SAT Math For Dummies - Buch.de · PDF fileMath tutor and author of Calculus II For Dummies ......
Transcript of SAT Math For Dummies - Buch.de · PDF fileMath tutor and author of Calculus II For Dummies ......
Mark ZegarelliMath tutor and author of Calculus II For Dummies and Basic Math & Pre-Algebra For Dummies
*SAT is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product.
• Score high on the mathematics sections of the SAT
• Identify question types (including those designed to confuse you)
• Manage your time during the exam
• Perfect your skills with three practice tests
Learn to:
*SAT MathMaking Everything Easier!™
Open the book and find:
• When to use a calculator (and which kinds are permitted)
• Advice on answering grid-in questions and word problems
• Reviews of the math topics you need to know
• The 411 on algebra, geometry, and more
• Straightforward explanations of statistics
• SAT problem-solving techniques
• How to work with decimals, fractions, and square roots
• Ten tips to take to the test
Mark Zegarelli holds degrees in math and English from Rutgers University. He is a math tutor and writer with over 25 years of professional experience. Zegarellli has written numerous Dummies books, including LSAT Logic Games For Dummies and Basic Math & Pre-Algebra Workbook For Dummies.
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Scoring well on the mathematics sections of the SAT isn’t guaranteed by getting good grades in algebra and geometry. Loaded with test-taking strategies, three practice tests, and hundreds of problems with detailed solutions and explanations, this is your go-to guide for making the best use of the limited time allowed and getting your best possible score!
• Demystify SAT math — get an overview of the SAT math topics you’ll need to know: arithmetic, algebra, geometry, coordinate geometry, and more
• Brush up on basics — refresh your knowledge of the basic math skills you’ll apply on the exam, and discover the formulas you’ll need to know
• Solve your way to success — pull together everything you’ve learned and discover how to approach and solve the types of word problems you’ll see on the actual exam
• Put your skills to the test — answer practice questions throughout the book, plus take three full-length practice tests with detailed answers and explanations at the end of the book
• Make the most of your time — discover the best ways to utilize your study time between now and exam day, as well as how to keep yourself on track on the big day
Manage your time and ace the mathematicssections of the SAT
*SAT Math
Zegarelli
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by Mark Zegarelli
*SAT MathFOR
DUMmIES‰
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*SAT Math For Dummies®
Published byWiley Publishing, Inc.111 River St.Hoboken, NJ 07030-5774www.wiley.com
Copyright © 2010 by Wiley Publishing, Inc., Indianapolis, Indiana
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 either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. 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, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for the Rest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the United States and other coun-tries, and may not be used without written permission. *SAT is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. All other trademarks are the property of their respective owners. Wiley Publishing, Inc., is not associated with any product or vendor mentioned in this book.
LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOM-MENDATIONS IT MAY MAKE. FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ.
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ISBN: 978-0-470-62085-4
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About the AuthorMark Zegarelli is the author of LSAT Logic Games For Dummies (Wiley) plus four other For Dummies books on basic math and pre-algebra, Calculus II, and logic. He holds degrees in both English and math from Rutgers University and is an SAT teacher and tutor.
Mark lives in Long Branch, New Jersey, and San Francisco, California.
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DedicationThis is for my dear friend Simon Stanley Marcus, with much gratitude for your boundless wisdom and presence.
Author’s AcknowledgmentsThis is my sixth For Dummies book, and again I enjoy the privilege of working with an edito-rial team that continues to inspire and call me to my best. Thank you to my Wiley editors: Chrissy Guthrie, Danielle Voirol, and Lindsay Lefevere. More thanks for my technical edi-tors, Amy Nicklin and Benjamin Wyss, for setting me on a better course whenever 2 + 2 = 5.
I really don’t know how to express proper gratitude for all of the wonderful people in my life who surround me with constant love, support, encouragement, and joy. But I want you to know that I feel truly blessed and fortunate to make my home here on Earth with all of you. So a very deep thank you to my family: Alan and Mary Lou Cary, Joe, Jasmine, and Jacob Cianflone, Deseret Moctezuma, Janet Rackham, Anthony and Christine Zegarelli, and Tami Zegarelli. And one more to my family of friends: Pete Apito, Bradley Averill, Joel Cohen, Chip DeCraene, Mark Dembrowski, Chris Demers, David Feaster, Rick Kawala, Michael Konopko, Al LeGoff, Brian London, Stephen McAllister, Lou Natale, Tom Nicola, Mark O’Malley, Tim O’Rourke, Christian Romo, Robert Rubin, Alison Sigethy, Rachel Silber, and Ken Wolfe.
And again, I must pay tribute to the kind folks at Maxfield’s House of Caffeine for providing a seemingly endless supply of coffee, bagels, bananas, and carrot juice.
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Publisher’s Acknowledgments
We’re proud of this book; please send us your comments at http://dummies.custhelp.com. For other comments, please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002.
Some of the people who helped bring this book to market include the following:
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Contents at a GlanceIntroduction ................................................................................1
Part I: Making Plans for This SATurday: An Overview of SAT Math .............................................................5Chapter 1: SAT Math Basics .......................................................................................................................7Chapter 2: Testing 1-2-3: SAT Math Test-Taking Skills ..........................................................................21
Part II: Did They Really Cover This Stuff in School? A Review of Math Skills .............................................................33Chapter 3: The Numbers Game: Arithmetic Review..............................................................................35Chapter 4: Return of the X-Men: Reviewing Algebra .............................................................................63Chapter 5: Picture Perfect: Reviewing Geometry ..................................................................................93Chapter 6: Functions and Coordinate Geometry .................................................................................139Chapter 7: From the Grab Bag: A Variety of Other SAT Math Skills ..................................................175
Part III: Your Problems Are Solved! SAT Problem-Solving Techniques ...............................................209Chapter 8: What’s in a Word? SAT Word Problems ............................................................................211Chapter 9: SAT Math Strategy ................................................................................................................237
Part IV: Practice Makes Perfect: SAT Math Practice Tests ..........249Chapter 10: Practice Test 1 ....................................................................................................................251Chapter 11: Answers and Explanations for Practice Test 1 ...............................................................269Chapter 12: Practice Test 2 ....................................................................................................................283Chapter 13: Answers and Explanations for Practice Test 2 ...............................................................303Chapter 14: Practice Test 3 ....................................................................................................................317Chapter 15: Answers and Explanations to Practice Test 3.................................................................335
Part V: The Part of Tens ...........................................................351Chapter 16: Ten Tips to Improve Your SAT Math Score ....................................................................353Chapter 17: Ten Tips to Be at Your Best on the SAT ..........................................................................357
Index ......................................................................................361
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Table of Contents
Introduction ................................................................................1About This Book ........................................................................................................................1Conventions Used in This Book ...............................................................................................2Foolish Assumptions .................................................................................................................2How This Book Is Organized ....................................................................................................2
Part I: Making Plans for This SATurday: An Overview of SAT Math..........................2Part II: Did They Really Cover This Stuff in School? A Review of Math Skills ..........3Part III: Your Problems Are Solved! SAT Problem-Solving Techniques ....................3Part IV: Practice Makes Perfect: SAT Math Practice Tests .........................................3Part V: The Part of Tens ..................................................................................................3
Icons Used in This Book ...........................................................................................................4Where to Go from Here .............................................................................................................4
Part I: Making Plans for This SATurday: An Overview of SAT Math ..............................................................5
Chapter 1: SAT Math Basics .................................................................................................7
Getting an Overview of the SAT Math Sections .....................................................................7Knowing What’s In: The Math You Need for the SAT ...........................................................8
Calculating with arithmetic questions ..........................................................................8Doing the algebra shuffle ................................................................................................9Go figure: Doing geometry ............................................................................................12Working with functions and coordinate geometry ....................................................14Rounding up some grab-bag skills ...............................................................................16
Knowing What’s Out: A Few Topics Not Covered on the SAT ...........................................18Building Your Problem-Solving Skills ....................................................................................19
Solving word problems ................................................................................................19Figuring out which tools to use....................................................................................19
Chapter 2: Testing 1-2-3: SAT Math Test-Taking Skills ................................................21
Knowing Both Types of SAT Math Questions ......................................................................21Answering multiple-choice questions .........................................................................21Responding to grid-in questions ..................................................................................22
Focusing on the Fine Print ......................................................................................................25Taking note of the Notes: General assumptions ........................................................25Regarding the Reference Information: Facts and formulas ......................................27
Getting the Timing Right .........................................................................................................27Calculating Your Way to Success: Calculators and the SAT ..............................................28
Choosing an acceptable calculator .............................................................................28Reviewing what you should absolutely, positively know
how to do on your calculator ...................................................................................29Considering other things that are good to know
how to do on your calculator ...................................................................................29Taking calculations step by step .................................................................................30Knowing the right time to use your calculator ..........................................................31
Putting the Flash Back in Flash Cards ...................................................................................31Using flash cards effectively .........................................................................................31Deciding what to put on flash cards ............................................................................32
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x SAT Math For Dummies
Part II: Did They Really Cover This Stuff in School? A Review of Math Skills ..............................................................33
Chapter 3: The Numbers Game: Arithmetic Review ......................................................35
Maintaining Your Integrity with Integers .............................................................................35Doing Some Digital Computing ..............................................................................................36The Space Between: Using Number Lines ............................................................................37Dividing and Conquering: Understanding Divisibility, Factors, and Multiples ...............38
Testing for divisibility ...................................................................................................39Factoring in knowledge of factors ...............................................................................40Multiplying your understanding of multiples.............................................................42
Understanding Percents .........................................................................................................43Converting between percents and decimals ..............................................................43Increasing your score (and decreasing your stress)
with percent increase/decrease problems .............................................................45Ratios: Making Comparisons ..................................................................................................47
Treating ratios as fractions ..........................................................................................48Proportions: Crossing paths with equal ratios ..........................................................49
Feeling Powerful with Exponents and Getting Rooted with Roots ....................................50Squaring up your knowledge of squares and square roots ......................................50Evaluating expressions with exponents and roots ....................................................51Fractional bases: Raising fractions to powers ...........................................................52Fractional exponents: Combining powers and roots ................................................53
Practice Problems for Arithmetic Review ............................................................................54Solutions to Practice Problems .............................................................................................57
Chapter 4: Return of the X-Men: Reviewing Algebra .....................................................63
Knowing the Algebra You Forgot to Remember ..................................................................63Vocab: A few choice words about algebra .................................................................64It is written: Knowing some algebra shorthand .........................................................65
Expressing Yourself with Algebraic Expressions ................................................................66Can I get your number? The value of evaluation .......................................................66Knowing the simple truth about simplifying ..............................................................67Taking the fear out of factoring....................................................................................70
Finding a Balance with Algebraic Equations ........................................................................72A lonely letter: Isolating the variable ..........................................................................73Doing away with fractions: Cross-multiplying to solve rational equations ............73Factoring to solve quadratic equations ......................................................................74Solving equations that have exponential variables ...................................................75Solving equations with radicals (roots) ......................................................................75Positive thoughts: Feeling confident with absolute value ........................................76
Solving Problems with More Than One Variable ................................................................77Solving an equation in terms of other variables ........................................................77Solving equations with extra variables .......................................................................78Solving a system of equations ......................................................................................78
Solving Inequalities .................................................................................................................80Solving basic inequalities .............................................................................................80Solving inequalities with absolute value.....................................................................81
Symbol Secrets: Working with New Notations ....................................................................83Practice Problems for Algebra ...............................................................................................84Solutions to Practice Problems .............................................................................................87
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Chapter 5: Picture Perfect: Reviewing Geometry ..................................................93Working All the Angles ............................................................................................................93
Crossing over with vertical angles ..............................................................................93Supplementary angles: Doing a one-eighty.................................................................94Going ninety: Right angles and complementary angles ............................................95Making matches: Parallel lines and corresponding angles ......................................96Sum of the angles in a triangle .....................................................................................97
Putting Triangles to the Test .................................................................................................98Touching base on the area of a triangle .....................................................................99Keeping right triangles cornered .................................................................................99Side shows: The triangle inequality...........................................................................103Getting familiar looks: Congruent and similar triangles .........................................104
Going for Four: Quadrilaterals .............................................................................................107Squares ..........................................................................................................................107Rectangles .....................................................................................................................108Parallelograms..............................................................................................................109
Rolling Along with Circles ....................................................................................................111From center stage: Radius and diameter ..................................................................111Finding the area of a circle .........................................................................................112Getting around to the circumference ........................................................................112Not quite full circle: Finding arc length.....................................................................113Touching on tangent lines ..........................................................................................114
Solidifying Your Understanding of Solid Geometry ..........................................................115Volume of a rectangular solid ....................................................................................115Volumes of a cylinder ..................................................................................................116Pyramids and cones ....................................................................................................117
Improving Your Geometric Perception ..............................................................................119Getting your head around rotations ..........................................................................119Adding a dimension: Getting other views of 3-D objects ........................................121Folding in information about surfaces ......................................................................122
Practice Problems for Geometry .........................................................................................124Solutions to Practice Problems ...........................................................................................132
Chapter 6: Functions and Coordinate Geometry ...........................................................139
Knowing How Mathematical Functions Function ..............................................................139Understanding the basic idea of a function ..............................................................139Solving functions with an input-output table ...........................................................140Using function notation ...............................................................................................140Functioning within certain limits: Finding the domain and the range ..................142
Coordinating Your Grasp of Coordinate Geometry ..........................................................144Getting to the point .....................................................................................................144Lining things up............................................................................................................145Feeling inclined to measure slope .............................................................................148
Graphing Linear Functions ...................................................................................................151Quadratic Functions ..............................................................................................................154
Solving quadratic equations .......................................................................................154Graphing quadratic functions ....................................................................................157
Transformations: Moving and Flipping Graphs .................................................................160Reflecting on reflections .............................................................................................161Shift happens: Moving left, right, up, or down .........................................................162
Practice Problems on Functions and Coordinate Geometry ...........................................163Solutions to Practice Problems ...........................................................................................170
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xii SAT Math For Dummies
Chapter 7: From the Grab Bag: A Variety of Other SAT Math Skills ..........................175
Lining Things Up with Sequences .......................................................................................175Setting up for Success: Set Theory ......................................................................................177
Understanding union and intersection .....................................................................177Knowing a few important sets of numbers ...............................................................178Intersections: Showing overlap with Venn diagrams ..............................................179
Thinking Logically: Logic Questions ...................................................................................180Statistically Speaking: Understanding Averages ...............................................................182
Knowing the three M’s: Mean, median, and mode ..................................................182Weighs and means: Finding weighted averages.......................................................184Finding the mean of algebraic expressions ..............................................................185
Figuring the Odds: Problems in Probability .......................................................................186Possible outcomes: Using your counting skills........................................................187What are the odds? Calculating probability .............................................................189On target: Visualizing geometric probability ...........................................................190
Seeing Is Believing: Interpreting Data from Graphs ..........................................................191Raising the bar with a bar graph................................................................................192Picturing data with a pictogram ................................................................................192Getting a slice of the pie chart ...................................................................................193Lining up information with a line graph....................................................................194Unscattering data with a scatterplot .........................................................................195
Practice Problems for Grab-Bag Skills ................................................................................198Solutions to Practice Problems ...........................................................................................204
Part III: Your Problems Are Solved! SAT Problem-Solving Techniques ................................................209
Chapter 8: What’s in a Word? SAT Word Problems .....................................................211
Solving Word Problems Using Equations ...........................................................................211Getting the groupings right: Translations with parentheses .................................212Translating equations that involve fractions ...........................................................213Choosing a variable to avoid fractions .....................................................................214Writing systems of equations: Using more than one variable ...............................215
Charting a Course: Drawing Charts to Solve Word Problems .........................................217Picturing Success: Sketching to Solve Word Problems ....................................................220
Distance drawings: Moving with a purpose .............................................................220Timelines: Avoiding algebra with a number line .....................................................222Spacing out: Uncovering hidden geometry ..............................................................223
Practice Word Problems .......................................................................................................225Solutions to Practice Problems ...........................................................................................228
Chapter 9: SAT Math Strategy ..........................................................................................237
Performing SA-Triage: How Difficult Is This Problem? .....................................................237Formulas for Success: Working with Math Formulas .......................................................238
Knowing the right formulas ........................................................................................238Answering formula questions ....................................................................................240
Plotting a Course to Answer Tough Questions ..................................................................242Tips and Tricks: Looking for Fast, Easy Approaches ........................................................243
Using the five resources at your service ..................................................................243Putting your brain to work .........................................................................................245
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xiii Table of Contents
Part IV: Practice Makes Perfect: SAT Math Practice Tests ...........249
Chapter 10: Practice Test 1 ...............................................................................................251
Section 1 .................................................................................................................................255Section 2 .................................................................................................................................260Section 3 .................................................................................................................................264
Chapter 11: Answers and Explanations for Practice Test 1 .......................................269
Solutions to Section 1 Questions .........................................................................................269Solutions to Section 2 Questions .........................................................................................273Solutions to Section 3 Questions .........................................................................................278Answer Key .............................................................................................................................281
Chapter 12: Practice Test 2 ...............................................................................................283
Section 1 .................................................................................................................................287Section 2 .................................................................................................................................292Section 3 .................................................................................................................................298
Chapter 13: Answers and Explanations for Practice Test 2 .......................................303
Solutions to Section 1 Questions .........................................................................................303Solutions to Section 2 Questions .........................................................................................307Solutions to Section 3 Questions .........................................................................................311Answer Key .............................................................................................................................315
Chapter 14: Practice Test 3 ...............................................................................................317
Section 1 .................................................................................................................................321Section 2 .................................................................................................................................326Section 3 .................................................................................................................................331
Chapter 15: Answers and Explanations for Practice Test 3 .......................................335
Solutions to Section 1 Questions .........................................................................................335Solutions to Section 2 Questions .........................................................................................340Solutions to Section 3 Questions .........................................................................................344Answer Key .............................................................................................................................349
Part V: The Part of Tens ............................................................351
Chapter 16: Ten Tips to Improve Your SAT Math Score ..............................................353
Study Diligently in Your Math Classes ................................................................................353Get Good at Doing Basic Calculations in Your Head .........................................................353Get Good at Using Your Calculator .....................................................................................354Study SAT-Specific Math Skills .............................................................................................354Study SAT-Specific Problem-Solving Skills .........................................................................354Get Comfortable Turning Words into Numbers ................................................................354Take Timed Practice Tests ...................................................................................................355Study from Your Timed Practice Tests ..............................................................................355Retake Your Timed Practice Tests ......................................................................................355Take the SAT More Than Once ............................................................................................355
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