Copyright ©2017, Prufrock Press Inc Edited by Lacy Compton ISBN-13: 978-1-61821-971-8 No part of this book may be reproduced, translated, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the publisher For more information about our copyright policy or to request reprint permissions, visit https://www.prufrock.com/permissions aspx At the time of this book’s publication, all facts and figures cited are the most current available All telephone numbers, addresses, and website URLs are accurate and active All publications, organizations, websites, and other resources exist as described in the book, and all have been verified The author and Prufrock Press Inc make no warranty or guarantee concerning the information and materials given out by organizations or content found at websites, and we are not responsible for any changes that occur after this book’s publication If you find an error, please contact Prufrock Press Inc Prufrock Press Inc P.O Box 8813 Waco, TX 76714-8813 Phone: (800) 998-2208 Fax: (800) 240-0333 http://www.prufrock.com The value of an education … is not the learning of many facts but the training of the mind to think —Albert Einstein Contents Acknowledgements Foreword to the Third Edition Introduction: Why Another SAT Book This Book’s Approach PART 1: GENERAL TEST-TAKING ADVICE AND STRATEGIES Chapter 1: What Is Math and Why Do I Need to Study It? Why You Should Never Say “I Hate Math!” Chapter 2: Test Preparation Strategies and Advice How Too Much “Strategy” Can Be Hazardous to Your Performance An Approach to Test Preparation Do Calculators Really Help? Too Much Memorization? Ubiquitous Algebra Fast Math Reading Comprehension PSAT Versus SAT SAT Versus ACT Final Tips for Better Learning PART II: SPECIFIC MATH TOPICS, TIPS FOR PROBLEM SOLVING, AND SAMPLE PR OBLEMS Chapter 3: Arithmetic Concepts 3.1: Integer 3.2: Odd/Even Numbers 3.3: Consecutive Integers 3.4: Number Line 3.5: Reciprocal (of a Number) 3.6: Factors (of a Number) 3.7: Common Factors 3.8: Greatest Common Factor (GCF) 3.9: Common Multiples (of Two or More Numbers) 3.10: Least Common Multiple (LCM) 3.11: Greatest Common Divisor (GCD) 3.12: Prime Numbers 3.13: Ratio 3.14: Percent 3.15: Proportion 3.16: Directly Proportional Quantities 3.17: Inversely Proportional Quantities 3.18: Sequence 3.19: Arithmetic Sequence 3.20: Geometric Sequence 3.21: Set 3.22: Union 3.23: Intersection 3.24: Venn Diagram 3.25: Factorials 3.26: Permutation 3.27: Combination 3.28: Absolute Value Chapter 4: Algebra 4.1: Monomial 4.2: Coefficient 4.3: Polynomial 4.4: Term 4.5: Like Terms 4.6: Exponents 4.7: Function 4.8: Quadratic Function 4.9: Domain 4.10: Range 4.11: Interpreting and Solving Equations/Inequalities 4.12: Cartesian Coordinate System 4.13: Straight Line 4.14: Slope 4.15: Distance 4.16: Midpoint 4.17: Complex Numbers Chapter 5: Geometry 5.1: Angles 5.2: Vertex 5.3: Acute Angles 5.4: Obtuse Angles 5.5: Vertical Angles 5.6: Supplementary Angles 5.7: Complementary Angles 5.8: Straight Angles 5.9: Right Angles 5.10: Exterior Angles 5.11: Polygon 5.12: Diagonal 5.13: Triangle 5.14: Acute Triangle 5.15: Obtuse Triangle 5.16: Right Triangle 5.17: Equilateral Triangle 5.18: Isosceles Triangle 5.19: Hypotenuse 5.20: Congruent Triangles 5.21: Similar Triangles 5.22: Altitude 5.23: Quadrilaterals 5.24: Parallelograms 5.25: Rectangles 5.26: Squares 5.27: Perimeter 5.28: Regular Polygon 5.29: Circle 5.30: Diameter 5.31: Radius 5.32: Arc 5.33: Tangent Line 5.34: Tangent to a Circle 5.35: Circumference 5.36: Central Angle 5.37: Area of a Sector and Arc Length 5.38: Solid Geometry Chapter 6: Trigonometry 6.1: Right Triangle 6.2: Unit Circle 6.3: Degrees to Radians Conversion 6.4: Trigonometric Functions 6.5: Miscellaneous Identities 6.6: Pythagorean Identities Chapter 7: Miscellaneous Topics 7.1: Average/Arithmetic Mean 7.2: Median 7.3: Mode 7.4: Weighted Average 7.5: Probability 7.6: Independent Events Chapter 8: Problems by Degree of Difficulty and Type 8.1: Easy Problems; Multiple Choice 8.2: Easy Problems; Open-Ended 8.3: Medium Problems; Multiple Choice 8.4: Medium Problems; Open-Ended 8.5: Hard Problems; Multiple Choice 8.6: Hard Problems; Open-Ended Chapter 9: Solutions to Problems in Chapter 8 Chapter 10: Additional Problems Chapter 11: Solutions to Problems in Chapter 10 Chapter 12: Test Analysis Sample Test Using the Online Test Chapter 13: Mind Games Chapter 14: Solutions to Mind Games Conclusion Appendix A Appendix B: Solutions to On Your Own Problems References About the Author Common Core State Standards Alignment Acknowledgements To the students I had the privilege to work with over the years, I am extremely grateful I want to thank them for putting up with my occasionally unconventional methods of teaching math, providing me with candid and useful feedback, and—most of all—for making my interaction with them translate into hours of fun and rewarding work In addition, I want to thank my children for fueling my passion for smart education; Dr MAC (Jean D’Arcy Maculaitis, Ph.D., President and founder of MAC Testing & Consulting, LLC) for being the catalyst for my involvement in various aspects of SAT preparation and—through her work and dedication of many years—providing me with a continuous source of inspiration; and my wife for her relentless support and encouragement Foreword to the Third Edition Given the tremendous number of SAT books on the market, the reception received by the first (2010) and the updated (2014) editions of this book far exceeded my expectations The fact that so many parents, students, and even educational organizations1 embraced the idea that math should not be a continuous exercise in memorization and applying trial-end-error using a calculator, but rather a quest to improve one’s analytic/critical thinking skills, was extremely encouraging and certainly played a major role in the decision to work on a new edition This new edition, of course, was created to take into account the major changes that the College Board implemented in its “New” SAT since March 16, 2016 When it comes to the need to update and publish a new edition of a test preparatory book like this, the author is always challenged to strike the right balance between novelty and material that can be reused (After all, as a colleague of mine jokingly put it, “the Pythagorean Theorem hasn’t really changed much since the publication of the previous edition!”) To provide some insight into the driving force behind and the “novelty” of this edition, the changes made by the College Board (in content and format) were summarized in Table 1 with additional specifics on the changes implemented starting March 2016 illustrated in Table 2 TABLE 1 “Old” Versus “New” SAT Category Total Time Components/Time/Number of Questions “Old” SAT 3 hours 45 minutes (25-minute essay is mandatory) • • • • Critical Reading/50 minutes/67 Writing/60 minutes/49 Math/70 minutes/54 Essay (mandatory)/25 minutes/1 “New” SAT 3 hours 50 minutes (50-minute essay is optional) • Evidence-Based Reading and Writing Reading Test/65 minutes/52 Writing and Language/35 minutes/44 Math (78%/22%; mc/oe) » No-calculators/25 minutes/20 (15/5; mc/oe) » Calculators/55 minutes/38 (30/8; mc/oe) Essay (optional)/50 minutes/1 Philosophy • Emphasis on general reasoning skills and vocabulary • Penalties for incorrect answers • Essay (mandatory) given at the beginning of the test; students are required to take a position on a presented issue • Stronger emphasis on reasoning and stronger focus on knowledge skills deemed important for college success and career readiness • Greater emphasis on the meaning of the words in extended context and how word choice shapes the meaning, tone, and impact • No penalties for wrong answers • Essay (optional) given at the end of the SAT; students are required to analyze a provided source text Score • Composite Score: (600–2400) • Area Scores: 3 (200–800; Critical Reading, Writing, and Math) • Test Scores: N/A • Cross-Test Scores: N/A • Subscores: N/A • Essay: Combined scores of two raters each scoring on a 1–6 scale • Composite Score: 400–1600 • Area Scores: 2 (200–800; Evidence-Based Reading and Writing + Math) • Test Scores: 3 (10–40; Reading, Writing and Language, Math) • Cross-Test Scores: 2 (10–40; Analysis in Science and History/Social Studies; based on selected questions from the two main areas) • Subscores: 7 (1–15; Command of