, but you do most of the question right, show your work, and make a mistake that lots of other students in the class make (so the grader can easily recognize it), you’ll probably get partial credit. If you do the same thing on the Math Subject Tests, you get one of the four wrong answers. But you don’t get partial credit for choosing one of the listed wrong answers; you lose a quarter-point. That’s the opposite of partial credit! Because the Math Subject Tests give the opposite of partial credit, there is a huge premium on accuracy in these tests.


One Point Over Another?

A hard question on the Math Subject Tests isn’t worth more points than an easy question. It just takes longer to do, and it’s harder to get right. It makes no sense to rush through a test if all that’s waiting for you are tougher and tougher questions—especially if rushing worsens your performance on the easy questions.


How Many Questions Should I Do?

Use the following charts to determine how many questions to do on your next practice test.

Math Level 1

Math Level 2

As you improve, your pacing goals will also get more aggressive. Once you take your next practice test and score it, come back to this chart and adjust your pacing accordingly. For example, if you initially scored a 550, but on your second test you scored a 610, then use the 610–650 line for your third test, and you may score a 700 (or even higher!).

Your Last Test

For “your last test,” use
your last Math Subject
Test if you’ve taken one,
or a previous SAT Math
score. (You can also use a
PSAT Math score: Append
a 0, so that a 55 is a 550.)
If you don’t know these
numbers, take a guess.

Personal Order of Difficulty (POOD)

You probably noticed that the previous chart doesn’t tell you which questions to do on the Subject Tests, only how many. That’s because students aren’t all the same. Even if a certain question is easy for most students, if you don’t know how to do it, it’s hard for you. Conversely, if a question is hard for most students but you see exactly how to do it, it’s easy for you. Most of the time, you’ll find lower-numbered questions easy for you and higher-numbered questions harder for you, but not always, and you should always listen to your POOD.

Develop a Pacing Plan

The following is an example of an aggressive pacing plan. You should begin by trying this plan, and then you should adapt it to your own needs.

First, do questions 1–20 in 20 minutes. They are mostly easy, and you should be able to do each one in about a minute. (As noted above, though, you must not go so quickly that you sacrifice accuracy.) If there is a question that looks more time-consuming, but you know how to do it, mark it so that you can come back to it later, but move on.

Second, pick and choose among questions 21–50. Do only questions that you are sure you can get right quickly. Mark questions that are more time-consuming (but you still know how to do them!) so that you can come back to them later. Cross out questions that you do not know how to do; you shouldn’t waste any more time on them.

Third, once you’ve seen every question on the test at least once and gotten all the quick points that you can get, go back to the more time-consuming questions. Make good choices about which questions to do; at this point, you will be low on time and need to make realistic decisions about which questions you will be able to finish and which questions you should give up for lost.

This pacing plan takes advantage of the test’s built-in order of difficulty and your POOD. You should move at a brisk but not breakneck pace through the easy questions so that you have enough time to get them right but not waste time. You should make sure that you get to the end of the test and evaluate every question, because you never know if you happen to know how to do question 50; it may be harder for most students than question 30, but it just may test a math topic that you remember very well from class (or this book). Delaying more time-consuming questions until after you’ve gotten the quick and easy points maximizes your score and gives you a better sense of how long you have to complete those longer questions, and, after some practice, it will take only a few seconds to recognize a time-consuming question.

QUESTION STRATEGY

It’s true that the math on the Math Subject Tests gets difficult. But what exactly does that mean? Well, it doesn’t mean that you’ll be doing 20-step calculations, or huge, crazy exponential expansions that your calculator can’t handle. Difficult questions on the Math Subject Tests require you to understand some slippery mathematical concepts, and sometimes to recognize familiar math rules in strange situations.

This means that if you find yourself doing a 20-step calculation, stop. There’s a shortcut, and it probably involves using one of our techniques. Find it.

Random Guessing

If you randomly guess on
five questions, you can
expect to get one right
and four wrong.
Your score for those five
questions will be:

This isn’t very helpful.

Process of Elimination (POE)

It’s helpful that the Math Subject Tests contain only multiple-choice questions. After all, this means that eliminating four answers that cannot possibly be right is just as good as knowing what the right answer is, and it’s often easier. Eliminating four answers and choosing the fifth is called the Process of Elimination (POE).

POE Guessing

If you correctly eliminate
two answer choices and
guess among the remaining
three, you have a one
-in-three chance of getting
the right answer. If you do
this on six questions, you
can expect to get two right
and four wrong.
Your score will be :
.

That’s not a lot for six
questions, but every
point helps.

POE can also be helpful even when you can’t get down to a single answer. Because of the way the SAT is scored (plus one raw point for a correct answer and minus a quarter-point for an incorrect answer), if you can eliminate at least one answer, it is to your advantage to guess.

So, the bottom line:


To increase your score on the Math Subject Tests, eliminate wrong answer choices whenever possible, and guess aggressively whenever you can eliminate anything.


There are two major elimination techniques you should rely on as you move through a Math Subject Test: Approximation and Joe Bloggs.

Approximation

Sometimes, you can approximate an answer:


You can eliminate answer choices by approximation whenever you have a general idea of the correct answer. Answer choices that aren’t even in the right ballpark can be crossed out.


Take a look at the following three questions. In each question, at least one answer choice can be eliminated by approximation. See whether you can make eliminations yourself. For now, don’t worry about how to do these questions—just concentrate on eliminating answer choices.

21. If  = 1.84, then x2 =

(A)   −10.40

(B)   −3.74

(C)   7.63

(D)   10.40

(E)   21.15

Here’s How to Crack It

You may not have been sure how to work with that ugly fractional exponent. But if you realized that x2 can’t be negative, no matter what x is, then you could eliminate (A) and (B)—the negative answers, and then guess from the remaining answer choices.

28. In Figure 1, if c = 7 and θ = 42˚, what is the value of a ?

(A)   0.3

(B)   1.2

(C)   4.7

(D)   5.2

(E)   6.0

Here’s How to Crack It

Unless you’re told otherwise, the figures that the Math Subject Tests give you are drawn accurately, and you can use them to approximate. In this example, even if you weren’t sure how to apply trigonometric functions to the triangle, you could still approximate based on the diagram provided. If c is 7, then a looks like, say, 5. That’s not specific enough to let you decide between (C), (D), and (E), but you can eliminate (A) and (B). They’re not even close to 5. At the very least, that gets you down to a 1-in-3 guess—much better odds.

Can I Trust The Figure?

For some reason, sometimes
ETS inserts figures
that are deliberately
inaccurate and misleading.
When the figure is wrong,
ETS will print underneath,
Note: Figure not drawn
to scale.” When you see
this note, trust the text
of the problem, but don’t
believe the figure, because
the figure is just there to
trick you.

37. The average (arithmetic mean) cost of Simon’s math textbooks was $55.00, and the average cost of his history textbooks was $65.00. If Simon bought 3 math textbooks and 2 history textbooks, what was the average cost of the 5 textbooks?

(A)   $57.00

(B)   $59.00

(C)   $60.00

(D)   $63.50

(E)   $67.00

Here’s How to Crack It

Here, once again, you might not be sure how to relate all those averages. However, you could realize that the average value of a group can’t be bigger than the value of the biggest member of the group, so you could eliminate (E). You might also realize that, since there are more $55 books than $65 books, the average must be closer to $55.00 than to $65.00, so you could eliminate (C) and (D). That gets you down to only two answer choices, a 50/50 chance. Those are excellent odds.

These are all fairly basic questions. By the time you’ve finished this book, you won’t need to rely on approximation to answer them. The technique of approximation will still work for you, however, whenever you’re looking for an answer you can’t figure out with actual math.

Joe Bloggs

What makes a question hard? Sometimes a hard question tests more advanced material. For example, on the Math Level 1, trig questions are relatively rare before about question 20. Sometimes a hard question requires more steps, four or five rather than one or two. But more often, a hard question has trickier wording and better trap answers than an easy question.

ETS designs its test around a person we like to call Joe Bloggs. (Joe Bloggs isn’t really a person; he’s a statistical construct. But don’t hold that against him.) When ETS writes a question that mentions “a number,” it counts on students to think of numbers like 2 or 3, not numbers like −44.76 or 4π. That instinct to think of the most obvious thing, like 2 or 3 instead of −44.76 or 4π, is called “Joe Bloggs,” and this instinct—your inner Joe Bloggs—is dangerous but useful on the Math Subject Tests.


Stop and Think

Anytime you find an answer choice immediately appealing on a hard question, stop and think again. ETS collects data from thousands of students in trial tests before making a question a scored part of a Math Subject Test. If it looks that good to you, it probably looked good to many of the students taking the trial tests. That attractive answer choice is almost certainly a trap—in other words, it’s a Joe Bloggs answer. The right answer won’t be the answer most people would pick. On hard questions, obvious answers are wrong. Eliminate them.


Joe Bloggs is dangerous because he gets a lot of questions wrong on the Math Subject Tests, especially on the hard questions. After all, these tests are testing students on math that they’ve already learned, but it somehow has to make students get wrong answers. It does that by offering answers that are too good to be true: Tempting oversimplifications, obvious answers to subtle questions, and all sorts of other answers that seem comforting and familiar. Joe Bloggs falls for these every time. Don’t be Joe Bloggs! Instead, eliminate answers that Joe Bloggs would choose, and pick something else!

43. Ramona cycles from her house to school at 15 miles per hour. Upon arriving, she realizes that it is Saturday and immediately cycles home at 25 miles per hour. If the entire round-trip takes her 32 minutes, then what is her average speed, in miles per hour, for the entire round-trip?

(A)   17.0

(B)   18.75

(C)   20.0

(D)   21.25

(E)   22.0

Here’s How to Crack It

This is a tricky problem, and you may not be sure how to solve it. You can, however, see that there’s a very tempting answer among the answer choices. If someone goes somewhere at 15 mph and returns at 25 mph, then it seems reasonable that the average speed for the trip should be 20 mph. For question 43, however, that’s far too obvious to be right. Eliminate (C). It’s a Joe Bloggs answer.

49. If θ represents an angle such that sin2θ = tanθ − cos2θ, then sinθ − cosθ =

(A)   −

(B)   0

(C)   1

(D)   2

(E)   It cannot be determined from the information given.

Here’s How to Crack It

On a question like this one, you might have no idea how to go about finding the answer. That “It cannot be determined” answer choice may look awfully tempting. You can be sure, however, that (E) will look tempting to many students. It’s too tempting to be right on a question this hard. You can eliminate (E). It’s a Joe Bloggs answer.


Keep Joe Bloggs in mind whenever you’re looking to eliminate answer choices and guess, especially on hard questions.


SO DO I HAVE TO KNOW MATH AT ALL?

The techniques in this book will go a long way toward increasing your score, but there’s a certain minimum amount of mathematical knowledge you’ll need in order to do well on the Math Subject Tests. We’ve collected the most important rules and formulas into lists. As you move through the book, you’ll find these lists at the end of each chapter.

The strategies in this chapter, and the techniques in the rest of this book, are powerful tools. They will make you a better test taker and improve your performance. Nevertheless, memorizing the formulas on our lists is as important as learning techniques. Memorize those rules and formulas, and make sure you understand them.

Using That Calculator

Behold the First Rule of Intelligent Calculator Use:


Your calculator is only as smart as you are.


It’s worth remembering. Some test takers have a dangerous tendency to rely too much on their calculators. They try to use them on every question and start punching numbers in even before they’ve finished reading a question. That’s a good way to make a question take twice as long as it has to.

The most important part of problem solving is done in your head. You need to read a question, decide which techniques will be helpful in answering it, and set up the question. Using a calculator before you really need to do so will keep you from seeing the shortcut solution to a problem.

Scientific or Graphing?

ETS says that the tests
are designed with the
assumption that most
test takers have graphing
calculators. ETS also says
that a graphing calculator
may give you an advantage
on a handful of questions.
If you have access
to a graphing calculator
and know how to use it,
you may want to choose
it instead of a scientific
calculator.

When you do use your calculator, follow these simple procedures to avoid the most common calculator errors.

·        Check your calculator’s operating manual to make sure that you know how to use all of your calculator’s scientific functions (such as the exponent and trigonometric functions).

·        Clear the calculator at the beginning of each problem to make sure it’s not still holding information from a previous calculation.

·        Whenever possible, do long calculations one step at a time. It makes errors easier to catch.

·        Write out your work! Label everything, and write down the steps in your solution after each calculation. That way, if you get stuck, you won’t need to do the entire problem over again. Writing things down will also prevent you from making careless errors.

·        Keep an eye on the answer choices to see if ETS has included a partial answer designed to tempt you away from the final answer. Eliminate it!

Above all, remember that your brain is your main problem-solving tool. Your calculator is useful only when you’ve figured out exactly what you need to do to solve a problem.

Set It Up!

Some questions on the
Math Subject Tests can be
answered without much
calculation—the setup
itself makes the answer
clear. Remember: Figure
out how to do the problem
with your brain; then do
the problem with your
calculator