Digital SAT Math Prep For Dummies, 3rd Edition: Book + 4 Practice Tests Online, Updated for the NEW Digital Format

Introduction

The first quarter of the 21st century is now history, but the SAT is still with us. And like most things that endure, the SAT has morphed more than once in order to stay current.

In its new format, the SAT is now almost exclusively an online test. This means that the days of carrying half a dozen well-sharpened yellow Number 2 pencils are something you’ll only hear about from older relatives.

What hasn’t changed (too much) is the math that you’ll need to know in order to get a good SAT math score. Math, for the most part, is still math. And that’s what this book is about.

About This Book

A lot of SAT prep books divide their attention between the Reading and Writing test and the Math test. This is fine as far as it goes, because you probably want to boost both scores. But in this book, I focus exclusively on math, math, and more math to help you achieve the best score you can on this — what can I say? — most often dreaded part of the test.

The SAT covers a variety of areas, including algebra, geometry, trigonometry, functions and graphs, and statistics and probability. This book focuses on those SAT topics and helps you get used to problem-solving so that you can turn facts and formulas into useful tools.

I wrote this book to give you the best possible advantage at achieving a good score on the math portion of your SAT. There’s no shortcut, but most of what you need to work on comes down to four key factors:

Knowing the basics inside and out

Strengthening SAT-specific math skills

Practicing answering SAT questions

Timing yourself taking SAT practice tests

For that last point, almost every example and problem here is written in SAT format — either as a multiple-choice question or as a student-produced response question. Chapters 3 to 15 contain math skills that are essential to the SAT. And, to give you that test-day experience, this book also includes two practice tests, with access to two additional online practice tests. That’s hundreds and hundreds of questions designed to strengthen your SAT muscle, so to speak.

This book also has a few conventions to keep in mind:

New terms introduced in a chapter, as well as variables, are in italics.

Keywords in lists and numbered steps are in boldface.

Foolish Assumptions

This is an SAT prep book, so my first assumption is that you or someone you love (your son or daughter, mom or granddad, or perhaps your cat) is thinking about taking the SAT sometime in the future. If not, you’re still welcome to buy the book.

My second assumption is that you’re currently taking or have in your life at some point taken an algebra course — even if you feel like it’s all a blur. Now, I wish I could tell you that algebra isn’t very important on the SAT — oh, a mere trifle, hardly a thought. But this would be like saying you can play NFL football without getting rushed by a bunch of 250-pound guys trying to pulverize you. It just ain’t so.

But don’t worry — this book is all about the blur and, more importantly, what lies beyond it. Read on, walk through the examples, and then try out the practice problems at the end of each chapter. I can virtually guarantee that if you do this, the stuff will start to make sense.

Icons Used in This Book

In this book, I use these four icons to signal what’s most important along the way:

Remember This icon points out important information that you need to focus on. Make sure you understand this information fully before moving on. You can skim through these icons when reading a chapter to make sure you remember the highlights.

Example Each example is a formal SAT-style question followed by a step-by-step solution. Work through these examples and then refer to them to help you solve the practice problems at the end of the chapter.

Tip Tips are hints that can help speed you along when answering a question. See whether you find them useful when working on practice problems.

Warning This icon flags common mistakes that students often make if they’re not careful. Take note and proceed with caution!

Beyond the Book

In addition to what you’re reading right now, this book comes with a free access-anywhere Cheat Sheet that includes tips to help you prepare for the math sections of the SAT. To get this Cheat Sheet, simply go to www.dummies.com and type SAT Math For Dummies Cheat Sheet in the Search box.

You also get access to three full-length online practice tests. To gain access to the online practice, all you have to do is register. Just follow these simple steps:

Register your book or ebook at Dummies.com to get your PIN (go towww.dummies.com/go/getaccess).

Select your product from the drop-down list on that page.

Follow the prompts to validate your product, and then check your email for a confirmation message that includes your PIN and instructions for logging in.

If you do not receive this email within two hours, please check your spam folder before contacting the Technical Support website at http://support.wiley.com or by phone at 877-762-2974.

Now you’re ready to go! You can come back to the practice material as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.

Your registration is good for one year from the day you activate your PIN.

Where to Go from Here

This book is organized so that you can safely jump around and dip into every chapter in whatever order you like. You can strengthen skills you feel confident in or work on those that need some attention.

If this is your first introduction to SAT math, I strongly recommend that you start out by reading Chapter 1. There, you find some simple but vital SAT-specific information that you need to know before you sit down in front of your computer to take the test.

If you’d like to start out by getting a sense of how ready you are for the SAT, skip forward to Chapter 16 and take a practice test. When you’re done, check out Chapter 17 to see how many questions you got correct, and to read through the answer explanation for every question you got wrong.

However, if it’s been a while since you’ve taken a math course, read Chapter 2, which covers the pre-algebra math skills you need to know before you proceed to the more difficult math later in the book. Chapters 3 and 4, which cover algebraic expressions and equations, can get your math brain moving again, and you may find that a lot of this stuff begins to look familiar as you go along.

Finally, if you read through a few chapters and feel that the book is moving more quickly than you’d like, go ahead and pick up my earlier book, Basic Math & Pre-Algebra For Dummies (John Wiley & Sons). There, I adopt a more leisurely pace and spend more time filling in any gaps in understanding you may find along the way.

Part 1

Getting Started with SAT Math

IN THIS PART …

Getting familiar with the new online SAT format

Reviewing important pre-algebra concepts you may have forgotten

Chapter 1

Welcome to SAT Math

IN THIS CHAPTER

Bullet Looking at the new SAT

Bullet Knowing how the SAT Math Test is organized

Bullet Getting familiar with the Reference tab

Bullet Using the Calculator tab

Bullet Understanding how to enter answers to student-produced response (fill-in-the-blank) questions

Bullet Identifying the math topics tested on the SAT

Bullet Knowing some basic strategies for attaining scores of 500, 600, or 700

Bullet Coming up with a basic plan of action for getting the SAT Math score you need

In this chapter, I give you a general overview of the new SAT. Then I provide you with a more thorough look at the SAT Math test, including the math topics you’re most likely to find there. I discuss the Reference and Calculator tabs you’ll have access to. I also help you avoid a few unnecessary errors you may encounter when entering answers to fill-in-the-blank questions.

After that, I give you a bit of essential SAT Math strategy that every student needs to know. I encourage you to think about your goal for the next SAT based on the level you’re currently working at.

Finally, I present three SAT success stories, in which three very different students who set and reached different SAT goals got into the colleges that they were aiming for.

Quick Overview of the New Online SAT Format

The SAT is a college readiness test and, in some U.S. states, is now being used as a skills test required for high school graduation.

It covers two main subjects in four sections: two Reading and Writing sections, followed by two Math sections. Each subject is scored on a 200-to-800 point scale, resulting in a composite SAT score from 400 to 1,600 points.

The big news is that starting in March of 2024, the SAT will be changing to an exclusively online format. In this section, I give you a basic overview of the SAT in its new online format.

Knowing how many questions you’ll face and how much time you’ll have to answer them

Remember Starting in March of 2024, the wise elders of the SAT are changing its format from a paper-based to online format.

The new version will comprise 98 questions (down from 154), for which you’ll be given a total of 134 minutes (down from 180). I’ve summed up the information you’ll need most in Table 1-1.

TABLE 1-1 Overview of the New SAT Format

This is all fairly good news for students. You’ll spend less time answering fewer questions, but you’ll have a little less time pressure, with tad more time to answer each question. To me, that looks like a win-win-win.

Understanding key changes found in the SAT online format

The new online SAT format provides a few important upgrades to the previous paper-based version of the SAT.

For starters, the SAT questions that you answer will be drawn from a large pool of SAT questions, rather than printed in booklets and distributed to students. This change means that the test you take will literally be different from the test that the person sitting next to you is taking — or that your best friend will take the following week.

One result of this change is that cheating becomes a lot more difficult. Students may (and probably will) still post SAT questions and answers online, and other students like yourself may still read them. But if you do, you’ll probably just end up figuring out how to answer a lot of questions that will never actually appear on your test. (Another word for this practice is studying, which virtually all teachers encourage, so please have at it!)

Furthermore, the SAT is now adaptive, which means that it adjusts its difficulty level depending upon how well you perform. So, after you complete your Section 1 Reading and Writing Test, an algorithm adjusts the type and difficulty levels of the questions that you’ll see on Section 2 just for you. Likewise, after you finish your Section 3 Math Test, a similar algorithm performs the same magic, adjusting the questions on Section 4.

The good news here is that this new feature makes it possible for the system to produce a reliable score using 56 fewer questions and using 41 fewer minutes of your valuable time compared with the paper-based test.

SAT Math Basics

In the new format, both SAT Math sections will include 22 questions.

That’s a total of 44 questions, each of which counts for 1 point on your raw score of correct answers (from 0 to 44). This raw score is converted to a scaled score (from 200 to 800), which becomes your SAT math score.

Using the Reference tab

The SAT math test includes a handy Reference list of formulas that you can use while taking the test, as shown in Figure 1-1. As you can see, this list includes a variety of geometric formulas for the area and circumference of a circle, the area of a rectangle and triangle, the Pythagorean Theorem, and other favorites.

FIGURE 1-1: The Reference list of formulas for the SAT.

To access this list at any time, click the Reference tab at the top right of the test page.

Tip Before taking the SAT, be sure you’re familiar with this Reference information, so you’ll remember it’s there when you’re working under time pressure on the actual test.

Clicking the Calculator tab

On the new version of the SAT, you can use a calculator on every question! This marks a big change from the previous paper-based version of the test, which had a No Calculator section.

The test itself provides an online calculator. To open it, click the Calculator tab at the top right of the page. The calculator is relatively self-explanatory, with a variety of calculator functions available to you. It also includes a graphing component to help you answer more difficult questions.

You can also use an SAT approved calculator, such as the one that you’re using in your current math class. It should go without saying (but I’ll say it anyway) that you’re not allowed to use any non-approved calculator — especially the one on your phone! — while you’re taking the SAT.

Answering student produced response (fill-in-the-blank) questions

The SAT math test includes 44 questions:

33 multiple-choice questions (Answer A through D)

11 student-produced response questions (sometimes called open-ended questions or fill-in-the-blank questions)

From your online adventures as a math student, you’re almost certainly familiar with both of these types of test questions.

Multiple-choice questions are just about foolproof. What can I say to be helpful? Don’t click B when you meant to click C.

Fill-in-the-blank questions aren’t all that much more difficult to handle. The new online SAT has, fortunately for you test takers out there, done away with the confusing grid that it used to employ in its paper-based format. (And we don’t miss it one bit!) So, in most cases, entering positive or negative integer answers should be a no-brainer.

Just a couple of words on entering rational numbers — that is, fractions and decimals:

Enter rational numbers that are greater than 1 (or less than –1) either as decimals or as improper fractions — never try to enter them as mixed numbers.

For example, suppose you want to enter the number 4.5. This decimal format works fine, or if you like, you can enter it as 9/2, which will appear as math . But don’t try to enter it as 4 1/2, because this will show up incorrectly as math .

Enter rational numbers that are repeating decimals either as fractions or by filling out the entire space available — don’t truncate them to save space.

For example, suppose you want to enter the number 2/3. This fractional format will show up correctly as math , or if you like you can enter it as a decimal in any of the following ways:

When entering a repeating decimal like this one, just make sure to fill in as many numbers as possible to fit in the space given — that is, four numbers and a decimal point. Don’t truncate (cut short) your answer in any of the following ways:

Doable, right?

Overview of SAT Math Topics

The math that’s covered on the SAT is very closely tracked to the math that’s covered in most U.S. high school math classes. So if you’re a current or recent U.S. high school student, you’re probably familiar with most of this curriculum.

The SAT breaks this down into four general areas of study: Algebra, Problem Solving and Data Analysis, Advanced Math, and Geometry and Trigonometry. In this section, I give you an overview of each of these topics.

Algebra

Algebra centers on the linear function math and other information covered in a typical high school Algebra I class. To answer SAT Math questions in this area, you’ll need to feel comfortable working with the following:

Evaluating, simplifying, and factoring algebraic expressions (Chapter 3)

Solving algebraic equations and inequalities (Chapter 4)

Working with linear functions in four complementary ways: words, tables, graphs, and equations (Chapter 5)

Solving systems of equations (both linear and non-linear), and identifying when linear systems have either no solution or infinitely many solutions (Chapter 6)

In Part 2, I cover all of these topics in depth.

Problem Solving and Data Analysis

Problem Solving and Data Analysis focuses on a relatively short list of problem-solving techniques:

Working with ratios, proportional equations, and percentages (Chapter 7)

Relying on a basic understanding of statistics and probability (Chapter 8)

Applying these techniques to information presented visually in tables and graphs (Chapter 9)

Part 3 focuses exclusively on these ideas.

Advanced Math

Advanced Math requires you to understand a core of information covered in high school Algebra II:

Working with functions using math notation, and knowing how to graph a core of basic functions and their most elementary transformations (Chapter 10)

Understanding how to work with and graph polynomials, especially linear, quadratic, cubic, and quartic polynomials (Chapter 11)

Graphing quadratic functions using standard, vertex, and factored forms (Chapter 12)

Graphing exponential and radical equations (Chapter 13)

Part 4 covers these Advanced Math topics.

Geometry and Trigonometry

Geometry and Trigonometry covers math that focuses on shapes and solids in two and three dimensions:

Solving problems using basic geometry and circles on the xy-plane (Chapter 14)

Working in-depth with right triangles, the Pythagorean theorem, and trigonometric ratios such as sine, cosine, and tangent (Chapter 15)

I provide this information in Part 5.

SAT Math Strategy

When it comes to doing well on the SAT, your test-taking strategy is a small but important piece of the puzzle. And this strategy also extends to knowing which questions to answer and which to skip, depending on the score you’re currently striving for.

In this section, I fill you in on this essential information.

Isn’t there some way to get a good SAT Math score without actually knowing math?

No.

I’d love to tell you otherwise, but no.

If the key to getting a great score were, say, choosing Answer C on every question, the name of this book would be Answer C Math For Dummies and it would be a lot shorter.

While you fully absorb that difficult truth, I will add that there’s a reasonable amount of strategy you should absolutely know before taking your first SAT. And while you may think that lots of students already know this stuff, plenty of others don’t — yet.

I don’t want you to be one of them.

So read on.

Is there a penalty for guessing?

If you have an older brother or sister who took the SAT before 2016, they may remember the old format, which had a penalty for filling in a wrong answer.

So please take note: The SAT in its current form has no penalty for filling in a wrong answer. This goes for all four sections, the Reading and Writing as well as the Math sections.

Obviously, then, you want to make sure that you fill in at least some answer for each multiple-choice question on the two Math sections. That’s 33 questions, so by pure chance, you can expect to get about 8 of these questions right just by making wild guesses.

Let’s take that thinking a step further: If you don’t fill in guesses for all the questions you don’t have time to think about, you’ll be competing against a ton of other students who are guessing. So, bottom line, you can’t afford not to guess every multiple-choice question you don’t know the answer to.

What about the fill-in-the-blank questions? Well, because these questions are entirely open ended, you don’t have much chance of answering them correctly with a wild guess. But if you have any idea what the answer might be, go ahead and enter it. Worst case, doing this won’t lose you any points.

Are some questions harder than others?

Generally speaking, SAT Math questions fall into three categories of difficulty: easy, medium, and hard. Both sections of the new SAT are identical in this regard. Table 1-2 shows the rough breakdown of questions by difficulty levels.

TABLE 1-2 Easy, Medium, and Hard Questions

Remember that every question counts for one point toward your raw score, which directly affects your scaled score (200–800). So, unlike the tests you take in school, the easiest and hardest questions on the SAT both have the same value.

Do I have to answer every question?

The short answer is, no, you don’t have to answer every SAT Math question to get a good score.

In fact, depending on your current performance level on practice tests, it may very well be to your benefit not to answer all of the questions.

This piece of strategy definitely goes against a lot of your training as a high school student. After all, in most of your classes, you can’t get an A or even a B on a test without answering just about all of the questions. If you only answer 75 percent and skip the rest, even if you answer perfectly, probably the best you can hope for is a C.

However, the situation with the SAT is entirely different.

On the SAT, you can get a 500 Math score by answering only about 60% of the questions on the test correctly. Think about it — a respectable score on the SAT would be a failing grade on a math test at school!

I dive more deeply into this aspect of strategy later in this section, when I ask you to consider your own personal starting point, path, and goal for the SAT.

For now — and this goes double if you’re a perfectionist — simply let go of the compulsive need to answer all 44 math questions on the SAT. Until you’re already scoring 740+ on your practice tests, answering all of the questions would likely be a poor allocation of your time. If you’re answering all of the questions, you’re probably rushing through questions that are within your reach, getting them wrong, and losing points you should be gaining.

So, how many questions should I answer?

The answer to this question depends on your current score, which I break down into three basic scenarios.

Clearing 500

Most colleges and universities prefer to enroll students who have an SAT composite score of at least 1,000, which is approximately 500 on both the Reading and Writing and the Math tests.

If you're Reading and Writing score is 550 or higher, you may be able to get away with a Math score that’s slightly less than 500. Even so, a good first goal would be to break 500 on the Math test.

To get this score, you need to answer about 26 of the 44 SAT math questions correctly. To this end, refer to Table 1-1, and then plan to do the following:

Attempt to answer all 14 easy questions correctly.

Choose 12 out of 16 medium questions to attempt to answer and guess the rest.

Guess on the 14 hard questions.

I know it seems weird to guess so many questions. But the SAT is different from the tests you take in