SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (25 page)

“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”
- S. Gudder

The Math questions on the SAT are a very mixed bag. The current version of the SAT features several different types of math; almost everything you could study in high school math is on there except calculus, trigonometry, and advanced statistics. On top of that, an individual question can be a combination of any of those areas, which often makes the questions hard to classify.

Some students cover all the basics of SAT
Math before they reach high school, and some students take geometry as seniors and never even have classes in algebra. For the first type of student, SAT Math concepts are almost forgotten; for the second type of student, they are just barely familiar.

In short, nobody I’ve ever met has felt completely comfortable with all the math on the SAT. I teach test-taking strategies for a living, I can answer
and explain every single question in the Blue Book, and I still don’t feel like I know a lot of math. Don’t let it bother you!

But that’s not all—mastering the key mathematical concepts that can appear on the SAT still won’t guarantee a high score. In fact, you probably know some students who are “math geniuses” who still don’t make perfect scores on the SAT Math section. You might even be one of those students yourself.

For those students—and for most students, actually—there’s something missing when it comes to SAT Math. There’s a key idea that they haven’t realized yet.

What idea is that?  It’s the fact that the SAT Math test is NOT a math test, at least not in the sense that you’re probably used to. The SAT Math section has very little to do with actual mathematical knowledge. Think of it as a logic test, or as a bunch of problem-solving exercises. Actually, the better you get at SAT Math, the more you’ll come to realize it’s just a game—and the more you come to see it as a game, the better you’ll get at it.

The truth is that the SAT Math section is primarily a test of your knowledge and application of mathematical definitions and properties. The calculations themselves aren’t complicated, as you’ll see when we go through some real test questions. The SAT could have made the calculations difficult, but the calculations themselves are always fairly easy, even on so-called “hard” questions. The only thing that makes SAT Math questions difficult is figuring out what they’re asking you to do in the first place.

So natural test-takers do better on “
SAT Math” because they focus on setting problems up, rather than automatically relying on formulas. Unfortunately, most test-takers never realize how different SAT Math is from school math, so they spend too much time trying to find complicated solutions to the problems on the SAT, as though the SAT were like a regular math test in high school. This is very frustrating, and results in low scores. It’s like trying to cook an omelet with a hammer.

Studying this
book will help you use the techniques that natural test-takers use to score well on SAT Math. More importantly, this Black Book will help you come to see the SAT “Math” test for what it really is: a reading and problem-solving test that happens to involve numbers!

Before we go any further, it’s important that you be in the right frame of mind when you approach SAT Math questions. As I’ve mentioned a couple of times so far, most SAT Math questions aren’t really “math” questions at all, at least not in the way you probably think of math questions. It’s important for us to understand why this is.

Put yourself in the College Board’s position for a moment. If you’re the College Board, your goal is to provide colleges and universities with useful, reliable data on their applicants’ abilities. It wouldn’t really make sense to have those applicants take a traditional test of advanced math, for two reasons:

o
Not all applicants will have taken the same math classes, so a traditional test wouldn’t be able to distinguish students who had never learned a certain type of math from students who had learned it and were bad at it.

o
More importantly, the high school transcript already does a pretty good job of indicating a student’s ability to answer traditional math questions.

A
traditional test of advanced math wouldn’t let the College Board provide very useful data to colleges and universities. And it wouldn’t make any sense to come up with a traditional test of basic math, either, because far too many test-takers would do very well on that, and the results would be meaningless.

The College Board’s solution to this pro
blem is actually kind of clever. They make sure that SAT Math questions only cover basic math topics, but they cover those basic topics in non-traditional ways. In this way, the College Board can be fairly certain that every test-taker has the potential to answer every question correctly—but only by thinking creatively, which keeps the results of the test interesting for colleges and universities.

In fact, let me say that last part again, in all caps,
and centered, because it’s super important:

SAT MATH QUESTIONS TEST BASIC MATH IDEAS IN STRANGE WAYS.

That idea is the thing that most test-takers don’t realize. It’s the thing that causes so many people to spend so much time practicing math for the SAT with so little result. The way to get better at SAT Math isn’t to learn advanced math, because advanced math ideas don’t appear on the test. The way to get better is to learn to take apart SAT Math questions so you can understand which basic ideas are involved in each question.

For this reason, you’ll often find that the most challenging SAT Math questions can’t be solved with any of the formulas you normally u
se in math class. In general, SAT Math questions avoid formal solutions. If anything, you might even say that answering SAT Math questions is kind of a creative process, because we never know exactly what the next question will involve, even though we can know the general rules and principles underlying its design.

Since the SAT Math section is all about basic math ideas presented in strange ways, there are two key areas of knowledge we’ll need to do well on the test:

o
Basic knowledge of arithmetic, geometry, and algebra (including some basic graph-related ideas)

o
A thorough understanding of the SAT’s unwritten rules, patterns, and quirks.

So you will need
some
math knowledge, of course, but you won’t need anything like trig, stats, or calculus, and you won’t have to memorize tons of formulas. Like I keep saying (and will continue to say), it’s much more important to focus on how the test is designed than to try to memorize formulas.

In a moment we’ll go through the “Math toolbox,” which is a list of
math concepts that the SAT is allowed to incorporate when it makes up questions. After that, we’ll get into the SAT’s unwritten rules of math question design.

In a moment, we’ll talk about how to attack the SAT Math section from a strategic perspective. But first, it’s important to make sure we know all the mathematical concepts the SAT is allowed to test (don’t worry, there aren’t that many of them).

If you already know all the concepts below, then you don’t need to go over them again. Instead, go to the next section and start learning how to attack the test.

If you’re not familiar with some of the concepts below, then take a few minutes to refresh yourself on them.

This concept review is designed to be as quick and painless as possible. If you feel that you’d like a little more of an explanation, the best thing to do is find somebody who’s good at math (a teacher, parent, or friend) and ask them to spend a little time explaining a few things to you.

This concept review might seem easier to you than the actual SAT Math section. That’s because the difficulty in SAT Math really comes from the setup of each problem, not from the concepts that the problem involves. The concepts in this review are the same concepts you’ll encounter in your practice and on the real test, but the real test often makes questions look harder than they really are by combining and disguising the underlying concepts in the questions.

For SAT
Math, it’s not that important to have a
thorough
understanding of the underlying concepts. All you need is a quick, general familiarity with a few basic ideas. So that’s all we’ll spend time on.

Please note that this list is similar in some ways to
chapters 14 through 18 of the College Board publication
The Official SAT Study Guide
, but my list is organized a little differently and presents the material in more discrete units. In addition, my list explains things in plainer language and omits some concepts that are redundant, making it easier to study.

As you’re going through this list, you may see concepts that aren’t familiar. Before you let yourself get confused, make sure you’ve read this list through TWICE. You’ll probably find that a lot of your confusion clears itself up on the second reading.

Also, please try to remember that the material in the Math Toolbox is pretty dry and technical, and that it’s not the focus of the proper strategic approach to the SAT. It’s just a set of basic ideas that need to be refreshed before we get into the more important stuff.

An integer is any number that can be expressed without a fraction, decimal, percentage sign, or symbol.

Integers can be negative or positive.

Zero is an integer.

Example:

These numb
ers are integers: -99, -6, 0, 8, 675

These numbers are NOT integers: pi,
96.7, 3/4

There are even integers and there are odd integers.

Only integers can be odd or even—a fraction or symbolic number is neither odd nor even.

Integers that are even can be divided by 2 without having anything left over.

Integers that are odd have a remainder of 1 when they’re divided by 2.

Example:

These are even integers: -6, 4, 8

These
are odd integers: -99, 25, 675

An even number plus an even number gives an even result.

An odd number plus an odd number gives an even result.

An odd number plus an even number gives an odd result.

An even number times an even number gives an even result.

An even number times an odd number gives an even result.

An odd number times an odd number gives an odd result.

Some integers have special properties when it comes to addition and multiplication:

Multiplying any number by 1 leaves the number unchanged.

Dividing any number by 1 leaves the number unchanged.

Multiplying any number by 0 results in the number 0.

Adding 0 to any number leaves the number unchanged.

Subtracting 0 from any number leaves the number unchanged.

It’s impossible, for purposes of SAT Math, to divide any number by 0.

SAT word problems are typically simple descriptions of one of the following:

Real-life situations

Abstract concepts

Example:

An SAT word problem about a real-life situation might look like this:

“Joe buys two balloons for three dollars each, and a certain amount of candy. Each piece of the candy costs twenty-five cents. Joe gives the cashier ten dollars and receives twenty-five cents in change. How many pieces of candy did he buy?”

An SAT word problem about an abstract concept might look like this:

“If
x
is the arithmetic mean of seven consecutive numbers, what is the median of those seven numbers?”

To solve SAT word problems, we have to transform them into math problems. These are the steps we follow to make that transformation:

Note all the numbers given in the problem, and write them down on scratch paper.

Identify key phrases and translate them into mathematical symbols for operations and variables. Use these to connect the numbers you wrote down.

Example:

In the phrase “two balloons for three dollars each,” the
each
part means we have to
multiply
the two balloons by the three dollars in order to find out how much total money was spent on the two balloons. 2 * 3 = 6. Six dollars were spent on the two balloons if they cost three dollars each.

After the word problem has been translated into numbers and symbols, solve it like any other SAT Math problem (see the
SAT Math Path in this chapter for more on that).

A number line is a simple diagram that arranges numbers from least to greatest.

The positions on a number line can be labeled with actual numbers or with variables.

Example:

This number line shows all the integers from -7 to 4:

On the SAT, number lines are drawn to scale and the tick marks are spaced evenly
unless the question notes otherwise.

To determine the distance between two numbers on a number line, just subtract the number to the left from the number to the right.

Example:

On the number line above, the distance between 1 and 3 is two units, which is the same thing as saying that 3 – 1 = 2.

On a number line, there is a DIFFERENCE between the distance that separates two numbers and the number of positions between them.

If you’re asked how many positions are BETWEEN two numbers on a number line, remember that you CANNOT answer this question by simply subtracting one number from the other—that’s how you would find the distance. You should actually
count
the positions—you’ll find the number of positions is one less than the difference you get when you subtract.

Example:

On the number line above, there are NOT two positions between the numbers 2 and 4, even though 4 – 2 = 2. There is only one position between the numbers 2 and 4, which is one less than the difference we get when we subtract the number 2 from the number 4.

On the SAT, the positions on a number line don’t have to represent whole numbers. They might represent groups of five numbers at a time, or hundredths, or any other consistent amount.

A number’s absolute value is the distance of that number from zero on the number line.

Example:

-4 and 4 both have an absolute value of 4. We signify the absolute value of a number with vertical lines on either side of the number:
|-4| = |4| = 4.

To square a number, multiply the number by itself.

Example:

Five squared is five times five, or 5 * 5, or 25.

To find the square root of a number, find the amount that has to be multiplied by itself in order to generate the number.

Example:

The square root of 25 is the amount that yields 25 when it’s multiplied by itself. As we just saw, that amount is 5. So the square root of 25 is 5.

When you square any number, the result is always positive. This is because a positive number times a positive number gives a positive result, and so does a negative number times a negative number.

Square roots on the SAT are always positive.

The SAT never asks about the square root of a negative number.

The SAT likes to ask about the squares of the numbers -12 through 12. Here they are:

Number
              Square

-12 or 12
              144

-11 or 11
              121

-10 or 10
              100

-9 or 9
                            81

-8 or 8
                            64

-7 or 7
                            49

-6 or 6
                            36

-5 or 5
                            25

-4 or 4
                            16

-3 or 3
                            9

-2 or 2
                            4

-1 or 1
                            1

0
                            0

While I don’t recommend using a calculator on the SAT if you can help it, remember that you can always find the square root of a number very easily on a good calculator.

A fraction is a special type of number that represents parts of a whole.

Fractions are written this way:

[number of parts being described in the situation]
[number of parts that the whole is divided into]

Example:

Imagine that we’re sharing a six-pack of soda cans. I really like soda, so I drink five of the cans. In this situation, I’ve had five of the six cans that make up the six-pack—I’ve had 5/6 of the six-pack.

The number above the fraction bar is called a
numerator
.

The number under the fraction bar is called a
denominator
.

When the numerator of a fraction is less than the denominator, the value of the fraction is less than 1.

When the numerator of a fraction is greater than the denominator, the value of the fraction is greater than 1.

Example:

1/2 is equal to one half, which is less than 1. 6/3 is equal to 2, which is greater than 1.

Any integer can be thought of as having the denominator 1 already underneath it.

Example:

7 is the same thing as 7/1.

A reciprocal is what you get if you switch the numerator and the denominator of a fraction.

Example:

The reciprocal of 2/3 is 3/2. The reciprocal of 7 is 1/7. (Remember that all integers can be thought of as having the denominator 1.)

To multiply two fractions, first multiply their numerators and write th
at amount as the numerator of the new fraction; then, multiply the denominators and write that amount as the denominator of the new fraction.

Example:

4/7 x 9/13 = 36/91

To divide fraction
a
by fraction
b
, we actually multiply fraction
a
by the RECIPROCAL of fraction
b
.

Example

4/7 divided by 9/13 = 4/7 x 13/9 = 52/63

Multiplying a non-zero integer by a fraction that’s less than 1 (that is, by a fraction where the numerator is less than the denominator) will give a result that is closer to zero on a number line than the original integer was. (Read this item again if you need to!)

Examples:

6 x 3/5 = 18/5, and 18/5 falls between 0 and 6 on a number line.

-7 x 2/9 = -14/9, and -14/9 falls between -7 and 0 on a number line.

Fraction
a
is equal to fraction
b
if you could multiply the numerator in
a
by a certain number to get the numerator in
b
, and you could also multiply the denominator in
a
by the same number to get the denominator of
b
.

Example:

3/5 is equal to 18/30 because 3 x 6 = 18 and 5 x 6 = 30. Here’s another way to write this:  3/5 x 6/6 = 18/30. Notice that 6/6 is the same thing as 1 (six parts of a whole that’s divided into six parts is the same thing as the whole itself). So all we really did here was multiply 3/5 by 1, and we know that doing this will give us an amount equal to 3/5.

For more on fractions, see the discussion of factors and multiples below.

The factors of a number
x
are the positive integers that can be multiplied by each other to achieve that number
x
.

Example:

The number 10 has the factors 5 and 2, because 5 * 2 = 10. It also has the factors 10 and 1, because 1 * 10 = 10.

“Common factors,” as the name suggests, are factors that two numbers have in common.

Example:

The number 10 has the factors 1, 2, 5, and 10, as we just saw. The number 28 has the factors 1, 2, 4, 7,
14, and 28. So the common factors of 10 and 28 are 1 and 2, because both 1 and 2 can be multiplied by positive integers to get both 10 and 28.

The multiples of a number
x
are the numbers you get when you multiply
x
by 1, 2, 3, 4, 5, and so on.

Example:

The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, and so on.

Remainders are what you get when you divide one number by another number and have something left over (this assumes you don’t use fractions or decimals to write the answer to your division problem).