SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (32 page)

The rules for SAT
Math problems are pretty much the same whether you’re looking at Multiple Choice questions or Student-Produced Response questions.

In the Writing and Critical Reading sections of the SAT, you can usually
‘fake’ your way past a few unfamiliar words in a particular question by using the proper technique. But if an SAT Math question asks you about the number of prime factors in a set, there’s typically no way to answer the question without knowing what prime factors and sets are. The questions typically don’t have any context in the way the questions on the Critical Reading section might. You pretty much have to know the terminology, which is why we discussed all the concepts in the Math Toolbox.

You DON’T have to know any
geometric formulas, but you do have to know when to use particular formulas that the College Board provides. For example, the SAT may want you to realize that you need to find the area of a triangle, but it won’t ask you to know the formula. The test provides every single geometry formula that you need to answer every question (even though it might not always seem that way to the untrained eye). These formulas either appear in the resource box at the beginning of each SAT Math section, or they appear in the question itself.

All the math on the SAT Math section is relatively easy.

In advanced high school math problems, the solution to one problem might involve complex graphs, trigonometric expressions, fraction bars, and pi; they’re very complex problems, and they have very complex answers.

On the SAT, the solution is much more likely to be a plain old number like 12
, because the actual calculations that we do for an SAT Math question are usually very basic. The most challenging part of an SAT Math question will typically be figuring out what the question is asking you to do in the first place; actually doing it usually isn’t that hard after that.

If that sounds a little confusing right now, don’t worry—it’ll make a lot more sense after we look at some examples of real SAT Math questions together.

You can assume that every drawing is done to scale EXCEPT when the test specifically says otherwise. This is a very useful fact, because it sometimes lets you answer questions just by measuring things
, or even eyeballing them—you don’t need any math at all.

In the Math Toolbox, we went over every single mathematical concept the SAT might throw at you. You’ll probably find that you’re familiar with most of them, if not all of them, and the rest are relatively straightforward. Once you know these concepts, you can rest assured that they will be enough to answer
every single real SAT Math question
.

Perhaps the most important rule of all, from a strategic perspective, is that EVERY SINGLE MATH QUESTION can be answered in less than 30 seconds.

This doesn’t mean that you’re going to get the question wrong if it takes you longer—it just means you aren’
t going about answering the question in the easiest way. When you’re looking for a way to solve the problem, just remember that every single question is simple, no matter how complicated it may seem at first. When we run into questions we can’t figure out at first, which is guaranteed to happen to everyone, we need to train ourselves so that our instincts are to try to make things simpler, not more complex.

Unless one of the answer choices is that the question hasn’t provided enough information, each question must have all the information you need to choose the correct answer choice—no matter how much it might seem like that isn’t true on some questions.

The College Board puts a lot of thought into the wrong answers that it offers you for every multiple-choice question. Those wrong answers aren’t just randomly generated. Instead, each one is the result of certain mistakes that the College Board thinks students will make on a particular question. Imagine that you try to solve an SAT Math question and end up with the number 15 as your answer. Then you look at the answer choices and see that they only include 10, 12, 18, 24, and 30. In that case, you would know right away that you had made a mistake on the question, and you’d be able to start over and try to solve the question correctly. From the College Board’s standpoint, this would be like letting you get away with a free mistake, because you’d be able to realize what you’d done wrong and fix it.

To keep that from happening, the College Board does its best to include 4 wrong answers to each question that try to anticipate the mistakes that you’re likely to make.

This might seem pretty mean-spirited on the part of the College Board, but we can actually use it to our advantage as test-takers. Since the College Board tries to come up with wrong answers to tempt us into making mistakes—and since it has to do this in standardized, repetitive ways, just like everything else it does—then we can learn to use the concepts and relationships that appear in the answer choices to get an idea of what the question is actually asking about.

This will make a lot more sense after we talk about common patterns that we’ll encounter in the answer choices, and
after we go through some solutions from the Blue Book together. We’ll cover some of those common patterns starting on the next page, and then we’ll do the Blue Book solutions a few pages after that.

Most of the hidden patterns on the SAT math section have to do with using the answer choices to help you check your answers, where that’s possible. Looking at the answer choices can reassure you that you have the right answer (or help show you your mistakes so you can correct them).

Many students are surprised to find out that these patterns appear reliably and consistently in real SAT Math questions from the College Board, because they really have very little to do with actual math rules—they’re purely related to the test-design principles required by the standardization of the test. So
before we get into these patterns, I’d like to take a moment to remind you why they’re a part of the SAT.

As we discussed at the beginning of this book, it’s important to remember that the SAT is not a normal test. It has a very specific purpose and must, therefore, follow very specific rules to make sure that questions are designed to test the same skills in the same underlying ways, without actually repeating the questions. It’s also important to remember that the SAT is predominantly a multiple-choice test, and that the multiple-choice format only requires a test-taker to separate right answers from wrong answers, rather than requiring you to provide correct answers on your own. This means that the relationships among right answers and wrong answers must remain constant for all real tests, because changing those relationships would involve changing the nature of the multiple-choice questions and breaking the standardization rules of the test.

Keep these things in mind as we discuss the patterns in the SAT Math section, and as you encounter real SAT Math questions in the future.

Very often, one of the wrong answer choices will be twice as much as the right answer choice, or half as much as it. This is especially true when the problem involves multiplying or dividing an amount by 2. If you solve a problem and get an answer like 18, a wrong answer choice like 36 might reassure you that you’re right.

Remember that this pattern is an indication that you’re probably right, not a confirmation that you’re definitely right. Also, it’s important not to get it backwards—in the same hypothetical example, the right answer might be 36, and the wrong answer might be 18! Be very aware of this useful pattern, but don’t rely on it exclusively.

One of the ways that the SAT will try to confuse you is by giving you a problem that involves two or three steps. When it does that, one of the wrong answers will often be the number that you would get if you stopped after one of the earlier steps. For example, a problem might ask you to find the price for pens by giving you the prices for different combinations of pencils and erasers. The problem might require you to figure out the price of pencils in order to figure out the price for
erasers, and one of the wrong answer choices would be the price of pencils. Because this wrong answer is actually a number that you found in the process of solving the problem, seeing it in as a wrong answer can reassure you that you’re on the right track.

As I said before, the SAT likes to give you problems that look complicated but have very simple solutions. The test often does this by showing you a rather complicated expression that you can simplify by substituting one thing for another. If you start looking for substitution opportunities, you’ll find them all over the test, and they’ll make your life easier. For example, the SAT will often reward you for substituting a difference of squares like
x
2
–
y
2
with the expression (
x – y
)(
x
+
y
). On the other hand, it might also reward you for going in the other direction, and realizing that (
x – y
)(
x
+
y
) can be restated as
x
2
–
y
2
.

Don’t worry if this sounds a little vague right now. We’ll see several examples of it when we go through real SAT questions from the College Board in a few pages.

Sometimes some or all of the answer choices in a math question will form a series. These series might be pretty easy to recognize in some cases, like if the answer choices are 7, 8, 9, 10, and 11. In other cases, the series might be less obvious, and it might be related to a concept in the question: if the question is talking about dividing some quantity by 4, then the answer choices might contain the series 3, 12, 48, because each number in the series is one fourth of the next number in the series.

The College Board seems to include series in the answer choices when it hopes that you’ll make a mistake and repeat a step in the solution one time too many or too few, ending up with one of the wrong answers in the series. In other words, if a question involves finding the perimeter of a triangle with sides of 5 units each, the answer choices might include the series 10, 15, 20, because the College Board is hoping you’ll either add 5 one time too few (ending up with 10) or one time too many (ending up with 20).

For this reason, when a series is involved in the answer choices, we’ll typically find that the correct answer isn’t the first or last number in the series. The College Board seems to like to put the correct answer near the middle of the series in order to allow you to make a mistake in either direction and still find a wrong answer that reflects your mistake.

Remember, as with the other patterns in this section, that this isn’t an unbreakable rule. So I’m not saying that we’ll never, ever find the right answer at the beginning or the end of a series; sometimes we will. I’m just saying that it’s more common to find it in one of the middle positions of a series, and that it helps to be aware of that.

The College Board likes to create wrong answers that incorporate elements of correct answers, probably in an attempt to make it harder for you to eliminate answer choices on the basis of a partial solution. In other words, if you’re working on a question where the answer choices are all algebraic expressions and you figure out that the correct answer should include the expression 2
r
along with some other stuff, then you’ll often find that a majority of the answer choices include 2
r
. This way, the College Board can try to force you to figure out the rest of the question in order to identify the correct answer.

While this can be an annoying thing for the College Board to do, you’ll find that you can actually use it to your advantage in many cases: after you think you’ve solved a question, if you see that the wrong answers seem to include a lot of the elements
in common with the choice that you like, you can often take that as a good sign that you’ve thought about the question correctly.

In other words, if the wrong answers seem to be imitating parts of the right answer, that’s typically a sign that you’ve understood the question correctly. (Notice I said
, “typically,” and not “always.”)

We’ll see several examples of this, and of the other SAT Math patterns, when we go through some questions from the Blue Book in a few pages.