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- Herausgeber: John Wiley & Sons
- Kategorie: Bildung
- Sprache: Englisch
- Veröffentlichungsjahr: 2024
Improve your score on the math section of the ACT
A good math score on the ACT exam can set you on the path to a number of rewarding college programs and future careers, especially in the STEM fields. ACT Math Prep For Dummies walks you through this challenging exam section, with simple explanations of math concepts and proven test-taking strategies. Now including access to an all-new online test bank—so you can hammer out even more practice sessions—this book will help you hone your skills in pre-algebra, algebra, geometry, trigonometry and beyond. Handy problem-solving tips mean you’ll be prepared for the ever-more-advanced questions that the ACT throws at students each year.
- Learn exactly what you’ll need to know to score well on the ACT math section
- Get tips for solving problems quicker and making good guesses when you need to
- Drill down into more complex concepts like matrices and functions
- Practice, practice, practice, with three online tests
If you’re a high school student preparing to take the ACT and you need extra math practice, ACT Math Prep For Dummies has your back.
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Table of Contents
Cover
Title Page
Copyright
Introduction
About This Book
Conventions Used in This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: Getting into the Act: An Overview of ACT Math Basics
Chapter 1: Reviewing ACT Math Basics
Getting an Overview of ACT Math
Taking Four Key Steps to ACT Math Success
What Should You Study? Knowing What’s on the ACT
You’re Off the Hook: Discovering What the ACT Doesn’t Cover
Chapter 2: Boosting Your Test-Taking Skills
Two-Timing the Test: Taking Two Passes to Answer Questions
To Guess or Not to Guess
Answering Multiple-Choice Questions
Calculating Your Way to Success: Calculators and the ACT
Chapter 3: Discovering Some Problem-Solving Strategies
Identifying Basic Math Skills You Need to Know
Getting Comfortable with Formulas
Sorting Through Word Problems
Part 2: Building Your Pre-Algebra and Elementary Algebra Skills
Chapter 4: Starting with the Basics: Pre-Algebra
Getting Back to Basics: Answering Basic Arithmetic Questions
Understanding Factors and Multiples
Perfecting Your Knowledge of Percents
Using Ratios and Proportions to Compare Quantities
Working with Powers and Square Roots
Chapter 5: It’s Elementary Algebra, My Dear Watson
Knowing the Two Big Ideas of Algebra
Discovering Six Choice Words about Algebra
Express Yourself: Working with Algebraic Expressions
Solving Everything but the Kitchen Sink
Lost in Translation: Translating Words into Equations
Chapter 6: Real-World Math: Statistics and Probability
Presenting Data in Tables and Graphs
Getting a Better-Than-Average Grasp of Averages
Looking at Likelihood: Probability
Chapter 7: Practice Problems for Pre-Algebra and Elementary Algebra
Practice Problems
Solutions to Practice Problems
Part 3: Digging In to Intermediate Algebra and Coordinate Geometry
Chapter 8: Moving to Intermediate Algebra
Knowing More or Less about Inequalities
Beating the System: Dealing with Systems of Equations
Keeping Things in Proportion: Direct and Inverse Proportionality
Working with Quadratic Equations and the Roots of Polynomials
Making Connections with Functions
Chapter 9: Coordinating a Path through Coordinate Geometry
Reviewing Some Basics of Coordinate Geometry
Giving Your Quads a Workout: Graphing Quadratic Functions
Recognizing Three Transformations
Taking on Advanced Equations
Chapter 10: Practice Problems for Intermediate Algebra and Coordinate Geometry
Intermediate Algebra and Coordinate Geometry Practice Problems
Solutions to Intermediate Algebra and Coordinate Geometry Practice Problems
Part 4: Visualizing Plane Geometry and Trigonometry
Chapter 11: Plain Talk about Plane Geometry
Knowing Your Angles
Working with Triangles
Breezing through Your Work with Quadrilaterals
Wheeling and Dealing with Circles
Examining Solid Geometry
Chapter 12: Trig and Beyond: ACT Advanced Math Topics
Trigonometry: Watching the Sines and Taking a Few Tangents
Reloading the Matrix
Logging Some Hours with Logarithms
Imagining the Possibilities with Imaginary Numbers
Chapter 13: Practice Problems for Geometry, Trig, and Advanced Math
Geometry, Trig, and Advanced Math Practice Problems
Solutions to Geometry, Trig, and Advanced Math Practice Problems
Part 5: Tracking Your Progress with Practice Tests
Chapter 14: Practice Test 1
Mathematics Test Answer Sheet
Mathematics Test
Chapter 15: Practice Test 1: Answers and Explanations
Mathematics Test
Answer Key for Practice Test 1
Chapter 16: Practice Test 2
Mathematics Test Answer Sheet
Mathematics Test
Chapter 17: Practice Test 2: Answers and Explanations
Mathematics Test
Answer Key for Practice Test 2
Part 6: The Part of Tens
Chapter 18: Ten Key Differences Between the ACT and the SAT
Differences in Scoring
Paper vs Online Format
Adaptive vs Non-Adaptive Format
Number of Math Tests
Test Organization
Existence of Fill-in-the-Blank Questions
Answer Multiple-Choice Format
Amount of Advanced Math
Difficulty of the Reading Test
Presence of the Science Test on the ACT
Chapter 19: Ten Items to Check Off before Taking the ACT
I Received My ACT Admission Ticket, and I Put It in a Safe Place
I’m 100 Percent Sure about the Date and Time of My ACT
I Know How to Get There, Too
I Purchased the Calculator I Intend to Use
I Feel Comfortable Using My Calculator
I Have a Backpack Ready to Go by the Door
I Picked Out My Clothes for the Morning
I’m Having a Relaxing Night before the Test
My Alarm Is Set for an Early Rise
I’m Focusing on My Breathing
Index
About the Author
Advertisement Page
Connect with Dummies
End User License Agreement
List of Tables
Chapter 4
TABLE 4-1 Converting Basic Percents to Decimals and Fractions
TABLE 4-2 The Square Numbers from to
Chapter 5
TABLE 5-1 Some Common Exponential Equivalents with a Base of 5
Chapter 8
TABLE 8-1: Four Types of Inequalities
Chapter 9
TABLE 9-1 Evaluating the Slope of a Line
TABLE 9-2: Common Polynomials
Chapter 11
TABLE 11-1: Sum of Interior Angles of Common Polygons
Chapter 12
TABLE 12-1 The Six Trigonometric Ratios
TABLE 12-2 Radian Measure for Common Angles
List of Illustrations
Chapter 9
FIGURE 9-1: The
xy-
graph is the basis for coordinate geometry.
FIGURE 9-2: A linear function is a straight line.
FIGURE 9-3: A quadratic function is graphed as a parabola.
FIGURE 9-4: Graphs of some common polynomials.
Chapter 11
FIGURE 11-1: A pair of intersecting lines results in only two distinct angles:
FIGURE 11-2: A line intersecting a pair of parallel lines results in only two d...
FIGURE 11-3: The formula for the area of a triangle is .
FIGURE 11-4: The 45-45-90 and 30-60-90 triangles.
FIGURE 11-5: Bisecting a square and an equilateral triangle.
FIGURE 11-6: Three common right triangles.
FIGURE 11-7: The four basic quadrilaterals: a square, a rectangle, a parallelog...
FIGURE 11-8: A tangent line forms a right angle with the radius.
FIGURE 11-9: A 45° arc of a circle.
FIGURE 11-10: A chord of a circle from
A
to
B.
Chapter 12
FIGURE 12-1: The 45-45-90 triangle and the 30-60-90 triangle.
FIGURE 12-2: A pair of similar 30-60-90 triangles.
FIGURE 12-3: The opposite, adjacent, and hypotenuse with respect to angle
x.
FIGURE 12-4: Graphs of and
,
with periods and amplitudes labeled.
FIGURE 12-5: Examples of three matrices.
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
About the Author
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ACT®Math Prep For Dummies®, 2nd Edition
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
Copyright © 2024 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.
Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc., and may not be used without written permission. ACT is a registered trademark of ACT, Inc. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc., is not associated with any product or vendor mentioned in this book.
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Library of Congress Control Number: 2024932621
ISBN 978-1-394-24226-9 (pbk); ISBN 978-1-394-24228-3 (ebk); ISBN 978-1-394-24227-6 (ebk)
Introduction
More than one-third of all graduating students in the United States — an estimated 36 percent — take the ACT (which, by the way, stands for American College Testing). If you're among this lucky bunch, you may be looking toward the future with a combination of anticipation and dread. You may be anticipating college, with its new experiences and first tastes of freedom, but dreading the hoops you have to jump through to get there. Of course, the ACT is one of these hoops. If you do well on this test, you're propelled to your first-choice college; if you do poorly, maybe not.
You're not alone. And I'm here to help. As the author of Basic Math and Pre-Algebra For Dummies (Wiley) and SAT Math For Dummies (Wiley), I've already helped thousands of would-be college students get through the arduous testing process and move into the exciting life that awaits them in college.
If you're willing to keep the faith and do the work in front of you, I'm confident that you can be among those who succeed on the ACT and take one step forward into your dreams. Let this book be your guide.
About This Book
Although you certainly want to do well on all four sections of the ACT (as well as the optional writing test), this book focuses exclusively on math. The reason I'm playing favorites is simple: For many students (and possibly you, too), math is the toughest section of the test. Because your composite score on the ACT is based in part on your math performance, you don't want math to drag down an otherwise good score.
The math you need to know to do well on the ACT is basically what's covered in high school: pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry. If you're thinking of college, the good news is that you've probably studied a lot of this material already. The bad news is that you may have forgotten it or never felt entirely comfortable with it in the first place.
This situation is perfectly normal, and most students who aren't math geniuses feel roughly the same as you. So your task is to find a realistic strategy to pull together the stuff you already know — even if you're not currently confident that you know it — and tie in the more advanced topics that may have eluded you. Luckily, you've come to the right place.
Everything in this book is designed to take the small amount of math you may remember and build on it so you can use it to tackle typical questions that appear on the ACT. If you've laid the groundwork in school and you're willing to practice, I'll help get you through the test.
My approach in this book is simple: practice, practice, and more practice. The focus is specifically on the types of questions that appear over and over again on the ACT. Even if math isn't your strong suit, when you become comfortable with this core material, you stand a much better chance of getting the kind of ACT score you want to see. And the best part is that you don't necessarily have to read this book from cover to cover. You can hop and skip around to whatever topics you need to work on most.
Conventions Used in This Book
Here are a few conventions to keep in mind as you make your way through this book:
Variables (such as
x
and
y
) and newly defined terms are in
italics.
Keywords in bulleted lists, action parts of numbered steps, and answers in the practice tests are in
bold.
Web sites are in
monofont
.
I alternate the range of the answer choices throughout the book. In one question, you see Choices (A), (B), (C), (D), and (E), and then in the one after it you see Choices (F), (G), (H), (J), (K). Because the ACT itself uses this pattern, I want you to become familiar with it before test day. And, yes, test-designers skip the letter
I
(so I do, too)
.
Why? Probably to avoid confusion because it looks like the number 1.
Foolish Assumptions
I'm going out on a limb here, but if you've bought this book, my first assumption is that you or somebody you know is thinking about taking the ACT. If not, you can certainly use it to improve your knowledge of high school math. And beyond that, it also makes a useful doorstop or something to stick under the leg of a very wobbly table.
My second assumption is that you've taken a year of algebra, whether in high school, junior college, or some other place. You don't actually have to feel like you know algebra. In fact, the point of this book is to build these very skills. But if you've at least sat through an algebra class, you probably know a lot more than you think you know. Really!
Icons Used in This Book
Throughout this book, I use the following four icons to help you keep track of the different kinds of information. Here's what each icon means:
This icon highlights essential information that you need to know, such as key ideas or formulas. Spending a few extra minutes studying these important points is time well spent. You also can use these icons to skim through a chapter you've already studied. Doing so helps solidify your understanding.
Each tip provides an insightful way to approach a question. You may find it especially helpful as a quick way to cut through a seemingly complicated problem.
This icon is like a flashing red light, drawing your attention to a sticky or subtle point that may trip you up if you're not aware of it. Take an extra moment to slow down and make sure you understand the point being made so it doesn't trip you up on the test.
When you see this icon, you know you're ready to put pencil to paper. This book includes dozens of examples of ACT questions, showing you how to answer them from start to finish. Use these examples to better understand a specific type of problem and then refer to them when answering the practice questions.
Beyond the Book
In addition to the book content, you can find valuable free material online. We provide you with a Cheat Sheet that serves as a quick checklist, including the basic supplies you need to draw, where to find inspiration, how to identify common drawing styles, and more. Check out this book's online Cheat Sheet by searching www.dummies.com for ACT Math Prep for Dummies Cheat Sheet.
Where to Go from Here
I've written this book as a complete reference to ACT math. You can start anywhere you like, dip in, clarify your understanding, and then hop back out with ease.
If you're completely unfamiliar with the ACT, I recommend that you begin with Chapter 1 before branching out. In that chapter, I outline the basics of the math portion of the ACT and provide an overview of the topics that are covered on the test.
If you're ready to get studying, flip to whatever chapter discusses the topic you need help with most. Chapters 4 through 13 provide a thorough review of the specific math skills that the ACT covers. You can focus on these chapters until you feel ready to take one of the practice tests in Part 5 or online. Alternatively, you can take a practice test first to find out which types of questions you can answer easily and which hang you up. To access the online tests, visit www.dummies.com/go/getaccess. Select your product, and then follow the prompts to validate your purchase. You’ll receive an email with your PIN and instructions.
Part 1
Getting into the Act: An Overview of ACT Math Basics
IN THIS PART …
Seeing an overview of what ACT math includes and excludes.
Learning some important test-taking and calculator skills.
Working with the basic four operations, negative numbers, order of operations (PEMDAS), and fractions, plus using formulas to solve problems and facing down word problems.
Chapter 1
Reviewing ACT Math Basics
IN THIS CHAPTER
Beginning with an overview of ACT math
Identifying four important steps to success
Studying the six math topics tested on the ACT
Figuring out what’s not tested
All across the United States — and especially in the Midwest, South, and Rocky Mountain states — high school juniors and seniors prepare for one of the most action-packed Saturdays of their lives. No, I don’t mean the prom, graduation day, or a really excellent date with someone their parents don’t know about and wouldn’t approve of if they did. No, they’re preparing for something even more fun than that: the ACT.
Well, okay, back on Earth, it turns out that at least some of this is true: It looks like you will be taking the ACT (which I figured out the moment you bought this book). In this chapter, I begin the work of helping you get ready for the most fun part of all: the 60-minute math test.
I start this chapter with a look at what’s on the math portion of the ACT. Then I provide a bit of advice about four general ways to improve your score. Finally, I list the six math topics that the ACT tests you on, including a breakdown of the specific skills within each topic that you should focus on to do well.
The chapter ends with a sigh of relief as you discover the math that you don’t need to worry about on the ACT. (However, the teacher of your current math class may still want you to know something about it, so don’t flush it from your memory just yet!)
Getting an Overview of ACT Math
The ACT contains five separate tests, which are always presented in this order:
English
Mathematics
Reading
Science
Writing (optional)
This book focuses exclusively on the second test, the ACT mathematics test. This math portion lasts for 60 fun-filled minutes and contains 60 questions. Generally speaking, questions appear roughly in order of difficulty.
The test is scored on a scale of 1 (lowest) to 36 (highest). As a rule of thumb, here’s how I think of ACT scores:
A score of 20 is about average, just about equivalent to a 500 math score on the SAT.
A score of 25 is a good, college-ready score, similar to a 600 math score on the SAT.
A score of 30 (or above!) is a great score, on par with a 700 math score on the SAT.
On the ACT, test graders don’t take off points for wrong answers — that is, you won’t be penalized for guessing. So keep an eye on the time and, just before your time is up, fill in answers to all 60 questions. Make as many educated guesses as you can, of course. But even wild guessing won’t hurt your score, so fill in those answers before time’s up!
Taking Four Key Steps to ACT Math Success
I like to give the following four important pieces of advice to those students who want to improve their ACT math scores:
Sharpen your basic math skills.
Get comfortable using your calculator.
Solidify your ACT-math-specific skills.
Take practice tests under timed conditions.
In this section, I give you an overview of these four steps. The rest of the book is devoted, one way or another, to working on them.
Sharpening your basic math skills
By the basics, I mean the math calculations that precede pre-algebra. For example:
Multiplication tables up to
Adding, subtracting, multiplying, and dividing fractions
Converting percents to decimals and vice versa
Converting common percents, such as 10%, 20%, 25%, 50%, and 75%, to fractions and vice versa
Working with negative numbers
Knowing the order of operations (PEMDAS: parentheses, exponents, multiplication and division, addition and subtraction)
When I say know this stuff, I mean that you should know it stone cold — the way you know your own name. If you’re wasting precious time trying to remember or calculate , you’ll benefit greatly from investing just a few hours to work with a set of flash cards designed to help you strengthen these skills. In Chapter 3, I cover these topics to refresh your memory.
Getting comfortable with your calculator
You may already be well aware (and grateful!) that calculators are allowed on the ACT. Moreover, because virtually everyone will be using one, you should consider a calculator not just optional, but mandatory.
First off, if you’re like me, you probably use your phone as your go-to calculator. Additionally, you may use an online graphing calculator (my favorite one is at desmos.com) for help while working on your homework. Unfortunately, on the ACT, you can’t use your phone or any other device that gives you online access, so you need either a scientific calculator or a graphing calculator.
If you already own a good calculator and are reasonably adept at using it, just use this quick rule of thumb to make sure it’s up to speed: Check to see whether it can do trigonometric calculations like sin x. If it can, it’s probably just fine; if not, you need to think about an upgrade well before the ACT so you have time to practice with it.
In Chapter 2, I give you some specifics about your calculator. And for those of you who are considering or who already own a graphing calculator, I recommend TI-83 Plus Graphing Calculator For Dummies (Wiley) and TI-89 Graphing Calculator For Dummies (Wiley), which are both written by C. C. Edwards.
Solidifying your ACT-specific math skills
Studying for the ACT should be mostly a review of skills that are covered in your math classes. However, no matter how hard you work in your classes, the material you covered two or three years ago may not be fresh in your mind. So focused preparation for the ACT can really pay off.
Parts 2, 3, and 4 (Chapters 4 through 13) provide a detailed review of the math that shows up most on the ACT. You can work through dozens of example problems and answer 108 practice questions specifically related to those topics.
Taking practice tests under timed conditions
Time is money, and money changes everything. So it’s not surprising that time changes everything — especially on the ACT, where you have only 60 minutes to answer 60 math questions. Purposefully working under low-stakes time pressure adds a useful dimension to your study, especially as you get closer to your test date when the stakes will be higher.
Part 5 of this book — Chapters 14 through 17 — contains two complete practice tests (and answers!) for you to try out, plus access to a third test online. I recommend that you take them under real test conditions. In other words, take them in one hour with the calculator you plan to use on the test and no additional help. You may start out taking the first test as a benchmark before you begin working on the rest of the problems in the book. Or, if you prefer, save all three tests until you feel confident answering questions with no time pressure — then start the clock running and see how you do.
What Should You Study? Knowing What’s on the ACT
The ACT covers six overall topics in math: pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry. In this section, I break down all these topics into manageable bits and discuss the individual skills included in each. Parts 2, 3, and 4 (Chapters 4 through 13) cover this material in depth, with plenty of example questions and practice problems.
Taking care of the basics in pre-algebra
Pre-algebra includes a variety of topics that prepare you for algebra. In this section, I discuss the specific pre-algebra skills that show up most on the ACT. And in Chapter 4, I focus on these types of questions, providing plenty of example questions and showing you how to answer them.
Basic arithmetic
You obviously need to know the four operations: addition, subtraction, multiplication, and division. You also want to feel comfortable working with negative numbers, fractions, and decimals. I cover some of this material in this book, but if you feel that you need a more thorough review, pick up Basic Math and Pre-Algebra For Dummies (Wiley) by yours truly.
Number sequences
A number sequence is a list of numbers arranged in a pattern. Here’s an example:
In this case, each number in the sequence is 3 greater than the number before it. An ACT question may ask you to find the next number or a missing number in a number sequence.
Factors and multiples
When one natural number is divisible by another, the smaller number is a factor of the greater number, and the greater number is a multiple of the smaller number. For example, 12 is divisible by 4, so
4 is a factor of 12.
12 is a multiple of 4.
To answer an ACT question, you may need to find all the factors of a number or the greatest common factor or the least common multiple among several numbers.
Fractions and Decimals
You probably first learned about fractions and decimals for the first time all the way back in fourth or fifth grade. If you’re like many students, fractions and decimals can still slow you down or, in some cases, derail you entirely from getting the right answer.
At a minimum, you should know how to add, subtract, multiply, and divide fractions and decimals - either by hand or using your calculator. To add and subtract fractions with different denominators, you may also need to know how to find a common denominator. And to complete a problem, you may need to simplify fractions or change an improper fraction to a mixed number.
Percents, ratios, and proportions
Like fractions and decimals, percents are a mathematical way of representing part of a whole. For example, 50 percent of something is half of it. A ratio is a mathematical comparison. For instance, if you have twice as many brothers as sisters, the ratio of brothers to sisters is 2 to 1, or 2:1. A proportion is an equation using two ratios. ACT questions may ask you to calculate something using percents and ratios, or you may have to set up a proportion to answer a question.
Powers (exponents) and square roots (radicals)
When you take a number to a power, you multiply that number by itself repeatedly. For example: (read three to the fourth power). In this case, 3 is the base (the number multiplied) and 4 is the exponent (the number of times the base is multiplied).
And when you take a root (also called a radical) of a number, you find a result that can be multiplied by itself repeatedly to produce the number you started with. The most common root is the square root — a result which, when multiplied by itself, produces the number you started with. For example, (read the square root of 25) because .
Powers and square roots are common math operations, and they show up a lot on all sorts of ACT questions.
Moving on to elementary algebra
Elementary algebra is essentially the algebra that’s covered in an Algebra I class. In this section, I go over the highlights of what skills the ACT expects you to remember and work with when answering questions. Chapter 5 covers these topics in greater detail with lots of examples.
Evaluating, simplifying, and factoring expressions
An expression is any string of numbers and symbols that makes mathematical sense. In algebra, you can do three common things with expressions:
Evaluate:
To
evaluate
an expression, you plug in the value of each variable and change the expression to a number. Be sure to follow the order of operations (exponents in the order they occur left to right, multiplication and division in the order they occur left to right, and addition and subtraction in the order they occur left to right). For example, here’s how you evaluate the expression , given that :
Simplify:
To
simplify
an expression, you remove parentheses and combine like terms to make the expression more compact. For example, here’s how you simplify the expression :
Factor:
To
factor
an expression, you find a factor that’s common to each term in the expression and pull it out of the expression using parentheses. For example, here’s how you factor
2x
out of the expression
:
Easier ACT questions may ask you to simply evaluate, simplify, or factor an expression. More difficult questions may require you to use these skills to handle more complex calculations.
Solving equations with one or more variables
Solving equations is the main point of algebra. You solve an equation by isolating the variable (commonly x) while keeping the equation in balance — that is, by making sure that in each step, you apply the exact same operation to both sides of the equation. Here are a few types of equations you need to know how to solve on the ACT:
Equations with fractions (rational equations), such as
Equations with square roots (radicals), such as
Equations with absolute values, such as
Equations with variables in the exponent, such as
Typically, an equation with more than one variable, such as , can’t be solved for a number. However, you can solve an equation with more than one variable in terms of the other variables in the equation. For example, here’s how you solve this equation for b in terms of a and c:
An ACT question may ask you to solve an equation in terms of other variables. Additionally, this skill is useful when working with math formulas.
Data and graphs
A graph is a visual representation of data. Common graphs include bar graphs, pie charts, line graphs, and pictograms. Graph reading is a basic but essential skill that you need for the ACT. A typical question may ask you to identify specific data given in a graph, or you may need to pull this data as a first step in a more complex calculation.
Basic statistics and probability
Statistics is the mathematical study of real-world information called data sets — lists of numbers that are objectively observed and recorded. Three common operations used on data sets are three types of averages called the mean, the median, and the mode. On the ACT, you need to know how to calculate all three.
Statistics and probability
Probability measures the mathematical likelihood that an event will occur. On the ACT, you may need to calculate simple, compound, or conditional probability.
Focusing on intermediate algebra
Intermediate algebra is the focus of a high school Algebra II class. In this section, I outline the essential intermediate algebra skills you need to be successful on the ACT. Later on, in Chapter 8, you can gain a solid understanding of this material.
Taking a look at inequalities
An inequality is a statement telling you that two math expressions aren’t equal. On the ACT, inequalities come in four basic varieties:
Greater than (>)
Less than (<)
Greater than or equal to (≥)
Less than or equal to (≤)
You solve inequalities using the same algebra rules you would use to solve equations — with the exception of a couple of twists (flip to Chapter 7 for details). The solution to an inequality is typically a range of answers expressed as a simpler inequality.
Working with systems of equations
A system of equations is made of two equations that are simultaneously true. On the ACT, a system of equations usually is limited to two variables. For example, take a look at this system:
You can solve a simple system of equations by the substitution method, isolating a variable in one equation and then plugging its equivalent into the other equation. For a more complicated system of equations, use the elimination (or combination) method by either adding or subtracting the two equations and solving the equation that remains.
Understanding direct and inverse proportionality
When two values, x and y, are directly proportional, a value, k, makes the following equation true:
Values that are directly proportional tend to rise and fall together. For example, when one value doubles, the other value also doubles.
When two values, x and y, are inversely proportional, a value, k, makes the following equation true:
Values that are inversely proportional tend to rise or fall opposite of each other. For example, when one value is multiplied by 3, the other value is divided by 3.
Examining quadratic equations
A quadratic equation is an equation in the form . You can solve a quadratic equation either by factoring or by using the quadratic formula:
The ACT almost certainly will have several questions that require you to work with quadratic equations.
Finding information about functions
A function is a mathematical connection between two values. Usually, the values are an input variable, x, and an output variable, y. In a function, when you know the value of x, the value of y is determined.
Typical ACT questions may ask you to use functions as models, to work with functional notation f(x), to simplify the composition of two functions, to find the inverse of a function, or to find the domain or range of a function.
Working with coordinate geometry
Coordinate geometry is geometry that occurs on the xy-graph. This topic overlaps with material introduced in both Algebra I and Algebra II classes. Here, I give you an overview of the basic information from coordinate geometry that you need to review to do well on the ACT. I go over these ideas in greater detail in Chapter 9.
Graphing linear functions
A linear function is any function of the form . For example:
Linear functions, which produce a straight line when graphed, are common on the ACT. Some of the skills you need to feel comfortable with include mastering the distance and midpoint formulas, finding the slope of a line, using the slope-intercept form to solve problems, and working with parallel and perpendicular lines.
Recognizing quadratic functions
A quadratic function is in the form . For example:
On the graph, a quadratic function produces a parabola — a curve that looks roughly like an arch (or a U). On the ACT, a question may ask you to pair up a quadratic function with its graph. More difficult questions may require you to find the axis of symmetry or the vertex of a parabola or to solve a quadratic inequality.
Transforming functions
A transformation of a function is a small change that affects that function in a predictable way. Typical transformations include reflections across the x-axis and y-axis as well as vertical and horizontal shifts. An ACT question may ask you to compare two similar functions and select the equation that transforms one into the other. Or a question may provide a function and a transformation and ask you to produce the resulting graph.
Grappling with higher-order polynomial functions and circles
More difficult ACT questions may include higher-order polynomials, such as cubic equations of the form , and graphs of circles. These questions are rather uncommon and require only a basic familiarity with the concepts.
Reviewing plane geometry
Plane geometry is the focus of a typical high school geometry class. In this section, I discuss the geometry that you’re likely to see on the ACT. Chapter 11 gives you a complete review of these topics.
Lines and angles
One common type of ACT question presents you with a figure that contains lines and angles and then asks you to find the value of a given angle. To answer this type of question, you need to know how to measure right angles, vertical angles, supplementary angles, the angles in a triangle, and the angles that result when two lines are parallel.
For example, an ACT question may show you a figure with some angles labeled and ask you to find the measure of an unlabeled angle. Or it may ask you to identify a pair of angles that are equal in measure.
Triangles
Virtually every ACT includes several questions about triangles. You may need to find the area of a triangle given the height and the base, use the Pythagorean theorem to work with right triangles, or work with the most common types of right triangles, such as the 3-4-5 triangle.
An ACT question may ask you to find the area of a triangle given the measurements of its height and base, or, turning this question around, it may ask you to find the height given the length of the base and the area of the triangle. ACT questions involving right triangles may ask you to identify the length of one side of a right triangle, given information about the other sides.
Quadrilaterals
A quadrilateral is a four-sided polygon. Basic quadrilaterals that you may encounter on the ACT include squares, rectangles, parallelograms, and trapezoids. You need to know how to find the area of all these, and, more generally, you must feel comfortable working with the formulas for these areas.
For example, an ACT question may give you the perimeter of a rectangle with additional information and ask you to find the area. Or it may give you information about some aspects of a parallelogram — such as its height and area — and ask you to calculate the length of its base.
Circles
Circles are quite common on the ACT. You need to know the formulas for finding the diameter, area, and circumference of a circle given its radius. Additionally, you should be able to work with tangent lines, arc length, and chords of circles.
An ACT question may ask you to find the circumference of a circle given its area. More difficult ACT questions may require you to combine other geometry formulas to measure the area of a triangle with one side that’s tangent to a circle or a chord of a circle.
Solid geometry
Solid geometry deals with geometry that occurs in three-dimensional space. A basic ACT question may require you to find the volume of a cube or box (rectangular solid). More advanced questions may ask you to work with more complicated solids, such as spheres, prisms, cylinders, pyramids, and cones.
Dealing with trigonometry and other advanced topics
The ACT includes questions about a few advanced math topics, including trigonometry. In this section, I go over these topics to make sure you’re prepared for them. For further details, check out Chapter 12.
Trigonometry
Trigonometry is the mathematics of triangles — most commonly right triangles. ACT questions cover basic trig information. For instance, you need to know how to find the six trig ratios of a triangle in terms of the opposite side, adjacent side, and hypotenuse. More advanced trig concepts deal with radian measure, graphs of trig functions, and some basic trig identities.
Matrices
A matrix is a grid of numbers with both a horizontal and a vertical dimension. Virtually every ACT has a question that asks you to recall basic information about matrices, such as adding or subtracting matrices, multiplying a matrix by a constant, or working with the determinant of a 2-by-2 matrix.
Logarithms
A logarithm is the inverse form of an exponent. Not every ACT includes a question about logarithms, but if you encounter this type of a question, knowing how to convert a logarithmic equation into an exponential equation is particularly helpful.
Imaginary and complex numbers
An imaginary number includes the value i where . A complex number is a number of the form . ACT questions about these types of numbers aren’t usually difficult. In fact, some basic information can help you to answer them.
You’re Off the Hook: Discovering What the ACT Doesn’t Cover
The ACT math test covers most of the topics you’re likely to find in a basic high school math curriculum. In fact, it’s more advanced than the SAT in its range of math topics. Fortunately, even the ACT doesn’t require you to know everything about math. Here are three easily identifiable areas of math that the ACT doesn’t cover:
Ellipses and hyperbolas:
The equations and graphs for ellipses and hyperbolas, often part of an Algebra II or a pre-calculus class, aren’t present on the ACT.
The value
e
and natural logarithms:
In a pre-calculus course, your teacher introduces you to the value
e
and its inverse function, the natural log. Both of these areas are essential for calculus, but you don’t need to worry about them for the ACT.
Calculus and beyond:
More and more high school students are taking one or even two years of calculus and other advanced math. On the ACT, you definitely don’t have to worry about limits, derivatives, integrals, or any other advanced concepts that you encounter in a calculus class.
This information comes as good news for most students. If you’re currently taking an advanced math class, of course, you still need to study to maintain your grades. (You don’t want to get a 36 on your ACT and then be rejected from your first-choice college because of low grades, right?) On the plus side, you may find many ACT questions easier than last night’s homework.
On the other hand, if your goal in life is to avoid as much math as possible going forward, then as the saying goes “You may already be a winner!” That is, if you’ve passed high school Algebra I and II, and Geometry, you should be in reasonably good shape.
In either case, use this book to review the topics you’re shaky on, solidify these skills with practice problems, and then take the practice tests in Part 5
