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California High School Exit Exam: Math by Jerry Bobrow, Ph.D.
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California High School Exit Exam: Math by Jerry Bobrow, Ph.D.
Contributing Authors George Crowder, M.A. Adam Bobrow, B.A. Dale Johnson, M.A.
CliffsTestPrep
®
California High School Exit Exam: Math by Jerry Bobrow, Ph.D.
Contributing Authors George Crowder, M.A. Adam Bobrow, B.A. Dale Johnson, M.A.
About the Author
Publisher’s Acknowledgments
Dr. Jerry Bobrow, Ph.D., is a national authority in the field of test preparation. As executive director of Bobrow Test Preparation Services, he has been administering the test preparation programs at over 25 California institutions for the past 27 years. Dr. Bobrow has authored over 30 national best-selling test preparation books, and his books and programs have assisted over two million test-takers. Each year, Dr. Bobrow personally lectures to thousands of students on preparing for graduate, college, and teacher credentialing exams.
Editorial Project Editor: Marcia L. Johnson Senior Acquisition Editor: Greg Tubach Copy Editor: Kathleen Robinson Production Proofreader: Arielle Mennelle Wiley Publishing, Inc. Composition Services
Author’s Acknowledgments My loving thanks to my wife, Susan, and my children, Jennifer, Adam, and Jonathon, for their patience and support in this long project. My sincere thanks to Michele Spence, former chief editor of CliffsNotes, for her invaluable assistance. I would also like to thank Marcia Johnson for final editing and careful attention to the production process. CliffsTestPrep® California High School Exit Exam: Math Published by: Wiley Publishing, Inc. 111 River Street Hoboken, NJ 07030-5774 www.wiley.com
Note: If you purchased this book without a cover, you should be aware that this book is stolen property. It was reported as “unsold and destroyed” to the publisher, and neither the author nor the publisher has received any payment for this “stripped book.”
Copyright © 2005 Jerry Bobrow, Ph.D. Published by Wiley, Hoboken, NJ Published simultaneously in Canada Library of Congress Cataloging-in-Publication Data Bobrow, Jerry. California high school exit exam—math / by Jerry Bobrow ; contributing authors, George Crowder, Adam Bobrow, Dale Johnson.— 1st ed. p. cm. — (CliffsTestPrep) ISBN 0-7645-5939-7 (pbk.) 1. Mathematics—Examinations, questions, etc. 2. California High School Exit Exam—Study guides. I. Crowder, George. II. Bobrow, Adam. III. Johnson, Dale. IV. Title. V. Series. QA43.B648 2004 510'.76—dc22 2004020172 ISBN: 0-7645-5939-7 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 1B/RV/RR/QU/IN 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 Legal Department, Wiley Publishing, Inc., 10475 Crosspoint Blvd., Indianapolis, IN 46256, 317-572-3447, or fax 317-572-4447. 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 RECOMMENDATIONS 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. Trademarks: Wiley, the Wiley Publishing logo, CliffsNotes, the CliffsNotes logo, Cliffs, CliffsAP, CliffsComplete, CliffsQuickReview, CliffsStudySolver, CliffsTestPrep, CliffsNote-a-Day, cliffsnotes.com, and all related trademarks, logos, and trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates. 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. For general information on our other products and services or to obtain technical support, please contact our Customer Care Department within the U.S. at 800-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002. Wiley also published its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Study Guide Checklist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Format of CAHSEE Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Questions Commonly Asked About CAHSEE Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 How You Can Do Your Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Positive Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Elimination Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Avoiding the Misread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
PART I: WORKING TOWARD SUCCESS Strategies for the Math Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Number Sense (Grade 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Standard Set 1.0: Working with Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Standard Set 2.0: Use Exponents, Powers and Roots, and Fractions with Exponents . . . . . 8 Statistics, Data Analysis, and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Grade 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Standard Set 1.0: Analyzing Statistical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Standard Set 2.0: Describing and Using Data Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Standard Set 3.0: Determining Probabilities and Making Predictions . . . . . . . . . . . . . . . . 16 Grade 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Standard Set 1.0: Collect, Organize, and Represent Data and Identify . . . . . . . . . . . . . . . 17 Measurement and Geometry (Grade 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Standard Set 1.0: Choose Appropriate Units of Measure and Conversions Between Measurement Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Standard Set 2.0: Compute Perimeter, Area, and Volume; Understand Effects of Scale Changes on These Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Standard Set 3.0: Know the Pythagorean Theorem, Understand Plane and Solid Shapes, Identify Attributes of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Algebra and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Standard Set 1.0: Using Algebraic Terminology, Expressions, Equations, Inequalities, and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Standard Set 2.0: Interpret and Evaluate Expressions Involving Powers and Simple Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Standard Set 3.0: Graph and Interpret Linear and Nonlinear Functions . . . . . . . . . . . . . . 46 Standard Set 4.0: Solve Simple Linear Equations and Inequalities (Rational Answers) . . . 46 Mathematical Reasoning (Grade 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Standard Set 1.0: Decide How to Approach Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Standard Set 2.0: Use Strategies, Skills, and Concepts to Solve Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Standard Set 3.0: Determine That a Solution Is Complete and Generalize to Other Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Algebra I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 5.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 6.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
CliffsTestPrep California High School Exit Exam: Math
Standard Set 7.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 8.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 9.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 10.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Standard Set 15.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A Quick Review of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Symbols, Terminology, Formulas and General Mathematical Information. . . . . . . . . . . . . . . 85 Common Math Symbols and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Math Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Important Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Math Words and Phrases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Mathematical Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Arithmetic Diagnostic Test (Including Number Sense, Probability, Statistics and Graphs). . . 90 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Arithmetic Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Place Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Rounding Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Estimating Sums, Differences, Products and Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Using Percents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Signed Numbers (Positive Numbers and Negative Numbers). . . . . . . . . . . . . . . . . . . . . 102 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Powers and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Squares and Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Scientific Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Parentheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Order of Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Some Basic Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Algebra Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Algebra Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Variables and Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Evaluating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Monomials and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Solving for Two Unknowns—Systems of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Basic Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Measurement and Geometry Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Measurement and Geometry Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Coordinate Geometry and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
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Table of Contents
PART II: FULL-LENGTH PRACTICE TESTS CAHSEE Practice Test #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Test #1—Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Reviewing Practice Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Review Chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Reasons for Mistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Number Sense (NS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Statistics, Data Analysis, Probability (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Algebra and Functions (AF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Measurement and Geometry (MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Algebra I (AL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Mathematical Reasoning (MR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
CAHSEE Practice Test #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Test #2—Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Reviewing Practice Test 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Review Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Reasons for Mistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Number Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Statistics, Data, Analysis, Probability (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Algebra and Functions (AF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Measurement and Geometry (MG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Algebra I (AI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Mathematical Reasoning (MR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
CAHSEE Practice Test #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Test #3—Answers and Explanations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Reviewing Practice Test 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Review Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Reasons for Mistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Number Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Statistics, Data Analysis, Probability (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Algebra and Functions (AF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Measurement and Geometry (MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Algebra I (AI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Mathematical Reasoning (MR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Arithmetic/Statistics and Probability Glossary of Terms . . . . . . . . . . . . . . . 265 Algebra Glossary of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Measurement and Geometry Glossary of Terms . . . . . . . . . . . . . . . . . . . . . 273 Final Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Final Preparation and Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Finishing Touches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
v
Preface We know that passing the CAHSEE Math is important to you! And we can help. As a matter of fact, we have spent the last thirty years helping over a million test takers successfully prepare for important exams. The techniques and strategies that students and adults have found most effective in our preparation programs at 26 universities, county offices of education, and school districts make this book your key to success on the CAHSEE Mathematics. Our easy-to-use CASHEE Mathematics Preparation Guide gives you that extra edge by: ■ ■ ■ ■ ■ ■ ■
Answering commonly asked questions Introducing important test-taking strategies and techniques Reviewing the California mathematics standards Analyzing sample problems and giving suggested approaches Providing a quick review of mathematics with diagnostic tests Providing three simulated practice exams with explanations Including analysis charts to help you spot your weaknesses
We give you lots of strategies and techniques with plenty of practice problems. There is no substitute for working hard in your regular classes, doing all of your homework and assignments, and preparing properly for your classroom exams and finals. But if you want that extra edge to do your best on the CAHSEE Mathematics, follow our Study Plan and step-by-stem approach to success on the CAHSEE. Best of luck, Jerry Bobrow, Ph.D.
vii
Study Guide Checklist Check off each step after you complete it. ❑
1. Read “Mathematics Study Guide” available from your school or from the California Department of Education (CDE).
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2. Review any information or materials available online at cde.ca.gov.
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3. Look over the format of CAHSEE Math (p. 1).
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4. Read “Questions Commonly Asked About CAHSEE Math” (p. 2).
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5. Learn how you can do your best (p. 3).
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6. Carefully read “Part I: Working Toward Success” focusing on the sample problems and suggested approaches.
■
Number Sense (pp. 7–16) Statistics, Data Analysis, and Probability (pp. 16–26) Measurement and Geometry (pp. 26–45) Algebra and Functions (pp. 45–60) Mathematical Reasoning (pp. 60–68)
■
Algebra I (pp. 68–83)
■ ■ ■ ■
❑
7. Read the introductory material in “A Quick Review of Mathematics.”
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8. Take the diagnostic test in “Arithmetic” (pp. 90–91) and review any basic skills that you need to refine (pp. 93–115).
❑
9. Take the diagnostic test in “Algebra” (pp. 115–116) and review any basic skills that you need to refine (pp. 117–139).
❑ 10. Take the diagnostic test in “Measurement and Geometry: (pp. 139–142) and review any basic skills that you need to refine (pp. 145–155). ❑ 11. Take “CAHSEE Practice Test #1” (pp. 159–178). After you take the test, check your answers and analyze your results using the “Answer Key” (pp. 181), the “Review Chart” (pp. 180), and the “Answers and Explanations” (pp. 182–192). Review any basic skills that you need to refine from “A Quick Review of Mathematics” (pp. 85–155). ❑ 12. Take “CAHSEE Practice Test #2” (pp. 193–210). After you take the test, check your answers and analyze your results using the “Answer Key” (pp. 213), the “Review Chart” (pp. 212), and the “Answers and Explanations” (pp. 214–223). Review any basic skills that you need to refine from “A Quick Review of Mathematics” (pp. 85–155). ❑ 13. Take “CAHSEE Practice Test #3” (pp. 225–247). After you take the test, check your answers and analyze your results using the “Answer Key” (pp. 251), the “Review Chart” (pp. 250), and the “Answers and Explanations” (pp. 252–264). Review your weak areas and then selectively review the strategies and samples in “Part I: Working Toward Success” (pp. 5–157). ❑ 14. Read “Finishing Touches” (pp. 277).
ix
Introduction Format of CAHSEE Math The test consists of 92 questions—80 of those questions actually count toward your score. The following areas are covered (not necessarily in this order): Number Sense (NS)
14 questions
Statistics, Data Analysis, Probability (P)
12 questions
Measurement and Geometry (MG)
17 questions
Algebra and Functions (AF)
17 questions
Mathematical Reasoning (MR)
8 questions
Algebra I (AI)
12 questions
Total questions that count on score
80 questions
Plus trial questions for future tests
12 questions
Total questions for the Mathematics test
92 questions
Because this is a new test, the number of questions and the types of questions might be adjusted slightly in later tests. Also note that the trial questions can be scattered anywhere on the exam. The actual CAHSEE Math exam is given in two sessions—46 questions in each session. Although there is no time limit on the test, the approximate working times are estimated as follows: Mathematics
Approximate Working Times
Number of Questions
Session 1
1 hour 30 minutes
46
Session 2
1 hour 30 minutes
46
Both sessions will be administered on the same day with a break between the two sessions. You should be allowed to take the time you need within the school day to finish the exam.
1
CliffsTestPrep California High School Exit Exam: Math
Questions Commonly Asked About CAHSEE Math Q: What does CAHSEE Math cover? A: CAHSEE Math tests state content standards in Grades 6 and 7, and Algebra I. The exam covers number sense (including computation), statistics, data analysis (graphs and charts), probability, measurement and geometry, mathematical reasoning, and algebra. Q: How much time do I have to complete the test? A: There is no time limit for the exam. If you do not complete the test in the time period given, simply ask the proctor for additional time. You should be allowed to take the time you need within the school day to finish the exam. Q: How is the exam administered: A: The exam is administered in two sessions. Each session will have 46 questions. In each session you are only allowed to work on the questions given in that session. The approximate working time for each session should be 1 hour and 30 minutes. Q: Can I use a calculator on the exam? A: No. Calculators are not allowed on the exam unless there is a modification specified on the student’s record. Q: Can I use scratch paper on the exam? A: No. The use of scratch paper is not permitted during the exam. All scratch work must be done in the question booklet. Q: When will I first take the test? A: You will take the exam for the first time in the second part of 10th grade. Q: What is a passing score? A: Raw scores (the actual number of correctly answered questions) are converted to scaled scores ranging from 250 to 450. A passing score is 350 or higher. Because this is a new exam, you might wish to check with your school district to confirm the passing scores. Q: When do I find out if I passed? A: Score reports are mailed to the school district and to your home about two months after you take the test. Q: What if I don’t pass the exam in 10th grade? A: You have several chances to take the test as a junior and senior. Q: How should I prepare? A: Keep up with your class work and homework in your regular classes. There is no substitute for a sound education. As you get closer to your exit-level tests, using an organized test-preparation approach is very important. Carefully follow the Study Plan in this book to give you that organized approach. It shows how to apply techniques and strategies and help focus your review. Carefully reviewing the Study Guide for each exit-level exam available from your school district or the California Department of Education (CDE) also gives you an edge in doing your best. Q: Should I guess on the tests? A: Yes! Because there is no penalty for guessing, guess if you have to. If possible, try to eliminate some of the choices to increase your chances of choosing the right answer. Q: Can I write in the test booklet? A: Yes. You can do your work in the test booklet. Use the test booklet for scratch paper and to mark problems or draw diagrams. Your answer sheet must not have any stray marks, but your test booklet can be marked up.
2
Introduction
Q: Is the test given in more than one language? A: No. The test is only given in English. All students must pass CAHSEE Math in English to be eligible to get a high school diploma. Q: How can I get more information? A: More information and released exam questions can be found on the CDE’s Web site.
How You Can Do Your Best A Positive Approach Because every question is worth the same number of points, do the easy ones first. To do your best, use this positive approach: ■ ■ ■
First, look for the questions that you can answer and should get right. Next, skip the ones that give you a lot of trouble. (But take a guess.) Don’t get stuck on any one of the questions.
Here’s a closer look at this system: 1. Answer the easy questions as soon as you see them. 2. When you come to a question that gives you trouble, don’t get stuck. 3. Before you go to the next question, see if you can eliminate some of the incorrect choices to that question. Then take a guess from the choices left! 4. If you can’t eliminate some choices, take a guess anyway. Never leave a question unanswered. 5. Put a check mark in your test booklet next to the number of a problem for which you do not know the answer and simply guess. 6. After you answer all the questions, go back and work on the ones you checked (the ones that you guessed on the first time through). Don’t ever leave a question without taking a guess. There is no penalty for guessing.
The Elimination Strategy Sometimes the best way to get the right answer is to eliminate the wrong answers. As you read your answer choices, keep the following in mind: 1. Eliminate wrong answer choices right away. 2. Mark them out in your test booklet. 3. If you feel you know the right answer when you spot it, mark it. You don’t need to look at all the choices (although a good strategy for some questions is to scan the choices first). 4. Try to narrow your choices down to two so that you can take a better guess. Getting rid of the wrong choices can leave you with the right choice. Look for the right answer choice and eliminate wrong answer choices.
Here’s a closer look at the elimination strategy.
3
CliffsTestPrep California High School Exit Exam: Math
Take advantage of being allowed to mark in your test booklet. As you eliminate an answer choice from consideration, make sure to mark it out in your test booklet as follows: A ? B C ? D Notice that some choices are marked with question marks, signifying that they are possible answers. This technique helps you avoid reconsidering those marked-out choices you have already eliminated and helps you narrow down your possible answers. These marks in your test booklet do not need to be erased!
Avoiding the Misread One of the most common errors is the misread, that is, when you simply misread the question. A question could ask, if 3x + x = 20, what is the value of x + 2? This question doesn’t ask for the value of x, but rather the value of x + 2. A question could ask, which of the following is the best estimate of 511 × 212? Here, you are looking for the best estimate. A question could be phrased as follows: What is the probability that a spinner will not stop on green if you spin it one time? The word not changes the preceding question significantly. To avoid misreading a question (and therefore answering it incorrectly), simply circle or underline what you must answer in the question. For example, do you have to find x or x + 2? Are you looking for what can happen or what cannot happen? To help you avoid misreads, circle or underline the questions in your test booklet in this way: If 3x + x = 20, then what is the value of x + 2? Which of the following is the best estimate of 511 × 212? What is the probability that the spinner will not stop on green if you spin it one time? (Sometimes the test has key words underlined for you.) And, once again, these circles or underlines in your test booklet do not have to be erased.
A Quick Review of Basic Strategies 1. 2. 3. 4. 5. 6.
4
Do the easy problems first. Don’t get stuck on one problem—they’re all of equal value. Eliminate answers—mark out wrong answer choices in your test booklet. Avoid misreading a question—circle or underline important words. Take advantage of being allowed to write in the test booklet. No penalty for guessing means “never leave a question without at least taking a guess.”
PART I
W O R K I N G TOWAR D SUCCESS This section emphasizes how to approach question types that you will be seeing on the CAHSEE Math. Sample problems are followed by complete explanations and important test-taking techniques and strategies. Read this section carefully. Underline or circle key techniques. Make notes in the margins to help you understand the strategies, suggested approaches, and question types. Part I includes a variety of samples and suggested approaches from each math category: Number Sense Statistics, Data Analysis, Probability Measurement and Geometry Algebra and Functions Mathematical Reasoning Algebra I
Strategies for the Math Test Before you take a careful look at the standards with samples and suggested approaches, let’s review some specific testtaking strategies for the Math Test. These strategies can be very helpful in solving a problem. Circle or Underline. Take advantage of being allowed to mark on the test booklet by always circling or underlining what you’re looking for. This ensures that you are answering the right question. Pull Out Information. Pulling information out of word problems often gives you a better look at what you’re working with; therefore, you gain additional insight into the problem. Work Forward. If you quickly see the method to solve the problem, then do the work. Work Backward. In some instances, it’s easier to work from the answers. Don’t disregard this method because it at least eliminates some of the choices and can give you the correct answer. Eliminate. From initial information, or using common sense, you might be able to eliminate some of the answers. If you can eliminate an answer, mark it out immediately in the question booklet. Substitute Simple Numbers. Substituting numbers for variables can often be an aid to understanding a problem. Substitute simple numbers because you have to do the work. Use 10 or 100. Some problems deal with percent or percent change. If you don’t see a simple method for working the problem, try using the values of 10 or 100, and see what you get. Be Reasonable. Sometimes you immediately recognize a simple method to solve a problem. If this is not the case, try a reasonable approach and then check the answers to see which one is most reasonable. Sketch a Diagram. Sketching diagrams or simple pictures can also be very helpful in problem solving because the diagram might tip off either a simple solution or a method for solving the problem. Mark in Diagrams. Marking in or labeling diagrams as you read the questions can save you valuable time. Marking can also give you insight into how to solve a problem because you have the complete picture clearly in front of you. Approximate. If it appears that extensive calculations are going to be necessary to solve a problem, check to see how far apart the choices are and then approximate. The reason for checking the answers first is to give you a guide to how freely you can approximate. Glance at the Choices. Some problems might not ask you to solve for a numerical answer or even an answer including variables. Rather, you might be asked to set up the equation or expression without doing any solving. A quick glance at the answer choices helps you know what is expected.
Number Sense (Grade 7) There are 14 problems involving number sense on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 1.0: Working with Rational Numbers ■ ■ ■ ■ ■ ■ ■ ■ ■
Use scientific notation—read, write, and compare. Perform operations with whole numbers, fractions, and decimals (terminating)—add, subtract, multiply, and divide. Apply the rules for powers and exponents. Convert fractions to decimals and percents. Compute with fractions, decimals, and percents. Estimate. Calculate percentage increase and decrease. Solve problems involving discounts, markups, commissions, and profits. Compute simple and compound interest. 7
Part I: Working Toward Success
Standard Set 2.0: Use Exponents, Powers and Roots, and Fractions with Exponents ■ ■ ■ ■ ■ ■ ■
Understand negative whole-number exponents. Multiply and divide exponents with a common base. Add and subtract fractions finding common denominators by using factors. Multiply, divide, and simplify rational numbers using exponent rules. Understand perfect square numbers. Estimate square roots. Define and determine absolute value.
Samples with Suggested Approaches 1. 2.9 × 103 = A. B. C. D.
.290 29 290 2,900
This is a straightforward conversion question. If you know how to do a problem, work forward carefully. The simplest method is probably to move the decimal point three places to the right and add zeros. Since this is the process when multiplying by a power of 10, simply move the decimal point the same number of places as the power of 10, as follows:
2.9 × 103 = 2.900
[three places to the right]
Another method is to first, change 103 to 10 × 10 × 10, which equals 1,000. Next, multiply 2.9 by 1,000, giving 2,900. The correct answer is D. Also, 2.9 × 103 is written in scientific notation. To change from scientific notation, simply move the decimal point according to the exponent of 10. 2. The speed of light is approximately 300,000,000 miles per second. How is this speed represented in scientific notation? A. B. C. D.
3.0 × 108 miles per second 3.0 × 109 miles per second 30 × 107 miles per second 300 × 106 miles per second
Underline or circle the key words in the question. The key words here are scientific notation. A number written in scientific notation is a number equal to or greater than 1, but less than 10 multiplied by a power of 10. So you can quickly eliminate Choices C and D because they do not start with a number between 1 and 10. Next, be aware of the number you are working with or converting, in this case 300,000,000. To change this number to scientific notation, simply place the decimal point to get a number between 1 and 10, and then count the zeros to the right of the decimal to get the power of 10.
8
Strategies for the Math Test
300,000,000 = 3 00,000,000 × 108 Since there are eight zeros, the power of 10 is 8. Therefore, this speed is correctly represented in scientific notation as 3.0 × 108 miles per second. The correct answer is A. 3. A family of four goes out to dinner at a restaurant. Three family members order appetizers at $5 each. Each family member orders an entrée at a cost of $14 each and a drink at $2 each. Two family members order dessert at a cost $6 each. If nothing else is ordered by the family, how much money do they spend at the restaurant in total? A. B. C. D.
$79 $83 $91 $96
You are looking for the total spent. In this case, the key word is actually pointed out for you. Next, pull out important information as follows: 3 appetizers at $5 each [3 × 5 = $15] 4 entrees at $14 each [4 × 14 = $56] 4 drinks at $2 each [4 × 2 = $8] 2 desserts at $6 each [2 × 6 = $12] Now simply add the amounts. 15 56 8 + 12 91 The total amount spent by the family is $91. The correct answer is C. 4. 7 - c 1 + 1 m = 8 2 4 1 A. 8 B. 1 4 1 C. 2 D. 5 8 This problem involves simply adding and subtracting fractions, but be careful. Since you have three fractions with different denominators, you must find the LCD (lowest common denominator). In this case each denominator divides into 8 evenly, so the LCD is 8. Next, you must convert each fraction to an equivalent fraction with a denominator of 8.
9
Part I: Working Toward Success
1 = 4 and 1 = 2 2 8 4 8 Now you have
7- 4+2 8 c8 8m
Work inside the parentheses first. 7- 6 =1 8 c8m 8 The correct answer is A. 5. To add the fractions 3 and 9 , you must find a common denominator. Which of the following is the prime5 20 factored form of the lowest common denominator (LCD)? A. B. C. D.
2×5 2 × 10 2×2×5 5 × 20
The key words are prime-factored form. You can quickly eliminate Choices B and D because they are not in prime-factored form—composed only of prime numbers. Working from the remaining answers, A and C, you can see that Choice A, 2 × 5 (which is 10), cannot be the common denominator for 3 and 9 . So eliminate 20 5 Choice A, leaving Choice C. The correct answer is C. If you want to work the problem forward, find the LCD of 5 and 20. Since 20 is a multiple of 5, the LCD is 20. The prime factors of 20 can be found using a factor tree as follows:
20 2 × 10 2 × 2 × 5 So the prime factorization of 20 is 2 × 2 × 5. 6. Which of the following results in a positive number? A. B. C. D.
(–2) × (–3) (–2) × (–3) × (–4) (–2) + (–3) + (–4) (–2) + (–3) + (4)
First, circle or underline what you are looking for, in this case, a positive number. If you get an answer that you are absolutely sure is right, you don’t need to look at the rest of the choices. In this case, Choice A is a negative times a negative, which is always a positive, so the correct answer must be A. There is no need to look at the other choices.
10
Strategies for the Math Test
For your information, the other choices work out as follows: Choice B: A negative times a negative times a negative is always negative. (An odd number of negative signs in multiplication or division is negative.) Choice C: Because you are adding only negative numbers, the answer must be negative. Choice D: (–2) + (–3) + (4) is the same as (–5) + (4), which is –1. 7. If John gets 3 out of 5 questions correct on his math test, what percentage does he get correct? A. B. C. D.
35% 40% 60% 66%
First, note what you are looking for, percentage correct. Next, pull out information. 3 out of 5 or 3 5 Since 3 = 60% , the correct answer is C. You should memorize some fraction-to-decimal-to-percent equivalents, 5 such as: 1 = .20 = 20% , 2 = .40 = 40% , and so on. 5 5 Another method is to simply work out the percentage. To find what percentage 3 out of 5 is, divide 3 (questions answered correctly) by 5 (questions attempted). This leaves you with a success rate of 0.6, or 60%. By the way, you can eliminate Choices A and B if you realize that 3 out of 5 is greater than 50%. 8. Little Billy weighed 60 pounds last year. This year he weighs 75 pounds. What is the percentage increase of Billy’s weight from last year to this year? A. B. C. D.
15% 20% 25% 75%
First, underline what you are looking for, percentage increase. To find percentage increase or decrease, use the following formula: Difference Starting point Pulling information out of the problem gives you: 75 - 60 = 15 = 1 = .25 = 25% 60 60 4 The correct answer is C.
11
Part I: Working Toward Success
Memorize some fraction-to-decimal-to-percent equivalents. A recommended list follows: 1 = .10 = 10% 10 2 = 1 = .20 = 20% 10 5 3 = .30 = 30% 10 4 = 2 = .40 = 40% 10 5 5 = 1 = .50 = 50% 10 2 6 = 3 = .60 = 60% 10 5 7 = .70 = 70% 10 8 = 4 = .80 = 80% 10 5 9 = .90 = 90% 10 1 = 1.00 = 100% 1 = .25 = 25% 4 3 = .75 = 75% 4 9. Carol’s factory produced 40,000 disks in 1987. In 1986, her factory produced 50,000 disks. What was the percent decrease in production from 1986 to 1987? A. B. C. D.
10% 20% 25% 30%
First, you are looking for percent decrease. To find percent decrease, set up a ratio by dividing the change in production between the two years by the total production in the starting year: 50, 000 - 40, 000 10, 000 1 = = 50, 000 50, 000 5 Now change 1 to a percent. 5 Divide 5 into 1, which gives .20 or 20%. The correct answer is B. 10. Patrice puts $600.00 in a bank. If her money earns 12% simple interest every year, how much interest does she make at the end of 5 years? A. B. C. D.
$60 $72 $300 $360
Underline what you are looking for, simple interest. The basic formula for simple interest is: Interest = principal × rate × time (The time is in years.)
12
Strategies for the Math Test
Next, pull out important information: Principal = $600 Rate = 12% or .12 Time = 5 years Now plug these numbers into the formula: Interest = (600)(.12)(5) = 360 The correct answer is D. 11. Two thousand dollars is deposited in a savings account that pays 6% annual interest compounded semiannually. To the nearest dollar, how much is in the account at the end of the year? A. B. C. D.
$2,060 $2,120 $2,122 $2,247
Although compound interest problems can take a little extra work, they are straightforward. You should work this problem step by step. A 6% annual interest rate, compounded semi-annually (every half year) is the same as a 3% semi-annual interest rate. At the end of the first half of the year, the interest on $2,000 at 3% is: $2,000 × .03 = $60 So the new balance at the end of the first half of the year is: $2,000 + $60 = $2,060 At the end of the first full year, the interest on $2,060 at 3% is: $2,060 × .03 = $61.80 ≈$62 So the new balance at the end of the first full year is: $2,060 + $62 = $2,122 The correct answer is C. Choice B is correct only if the problem involves simple interest, not interest compounded semi-annually. 12. The regular price of a baseball glove at Big Three Sporting Goods is $50. If the glove is on sale for 30% off, what is the sale price of the baseball glove? A. B. C. D.
$20 $25 $30 $35
13
Part I: Working Toward Success
First, you are looking for the sale price. Next, pull out information. Regular price: $50 Sale: 30% off Now multiply the regular price by the percent off to get the amount saved. 50 × 30% = 50 × .30 = 15 Since the regular price is $50 and the amount saved is $15, the sale price must be $35. The correct answer is D. -3 13. 10 - 6 10 A. B. C. D.
= 10–3 10–2 102 103
This is a straightforward mechanical problem. You must know the rules for dividing numbers of the same base with exponents. When you divide numbers with exponents and the bases of the numbers are the same, then you keep the same base and subtract the exponents. For example, xa divided by xb is xa–b. -3 In this case, 10 - 6 = 10 - 3 ' 10 - 3 ] - 6 g = 10 - 3 + 6 = 10 3 . 10
The correct answer is D. 14. (63)9 = A. B. C. D.
66 612 627 6729
This is another straightforward mechanical problem. You must know the rule for taking the power of a number that already has an exponent. The rule is to simply multiply the exponents. In algebraic terms, (xa)b = xab. Using the numbers in the problem you have: (63)9 = 6(3 x 9) = 627 The correct answer is C. 15. (3)–2 = A. B. C. D.
14
–6 -1 9 1 6 1 9
Strategies for the Math Test
Similar to the previous problem, this is a straightforward mechanical problem. You must know the rule for negative exponents. To remove the negative sign in front of the exponent, drop the number and the exponent under the number 1 in a fraction. In algebraic terms, x - a = x1a . Using the numbers in the problem you have: 3 - 2 = 12 = 1 9 3 The correct answer is D. 16. 55 × 53 = A. B. C. D.
58 515 258 2515
To answer this straightforward mechanical problem, you must know the rule for multiplying numbers of the same base with exponents. To multiply numbers of the same base with exponents, simply keep the same base number and add the exponents. In algebraic terms, ax × ay = a(x + y). Using the numbers in the problem you have: 55 × 53 = 5(5+3) = 58 The correct answer is A. 17. The square root of 90 is between A. B. C. D.
9 and 10 10 and 11 11 and 12 12 and 13
This problem is most easily answered by working backward, from the answers. Start with Choice A, 9 and 10. If you square 9, that is, 9 × 9, you get 81, which is below 90. Next, try squaring the second number, 10, and you get 100. Since 90 is between 81 and 100, the square root of 90 is between 9 and 10. The correct answer is A. You can work this problem forward by approximating the square root of 90. First, find the closest perfect square number below 90. That is 81. Next, find the closest perfect square number above 90, which is 100. Since 90 is between 81 and 100, it falls somewhere between 9 and 10. 9
10
81 < 90 < 100 18. If x = 6, what is the value of x? A. B. C. D.
–6 or 0 –6 or 6 0 or 6 0 or 12
15
Part I: Working Toward Success
You can work this problem forward, using the definition of absolute value. If you know that absolute value refers to actual distance on a number line, and not direction, then it is evident that x can be –6 or 6. The correct answer is B. You can also work this problem by plugging in the answers, but you still need to know how to work with absolute values. 19. What is the absolute value of –9? A. B. C. D.
–9 –3 1 9 9
This question is just testing your knowledge of the definition of absolute value. Since absolute value refers to actual distance on a number line, and not direction (positive or negative), the absolute value of –9 is 9. The correct answer is D.
Statistics, Data Analysis, and Probability There are 12 problems involving statistics, data analysis, and probability on the CAHSEE. These problems are grouped together. The areas covered include:
Grade 6 Standard Set 1.0: Analyzing Statistical Measurements ■ ■ ■
Mean Mode Median
Standard Set 2.0: Describing and Using Data Samples ■ ■
Identify claims. Evaluate validity of claims.
Standard Set 3.0: Determining Probabilities and Making Predictions ■ ■ ■ ■ ■ ■
16
Represent outcomes for compound events; Express probability of each outcome. Use ratios, proportions, and decimals (between 0 and 1) and percentages (between 0 and 100) to represent outcomes. Verify that probabilities are reasonable. Know that if P is the probability of an event occurring, then 1 – P is the probability of an event not occurring. Understand independent and dependent events and how they are different.
Strategies for the Math Test
Grade 7 Standard Set 1.0: Collect, Organize, and Represent Data and Identify Relationships ■ ■ ■
Know various forms of data displays. Display data and compare data. Describe and display data on a scatterplot.
Samples with Suggested Approaches 1. The Chicago Bulls scored 98, 110, 112, and 120 points in four consecutive games. What is the mean score for the Chicago Bulls in these four games? A. B. C. D.
110 112 120 440
Underline the key words mean score. Now that you know what you are looking for, pull out important information. 98, 110, 112, 120 To find the mean score, you must find the total of the scores and then divide that total by the number of games played. 98 + 110 + 112 + 120 = 440 total points Since four games were played, divide 440 by 4, giving a mean score of 110. The correct answer is A. 2. The following table shows Teresa’s scores on her first-semester economics exams.
Date September 27 October 6 October 12 November 2 November 14 November 21 December 4 December 13
Score 90% 80% 90% 87% 83% 91% 88% 96%
What is her median score on the economics exams? A. 87 1 2 B. 88 C. 89 D. 90
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Part I: Working Toward Success
Focus on what you are looking for—in this case, the median score. To find the median, first put the scores in numerical order. 80, 83, 87, 88, 90, 90, 91, 96 Next, count in halfway. Since there are an even number of scores, you must take the average of the two middle scores. The average of 88 and 90 is 89. The correct answer is C. 3. The following chart shows the amount of money collected during a school’s magazine sale.
Magazine Sale Income Cost of magazine
$10
$12
$15
$19
Number sold
18
9
5
5
Which of the following is the mode of the cost of the magazines sold during the magazine sale? A. B. C. D.
$5 $10 $12 $15
You are looking for the mode. The mode is the cost that appears the most. Eighteen $10 magazines were sold. The correct answer is B. 4. The following graph shows the acceleration test results of the Roadster II. Acceleration Test Results of the Roadster II 100 90 80
miles per hour
70 60 50 40 30 20 10 0 1
2
3
4
5
seconds
6
7
8
According to the preceding line graph, the Roadster II accelerated the most between A. B. C. D.
18
1 and 2 seconds 2 and 3 seconds 3 and 4 seconds 4 and 5 seconds
Strategies for the Math Test
The key words here are accelerated the most. To answer this question, you must understand how the information is presented. The numbers on the left side of the graph show the speed in miles per hour. The information at the bottom of the graph shows the number of seconds. The movement of the line can give important information and show trends. The more the line slopes upward, the greater the acceleration. The greatest slope upward is between 3 and 4 seconds. The Roadster II accelerates from about 40 to about 80 miles per hour in that time. The correct answer is C. The following graph shows how John spends his monthly paycheck.
20% in the bank
25% entertainment
20% car and bike repair
15% his hobby 10% misc. items
10% school supplies
How John Spends His Monthly Paycheck
5. John spends one quarter of his monthly paycheck on A. B. C. D.
his hobby car and bike repair entertainment school supplies
Focus on what you are looking for, one quarter of his monthly paycheck. To answer this question, you must be able to read the graph and apply some simple math. The information is given in the graph. Each item is given along with the percent of money spent on that item. Because one quarter is the same as 25%, entertainment is the answer you are looking for. The correct answer is C. Use the preceding circle graph to answer the following question: 6. If John receives $100 on this month’s paycheck, how much does he put in the bank? A. B. C. D.
$2 $20 $35 $60
To answer this question, you must again read the graph carefully and apply some simple math. John puts 20% of his income in the bank. Twenty percent of $100 is $20. So, he puts $20 in the bank. The correct answer is B.
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Part I: Working Toward Success
Number of Delegates Committed to Each Candidate
Candidate 1
Candidate 2
Candidate 3
Candidate 4
0
200
400
600
800
Delegates
7. Based on the preceding graph, Candidate 1 has approximately how many more delegates committed than Candidate 2? A. B. C. D.
150 200 250 400
To understand this question, you must be able to read the bar graph and make comparisons. The graph shows the “Number of Delegates Committed to Each Candidate,” with the numbers given along the bottom of the graph in increases of 200. The names are listed along the left side. Candidate 1 has approximately 800 delegates (possibly a few more). The bar graph for Candidate 2 stops about three quarters of the way between 400 and 600. Now, consider that halfway between 400 and 600 is 500. So, Candidate 2 is at about 550. 800 – 550 = 250 The correct answer is C.
20
Strategies for the Math Test
8. The following graph shows the average decibel levels associated with various common sources of noise. Listening to high levels of noise for a long time can damage the eardrum and cause loss of hearing.
Decibel Levels of Common Noise Sources 130 120 110 100
decibels
90 80 70 60 50 40 30 20 10 gunfire
nearby thunder
passing train
nightclub
orchestra
factory
noisy office
shout
0
According to the preceding graph, which of the following sources of noise is most likely to damage the eardrum? A. B. C. D.
an orchestra a passing train nearby thunder gunfire
To answer this question, you must be able to read the graph and understand the information included. The decibels of noise are listed along the left-hand side in increases of 10. The common sources of noise are listed along the bottom of the graph. Since gunfire has the highest decibel rating, it is the loudest of the choices, and is therefore the most likely to damage the eardrum. The correct answer is D.
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Part I: Working Toward Success
9. The following diagram shows a spinner that is equally divided into four sections.
red
yellow
green
blue
In spinning the spinner only once, what is the probability of spinning red, yellow, or blue? A. B. C. D.
1 4 1 3 1 2 3 4
Underline what you are looking for, probability of red, yellow, or blue. To set up this probability problem, remember the formula: number of desired outcomes = probability total possible outcomes There are 3 chances out of 4 total possibilities of spinning either red, yellow, or blue. Thus, the probability is 3 . 4 The correct answer is D.
22
Strategies for the Math Test
10. Use the equally spaced spinner in the following figure to answer this question.
8
1
7
2
6
3 5
4
In one spin, what is the probability of getting a prime number? A. B. C. D.
1 8 1 2 5 8 3 4
There are four prime numbers on the spinner—2, 3, 5, 7. (1 is not a prime number.) Since there are 8 possibilities, the probability is 4 out of 8 or 1 . 2 The correct answer is B. 11. Three green, four red, and seven yellow marbles are placed in a bag. If a yellow marble is selected first and not replaced in the bag, what is the probability of selecting a yellow marble on the second draw? A. B. C. D.
3 7 4 7 1 2 6 13
First, focus on what you are looking for, probability of a yellow marble on the second draw. Since there are a total of 14 marbles and 7 are yellow, then the probability of selecting a yellow marble on the first draw is 7 . If a 14 yellow marble is selected on the first draw, then 6 yellow marbles are left out of 13 total marbles. The correct answer is D, 6 . 13
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Part I: Working Toward Success
12. What is the probability of tossing a penny twice so that both times it lands heads up? A. B. C. D.
1 8 1 4 1 3 1 2
The probability of throwing a head in one throw is: chances of a head =1 total chances ^1 head + 1 tailh 2 Since you are trying to throw a head twice, multiply the probability for the first toss, 1 , by the probability for 2 the second toss (again 1 ). Thus, 1 # 1 = 1 , and 1 is the probability of throwing heads twice in two tosses. 2 2 2 4 4 Another way of approaching this problem is to look at the total number of possible outcomes:
1. 2. 3. 4.
First Toss H H T T
Second Toss H T H T
Thus, there are four different possible outcomes. There is only one way to throw two heads in two tosses. Thus, the probability of tossing two heads in two tosses is 1 out of 4 total outcomes, or 1 . The correct answer is B. 4 13. Brenda is at a baseball hat store. In this store, there are only five different choices for hat color: red, orange, yellow, green, and blue. There are only four different choices for teams: Dodgers, Indians, Yankees, and Cubs. Each team hat is made in each color. If Brenda is handed a hat from the store at random, what is the probability that it is an orange Cubs hat? A. B. C. D.
1 25 1 20 1 5 1 4
You are looking for an orange Cubs hat. The probability that the hat is orange is 1 . The probability that the hat 5 is a Cubs hat is 1 . To find the probability that the hat is both an orange hat and a Cubs hat, you multiply the 4 probability of each, 1 # 1 = 1 because these are independent probabilities. Therefore, you know that the 4 5 20 probability that the hat is an orange Cubs hat is 1 . The correct answer is B. 20
24
Strategies for the Math Test
14. Bob Newhart is the host of a game show that has four contestants. Bob has four cards behind his back. One is blue, one is green, one is yellow, and one is orange. Each contestant can make only one selection and must keep that selection. The contestant who selects the green card wins a new sports car. The first contestant, Koji, selects a card. If the card he selects is yellow, what is the probability that the second contestant, Keiko, wins the new sports car? A. B. C. D.
1 8 1 4 1 3 1 2
To answer this question, you must follow the information given carefully. When Koji picks a card, the probability of his getting the green card is 1 . Since he eliminates one card that is not 4 the green card, there are only three cards remaining, and one of them is green. Therefore, the probability that Keiko picks the green card and wins a new sports car is 1 . The correct answer is C. 3 15. A small container is filled with buttons. Each button is the same size and shape. The container has six yellow buttons, three green buttons, five blue buttons, and four brown buttons. Ricardo takes out three green buttons and does not put them back into the container. If Ricardo selects another button at random, what is the probability that the button is yellow? A. B. C. D.
1 18 1 6 2 5 3 5
You should first focus on what you are looking for, the probability of selecting a yellow button (after the three buttons are removed). Next, pull out important information: 6 yellow 3 green 5 blue 4 brown Note the action in the problem—three green buttons are removed. Now you should carefully attack the problem. A total of 18 buttons are in the container at the start. If the 3 green buttons are removed, only 15 buttons remain in the container. You know that 6 are yellow, 5 are blue and, 4 are brown. Therefore, the probability of picking a yellow button out of those 15 in the container is 6 , which reduces to 2 . The correct answer is C. 15 5
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Part I: Working Toward Success
3 2 1
50 40 (per person)
C.
$ Spent weekly at market
A.
No. of cars per household
16. Which of the following scatterplots represents a positive correlation?
30 20 10 10
40
D.
30 20 10
10
20
30
40
No. of cigarettes smoked
B.
No. of people inside the club
1 2 3 4 No. of drivers per household
20
30 40 No. of people
50
60
50 40 30 20 10 10
20
30
40
As the number of cars increases, the number of drivers increases steadily. When the dots show a pattern going up to the right, the scatterplot shows a positive correlation. The correct answer is A.
Measurement and Geometry (Grade 7) There are 17 problems involving measurement and geometry on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 1.0: Choose Appropriate Units of Measure and Conversions Between Measurement Systems ■ ■ ■ ■ ■
Compare weights, capacities, geometric measures, times, and temperatures in different measurement systems. Understand scale models and drawings. Solve problems using measures expressed as rates and products. Check units of solutions. Check reasonableness of answers.
Standard Set 2.0: Compute Perimeter, Area, and Volume; Understand Effects of Scale Changes on These Measures ■
■ ■
26
Find the perimeter and area of basic two-dimensional figures—triangles, parallelograms, rectangles, squares, and circles. Find the surface area and volume of basic three-dimensional figures—prisms and cylinders. Estimate and compute the area of complex and irregular two- and three-dimensional figures using basic geometric shapes.
Strategies for the Math Test
■
■ ■ ■
Compute the perimeter, total surface area, and volume of basic three-dimensional objects made from rectangular solids. Understand how scale factor affects the surface area and volume of a solid. Relate measurement changes involving scale drawings. Convert between units.
Standard Set 3.0: Know the Pythagorean Theorem, Understand Plane and Solid Shapes, Identify Attributes of Figures ■ ■ ■ ■ ■
Plot simple figures on coordinate graphs. Determine dimensions and areas of simple figures on coordinate graphs. Translate and reflect images on coordinate graphs. Use the Pythagorean theorem and its converse. Understand the congruence of figures and the relationships between sides and angles. The following formulas are not provided on the exam, you should know them.
Area of a parallelogram: A = bh (where b is the base and h is the height) Area of a triangle: c A = 1 bh m 2 Volume of a rectangular solid: V = lwh (where l is the length, w is the width, and h is the height) Circumference of a circle: C = πd (where d is the diameter) Other formulas are provided.
Samples with Suggested Approaches 1. One meter is— A. B. C. D.
10 centimeters 100 decimeters 1,000 millimeters 10,000 kilometers
You should know some of the basic conversions: 1,000 millimeters = 1 meter. The correct answer is C. Other basic conversions are as follows: 10 decimeters = 1 meter 100 centimeters = 1 meter 1,000 millimeters = 1 meter 1 kilometers = 1 meter 1000 or 1 kilometer = 1,000 meters
27
Part I: Working Toward Success
2. Jillian weighs 60 kilograms (kg). Approximately how many pounds (lbs) does she weigh? (1 kg ≈ 2.2 lbs) A. B. C. D.
25 lbs 30 lbs 120 lbs 132 lbs
First, you are looking for how many pounds. Next, pull out important information: Jillian: 60 kg 1 kg ≈ 2.2 lbs Now, since 1 kg is approximately 2.2 lbs, you should multiply 60 by 2.2 lbs. 60 × 2.2 = 132 lbs So Jillian weighs approximately 132 lbs. The correct answer is D. 3. Jan Ove practices ping pong for 5 hours every day. How many minutes does Jan Ove practice in a week? A. B. C. D.
35 300 420 2,100
You are looking for minutes per week. First, change hours per day to minutes per day. Since Jan Ove practices 5 hours every day, multiply the hours by 60 to get the minutes per day. He practices 300 minutes every day. Since he practices every day, he practices 7 days a week. To find out the number of minutes he practices per week, simply multiply the minutes per day by the number of days in a week. 7 days × 300 minutes = 2,100 minutes per week The correct answer is D. 4. If a car is traveling at 1 mile per minute, how many miles per hour is it traveling? A. B. C. D.
1 60 1 60 100
Note that you are looking for miles per hour. Since the car is going 1 mile per minute and there are 60 minutes in an hour, the car is going 60 miles per hour. The correct answer is C. Choices A and B can quickly be eliminated because they are not reasonable.
28
Strategies for the Math Test
5. Daniel read nine comic books at a rate of six comic books per hour. If each comic book was the same number of pages and took the same amount of time to read, how long did it take him to read all nine comic books? A. B. C. D.
30 minutes 54 minutes 90 minutes 150 minutes
First, note that you are looking for how long it took him to read nine comic books. Next, pull out important information. 9 comic books 6 comic books per hour Now you can simply divide the number of comic books by the rate (comic books per hour) and get: 9 =13 =11 2 6 6 Since 1 1 hours is 90 minutes, the correct answer is C. 2 Another method is as follows: If Daniel reads 6 books in an hour, that means he reads 1 book every 10 minutes (1 hour = 60 minutes). Therefore, if he is reading 9 books, it takes 9 × 10 minutes, or 90 minutes. 6. Use the following diagram to answer the question that follows. C
D r O
A
10
B
Circle O is inscribed in square ABCD as shown in the preceding figure. The area of the shaded region in square units is approximately— (A = πr2 and π ≈ 3.14) A. B. C. D.
10 25 30 50
Underline or circle the words area of the shaded region. There are several approaches to this problem. One solution is to first find the area of the square. 10 × 10 = 100
29
Part I: Working Toward Success
Then subtract the approximate area of the circle. The radius of the circle is half the length of one side of the square. So,
A = π(5)2 = 25π or about 75 (3 × 25)
Now,
100 – 75 = 25.
Therefore, the total area inside the square but outside the circle is approximately 25. One quarter of that area is shaded. Therefore, 25 is approximately the shaded area. The closest answer is 10, Answer A. 4 A more efficient method is to first find the area of the square. 10 × 10 = 100 Then divide the square into four equal sections as follows. D
C
A
10
B
Since a quarter of the square is 25, the only possible answer choice for the shaded area is 10. The correct answer is A.
C
7. In the preceding diagram, a square and a circle intersect. If C is the center of the circle, what percent of the circle remains unshaded? A. B. C. D.
25% 35% 45% 50%
You are looking for the percent unshaded. Because one angle of a square is 90˚ and the circle is 360˚, the percent is 90 or 1 = 25% . The correct answer is A. 4 360
30
Strategies for the Math Test
8. The two figures drawn below are similar. What is the height, h, of the second figure in inches?
h 4″ 8″
A. B. C. D.
24″
4 inches 8 inches 12 inches 16 inches
To find the height, h, of the second figure, you should set up a proportion. 4= h 8 24 Since 8 goes into 24 three times, the second figure is three times the size of the first. So 3 times 4 is 12. The correct answer is C. Or you can solve this by cross-multiplying 96 = 8h 4 8
h 24
Then divide by 8 96 = 8h 8 8 12 = h Or reduced Re duced 1= h 2 24 24 = h 2 12 = h 9. Genevieve has a fish tank custom made. This fish tank holds 48 gallons of water. She needs a tank that holds only 24 gallons of water. Which of the following changes should be made, based on the dimensions of the original tank, to satisfy Genevieve’s needs while she is custom making a new fish tank? A. B. C. D.
Divide length, width, and height by 2. Divide only the length by 0.5. Multiply the width and length by 0.5. Multiply only the height by 0.5.
31
Part I: Working Toward Success
Since the volume of the tank equals length times width times height (V = lwh), you only need to make one of those dimensions half the size to cut the entire volume in half. To make a dimension half of its original size, you can multiply by 0.5 or divide by 2. The correct answer is D. School
6 miles
First stop
8 miles
Second stop
10. A bus leaves from school and makes two stops. The first stop is 6 miles south of school, and the second stop is 8 miles east of the first stop. If the bus drives in a straight line back to school from the second stop, how many total miles is it traveling after school? A. B. C. D.
5 10 14 24
First, you are looking for total miles. Next, focus on the diagram given. Your plan should be to find the longest side (hypotenuse) of the right triangle, and then total the lengths of all the sides. You might recognize that this right triangle is in the ratio 3–4–5 (6–8–?), so the length of the longest side is 10. Now, simply add the lengths of the sides: 6 + 8 + 10 = 24 So the correct answer is D. If you don’t spot the 3–4–5 right triangle relationship, you can use the Pythagorean theorem to find the length of the hypotenuse: a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 100 = c2 10 = c Then add the other two sides to get 24.
32
Strategies for the Math Test
6
6
4
11. Which figure is similar but not congruent to the figure shown? A. 5
5
6
B. 3
3
2
C. 4
5
3
D.
6
6
4
The key here is that you are looking for similar figures. Since similar means that the figures have the same shape (same angle measurements and proportional corresponding sides), Choice B is the only correct answer. Notice that Choice D is exactly the same—congruent, not similar. The correct answer is B.
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Part I: Working Toward Success
12. In the following figure, the length of line segment AB is 8 centimeters (cm). B
C
8 cm
A
D
What is the radius of the circle inscribed in square ABCD? A. B. C. D.
4 cm 8 cm 4π cm 8π cm
Focus on the word radius. Because the circle is inscribed in the square, the diameter of the circle is the length of one side of the square, 8 cm. B
8 cm
A
C
8
D
Since the radius of a circle is half of the diameter of the circle, the radius is 4 cm. The correct answer is A. 13. Two-centimeter cubes are arranged as shown in the following figure.
What is the total surface area? A. B. C. D.
34
4 cm2 16 cm2 18 cm2 72 cm2
Strategies for the Math Test
This problem should be done in parts. First, you are looking for total surface area. After analyzing the figure, it is clear that there are four cubes that each have six faces. The three cubes that are on the outside (two on each side and one on top), each have five faces showing. Each face is 4 cm2 (2 × 2). Since there are three cubes with five faces showing, and each face is 4 cm2, part of the surface area is (3 × 5 × 4 cm2 = 60 cm2). Next, you need to include the forth cube that is the most central. This cube is the only cube touching all the other three, and this cube has only three faces showing, so the surface area that shows is 12 cm2 (4 cm2 × 3). The total surface area is: 60 cm2 + 12cm2 = 72 cm2 The correct answer is D. Another method to solve this problem is to count the number of faces that are showing: 3 bottoms, 12 sides, 3 tops (count carefully). Since each face is 4 cm2, you have 18 × 4 cm2, which equals 72 cm2.
6 3 4 8
14. What is the area of the shaded region in the preceding figure, in square units? A. B. C. D.
9 square units 12 square units 16 square units 18 square units
Note that you are looking for the area of the shaded region. Your plan should be to find the area of each triangle and then subtract the area of the smaller triangle from the larger triangle. First find the area of the larger triangle, triangle ABC. Since the area of a triangle is 1 bh , the area of the larger 2 triangle is: 1 6 8 = 3 8 = 24 2 ^ h^ h ^ h^ h Next, find the area of the smaller triangle. 1 3 4 = 1 12 = 6 2 ^ h^ h 2 ^ h Now subtract. 24 – 6 = 18 So the area of the shaded region is 18 square units. The correct answer is D.
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Part I: Working Toward Success
15. What is the maximum number of milk cartons, each 2 inches wide by 3 inches long by 4 inches tall, that can fit into a cardboard box with inside dimensions of 16 inches wide by 9 inches long by 8 inches tall? A. B. C. D.
18 20 24 48
Drawing a diagram, as shown in the following figure, might be helpful in envisioning the process of fitting the cartons into the box. If possible, whenever a figure is mentioned but not drawn, draw it. Notice that 8 cartons fit across the box, 3 cartons deep and 2 stacks high.
8″ 4″ 2″
3″
9″
16″
The correct answer is D. 16. As shown in the following figure, camp counselor Craig builds a footbridge from the summer camp to the lake so that the campers don’t have to crawl down a perpendicular 5-foot cliff and then trudge through 12 feet of swamp.
Summer Camp foo
5ft
t br idg e
swamp 12ft
lake
How long (in feet) is the footbridge? A. B. C. D.
17 15 14 13
To solve this, you need to find the hypotenuse (the footbridge) of the right triangle by using the Pythagorean theorem: a2 + b2 = c2 (5)2 + (12)2 = c2 25 + 144 = c2 169 = c2 13 = c
36
Strategies for the Math Test
So the correct answer is D. You might recognize the ratio 5:12:13 as a Pythagorean triple. This saves you the work of using the Pythagorean theorem. 17. The scale drawing of a football field follows. The scale is 1 centimeter (cm) = 8 meters (m).
6 18 cm
13 cm
What is the width of the football field in meters? A. B. C. D.
61 8 13 3 4 49 110
You must find the width of the football field. To find this width, you can set up a proportion. Set up the proportion like this: 1 1 cm = 6 8 cm x 8 meters To solve this problem, cross multiply: x = 8 c6 1 m 8 1
1 = 68 8 x x = 49 The correct answer is C.
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Part I: Working Toward Success
20 8 16
9
3
18. As shown in the preceding figure, two right triangles have been removed from the corners of the square. What is the area of the remaining shaded figure in square units? A. B. C. D.
280 340 360 380
You are looking for the area of the shaded figure. First, find the area of the square: 20 × 20 = 400. Next, use the dimensions given to find the dimensions of each triangle, and then find the area of each triangle. Since each side of the square is 20, by subtracting the dimensions given, you have the following: 20 8 16 20 12 4 9
3
The area of the larger triangle is: 1 bh = 1 ^ 9h^12 h = 1 ^108h = 54 2 2 2 The area of the smaller triangle is: 1 bh = 1 ^ 3h^ 4 h = 1 ^12 h = 6 2 2 2 Finally, subtract the sum of the areas of the triangles from the area of the square: 400 – (54 + 6) = 400 – 60 = 340 The correct answer is B.
38
Strategies for the Math Test
8 9 5
14
19. What is the area of the preceding figure in square units? A. B. C. D.
60 70 82 94
Underline or circle the words area of the figure. Since this is an irregular figure, use common shapes to work it out. Start by finding the area of the rectangle, which is length times width: 14 × 5 = 70 So, the area of the rectangle is 70 square units. At this point, you can eliminate Choices A and B. The area must be greater than 70. Now you need to find the dimensions of the triangle. By using the dimensions given and subtracting from the length, you get the following dimensions:
4 8 6 5
5
14
You only need the base and height of the triangle to find the area. The height is 4 and the base is 6. A = 1 bh 2 A = 1 ^ 6 h^ 4 h 2 A = 1 ^ 24 h = 12 2 Now add the areas of the rectangle and triangle, 70 + 12 = 82. The correct answer is C.
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Part I: Working Toward Success
y
B
C
A
D x
20. Which of the following rectangles can result from reflecting rectangle ABCD across the x-axis? A.
C.
y
y
x
B.
C
B
C
A
D
A
D
D.
y
x
40
x
B
B
C
A
D
y
B
C
A
D
x
Strategies for the Math Test
When you reflect a figure across the x-axis, the x-coordinates remain the same, but the y-coordinates become the negative of the original coordinates. So, the rectangle moves below the x-axis, but is still the same distance from the x-axis and the y-axis. You get a mirror image on the other side of the x-axis. The correct answer is C. y
B
C
A
D
A
D
B
C
x
Although Choice A shows the rectangle below the x-axis, it is not the same distance from the x- or y-axes. 21. Judith made a drawing of her school, using the scale of: 1 centimeter (cm) on the model = 12 feet (ft) in real life If the length of the main corridor of her school is 180 ft, how long should Judith have drawn the corridor on the scale drawing? A. B. C. D.
12 cm 15 cm 18 cm 20 cm
You must find how long the corridor should be on the scale drawing. To find this length, you need to set up a proportion. Set up the proportion like this: 1 cm = x 12 ft 180 ft To solve this problem, cross multiply: 180 = 12x 1 cm x = 12 ft 180 ft Then divide by 12.
180 = 12x 12 12 15 = x
So the corridor on the scale drawing should be 15 cm long. The correct answer is B.
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Part I: Working Toward Success
B A
C
22. Two small circles with the same radius, and with centers A and C, are inscribed in a large circle whose center is at B, as shown in the preceding diagram. If the distance from A to C is 10 cm, what is the radius of the large circle? A. B. C. D.
5 cm 10 cm 20 cm 25 cm
You are looking for the radius of the large circle. To solve this problem, you need to use your knowledge of radii and diameters. If AC = 10 cm, and the small circles have the same radius, then AB = 5. So the radius of circle A is 5 cm. Its diameter is double its radius, or 10 cm, which also happens to be the radius of the large circle. This can be more easily seen in the following diagram.
B A 5
The correct answer is B.
42
C 5
5
5
Strategies for the Math Test
23. The following diagram shows the dimensions of a piece of paper. 8 1/2″
11″
What is the radius (in inches) of the largest circle that can be drawn on the piece of paper? A. B. C. D.
11 81 2 51 2 41 4
You are looking for the radius. Without going off the paper, the diameter of the largest circle is 8.5 inches. 8 1/2″ 8 1/2″ 11″
Because the radius is half the diameter: 81 # 1 =4 1 2 2 4 17 1 17 OR ; # = =4 1E 2 2 4 4 The correct answer is D. The common mistake is B, which is the diameter.
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Part I: Working Toward Success
3″
4″ Cube A
Cube B
24. The two cubes shown in the preceding figure have edges of 3 in. and 4 in. What is Volume of cube A ? Volume of cube B 3 A. 4 B. 9 16 C. 27 64 81 D. 256 First find the volume of cube A, which is 3 × 3 × 3 = 27. Next, find the volume of cube B, which is 4 × 4 × 4 = 64. Finally, set up the ratio: Volume of Cube A = 27 Volume of Cube B = 64 The correct answer is C. 25. Which term best describes the polygon that is formed from the points (1,2), (1,4), (3,2), (3,4)? A. B. C. D.
pentagon triangle square rectangle
Here, you need to make a quick sketch of a grid and plot these points on the grid as follows:
4 3 2 1 1
2
3
4
From this sketch, it is easy to see that the figure is a 2 × 2 square. The correct answer is C. You can eliminate Choice A because you need 5 points for a pentagon (five sides). You can eliminate Choice B because you need three sides for a triangle (and three of these points aren’t on a straight line to give you three sides).
44
Strategies for the Math Test
26. The following diagram shows the dimensions of a cylinder.
10′ 12′
Which of the following is the best estimate of the volume (in cubic feet) of the cylinder? (π ≈ 3.14 and volume = πr2h) A. B. C. D.
3,600 4,500 7,700 14,400
To find the volume of a cylinder with its sides perpendicular to its base, first find the area (A) of its base (a circle): π times (radius)2
[A = πr2]
Then, multiply this number by the container’s height. So, Volume = π × (radius)2 × height = 3.14(10)2(12) = 3.14(100)(12), or approximately 3(100)(12) = 3,600 The correct answer is A.
Algebra and Functions There are 17 problems involving algebra and functions on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 1.0: Using Algebraic Terminology, Expressions, Equations, Inequalities, and Graphs ■ ■ ■ ■ ■ ■
Use variables and appropriate operations. Write expressions, equations, and inequalities. Know the order of operations. Evaluate expressions. Represent relationships graphically. Interpret graphs and parts of graphs.
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Part I: Working Toward Success
Standard Set 2.0: Interpret and Evaluate Expressions Involving Powers and Simple Roots ■ ■ ■ ■ ■
Interpret positive whole-number powers. Interpret negative whole-number powers. Simplify and evaluate expressions with exponents. Multiply and divide monomials. Take powers and extract roots.
Standard Set 3.0: Graph and Interpret Linear and Nonlinear Functions ■ ■ ■ ■
Graph functions of the form y = nx2 and y = nx3. Graph linear functions. Know that slope is the ratio of rise over run. Plot the value of quantities with the same ratios.
Standard Set 4.0: Solve Simple Linear Equations and Inequalities (Rational Answers) ■ ■ ■ ■
Solve two-step linear equations and inequalities with one variable. Interpret solutions in context. Verify the reasonableness of the results. Solve multi-step problems involving rate, speed, distance, and time or direct variation.
Samples with Suggested Approaches 1. Which inequality expresses the following relationship: A number, n, multiplied by 3 is less than or equal to 12? A. B. C. D.
3n ≤ 12 3n ≥ 12 3n < 12 3n > 12
Spread out the information given, and then work carefully, step by step. A number, n, multiplied by 3, is less than or equal to 12. n
×
3
≤
12
≤
12
Rearranging slightly: 3n 3n shows n multiplied by 3. < means less than but “≤” means less than or equal to, which is what you are looking for. Therefore, 3n ≤ 12 expresses the relationship. The correct answer is A.
46
Strategies for the Math Test
2. A stamp collector has 3,000 foreign stamps. She then sells x number of her foreign stamps and purchases 4y number of foreign stamps. Which of the following gives an expression representing the number of foreign stamps she has following her purchase? A. B. C. D.
3000 x+y 3,000 – x + y 3,000 − 4x + y 3,000 − x + 4y
You are looking for the expression that represents the number of stamps she has following her purchase. Take the information given, and follow the steps that she takes. The stamp collector starts with 3,000 foreign stamps. She then sells x number of foreign stamps, which are represented by 3,000 – x foreign stamps. Then, she purchases 4y foreign stamps. Since she has purchased 4y foreign stamps, she is adding 4y foreign stamps to her collection, so she now has 3,000 – x + 4y foreign stamps. The correct answer is D. 3. This year Tina’s father is 39 years old. He is triple her age. Which equation can be used to determine Tina’s age? A. B. C. D.
39 = 3T 39 = 3 + T 39 = T – 3 39 = T ÷ 3
Triple means multiply by three, and the equation in Choice A describes this: 39 = 3T (39 is 3 times Tina’s age) The correct answer is A. 4. Tom is just 4 years older than Fran. The total of their ages is 24. What is the equation for finding Fran’s age? A. B. C. D.
x + 4x = 24 x + 4 = 24 4x + 4 = 24 x + (x + 4) = 24
If Tom is 4 years older than Fran, and we let x represent Fran’s age, Tom’s age must be 4 years more, or x + 4. Therefore, because the total of their ages is 24: Fran’s age + Tom’s age = 24 x
+
(x + 4) = 24
The correct answer is D. 5. If x = 3 and y = 5, then 6x2– 4y = A. B. C. D.
16 34 54 60
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Part I: Working Toward Success
To evaluate an expression, simply plug in the given numbers or values. These types of problems are usually easy to solve as long as you are careful in your calculations and understand the order of operations. Plugging in the values given for x and y: 6x2 – 4y = 6(3)2 – 4(5) = 6(9) − 4(5) = 54 – 20 = 34 The correct answer is B. Remember, the order of operations is Parentheses Exponents Multiplication or Division Addition or Subtraction A good tool for remembering the order of operations is PEMDAS. 6. Use the following graph to answer the next question.
% attendance drop
30 20 10 0 0
10
20
30
40
degree drop in temperature
According to the graph, if the temperature falls 35°, what percentage does the attendance drop? A. B. C. D.
10% 20% 30% 40%
When referring to a graph, be sure to understand the information given. Pay special attention to the labels on the graph itself. In this case, note that on the graph a 35° drop in temperature (horizontal line) correlates with a 20% attendance drop (the fourth slash up the vertical line). The correct answer is B.
48
Strategies for the Math Test
7. Annette does a school-wide survey and publishes her results, as shown in the following graph. Schoolwide Eye Color Survey
30% blue
40% brown
5%
20% hazel 5%
green other
If 62 people at Annette’s school have hazel eyes, how many have brown eyes? A. B. C. D.
62 93 124 155
First circle or underline the words brown eyes. Before doing any work, take a careful look at what information you are given. According to the graph, 20% of the people at Annette’s school have hazel eyes, while 40% have brown eyes. This means that there are twice as many brown-eyed people as there are hazel-eyed people. 62 people have hazel eyes. 2 × 62 people have brown eyes. So, 2 × 62 = 124 people who have brown eyes. The correct answer is C.
Distance in miles
8. The following graph shows the relationship between time and distance for four male distance runners. Runner A
4 3 2
Runner C Runner B
1
Runner D 10 20 30 40 Time (minutes)
50
According to the preceding graph, after 30 minutes of running, runner A is approximately how many miles ahead of his closest competitor? A. B. C. D.
1 2 2.5 3
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Part I: Working Toward Success
Distance in miles
Note what you are looking for and what information you are given. Since time is represented on the x-axis of this graph, you go right on the x-axis to the 30-minute mark. Then you find runner A at that time. After 30 minutes, he has run about 3.5 miles. Next you should find his closest competitor, runner C. After 30 minutes, he has run about 2.5 miles. Therefore, runner A is about 1 mile ahead of his closest competitor. The correct answer is A. Runner A
4 3 2
Runner C Runner B
1
Runner D 10 20 30 40 Time (minutes)
50
No. of moviegoers
70 60 50 40 30 20 10 1-10
11-20 21-30 31-40 41-50 51-60 Age (years)
9. The preceding graph shows the age of moviegoers. Approximately how many moviegoers are 30-years old or younger? A. B. C. D.
30 50 100 130
Underline or circle the words 30-years old or younger. To find how many moviegoers are 30-years old or younger, you need to read the graph carefully. You need to add up the number of moviegoers ages 1–10, 11–20, and 21–30. This gives you: 50 + 50 + 30 = 130 The correct answer is D.
50
Strategies for the Math Test
10. x4y4 = A. B. C. D.
4xy (xy)4 (xy4)4 xy8
This mechanical problem is testing your knowledge of the rules of exponents. Since x4y4 = x × x × x × x × y × y × y × y, which is the same as (xy) × (xy) × (xy) × (xy), you can write x4y4 as (xy)4. The correct answer is B. You can also work this problem from the answer choices. 11. a _ 9x 2 i k = 2
A. B. C. D.
3x 9x 9x2 81x4
If you take the square root of any number, n, and then square it, you are left with n. The action of squaring a term is cancelled out by taking the square root of that term. In this case you take the square root of the term 9x2 and then square it, leaving you with 9x2. The correct answer is C. 12.
25x 8 = A. B. C. D.
5x2 5x4 25x2 25x6
Since 25 = 5 and x 8 = x4, 25x 8 = 5x4. Another method of solving this problem is to note that 25x 8 = 5x 4 # 5x 4 , so 25x 8 = 5x4. The correct answer is B. 13. Simplify the expression shown below: (9s3t5)(3st4) A. B. C. D.
12s3t9 12s4t9 27s4t9 27s4t20
To answer this mechanical question, you must know the rules for multiplying monomials. First, multiply the numbers together, 9 × 3 = 27. Then, work with the variables: s3 × s = s4 (Keep the same base and add the exponents: s = s1.) Likewise, t5 × t4 = t9. +
+
(9s3t5)(3s1t4) = 27s4t9 ×
The correct answer is C.
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Part I: Working Toward Success
y 5 4 3 2
B
1 -5
-4
-3
-2
1
-1
2
3
4
5
x
-1
A
-2 -3 -4 -5
14. In the preceding graph, what is the slope of the line that passes through points A and B? A. B. C. D.
3 2 1 3 1 2 2 1
Probably the simplest method is to notice that you must go up three and across two to get from point A to point B. Since slope is rise/run, you get 3/2. The correct answer is D. y 5 4 3 2 2 -5
-4
-3
-2 3
A
B 1
2
3
4
5
-1 -2 -3 -4 -5
Another method to find the slope of the line is to use the slope formula: slope =
52
_ y 2 - y 1i ^ x 2 - x 1h
x
Strategies for the Math Test
Using the coordinates of A (–1,–2) and B (1,1), simply plug into the formula:
_ y 2 - y 1i ^1h - ^ - 2 h 1 + 2 3 = = = ^ x 2 - x 1h ^1h - ^ -1h 1 + 1 2 y . . . 4 3 2 1 -3
-2
-1
-1
1
2
3
4
. . .
. . .
x -4
-2 -3 -4 . . .
15. What is the slope of the line shown in the preceding graph? A. B. C. D.
–3 -1 3 1 3 3
First, you can eliminate Choices A and B because negative slopes go down to the right, and the line in the preceding graph goes up to the right. The simplest method of solving this problem is noticing that the line crosses the x-axis at –1, so you go to the 3 right one and up three (where the line crosses the y-axis). Since you want rise run you get 1 or just 3. The correct answer is D. Another method is to plug into the slope formula: slope =
_ y 2 - y 1i ^ x 2 - x 1h
as in the previous problem. slope =
^ 3h - ^ 0 h 3 = =3 ^ 0 h - ^ -1h 1
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Part I: Working Toward Success
16. The slope of the following line is - 3 . 4 y
a 8
x
What is the value of a? A. B. C. D.
2 4 6 10
Since the slope of the line is rise run , you can set up a proportion to find the value of a. - 3 = -a 4 8 (Cancel the negatives from each side.) You can multiply both sides by 8, and you get: ^8h^ 3h =a 4 24 = a 4
The correct answer is C.
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Strategies for the Math Test
17. What is the equation of the line on the following graph? y 4 3 2 1
x -4
-3
-2
-1
-1
1
2
3
4
-2 -3 -4
A. B. C. D.
y = 2x + 4 y = 2x – 4 y = 2x + 2 y = 2x – 2
A careful review of this graph can save you a lot of work. The line crosses the y-axis at –2, so the y-intercept is –2. The y-intercept form of the equation is y = mx + b, where b is the y-intercept. The equation of the preceding line must end in –2, y = mx–2 because –2 is the y-intercept. The only equation that has a –2y-intercept is D. So even though you didn’t work out the equation of the line, you do know that it can’t be Choices A, B, or C. Also, the slope of each equation is the same c 2 m ; the only difference is the y-intercept. The correct answer is D. 1
55
Part I: Working Toward Success
18. Which graph best represents the equation y = x2? y
y
x C.
A.
y
B.
x
y
x D.
x
A careful look at y = x2, let’s you know that this is not a linear equation (not a straight line). So you can immediately eliminate Choice C. Also, y can never be negative, so eliminate Choices A and D. You are left with Choice B, the correct answer. You can actually plot this equation from the beginning. You start by plugging in numbers.
y = x2 x
y
0 1 -1 2 -2
0 1 1 4 4
Finally, you plot these points and see the shape of the graph. Fortunately, with this particular problem, all this work was not necessary. The correct answer is B.
56
Strategies for the Math Test
19. The Make an Orphan Happy Foundation donates toys in groups of five with each toy valued at $2. The following table shows the number of toys donated and the value of the donations. Number of toys donated 5 10 15 20 25
Value ($) 10 20 30 40 50
Which of the following is a graph of the information given in the table? A.
C.
Value ($)
Value ($)
Number of toys donated
B.
Number of toys donated
D.
Value ($)
Value ($)
Number of toys donated
Number of toys donated
As the number of toys donated increases, the value increases in direct proportion. The only graph that shows this straight-line increase is graph B. The correct answer is B. 20. The number of students in a classroom is represented by x in the inequality 3x – 50 ≤ 100. Which phrase most accurately describes the number of students in this classroom? A. B. C. D.
at most 50 at least 50 more than 50 less than 50
To answer this question, you must solve the inequality. 3x – 50 ≤ 100
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Part I: Working Toward Success
First add 50 to both sides of the inequality. 3x - 50 # 100 + 50 + 50 3x # 150 Next, divide both sides by 3. 3x # 150 3 3 x ≤ 50
So,
This is read “x is less than or equal to 50.” This also means that x is at most 50. The correct answer is A. 21. If x + 6 = 9, then 3x + 1 = A. B. C. D.
3 9 10 34
You should first circle or underline the term 3x + 1 because this is what you are solving for. Solving for x leaves x = 3. Then substituting this value for x into 3x + 1 gives 3(3) + 1, or 10. The most common mistake made in solving this problem is to solve for x, which is 3, and mistakenly choose A as the answer. But, you are solving for 3x + 1, not just x. You should also notice that most of the other choices are possible answers if you made common or simple mistakes. Make sure that you are answering the right question. The correct answer is C. 22. Solve for x. 4x + 3 = 15 A. B. C. D.
3 4 3 4 5
First, note that you are solving for x. Now, follow the normal procedures for solving a simple equation. Subtract 3 from each side. 4x + 3 = 15 - 3 -3 4x = 12 Now divide both sides by 4. 4x = 12 4 4 So, The correct answer is B.
58
x=3
Strategies for the Math Test
You can also answer this question by plugging in the answer choices. Choice A: 3 4
4x + 3 = 15 4 c 3 m + 3 ? 15 4 3 + 3 ≠ 15
Choice B: 3
4(x) + 3 = 15 4(3) + 3 ? 15 12 + 3 = 15
Eliminate A.
The correct answer is B. 23. In a movie the weight of a monster varies directly with the height of the monster. A 10-foot tall monster weighs 280 pounds. How tall is a monster that weighs 322 pounds? A. B. C. D.
11 feet tall 11 1 feet tall 2 12 feet tall 13 feet tall
You are given a direct variation, so you can set up the following proportion: height 10 foot tall x = = weight 280 pounds 322 pounds Make the numerators the height of the monsters and the denominators the weights. Since you don’t know the height of the 322-pound monster, you should use x to represent that monster’s height. Multiply both sides by 322 to solve for x. ^10 h^ 322 h =x 280
3220 = x 280 Dividing by 280 gives: 11 1 = x 2 The correct answer is B. 24. Jennifer has 750 pounds of furniture to move into her new apartment. She has been working for 3 days and has moved 150 pounds of furniture. If she continues to move furniture at the same rate, how many days in total does it take her to move all her furniture into her new apartment? A. B. C. D.
5 12 15 18
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Part I: Working Toward Success
You are looking for how many days in total. If Jennifer moves 150 pounds of furniture in 3 days, she is moving 50 pounds of furniture a day (150 ÷ 3 = 50). Since she has a total of 750 pounds that she has to move at a rate of 50 pounds of furniture a day, you divide 750 by 50 and find that it takes her 15 days to move all her furniture into her new apartment. The correct answer is C. You can solve this problem using the following proportion: 3 = x 150 750 2250 = x 150 15 = x
Mathematical Reasoning (Grade 7) There are eight problems involving number sense on the CAHSEE. These problems are grouped together and incorporate skills from number sense, statistics, data analysis, probability, measurement and geometry, and algebra. The areas covered include:
Standard Set 1.0: Decide How to Approach Problems ■ ■ ■ ■ ■ ■
Analyze problems. Identify relationships. Distinguish relevant from irrelevant information. Identify missing information. Prioritize information. Observe patterns.
Standard Set 2.0: Use Strategies, Skills, and Concepts to Solve Problems ■ ■ ■ ■
Estimate to verify that an answer is reasonable. Estimate unknown values graphically. Solve problems using logical reasoning. Solve problems using arithmetic and algebraic techniques.
Standard Set 3.0: Determine That a Solution Is Complete and Generalize to Other Situations ■ ■ ■
60
Develop generalizations from results. Generalize from strategies used in other problems. Apply generalizations to new situations.
Strategies for the Math Test
Samples with Suggested Approaches 1. Andrea rode her bike for 3 hours at a rate of 15 miles per hour. What is the correct method to find the total miles that Andrea traveled? A. B. C. D.
Add 15 and 3. Divide 15 by 3. Multiply 15 by 3. Multiply 60 by 15.
Circle or underline the words total miles traveled. To find her total miles traveled, multiply 15 by 3. The actual total of miles traveled is 45 miles. Does this method give you a reasonable answer? Yes, if she travels 15 miles in one hour, then she travels 30 miles in 2 hours and 45 miles in 3 hours. The correct answer is C. 2. If x is an even number, then which of the following is true about x ? 2 A. It is an even number. B. It is an odd number. C. It is a multiple of 2. D. It can be odd or even. When given a situation using types of numbers (odd, even, negative, and so on), try some simple numbers. If x is 2 (which is an even number), then x is 1, which is an odd number. If x is 4, then x is 2, which is an even number. 2 2 The correct answer is D. 3. Which is the best estimate of 931 × 311? A. B. C. D.
2,700 27,000 270,000 2,700,000
First, check the answer choices to see how far apart they are. This gives you an indication of how close your approximation should be. In this case the choices are far apart, so your approximation does not need to be too accurate. As a matter of fact, these choices differ only by the number of zeros after 27. So be careful that you have the correct number of zeros in your estimate. Round each number to the nearest hundred.
932 × 311 900 × 300 = 270,000 So the best estimate is 270,000. The correct answer is C. 4. Which of the following is the best estimate of 595 ÷ 184? A. B. C. D.
2 3 4 5
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Part I: Working Toward Success
When you check the choices in this question, they seem close, but they still give you plenty of room to estimate.
595 ÷ 184 600 ÷ 200 = 3 So the best estimate is 3. The correct answer is B. 5. Andre is fighting the battle of the bulge. He is counting calories consumed and calories burned. Today, Andre consumed 280 calories from fruits and vegetables, 530 calories from meat, 125 calories from cereals, and 275 calories from dairy products. Andre burned 980 calories today. Which expression can be used to express the BEST estimate of his net calorie count for the day? A. B. C. D.
300 + 500 + 100 + 300 – 1,000 300 + 500 + 200 + 300 – 1,000 300 + 600 + 100 + 300 – 1,000 300 + 500 + 100 + 200 – 1,000
Each choice is rounded off to the nearest hundred. Also, you are looking for the best expression. To answer this question, round off each number and set up the expression as follows:
280 + 530 + 125 + 275 − 980 300 + 500 + 100 + 300 − 1,000 The correct answer is A. 6. Manuel purchases tickets to an amusement park for his family. Children’s tickets cost half price, and adult tickets cost the full price of $8.00 each. Which of the following expressions represents the total dollars spent for tickets if there are 2 adults and 6 children in Manuel’s family? A. B. C. D.
2(8) + 3(4) 2(4) + 6(6) 2(8) + 4(4) 2(8) + 6(4)
In this question you are being asked to set up an expression. Focus on key words and information given. From the information given you can figure out the total spent for adult and children’s tickets: 2 adults × $8 for each ticket 6 children × $4 for each ticket (Note: Half the price of $8 = $4.) So, the total spent for tickets was 2(8) + 6(4). The correct answer is D.
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Strategies for the Math Test
7. Union High School is planning to order yearbooks for the student body. The following chart shows the percent of students in each class that usually order yearbooks. Class
Percent
Freshman Sophomore Junior Senior
40% 60% 85% 95%
Considering the preceding information, if there are 500 sophomores and 600 juniors in the school, what additional information is necessary to find out how many more sophomores than freshman will probably order yearbooks? A. B. C. D.
the number of seniors in the school the number of juniors not ordering yearbooks the number of freshman in the school the number of sophomores not ordering yearbooks
Since the question asks for the additional information necessary, and focuses on the difference between freshman and sophomores, you need to know the number of freshman in the school. The correct answer is C. 8. A Zowie battery lasts 60 hours and costs $1.20. A Rayvox battery lasts 75 hours and is sold in a package at a cost of $4.99 per package. To determine whether a Zowie battery or a Rayvox battery is the better buy, which additional piece of information is needed? A. B. C. D.
Zowie batteries cost less than Rayvox batteries. Rayvox batteries last 25% longer than Zowie batteries. Rayvox batteries are sold only in packages containing three batteries. Zowie batteries are sold in packages containing only one battery.
To compare the cost of each battery, you must know how many Rayvox batteries are in the package that costs $4.99. You do not have to know which battery is the better buy. You only have to know what information you need to determine which is the better buy. The correct answer is C.
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Part I: Working Toward Success
9. The following graph shows the values of stocks and bonds for the first three quarters of the current year.
Values of Stocks and Bonds for Current Year
(In Thousands of Dollars)
70 60 50 Stocks Bonds
40 30 20 10 0
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
Of the following, which is the best prediction of the possible value of stocks in the fourth quarter? A. B. C. D.
$50,000 $60,000 $70,000 $80,000
The question is asking about the value of stocks, so make sure to underline the words prediction, value, stocks, and fourth quarter. Focus on stocks. Now, take a careful look at the value of stocks in each quarter—the first quarter is $20,000, the second quarter is $30,000, and the third quarter is $50,000. The increase is $10,000 from the first quarter to the second quarter, and $20,000 from the second quarter to the third quarter. It is reasonable to predict that the increase will be $30,000 from the third quarter to the fourth quarter (increases of $10,000, $20,000, and $30,000), so the value of stocks in the fourth quarter can be $30,000 more than the third quarter (50,000 + 30,000), or $80,000. Since not a lot of data is given (only three quarters), other predictions are possible, but D is the best choice given. The correct answer is D.
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Strategies for the Math Test
10. The following graph shows the average lamp prices for the years listed. Average Lamp Prices (and Projection) 40 35
Price (in dollars)
30 25 20 15 10 5 0
1970
1990
2010
Which of the following was the most probable average lamp price in 1970? A. B. C. D.
$5 $8 $14 $20
Underline the words average lamp price in 1970. Now, continue the slope of the line back to the left until you get to 1970. The most probable average price is $14. Using the edge of your answer sheet helps you follow the slope more accurately. The correct answer is C.
Verbal SAP Scores
11. According to the line of best fit shown on the following scatterplot, approximately how many hours of extra reading does a student have to do every month to score a 600 on the verbal part of the SAPs?
Line of best fit
800 600 400 200 10
20
30
40
50
60
No. of hours spent doing extra reading per month
A. B. C. D.
20 30 50 60
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Part I: Working Toward Success
First, focus on the line of best fit. Again, use the edge of your answer sheet to most accurately continue the line. (You can also use the side of your pencil.) The point that is on the line of best fit and is also across from the verbal score of 600 is directly above the 30 representing the number of extra hours of reading every month. The correct answer is B. 12. The following circle has a radius, r.
r
If r < 9, which of the following cannot be the area of the circle? A. B. C. D.
3π 9π 18π 81π
The only way the area of the circle can be 81π is if r = 9, but r is less than 9; therefore the area cannot be 81π. The correct answer is D.
x
y
4 6 3 5 2
8 12 6 10 4
13. In the table above, showing corresponding values of x and y, which equation represents the relationship? A. B. C. D.
x=y+4 x= 1 y+6 2 x = 2y x= 1 y 2
A careful look at the table reveals that in each case, 2 × x gives y (4 and 8, 6 and 12, 3 and 6, and so on). So x is 1 y . The correct answer is D. If the x values are put in proper order (2, 3, 4, 5, 6), this relationship is much easier 2 to recognize. 14. How many numbers are less than 100 and are divisible by 2, 3, and 7? A. B. C. D.
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2 3 4 5
Strategies for the Math Test
For a number to be divisible by 2, 3, and 7, it must be a multiple of 2, 3, and 7. Simply multiply 2, 3, and 7, and you get 42, which is a multiple of these three numbers. Double 42 and you get 84, another multiple of 2, 3, and 7. So there are two numbers less than 100 that are divisible by 2, 3, and 7. The correct answer is A. 15. Tina is trying to find the y-intercept for the linear equation. x+y–6=0 Step 1: Add 6 to each side: x + y = 6 Step 2: Subtract x from each side: y = –x + 6 Step 3: Substitute 0 for x: y = –(0) + 6 Step 4: The y-intercept is 6: y = 6 Tina’s method shows that the y-intercept in the equation y = mx + b is which of the following? A. B. C. D.
m x y b
Tina’s method was to put the equation into the form y = mx + b, where m is the slope and b is the y-intercept. She substituted 0 for x, to find that the line represented by this equation crosses the y-axis at 6. So 6 is the y-intercept, represented by b. The correct answer is D. Adam eats 2 boxes of cookies every week. At this rate, how long does it take Adam to eat 18 boxes of cookies? 16. The same mathematical processes used to solve the preceding problem can be used to solve which of the following problems? A. B. C. D.
Right Says Fred records 2 hit records every month. How many hit records do they record in their 3-month career? Planet Jackson has saved 300 lives so far in her 18-month career as a super hero. How many lives does she save each week? Bill Plates eats 2 boxes of cookies every week. At this rate, how many boxes of cookies does he eat in 4 weeks? Lil’ Chill raps 2 hours every day. At this rate, how long does it take Lil’ Chill to rap for a total of 24 hours?
The process used to answer the original question involves dividing the total number of boxes of cookies that need to be eaten (18) by the rate that Adam eats cookies at (2 boxes per week). In Answer D, Lil’ Chill’s case, you also need to divide the total number of hours needed (24) by the rate at which he raps (2 hours a day). The correct answer is D. Eight out of ten dentists surveyed recommend Popsodent toothpaste. 17. Which of the following is a valid conclusion based on the information given? A. B. C. D.
Two out of ten dentists recommend another brand of toothpaste. Popsodent is the best-tasting toothpaste. More patients use Popsodent toothpaste than any other brand. At least one dentist surveyed could have recommended another brand.
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Part I: Working Toward Success
The fact that eight out of ten dentists surveyed recommend Popsodent toothpaste does not tell us anything about the other dentists. They might not recommend any toothpaste, so eliminate Choice A. Choice B, taste, is never addressed. For Choice C to be a valid conclusion, you have to assume that patients follow the recommendations of their dentists. Also, what about patients who go to dentists other than those surveyed. Since two dentists surveyed didn’t recommend Popsodent, they could have recommended another brand. The correct answer is D.
Algebra I There are 17 problems involving Algebra I on the CAHSEE. These problems are grouped together. The areas covered include:
Standard Set 2.0 ■
Understand opposites, reciprocals, and square roots.
Standard Set 3.0 ■
Solve equations and inequalities involving absolute values.
Standard Set 4.0 ■
Simplify expressions before solving linear equations and inequalities in one variable.
Standard Set 5.0 ■ ■
Solve problems with many steps including linear equations and linear inequalities in one variable. Provide justification for each step.
Standard Set 6.0 ■
Graph linear equations and compute x- and y-intercepts.
Standard Set 7.0 ■ ■
Verify that a point lies on the line of an equation given. Derive linear equations.
Standard Set 8.0 ■
Understand the concept of parallel lines and the slopes of those lines.
Standard Set 9.0 ■ ■ ■ ■
Solve a system of two linear equations in two variables algebraically. Interpret answers graphically. Solve a system of two linear inequalities in two variables. Sketch solution sets.
Standard Set 10.0 ■
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Use operations with monomials and polynomials—add, subtract, multiply, and divide.
Strategies for the Math Test
Standard Set 15.0 ■
Solve rate, work, and percent mixture problems algebraically.
Samples with Suggested Approaches 1. If x = –4, then –2x = A. B. C. D.
–8 -1 8 1 8 8
If x = –4, then substituting –4 into –2x gives –2(–4) = –(–8). The negative of a negative number is positive, so the answer is positive 8. The correct answer is D. 2. If –x = 3, then x = A. B. C. D.
–3 -1 3 1 3 3
Since –x means the opposite of x, and –x = 3, then x must be –3, the opposite of 3. Or you can simply multiply each side of the original equation by –1, as follows: –x = 3 (–1)(–x) = 3(–1) x = –3 The correct answer is A. 3. If the area of a rectangle with a width of 5 inches (in) is 40 in2, what is the perimeter of the rectangle? A. B. C. D.
26 in 40 in 80 in 100 in
If a geometric figure is described, but not drawn for you, draw the figure and label the information given.
5 in
40 in2
This gives you insight into how to work the problem. Be sure to underline or circle the word perimeter since that is what you are looking for.
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Part I: Working Toward Success
To find the perimeter of a rectangle, you must find the sum of the lengths of all the sides. You know that the area of the rectangle is 40 in2 and that the width is 5 in, so you can divide 40 by 5 to find the length of the rectangle (8 in). Now finish labeling the figure and total the sides. 8 in
5 in
40 in2
5 in
8 in Since 5 + 8 + 5 + 8 = 26, the correct answer is A. You can also work the problem as follows: Because you know that opposite sides are equal in a rectangle, the perimeter is 2(5) + 2(8) = 26. 4. If x is an integer, what is the solution to 5 x = 30 ? A. B. C. D.
{5, 6} {5, –6} {0, 6} {–6, 6}
Integers are positive and negative whole numbers and zero. One way to answer this question is to work from the choices given by plugging them in. Another method is to actually solve the problem as follows: 5 x = 30 Divide each side by 5
5 x = 30 5 5
Then
x =6
So
x = 6 or –6
The correct answer is D. 5. Given that x is an integer, which of the following is the solution set for x + 2 < 4 ? A. B. C. D.
{0} {0,1} {–2, –1, 0, 1} {–5, –4, –3, –2, –1, 0, 1}
Integers are positive and negative whole numbers and zero. To solve an inequality when you have an absolute value sign, you must form two equations. In this case your two inequalities are: x + 2 < 4 and x + 2 > –4
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Strategies for the Math Test
In the second inequality, when you take the negative, you change the direction of the sign. Now, solve each inequality by subtracting 2 from each side. x+2< 4 - 2 -2 x < 2 and x+2> -4 - 2 -2 x > -6 This can also be written –6 < x < 2; that is, x is between –6 and 2. Since x is an integer, x can be –5, –4, –3, –2, –1, 0, or 1. The correct answer is D. You can also find the solution set by working from the answers. Start by plugging in a number from Choice D (since it has the most members) that is different from numbers in the other choices. If the number works, then the answer is D; if not, then you can eliminate D. -5 + 2 < 4
Try –5
-3 < 4 3 –3(x – 2)? A. B. C. D.
6x > 13 8x > –12 12x > 13 12x > –13
Again, notice that the answer choices are simplified, but not completely solved. The x’s and numbers are sorted to each side of the inequality sign. So, do only as much work as is necessary. First, simplify the right side of the inequality. 9x – 7 > –3(x – 2)
9x − 7 > -3(x − 2) You get: 9x – 7 > –3x + 6
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Strategies for the Math Test
Next, get all the x’s on one side by adding 3x to both sides. 9x - 7 > - 3x + 6 + 3x + 3x 12x - 7 > 6 Now add 7 to both sides. 12x - 7 > + 6 +7 +7 12x > 13 The correct answer is C. 9. Solve for x. 4(2x + 3) < –2(3x – 4) A. B. C. D.
x < - 27 x0 x> 7 2
First, simplify each side by multiplying through the parentheses. 4(2x + 3) < –2(3x–4) You get:
4(2x + 3) < -2(3x − 4) 8x + 12 < –6x + 8
Next, add 6x to each side. 8x + 12 < - 6x + 8 + 6x + 6x 14x + 12 < 8 Now subtract 12 from each side. 14x + 12 < 8 - 12 - 12 14x < -4 Finally, divide each side by 14. 14x < - 4 14 14 So
x < -4 14
This reduces to x < -72 . The correct answer is A.
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Part I: Working Toward Success
10. Sasha solved the inequality 5x – 5 ≥ 2x + 4 using the following steps. 5x – 5 ≥ 2x + 4 Step 1: 3x – 5 ≥ 4 Step 2: 3x ≥ 9 Step 3: x ≥ 3 Sasha did which of the following to get from step 1 to step 2? A. B. C. D.
subtracted 5 from each side added 5 to each side divided each side by 3 subtracted 2x from each side
Focus on the question. What did she do to get from step 1 to step 2. To get from step 1 to step 2, Sasha added 5 to each side. 3x - 5 $ 4 +5 +5 3x $ 9 The correct answer is B. 11. Which of the following statements describes the x-intercept? A. B. C. D.
the point where a line crosses the y-axis a line that is parallel to the x-axis the point where a line crosses the x-axis a line that is parallel to the y-axis
The x-intercept is the point where the line crosses the x-axis, the coordinates (x, 0). The correct answer is C.
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Strategies for the Math Test
y
x
12. Which of the following is the equation of the line shown in the preceding graph? A. B. C. D.
y = 2x + 1 y = 2x + 2 y = –2x – 1 y = –2x – 2
A quick look at the choices lets you eliminate some by looking for the y-intercept first. Since the line crosses the y-axis at –1, the y-intercept on the graph is –1. You can eliminate Choices A, B, and D. The correct answer is C. In this question, you don’t even need to deal with the slopes. But if you are working with the slopes, you can eliminate Choices A and B immediately because they are positive. Since the line goes down to the right, the slope is negative.
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Part I: Working Toward Success
13. Which of the following is a graph of y = 3x – 2? y
y
x C.
A.
y
B.
x
y
x D.
x
Since the equation given is in slope-intercept form (y = mx + b, where m is the slope and b is the y-intercept), pay special attention to the slope and y-intercept. In this equation, y = 3x–2, the slope is 3 and the y-intercept is 1 –2. The line must cross the y-axis at –2, so first eliminate any choices that do not cross the y-axis at –2. Eliminate Choices B and D. Next, eliminate Choice C because, upon careful examination, you can see that the slope is 1 , 1 not 3 . The correct answer is A. 1 14. Which of the following points lies on the line 3x – 2y = 12? A. B. C. D.
(0, –6) (0, 0) (4, 6) (6, 0)
Probably the best method to answer this question is to work from the choices given. Simply plug in the coordinates as values of x and y to see which make the equation true.
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Strategies for the Math Test
3x – 2y = 12 Choice A (0,–6)
3(0)−2(–6) = 12 0 + 12 = 12
Since (0, –6) satisfies the equation, this point must lie on the line. The correct answer is A. 15. What is the slope of the line parallel to the line y = 2x - 1 ? 2 A. –2 B. - 1 2 1 C. 2 D. 2 Underline or circle the words slope of the line parallel to. Your focus is first to find the slope. Because the equation is in slope-intercept form, y = mx + b, where m is the slope, it is easy to see that the slope is 2. Parallel lines have the same slope. Therefore, the slope of a line parallel to the given line is also 2. The correct answer is D. 16. What is the slope of the line identified by 3y = 4(x + 3)? A. B. C. D.
4 3 4 3 3 4
To answer this mechanical question, simply change the equation to slope-intercept form: y = mx + b 3y = 4(x + 3) 3y = 4x + 12 Next divide each side by 3, which leaves: y= c4m x+ 4 3 So the slope is 4 . The correct answer is C. 3 17. Which of the following points is the y-intercept of the line 3x + 2y = 6? A. B. C. D.
(2, 0) (2, 1) (0, 2) (0, 3)
Probably the fastest method to answer this question is working from the answers. First, note that you are looking for the y-intercept. The y-intercept is where the line crosses the y-axis, so x must be 0. Eliminate Choices A and B because the x coordinate is not 0. Next, plug in Answers C and D to see which is true for the equation.
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Part I: Working Toward Success
Choice C (0, 2): 3x + 2y = 6 3(0) + 2(2) ? 6 0+4≠6 So you can eliminate C. The correct answer is D. Choice D (0, 3): 3x + 2y = 6 3(0) + 2(3) ? 6 0+6=6 Another method for solving this problem is to change the equation of the line to slope-intercept form, y = mx + b, where b is the y-intercept. 3x + 2y = 6 Subtract 3x from each side. 3x - 2y = 6 - 3x - 3x 2y = - 3x + 6 Divide both sides by 2. 2y - 3x 6 = + 2 2 2 y = c -3 m x + 3 2 So the y-intercept is 3. x+y=6 ) 2x - y = 3
18.
Which of the following ordered pairs is the solution to the system of equations shown above? A. B. C. D.
(2, 4) (5, 1) (3, 3) It cannot be determined.
In this question you can either solve the system of equations or plug in the answer choices. Let’s solve the system of equations. First, combine the equations to eliminate a variable: x+y=6 (+) 2x - y = 3 3x =9
78
Strategies for the Math Test
Now, dividing both sides by 3 leaves x = 3. Since x = 3, simply plug x into either equation and solve: x+y=6 (3) + y = 6 So y = 3. The correct answer is C, (3, 3). The other method of plugging in the answers goes like this: Since Choices A, B, and C each total 6, you only need to plug them into the second equation to find the right answer. Choice A, (2,4): 2(2) − (4) ? 3 4 − 4 ≠3 Eliminate Choice A. Choice B, (5,1): 2(5) − (1) ? 3 10 − 1 ≠ 3 Eliminate Choice B. Choice C, (3,3): 2(3) − (3) ? 3 6−3=3 Choice C is correct. Although it is not the correct answer for this problem, note that some problems have the answer choice of “It cannot be determined,” “No solution,” or “No intersection.” 7x + y = 8 ) x + 3y = 4
19.
What is the solution to the system of equations shown above? A. B. C. D.
(0, 0) (0, 8) (1, 1) (3, 4)
Again, to answer this question, you can work from the choices and plug in answers or solve algebraically. If you plug in the choices, simply take one of the equations and see if the solutions work.
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Part I: Working Toward Success
7x + y = 8 Choice A, (0, 0):
7(0) + 0 ? 8 0≠8
Eliminate Choice A. Choice B, (0, 8): 7(0) + (8) = 8 Choice B is possible. Choice C, (1, 1): 7(1) + (1) ? 8 7 + 1 =8 Choice C is possible. Choice D, (3, 4): 7(3) + (4) ? 8 21 + 4 ≠ 8 Eliminate Choice D. Now, take Choices B and C and plug them into the second equation. x + 3y = 4 Choice C, (1, 1): 1 + 3(1) ? 4 1+ 3 =4 The correct answer is C. If you decide to solve this problem algebraically, you solve for one variable first by eliminating the other one. Multiply the top equation by 3: 3(7x + y = 8) 21x + 3y = 24 Subtract the second equation from the first: (-)
80
21x + 3y = 24 x - 3y = 4 20x = 20
Strategies for the Math Test
Now, divide both sides by 20. 20x = 20 20 20 x=1 Now that you have your x value, plug it back into either original equation. 7x + y = 8 7(1) + y = 8 7+y=8 y=1 Now, you know that x = 1 and y = 1. You have the solution: (1, 1). 20. Simplify. (x2 + 7x – 7) − (4x2 – 6x – 1) A. B. C. D.
3x2 + x – 6 –3x2 + x – 8 –3x2 – 13x – 6 –3x2 + 13x – 6
First, take a look at the choices to see if they are simplified completely, that is, all operations are complete and all like terms are combined. In this case each choice is simplified completely. So simplify: (x2 + 7x – 7) − (4x2 – 6x – 1) First, multiply through the right parentheses. x2 + 7x – 7 – 4x2 + 6x + 1 Now, combine like terms.
x2 + 7x − 7 − 4x2 + 6x + 1
–3x2 + 13x – 6 The correct answer is D.
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Part I: Working Toward Success
x+2
21. The square shown above has a side of length x + 2 units. The area of the square can be represented by which of the following expressions? A. B. C. D.
x2 + 2 x2 + 4 x2 + 4x + 4 x2 + 2x + 4
To find the expression representing the area of the square, you must first note that all sides of a square are equal. So, if one side is x + 2 units, all sides are x + 2 units. Mark the diagram as follows: x+2 x+2
Now, simply multiply (x + 2) by (x + 2). You can use the FOIL method (first, outer, inner, last).
(x + 2) (x + 2) x2 + 2x + 2x + 4 x2 + 4x + 4 The correct answer is C. 22. Mr.Tuchman can paint 30 surfboards in an hour. Mr. Christianson can paint 60 surfboards in an hour. If they are both painting surfboards, how long does it take them to paint a total of 45 surfboards? A. B. C. D.
30 minutes 45 minutes 60 minutes 90 minutes
You are looking for how long it takes to paint 45 surfboards if both men are painting. You can work this problem from the answers. First, try Answer A, 30 minutes. If Mr. Tuchman can paint 30 surfboards in an hour, then he can paint 15 surfboards in half an hour. If Mr. Christianson can paint 60 surfboards in an hour, then he can paint 30 surfboards in half an hour. So, together they can paint 45 surfboards in 30 minutes. The correct answer is A.
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Strategies for the Math Test
You can also work this problem algebraically. If t is the number of hours, and Mr. Tuchman paints at a rate of 30 surfboards an hour, this can be expressed as 30t. If Mr. Christianson paints at a rate of 60 surfboards an hour, then this can be expressed as 60t. They both have to work to paint a total of 45 surfboards, so you can set up the equation 30t + 60t = 45. Now, solve as follows: 30t + 60t = 45 90t = 45 Dividing by 90 gives
t = 45 90
So, t = 1 hour, or 30 minutes. The correct answer is A. 2
83
A Quick Review of Mathematics The following pages are designed to give you a quick review of some of the basic skills used on CAHSEE Math: arithmetic, algebra, measurement and geometry, properties of numbers, simple probability and statistics, and graphs. Before beginning the diagnostic review tests, it is wise to become familiar with basic mathematics terminology, formulas and general mathematical information. These topics are covered first in this chapter. Then proceed to the arithmetic diagnostic test, which you should take to spot your weak areas. Then use the arithmetic review that follows to strengthen those areas. After reviewing the arithmetic, take the algebra diagnostic test and again use the review that follows to strengthen your weak areas. Next, take the measurement and geometry diagnostic test and carefully read the complete measurement and geometry review. Even if you are strong in arithmetic, algebra, and measurement and geometry, you might wish to skim the topic headings in each area to refresh your memory about important concepts. If you are weak in math, you should read through the complete review.
Symbols, Terminology, Formulas and General Mathematical Information Common Math Symbols and Terms Symbol References: =
is equal to
≠
is not equal to
>
is greater than
1 > 1 , so in estimating the sum, Answer A is too high, and C is too low. 1 is a little smaller than 1 and a 2 3 4 3 2 little larger than 1 , so Answer B is a good estimate. Answer D is incorrect because the three sums are not equally 4 close to the exact sum, 1 1 . 12
222
Test #2—Answers and Explanations
77. C. Calculators don’t make mistakes, but people do, and Chris certainly did. Answer A is incorrect. Answer B is incorrect because adding the balance with $120.20 does not make such a large amount. Answers C and D both start out good, but D’s explanation for the mistake does not account for the error. 78. B. If each person eats 50 square inches, then four people eat 200 square inches. Find that point on the y-axis and trace horizontally until you run into the curve. Then trace vertically down to the x-axis. You should be at the point that represents a diameter of 16 inches. 79. C. If one block is left over when two children share the blocks, the number of blocks must be odd. If none are left over when three or five children share them, the number must be a multiple of 3 and 5. Answers C and D (15 and 30) are both multiples of 3 and 5; however, 30 is an even number and must be eliminated. 80. D. The example given requires the addition of equal groups, which is (more simply) multiplication. Answer A requires division, Answer B requires addition of unequal groups, and Answer C requires subtraction. Only Answer D also involves multiplication.
223
CAHSEE Practice Test #3 Directions: Mark only one answer to each question on your answer sheet. If you change an answer, make sure that you erase the previous mark completely. Notes: (1) Figures that accompany problems are drawn as accurately as possible EXCEPT when it is stated that a figure is not drawn to scale. All figures lie in a plane unless noted otherwise. (2) All numbers used on the exam are real numbers. All algebraic expressions represent real numbers unless stated otherwise.
1. Which of the following has the greatest value? A. B. C. D. 2.
1.99 × 10–1 9.19 × 10 2 1.19 × 10 3 9.91 × 10–4
3 + 3-2 = 14 c 8 7 m A. B. C. D.
4 15 17 56 9 28 17 14
3. At 6 p.m. in Alaska the thermometer read –3°F. By midnight the temperature had dropped 18°. What was the reading on the thermometer at that time? A. B. C. D.
–21° –18° –15° 15°
4. Sonia earns $300, pays 1 of the money in taxes, 5 and keeps the rest. How much money does Sonia keep? A. B. C. D.
$50 $60 $240 $250
One dollar is worth about .81 of a Euro (European Economic Unit). 5. Which of the following statements is consistent with the information in the box? A. B. C. D.
A dollar is worth less than half a Euro. A dollar is worth about four-fifths of a Euro. Eighty-one dollars are worth about 100 Euros. A Euro is worth about 81 cents.
6. By halftime the Wolverines had scored 49 points, 18 by their center. Approximately what percent of the team’s points were scored by other players? A. B. C. D.
17% 37% 63% 83%
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Part II: Full-length Practice Tests
7. In 30 years the price of premium gasoline has increased from about $.25 a gallon to about $2.75 a gallon. What has been the percent increase in the price of premium gasoline? A. B. C. D.
100% 250% 1,000% 2,500%
8. An $80 jacket is on sale for 40% off. With a coupon, John saves an additional 15% off the reduced price. How much does John save off the original price of the jacket? A. B. C. D.
$36.00 $39.20 $40.80 $44.00
9. A movie star is paid 5% of the gross ticket sales above $30,000,000. If the movie sells $50,000,000 worth of tickets, how much does the actor earn? A. B. C. D.
$1,000,000 $1,500,000 $2,500,000 $20,000,000
_3 4i
3
12.
34 # 32 A. B. C. D.
B. C. D.
–9 - 1 27 1 27 1 9
11. Which of the following is the prime-factored form of the lowest common denominator of 5 + 3? 6 8 A. B. C. D.
226
2×1 2×2×2×3 2×3×2×2×2 6×8
1 2 32 36
13. The square of a whole number is between 2,500 and 3,600. The number is between A. B. C. D.
30 and 40 40 and 50 50 and 60 60 and 70
14. If x = 5, then x = A. B. C. D.
–5 5 –5 or 5 1 or 5 5
15. The following table shows sales figures for five salesmen:
10. Which of the following is equivalent to (3)–3? A.
=
Salesman
Sales
Salesman #1
$20,000
Salesman #2
$21,000
Salesman #3
$19,000
Salesman #4
$21,000
Salesman #5
$25,000
A sixth salesman joins the company and records $55,000 in sales. Which of the measures of central tendency does not change when the new salesman’s figures are added to the data? A. B. C. D.
the mean the median the mode neither the median nor the mode
CAHSEE Practice Test #3
16. Tom has an average (mean) of 80% for six history tests. His average for the first three of those tests was 72%. He scored 90% on his fourth test, and 88% on his fifth test. What was his score on the sixth test? A. B. C. D.
17. The sixth grade is having a raffle ticket sale to raise money. The following table shows sales figures for the five classes: Teacher
70% 76% 80% 86%
Ticket Sales
Mr. Navarete
480 tickets
Ms. Green
650 tickets
Ms. Schuman
520 tickets
Ms. Petrie
290 tickets
Mr. Frankl
610 tickets
Mr. Peters
140 tickets
What is the median number of tickets sold? A. B. C. D.
405 480 500 520
CAHSEE Practice Test #3 GO ON TO THE NEXT PAGE 227
Part II: Full-length Practice Tests
18. Researchers surveyed algebra students to determine how much time they spent on daily homework. No time 25%
1 hour or more 43%
Some time, but less than 15 minutes 13%
15 minutes to half an hour 19%
According to the preceding circle graph A. B. C. D.
More than half the students worked less than half an hour. More than half the students either did no homework or spent an hour or more on homework. Students are not spending enough time on math homework. Less than half the students worked half an hour or more.
19. Karl rolls a fair number cube and flips a fair coin. The possible outcomes are shown in the following figure. 1
Heads Tails
2
Heads Tails
3
Heads Tails
4
Heads Tails
5
Heads Tails
6
Heads Tails
What is the probability that he rolls an odd number and flips heads? A. 1 6 B. 1 4 C. 1 2 D. 1
228
20. Mr. Garvitch buys a dozen donuts. There are 3 jelly donuts, 2 chocolate donuts, 2 crumb donuts, 2 sugar donuts, a buttermilk donut, and 2 glazed donuts. If he randomly selects a donut, what is the probability that it is not a jelly donut or a buttermilk donut? A. B. C. D.
1 3 2 3 3 4 11 12
CAHSEE Practice Test #3
21. Pancho spins the following spinner one time:
23. The following graph represents information on the siblings of 500 students at an elementary school.
Red Green
4 or more siblings 12%
Green
no siblings 18%
Blue Red
3 siblings 17%
What is the probability that the pointer lands on a green section? A. B. C. D.
33.3% 40% 50% 60%
2 siblings 23%
How many students had no siblings or only one sibling?
22. The fair spinner shown in the following figure has landed on A four times in a row. How does this affect the theoretical probability of an outcome of A on the next spin?
A
A.
B.
D.
A. B. C. D.
18 students 30 students 48 students 240 students
B
To answer this question, more needs to be known about the spins that preceded the last four spins. Because A has come up several times in a row, an outcome of A is more likely than an outcome of B. Because A has come up several times in a row, an outcome of A is less likely than an outcome of B. Outcomes of the previous spins have nothing to do with the outcome of the next spin.
CAHSEE Practice Test #3
C.
1 sibling 30%
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Part II: Full-length Practice Tests
Biology scores
24. The following graph shows Dontrey’s scores on biology tests. 100 90 80 70 60 50 40 30 20 10 0 1st test
2nd test
3rd test
4th test
5th test
6th test
Biology tests
Dontrey’s scores decreased the most from A. B. C. D.
the 2nd to the 3rd test the 3rd to the 4th test the 4th to the 5th test the 5th to the 6th test
25. Which scatter plot shows a negative correlation between the two variables? B.
0
Days of Training
Price per Day
A. 7 6 5 4 3 2 1
20 15 10 5 0
20
40
20
40
60
80
Number of Animals
60
Number of Days C.
D. 60
20
Miles Run
Price per Call
30
10
0
5
10
Number of Phone Calls
40
20
15 0
10
20
Number of Students
230
30
CAHSEE Practice Test #3
26. Which statement best describes the relationship between the price of cherries and the pounds of cherries purchased by shoppers?
Number of pounds sold (in 100,000s)
Cherry Price and Cherry Purchases 200
27. Which of the following expressions represents the statement, “Four less than one-third the product of 10 and n”? A. B.
150
C.
100
D.
50 0 $0.00
28. Which equation represents the statement, “The sum of five and y is seven more than x”? $1.00
$2.00
$3.00
$4.00
$5.00
Price per pound
A. B. C. D.
10 + n - 4 3 10n - 4 3 1 ^10n - 4 h 3 1 ^10 + n - 4 h 3
The number of pounds purchased increases as the price decreases. The number of pounds purchased increases as the price increases. The number of pounds purchased decreases as the price decreases. There is no relationship between the price and the pounds purchased.
A. B. C. D.
y=2+x 2+y=x 5+x=7+y y=2+x
29. If x = 2 and y = –3, then xy(10 – y) = 3 A. B. C. D.
–26 –14 14 26
CAHSEE Practice Test #3 GO ON TO THE NEXT PAGE 231
Part II: Full-length Practice Tests
Palm Springs And Los Angeles– Average Monthly High Temperatures 110 100 Los Angeles Palm Springs
90 80 70
Ja n Fe uary br ua M ry ar ch Ap ril M ay Ju ne Ju A ly Se ug pt us em t O ber N cto b ov e e r De mb ce er m be r
60
30. According to the preceding graph, which month shows the least difference between high temperatures in Los Angeles and Palm Springs? A. B. C. D.
January July August December
31. The following table shows the distribution of grades for students who take algebra during first period versus students who take algebra during sixth period. Grades
1st period
6th period
A
10
2
B
13
7
C
25
9
D
30
10
F
22
22
According to the table, what is the difference for the two periods in the percentage of students receiving a grade of F? A. B. C. D.
232
no difference 11% difference 22% difference 44% difference
CAHSEE Practice Test #3
Number of Models
Gas Barbecue Grills 25 20 15 10 5 0 $50–100
$101–200 $201–300 $301–400 $401–5,000
Price Range
32. The preceding graph shows how many different models of gas barbecues are marketed for various price ranges. How many models are available for more than $300? A. B. C. D.
33.
15 30 44 73
x 3 y-2 = x 2 y-3 A. B. C. D.
xy x y xxxyy xxyyy y x
34. Simplify the expression (2a2b)2(3ab5). A. B. C. D.
6a3b6 12ab5 12a5b7 36a6b12
CAHSEE Practice Test #3 GO ON TO THE NEXT PAGE 233
Part II: Full-length Practice Tests
35. Which of the following could be the graph of y = 1 x 2 ? 2 y A.
y B.
4
4
2
–4
2
0
–2
x 2
4
–4
0
–2
–2
–2
–4
–4
y C.
D.
234
0
2
4
4
2
–2
4
y
4
–4
x 2
2
x 2
4
–4
0
–2
–2
–2
–4
–4
x
CAHSEE Practice Test #3
36. What is the slope of the line shown in the following graph? y 6
4
2
–4
x
0
–2
2
4
6
–2
A. B. C. D.
-3 2 -2 3 2 3 3 2
37. What is the equation of the following graph? y 6
4
2
–6
–4
0
–2
x 2
4
6
–2
CAHSEE Practice Test #3
–4
–6
A. B. C. D.
y= 3 x-5 4 y= 4 x-5 3 y = - 5x - 3 4 4 y = 5x 3
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Part II: Full-length Practice Tests
38. According to the following graph, how fast is the airplane flying in miles per hour? Airspeed
Distance (miles)
2500
42. Joe drives a total of 20 miles to get to work. He drives 3 miles to get to the freeway in 9 minutes, 16 miles on the freeway in 18 minutes, and the last mile in 3 minutes. What is his average rate of speed for the whole trip?
2000
A.
1500
B. C. D.
1000 500 0 0
1
2
3
4
5
Time (hours)
A. B. C. D.
300 400 500 600
43. The Valdez family drives 720 miles on a trip. Their car averages 24 miles per gallon of gas, and they pay an average of $2.40 per gallon. How much do they spend on gas? A. B. C. D.
39. Solve for x. 8- x –7 x9
44. If a package weighs 7 pounds, 8 ounces, which would be the closest to its metric weight?
5 13 15 20
A. B. C. D.
3.405 kilograms 340.5 grams 7500 grams 7500 kilograms
On the Fahrenheit thermometer, water freezes at 32º and boils at 212º. On the Celsius thermometer, water freezes at Oº and boils at 100º. 45. Based on the information, which of the following statements is correct?
41. Which number line shows the solution to the inequality 3 + 3x ≤ –6? A.
A. B.
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
C.
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
D.
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
B. C. D.
236
$30.00 $57.60 $72.00 $720.00 454 grams ≈ 1 pound 1000 grams = 1 kilogram 16 ounces = 1 pound
40. Tania is saving money for a CD player that costs $100. She has $35, and each week she saves her $5 allowance. She calculates her total savings by using the equation S = 5W + 35, where S is her total savings, and W is the number of weeks she saves her allowance. According to her equation, how many weeks must she save before she has enough money for the CD player? A. B. C. D.
2 of a mile per minute 3 1.5 miles per minute 50 miles per hour 60 miles per hour
Each degree Fahrenheit is equivalent to about 2 degrees Celsius. Each degree Celsius is equivalent to about 2 degrees Fahrenheit. Degrees Fahrenheit and degrees Celsius are roughly equivalent. Based on this information, no relationship can be established between the two systems.
CAHSEE Practice Test #3
46. John is making a drawing of a tennis court, using a scale of 1 inch = 12 feet. On the drawing, how long should he make the 78-foot length of the court? 21‘
21‘
18‘ 12‘
POST
SIDE LINE
4‘6“
SERVICE LINE
ALLEY LINE
CENTER SERVICE LINE
13‘6“
13‘6“
NET
SINGLES
DOUBLES
36‘ 27‘
21‘
BACK SCREEN
18‘
BASE LINE
21‘
42‘ 78‘ SIDE SCREEN
A. B. C. D.
936 inches 65 inches 6.5 inches 1 inch
47. If it takes 8 days for one person to build a fence, how long does it take with three people on the job? A. B.
A. B. C. D.
4 hours 4 hours, 6 minutes 4 hours, 10 minutes 4 hours, 30 minutes
CAHSEE Practice Test #3
C. D.
3 day 8 2 2 days 3 4 days 24 days
48. Sam walks 12 1 miles at a speed of 3 miles per 2 hour. How long does it take him to walk that distance?
GO ON TO THE NEXT PAGE 237
Part II: Full-length Practice Tests
15 inches 9 inches 7.2 inches
12 inches
49. What is the area of the preceding triangle? A. B. C. D.
36 sq. inches 43.2 sq. inches 54 sq. inches 108 sq. inches
50. Moira is constructing a cylinder from the net shown. What is the approximate volume of the cylinder?
51. A circle with a radius of 5 centimeters is inscribed inside a square as shown in the following figure.
6 in.
3 in.
What is the perimeter of the square?
A. B. C. D.
238
18 cubic inches 54 cubic inches 113 cubic inches 169 cubic inches
A. B. C. D.
20 centimeters 25 centimeters 40 centimeters 100 centimeters
CAHSEE Practice Test #3
52. What is the volume of the following solid figure? (Volume of rectangular solid = lwh) (Volume of triangular pyramid = 1 × base area 2 × height)
53. The area of the shaded triangle is 4 sq. units. What is the area of square ABCD?
8 ft. 6 ft.
A. B. C. D.
A
B
D
C
32 square units 48 square units 56 square units 64 square units
54. Jeremy built the following shape from cubes. 3 ft.
A. B. C. D.
4 ft.
72 cubic feet 84 cubic feet 96 cubic feet 576 cubic feet Ron wants to make the same shape, but twice as high, twice as wide, and twice as long. How many cubes does Ron need to make the shape? A. B. C. D.
54 cubes 108 cubes 162 cubes 216 cubes
CAHSEE Practice Test #3 GO ON TO THE NEXT PAGE 239
Part II: Full-length Practice Tests
55. The following piece of wood measures 8ft. by 6in.
56. On the following coordinate grid, the distance between 0 and 1 is one unit. y
6 in.
B
8 ft. 6
What is the area of the piece of wood in square feet? A. B. C. D.
4
4 sq. ft. 8.5 sq. ft. 48 sq. ft. 576 sq. ft.
C A
2
0
2
D 4
x 6
What is the area of square ABCD? A. B. C. D.
240
25 sq. units 36 sq. units 49 sq. units 50 sq. units
CAHSEE Practice Test #3
57. Rectangle ABCD is drawn on the coordinate grid as shown in the following figure. y
A
D
B
C
(2,2)
(10,2) x
0
If the area of rectangle ABCD is 24 sq. units, what are the coordinates of point A? A. B. C. D.
(5, 5) (2, 3) (2, 5) (2, 8)
58. Triangle ABC is drawn on the following coordinate grid. y
8 B 6 4 2 A 0
–2
C
x 2
4
6
8
10
12
14
CAHSEE Practice Test #3
–2
What is the length of side BC? A. B. C. D.
8 units 10 units 12 units 14 units
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Part II: Full-length Practice Tests
59. Right triangle ABC is created by positioning three squares as shown in the following figure. D
F
A
Area = 16 sq. units
G
A. B. C. D.
242
1 unit 3 units 4 units 9 units
c
b
C
H
What is the length of triangle side a?
Area = 25 sq. units
a
B
I
E
CAHSEE Practice Test #3
60. Which figure contains two congruent triangles?
62. What is the solution set for the equation 3x - 3 = 15 ? A. B. C. D.
A.
{–4} {6} {–4, 6} {–6, 6}
63. Which equation is the same as x - 2 = 6 ? 5 3 A. 3x – 10 = 90 B. 5x – 6 = 90 C. 3x – 6 = 6 D. 5x – 6 = 6
B.
64. Which equation is equivalent to 1 ^ 4x - 12 h - 1 ^12 - 3x h = 5 ? 2 3 A. B. C. D.
C.
–x – 2 = 5 –x – 10 = 5 3x – 2 = 5 3x – 10 = 5
65. Francisco solved the equation 2 x - 6 = 4 using 3 the following steps: Given: 2 x - 6 = 4 3 Step 1: 2 x = 10 3
D.
Step 2: x = 15 2
-1 4 61. What number is equivalent to ^ 4 h ^ 2 h c 1 m ? 2
A. B. C. D.
–32 –16 1 4
To get from Step 1 to Step 2, Francisco— A. B. C.
CAHSEE Practice Test #3
D.
multiplied both sides by 2 3 multiplied both sides by 3 2 3 divided both sides by 2 1 added to both sides 3
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Part II: Full-length Practice Tests
y 6
4
2
–8
–6
–4
0
–2
x 2
4
6
8
–2
–4
–6
66. What is the equation of the line shown in the preceding graph? A. y = 3 x - 3 4 B. y = - 3 x - 3 4 4 C. y = x + 4 3 D. y = - 4 x + 4 3 67. What are the x- and y-intercepts for –2x + 3y = 6? A. B. C. D.
x-intercept: (2, 0); y-intercept: (0, –3) x-intercept: (0, –3); y-intercept: (2, 0) x-intercept: (0, 2); y-intercept: (–3, 0) x-intercept: (–3, 0); y-intercept: (0, 2)
69. What is the slope of a line that is parallel to the graph of 5x + 10y = 7? A. B. C.
68. Which of the following points lies on the line y = - 1 x? 2 A. B. C. D.
244
(–6, –3) (–3, –6) (6, –3) (–3, 6)
D.
–2 -1 2 1 2 2
CAHSEE Practice Test #3
70. Which graph represents the solution to the following system of equations? -x+y=3 ) - 4x + 2y = 6 y
y B.
8
8
A.
–4
–2
6
6
4
4
2
2
0
x 2
4
–4
–2
0
y C.
x 2
4
2
4
y
8
8
D.
–4
–2
6
6
4
4
2
2
0
x 2
A. B. C. D.
–x 4x2 – x 4x2 + 7x 7x
–4
–2
0
x
72. How much more time does it take a car traveling at an average of 60 miles per hour to cover a distance of 200 miles than it does for a car traveling at a rate of 75 miles per hour? A. B. C. D.
40 minutes 1 hour, 20 minutes 2 hours, 40 minutes 3 hours, 20 minutes
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CAHSEE Practice Test #3
71. Which of the following is equivalent to 3x – 2x(x – 2) + 2x2?
4
Part II: Full-length Practice Tests
Paula needs 180 units to graduate from college. She plans to earn 45 units each year to graduate in 4 years. How many classes should Paula take each year to stick to her plan?
76. Jennifer is keeping a running total of her purchases at a store to estimate her bill. So far her cart contains:
73. What other information is needed in order to solve this problem? A. B. C. D.
Paula’s grade point average the cost of each class the number of units earned per class Paula’s academic major
Three men go fishing, then divide the day’s catch. Each man takes one third of the fish. One fish is left over, which they toss to a hungry cat. 74. If each man takes x fish, which expression can be used to find the total number of fish the men caught? A. B. C. D.
x = b, 3
246
CDs
$12.39
Chicken
$19.75
Computer Printer
$145.99
Which of the following expressions gives the best estimate of Jennifer’s checkout bill? A. B. C. D.
30 + 10 + 20 + 200 30 + 12 + 20 + 100 30 + 10 + 20 + 150 40 + 10 + 20 + 100
a and b are positive integers.
Which of the following conclusions can be made from the preceding information? A. B. C. D.
Price $34.27
77. Jones converts fractions to decimals to do problems on the calculator. To add 1 + 1 + 1 , 3 3 3 he first enters 1 ÷ 3 on the calculator, then rounds the decimal quotient to .3. He then adds .3 + .3 + .3 and determines that the sum of the 3 fractions is .9, or 9 10
3x +1 3(x + 1) x -1 3 x-1 3
75. x = a, 2
Item Books
x is a multiple of 4. x is a fraction. x is a negative number. x is evenly divisible by 2 and by 3.
Which of the following best explains the mistake Jones made? A. B. C. D.
He misplaced the decimal point in the sum. Rounding introduced a serious error into what should be an exact computation. He should have multiplied .3 by 3. 1 + 1 + 1 = 3 , not 3 3 3 3 9 10
CAHSEE Practice Test #3
78. A market research company sent out 500 questionnaires to determine customers’ movie preferences. The following graph shows data from the questionnaires that were returned.
79. The following table shows values for x and the corresponding values for y.
Movie Preferences Other Drama (10) (20) Horror (20) Comedy (130)
x
y
0
3
1
5
5
13
13
29
Which of the following represents the relationship between x and y? A. B. C. D.
y = 2x + 3 y=x+3 y = 5x y=x+4
Action (140)
How many people did not return the questionnaire? A. B. C. D.
It cannot be determined from the graph. 180 320 500
80. A combination of 10 nickels and dimes has total value of $.75. To determine the number of dimes, Juliet writes the equation: .10x + .05(10 – x) = .75 If, instead, there were a total of 13 coins, which of the following equations shows the correct adjustment to Juliet’s equation? A. B. C. D.
.13x + .05(10 – x) = .75 .10x + .05(13 – x) = .75 .13x + .05(13 – x) = .75 .10x + .05(13) = .75
CAHSEE Practice Test #3
STOP 247
Test #3—Answers and Explanations Reviewing Practice Test 3 Review your simulated CAHSEE Math practice examination by following these steps: 1. Check the answers you marked on your answer sheet against the answer key that follows. Put a check mark in the box following any wrong answer. 2. Fill out the Review chart (p. 250). 3. Read all the explanations (pp. 252–264). Go back to review any explanations that are not clear to you. 4. Fill out the Reasons for Mistakes chart on p. 250. 5. Go back to the “Math Review” section and review any basic skills necessary before taking the next practice test. Don’t leave out any of these steps. They are very important in learning to do your best on CAHSEE Math.
249
Part II: Full-length Practice Tests
Review Chart Use your marked answer key to fill in the following chart for the multiple-choice questions. Possible Number Sense (NS)
(1–14)
14
Statistics, Data Analysis, Probability (P)
(15–26)
12
Algebra and Functions (AF)
(27–43)
17
Measurement and Geometry (MG)
(44–66)
17
Algebra I (AI)
(67–72)
12
Mathematical Reasoning (MR)
(73–80)
8
Totals
Completed
Right
Wrong
80
Reasons for Mistakes Fill out the following chart only after you have read all the explanations that follow. This chart helps you spot your strengths, weaknesses, and your repeated errors or trends in types of errors. Total Missed
Simple Mistake
Misread Problem
Lack of Knowledge
Number Sense (NS) Statistics, Data Analysis, Probability (P) Algebra and Functions (AF) Measurement and Geometry (MG) Algebra I (AI) Mathematical Reasoning (MR) Totals
Examine your results carefully. Reviewing the preceding information helps you pinpoint your common mistakes. Focus on avoiding your most common mistakes as you practice. The Lack of Knowledge column helps you focus your review in the “Math Review” section. If you are missing a lot of questions because of lack of knowledge, you should go back and spend extra time reviewing the basics.
250
Test #3—Answers and Explanations
Answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
C (NS) B (NS) A (NS) C (NS) B (NS) C (NS) C (NS) B (NS) A (NS) C (NS) B (NS) D (NS) C (NS) C (NS) D (P) D (P) C (P) B (P) B (P) B (P)
21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
C (P) D (P) D (P) D (P) C (P) A (P) B (AF) D (AF) A (AF) D (AF) C (AF) B (AF) A (AF) C (AF) B (AF) B (AF) A (AF) B (AF) D (AF) B (AF)
Number Sense (NS)
14
Statistics, Data Analysis, Probability (P)
12
Algebra and Functions (AF)
17
Measurement and Geometry (MG)
17
Algebra I (AI)
12
Mathematical Reasoning (MR)
A (AF) A (AF) C (AF) A (MG) B (MG) C (MG) B (MG) C (MG) C (MG) D (MG) C (MG) B (MG) D (MG) D (MG) A (MG) A (MG) C (MG) B (MG) B (MG) B (MG)
61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
C (AI) C (AI) A (AI) D (AI) B (AI) A (AI) D (AI) C (AI) B (AI) A (AI) D (AI) A (AI) C (MR) A (MR) D (MR) C (MR) B (MR) B (MR) A (MR) B (MR)
8
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Answers and Explanations Number Sense 1. C. Since the first factor in each of these expressions is a number between 1 and 10, concentrate on the second factor (10 raised to some power). When 10 is raised to a negative exponent, it becomes a fraction: 10 -1 = 1 10 10 - 2 = 1 100 Therefore, the expressions in Answers A and D are both very small. B is equivalent to 919, but C is equivalent to 1,190, so it is the largest. 2. B. The least common denominator for 8 and 7 is 56, so: 3 + 3-2 = 14 c 8 7 m 3 + 21 - 16 = 3 + 5 14 c 56 56 m 14 56 Fourteenths can be converted into fifty-sixths: 12 + 5 = 17 56 56 56 3. A. When temperature drops, it is like subtracting a positive, so we can write an equation to solve the problem: –3˚ – 18˚ = –21˚ 4. C. Taxes = 1 × $300 = $60. The “rest of the money” = $300 – $60 = $240 5 5. B. It’s easy to get confused when evaluating these statements because several of them are merely the reverse of the correct relationship between the dollar and the Euro. The key thing to remember is that the statement says the Euro is worth more than the dollar; therefore, it’s going to take more dollars to equal fewer Euros. Rule out Answers C and D on that basis. Rule out Answer A since .81 is more than half (.50). 4 is equal to .8, which is 5 close to .81, so Answer B is the most accurate. 6. C. The rest of the team scored 49 – 18 = 31 points, so we can set up a proportion to solve the problem: 31 = x 49 100 Since 49 is close to half of 100, to estimate, just double 31; so 31 is close to 62% of 49. The only answer close to that is C. 7. C. When calculating percent change, use the formula: change in price percent = 100 starting price The change in price = $2.75 – $.25 = $2.50; the starting price = $.25, so: $2.50 = x $.25 100 To solve a proportion, you can cross-multiply and set the products equal:
Then:
252
$250 = $.25x $250 $.25x = $.25 $.25 x = 1,000
Test #3—Answers and Explanations
8. B. Forty percent of 80 is $32. John saves 15% of the reduced price. The reduced price is $80 – $32 = $48. Fifteen percent of $48 is $7.20. So, John saves $32 + $7.20 = $39.20. 9. A. The star makes 5% of the sales above $30,000,000. Those sales are $50,000,000 – $30,000,000 = $20,000,000. Five percent of $20,000,000 is $1,000,000. 10. C. This problem can be rewritten: -3 1 1 1 ^ 3h = 3 = 3 $ 3 $ 3 = 27 3
11. B. The lowest common denominator of 6 and 8 is 24: It is the smallest number that is divisible by both 6 and 8. Twenty-four can be factored as 4 × 6. Because neither 4 nor 6 are prime numbers, they must be factored as well: 4 × 6 = (2 × 2)(2 × 3) = 2 × 2 × 2 × 3 Answer D is incorrect for two reasons: Neither 6 nor 8 are prime numbers; therefore, this cannot be the prime factored form. Moreover, 48 (the product of 6 and 8) is a common denominator, but not the lowest common denominator. 12. D. If you forget the rules for exponents c 3 ' 3 = 3 m , you can rewrite the problem to find the solution: 12 6 6 _3 4 i
3
34 # 32
=
34 # 34 # 34 = 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 # 3 = 3 # 3 # 3 # 3 # 3 # 3 = 36 3#3#3#3#3#3 3#3#3#3#3#3 1 13. C. Both 2,500 and 3,600 are perfect squares. 2500 = 25 × 100 = 5 × 10 = 50 3600 = 36 × 100 = 6 × 10 = 60 Pick any number between 50 and 60, multiply it by itself, and that product falls between 2,500 and 3,600. 14. C. The absolute value of a number is always positive: It is the distance between a particular number and zero on the number line. From 0 to 5 is a distance of 5 units; however, from –5 to 0 is also a distance of 5 units. Therefore, x could be either –5 or 5.
Statistics, Data Analysis, Probability (P) 15. D. To solve this problem, first arrange the sales figures in numerical order: $19,000, $20,000, $21,000, $21,000, $25,000 Now determine the mean, median, and mode for this data set: Mean: $21,200 Median: $21,000 Mode: $21,000 Now add the sixth value to the data set: $19,000, $20,000, $21,000, $21,000, $25,000, $55,000 The mode does not change because $21,000 is the only repeated data value. The median is the mean of the two middle terms; however, because the two middle terms are the same value, the median is still $21,000. Only the mean is affected in this case by the addition of a relatively extreme data value.
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16. D. Since Tom had a mean of 80% after six tests, he must have had a total of 480 points because 480 ÷ 6 = 80. Tom’s average after three tests was 72%, so after three tests he had 216 points because 3 × 72 = 216. When his fourth and fifth tests are added to this, the result is 394. So his sixth test had to be the difference between the total points he needed (480) and the points he had after five tests (394). 480 – 394 = 86 17. C. To find the median, first arrange the data values in numerical order: 140, 290, 480, 520, 610, 650 When there are an even number of data values, the median is the mean of the middle two values: 480 + 520 = 500 2 18. B. Answer A is false; less than half the students worked less than half an hour. Answer B is correct; 20% of the students did no homework, and 35% of the students worked for an hour or more. That is a total of 55%, which is more than half. Answer C is a subjective statement, not a mathematical observation. Answer D is backward: More than half the students worked more than half an hour. 20% + 35% = 55% 19. B. Half the time an odd number is rolled, and half the time heads are flipped. 1 # 1 = 1 . This can also be seen 2 2 4 on the tree diagram, which shows 12 possible outcomes, each of which is equally likely. The favorable outcomes are 1 with heads, 3 with heads, and 5 with heads. Three favorable outcomes out of a total of 12 outcomes is one-fourth. 20. B. Each of the 12 donuts is equally likely to be picked. Eight of the 12 donuts are neither a jelly donut nor a buttermilk donut. 8 =2 12 3 21. C. The trick to this problem is that the sections are not all the same size. The small green section is 1 of the 8 circle’s area, whereas the large green section is 3 of the circle’s area. 1 + 3 = 4 = 1 Because green represents 8 8 8 8 2 half the area of the circle, it can be expected to occur half the time. 22. D. Spins are independent events; that is to say, one spin has nothing to do with the preceding spins. Answers A, B, and C either imply or directly state that the probability of the outcome of the next spin depends on the outcomes of the previous spins, which is false. Only Answer D correctly explains the independent relationship of the spins. 23. D. Siblings are brothers and sisters, but even if the word is unfamiliar to you, don’t let it throw you: You can still solve the problem. Eighteen percent of the students have no siblings, and 30% have one sibling, so 48% of the students have no siblings or one sibling. That means that out of 100 children, 48 are in this situation. Out of 500 children, five times as many are in this situation. 48 × 5 = 240 children 24. D. To find the greatest decrease, look for the portion of the line graph that shows the greatest (steepest) negative slope. In this case, it occurs when Dontrey’s score falls from 90 to 70. Though 70 is not his lowest score, the change from 90 to 70 is greater than the change from 70 to 60. 25. C. What might make this question tricky is that no information is given on the variables to help you puzzle out the relationship. A scatter plot that shows a negative correlation presents an array of points that fall into a line sloping downward from left to right. This represents an inverse relationship between the two variables: When one increases, the other decreases, and vice versa.
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Test #3—Answers and Explanations
26. A. The amount of cherries purchased decreases as the price increases; however, none of the answer choices express this exact relationship. However, the converse of the statement is also true: The number of pounds purchased increases as the price decreases.
Algebra and Functions (AF) 27. B. Frequently the relationship expressed first in a verbal description is the operation that must be performed last. In this case, the first relationship to establish is “the product of 10 and n.” Product means the result of multiplication, which can be expressed as 10n. One-third of 10n can either be expressed as 10n or 1 (10n) . From this quantity, we 3 3 must then subtract 4. Though Answers B and C appear similar, the order of the operations is different, and the result is different. To appreciate this, substitute the value of 1 for n. 10n - 4 = 10 $ 1 - 4 = 3 1 - 4 = - 2 3 3 3 3 1 ^10n - 4 h = 1 ^10 $ 1 - 4 h = 1 ^ 6 h = 2 3 3 3 28. D. The “sum of five and y” is 5 + y. “Seven more than x” is 7 + x. Because they are equal, we can say that 5 + y = 7 + x. However, that answer choice is not available. By subtracting five from each side, we can simplify the relationship. y=2+x 29. A. To evaluate this expression, plug the given values in for the variables, and perform the operations: xy _10 - y i = c 2 m^ - 3h^10 - - 3h = ^ - 2 h^13h = - 26 3 30. D. To find the least difference, look for the place where the two lines are the closest, which has to be January or December. Because they are slightly closer in December, the correct answer is D. Answers B and C are the months during which the highs have the greatest difference. 31. C. A total of 100 students are in first period, and 22 of them received an F, which is 22% of those students. However, only 50 students are in sixth period. Twenty-two of those students represents 44% of the sixth-period students. The difference in the failure rate is: 44% – 22% = 22% 32. B. Gas grills selling for more than $300 are shown by the two far-right columns, each of which represents a quantity of 15 models. Therefore, the total number of models available at this price is 30. 33. A. Raising a variable to a positive exponent can be represented as repeated multiplication. When the variable is raised to a negative exponent, that is the inverse of multiplication, which is division. We can rewrite the expression: x 3 y - 2 xxxyyy = xxyy = xy x 2 y -3 34. C. The trick is to note that the first term is squared. It might be helpful to rewrite the expression: (2a2b)2(3ab5) = (2a2b)(2a2b)(3ab5) = (2 × 2 × 3)(aaaaa)(bbbbbbb) = 12a5b7 35. B. When x is raised to a power of two, its graph is a parabola, which is the form of the line in Graphs A and B. However, when the coefficient of x is negative, the parabola opens downward, as in Graph A, so you must eliminate it. Graphs C and D are linear functions and do not contain a squared variable.
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36. B. Because the line slopes downward from left to right, it has a negative slope—therefore you can eliminate Answers C and D. Slope is defined as: vertical shift horizontal shift In this case, the line goes down two units for every three units it moves to the right: -2 = - 2 3 3 37. A. This line goes up 3 units for every 4 units it moves to the right, which gives it a slope of 3 . It crosses the 4 y-axis at –5, which gives it a y-intercept of –5. Its equation is then written in the slope-intercept form as: y= 3 x-5 4 Once you spot that the y-intercept is –5, you could eliminate Choices C and D. Or, if you spot that the slope is 3 4 the correct answer is A. It is the only choice with a slope of 3 . 4 38. B. The easiest way to solve this problem is to look for a place where the graphed line crosses an intersection on the grid. This happens at 2.5 hours, and at 5 hours. At the 5 hours mark, the airplane has traveled 2,000 miles. 2,000 ÷ 5 = 400 39. D. To solve the inequality: 8- x 9 40. B. To solve this equation: S = 5W + 35 100 = 5W + 35 –35 = –35
Subtract 35 from each side to isolate the variable.
65 = 5W 65 = 5 W 5 5
Divide each side by 5.
W = 13 The correct answer could also be easily determined by substituting the answer choices into the equation.
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Test #3—Answers and Explanations
41. A. To solve this inequality: 3 + 3x ≤ –6 –3 = –3
Subtract 3 from each side to isolate the variable.
3x ≤ –9 3x # - 9 3 3
Divide each side by 3 to clear the coefficient.
x ≤ –3 So, we are looking for the number line that includes the point –3 and all points that are smaller than –3, which are the points to the left of that number. That is Answer A. 42. A. We are comparing miles to time, either minutes or hours. Since the information is given in minutes, it’s best to start with minutes: 3 + 16 + 1 = 20 = 2 9 + 18 + 3 30 3 So Joe drives 2 miles in every 3 minutes. Or, Joe drives 2 of a mile per minute. Since some of the answer choices 3 are given in terms of miles per hour, perhaps we’d better convert this to mph to be sure they are incorrect. 2 # 20 = 40 = 40 miles for 60 minutes, or 40 miles per hour 3 20 60 43. C. Dividing the distance traveled by the miles per gallon tells how many gallons of gasoline the family uses: 720 ÷ 24 = 30 gallons Now, multiplying the number of gallons by the price per gallon tells how much money they spend on gas: $2.40 × 30 = $72.00 This might seem like going in circles because of the numerical similarity between miles per gallon (24) and the price of gasoline ($2.40).
Measurement and Geometry (MG) 44. A. Eight ounces = 1 of a pound because 8 = 1 . So 7 pounds, 8 ounces equals 7.5 pounds. To convert pounds to 2 16 2 grams, multiply by 454: 7.5 pounds × 454 grams per pound = 3,405 grams However, this answer is not one of the choices that are available. Answers B and C, which are both expressed in terms of grams, are clearly incorrect and can be eliminated. That leaves Answers A and D to consider, both of which are expressed in terms of kilograms. To convert grams to kilograms, divide by 1,000: 3,405 grams ÷ 1,000 grams per kilogram = 3.405 kilograms
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Part II: Full-length Practice Tests
45. B. The change in temperature necessary to cause a state change in water from boiling to freezing is the same, no matter which system is used to measure it. In Fahrenheit, that change in temperature is equal to the difference in the two temperatures: 212˚ – 32˚ = 180˚F In Celsius: 100˚ – 0˚ = 100˚C So: 180˚F = 100˚C If it takes fewer degrees Celsius to equal the degrees Fahrenheit, a Celsius degree must be bigger than a Fahrenheit degree. 46. C. The length of the court is 78 feet. If 1 inch = 12 feet, then divide 78 by 12 to see how long to make the length: 78 feet ÷ 12 feet/inch = 6.5 inches 47. B. A job that can be done in 8 “man-days” can be completed by 8 men in one day, or 1 man in eight days. The job is a product of the number of men and the number of days. So we can ask: 3 men × how many days = 8 man days To solve for days, divide both sides by 3 men: how many days = 8 days = 2 2 days 3 3 48. C. All the answers include 4 hours since it takes 4 hours to walk 12 miles. The question is, how long does it take to walk the extra half mile? Eliminate Answer A since it allows no time whatsoever. Three miles per hour equals 6 half-miles per hour; so to walk half a mile takes one-sixth of an hour. One-sixth of an hour is 10 minutes. 49. C. The area of a triangle = 1 base × height. 2 Any side of a triangle can be used as the base; the important consideration is that the height is a line drawn perpendicular to the base from the opposite vertex. So in this drawing, use 15 inches as the base and 7.2 inches as the height: 1 15 inchesh^ 7.2 inchesh = 54 square inches 2^ 50. D. The formula for the volume of a cylinder is base × height. The base of a cylinder is a circle; the formula for the area of a circle is πr2. The radius of the circle shown is 3 inches, and the height of the cylinder is 6 inches, so to calculate the volume: base × height = πr2 × 6 in. = (3.14)(32)(6) ≈ 169.56 in.3 51. C. If the radius of the circle is 5 inches, the diameter of the circle is twice that, or 10 inches. The diameter of the circle is equal to the length of one of the sides of the square. To find the perimeter of the square, multiply the length of one side by 4: 10 in. × 4 = 40 in.
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Test #3—Answers and Explanations
52. B. The solid figure is formed by combining a rectangular prism with a triangular prism.
2
4
3 8 ft.
6 ft.
4 ft.
3 ft.
Volumerectangular prism = length × width × height = (3 ft.)(4 ft.)(6 ft.) = 72 ft.3 Volumetriangular prism = 1 × base area × height = 1 (2)(3)(4) = 12 ft.3 2 2 The total volume of the figure is the sum of the two volumes: 72 ft.3 + 12 ft.3 = 84 ft.3 53. D. The large square, ABCD, is subdivided into several different polygons whose areas are related. The area of the small, shaded triangle is one-half the area of the small square, the parallelogram, and the medium-sized triangle. So: 2 small triangles + small square + parallelogram + medium triangle = 8(area of small triangle) = 8(4 sq. units) = 32 sq. units This is half the area of the large square, so: Arealarge square ABCD = 2(32 sq. units) = 64 sq. units Another method would be to break the figures in half of the triangle into small congruent triangles and count them as follows: A
B 1
4 2
3 7
5 6 8
D
C
8 small triangles times 4 sq. units = 32 sq. units. Doubling this would give 64 sq. units.
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Part II: Full-length Practice Tests
54. D. The length, width, and height as shown are each 3; so if Ron doubles each of the dimensions, he needs 6 × 6 × 6 = 216 cubes. 55. A. Six inches is half a foot, so the area of the board is 8 ft. × .5 ft. = 4 ft.2 56. A. To find the area of the square you will need to find the side and then square it. By using the coordinate grid you can make a right triangle with the origin at (0,0); one leg goes to point D (4,0), and the other goes to point A (0,3).
B 6
4
C A
2
0
2
D 4
6
So the lengths of the legs of the right triangle are 3 and 4. If you know Pythagorean Triples you would know that the hypotenuse is then 5, as this is a 3-4-5 right triangle. Otherwise use the Pythagorean theorem: a2 + b2 = c2 (3)2 + (4)2 = c2 9 + 16 = c2 25 = c2 5 = c2 Since one side of the square is 5, the area is 5 × 5 or 25 square units. 57. C. The length of Side DC is equal to the difference in x-coordinates, or: 10 – 2 = 8 units Therefore, to have an area of 24 units, the length of Width AD has to be 3 units. The x-coordinate of Point A is the same as the x-coordinate of Point D since AD is parallel to the y-axis. Its y-coordinate has to be 3 units above that of Point D. That location is (2,5) on the coordinate grid. 58. B. ABC is a right triangle with legs measuring 6 units and 8 units. Side BC is the hypotenuse of the triangle, and its length is equal to the square root of the sum of the squares of the two legs: BC = 6 2 + 8 2 = 36 + 64 = 100 = 10 units 59. B. By the Pythagorean theorem: AreaBCHI = AreaABED – AreaACGF = 25 – 16 = 9 units2 Since BCHI is a square, the length of one of its sides is the square root of its area: 9 units2 = 3 units = a 60. B. Segments marked with the same number of hash marks are congruent to each other. Figure B is a parallelogram divided into two triangles by its diagonal. The hash marks show that each triangle has two sides that are congruent to two sides in the other triangle. The diagonal forms the third side of each triangle.
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Test #3—Answers and Explanations
Algebra I (AI) 61. C. Rewriting this equation might help to simplify it. The base gives the factor, the exponent tells how many times to use it as a factor. A positive number raised to a negative exponent becomes a fraction, not a negative number. 2
-1 4 1 1 1 1 1 1 16 ^ 4 h ^ 2 h c 2 m = c 4 m^ 2 $ 2 $ 2 $ 2 hc 2 $ 2 m = 4 $ 16 $ 4 = 16 = 1
62. C. The easiest way to deal with an absolute value equation like this might be to substitute the answer choices into the equation and see which ones work. Absolute value is always positive, so the total quantity within the absolute value bars can be positive or negative. What makes this equation tricky is that the entire left side of the equation is enclosed within absolute value bars. In essence, what it’s saying is that 3x – 3 is equal to either 15, or to –15. So you need to solve two equations: 3x – 3 = 15
3x – 3 = –15
+3 = +3
+3 = +3
3x = 18
3x = –12
3x = 18 3 3
3x = - 12 3 3
x=6
x = –4
63. A. To clear the fractions on the left side of the equation, multiply each side of the equation by the least common denominator of the fractions. The least common denominator of 5 and 3 is 15, so: x - 2 =6 5 3 15 c x - 2 m = 15 ^ 6 h 5 3 15 $ x - 15 $ 2 = 90 3 5 3x – 10 = 90 64. D. To simplify the left side of the equation, you must distribute the multiplication by each factor outside the parentheses to the terms within the parentheses, then combine like terms: 1 4x - 12 h - 1 ^12 - 3x h = 5 2^ 3 1 1 1 1 c m^ 4x h - c m^12 h + c - m^12 h + c - m^ - 3x h = 5 2 2 3 3 2x – 6 – 4 + x = 5 3x – 10 = 5 65. B. At the end of Step 1, Francisco has isolated the variable. His objective now is to get a coefficient of 1. To get a coefficient of 1, divide both sides by the current coefficient. However, none of the answer choices say, “Divided both sides by 2 .” Multiplication and division are inverse operations, so instead of dividing, multiply by the 3 inverse of 2 , which is known as the reciprocal. Multiplying both sides by 3 (the correct answer) is the same as 3 2 dividing both sides by 2 . 3
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66. A. The answer choices are all given in the slope-intercept form, where the coefficient (number in front) of x is the slope, and the second number is the y-intercept (or the point on the y-axis that the line passes through). Lines that slope upward from left to right have a positive slope, so eliminate Answers B and D, which show a negative slope. This line passes through the y-axis at –3, so it has a y-intercept of –3. With this piece of information, you can eliminate Answer C, leaving Answer A as the only possibility. To confirm the exact slope, observe that the line rises 3 units for every 4 units it moves to the right, which is a slope of 3 . 4 67. D. The y-intercept is the point where the line crosses the y-axis, and it always has an x-coordinate of 0. Likewise, the x-intercept is the point where the line crosses the x-axis, and it always has a y-coordinate of 0. Knowing this eliminates Answers B and C, which have this relationship backward. To solve for the y-intercept, set x = 0 and solve the equation: –2x + 3y = 6 –2(0) + 3y = 6 3y = 6 y=2 So the y-intercept is (0, 2). To solve for the x-intercept, set y = 0 and solve the equation: –2x + 3y = 6 –2x + 3(0) = 6 –2x = 6 x = –3 So the x-intercept is (–3, 0). 68. C. It’s a good idea to visually inspect the equation to see whether any relationships can be taken advantage of to eliminate answers. In this case, there are: y = - 1 x . The signs for x and y are opposite, so whenever x is positive, 2 y is negative, and vice versa. On that basis, eliminate Answers A and B, which have both x- and y-coordinates as negative. At this point, perhaps substitute the coordinates into the equation to see which pair satisfies the equation. Using the coordinates from Answer C: - 3 = - 1 ^6h 2 This is true, so those coordinates do represent a point on the line. 69. B. Lines that are parallel have the same slope. The difficulty is that, in this problem, the equation of a line is given in the standard form, not the slope-intercept form (which is y = mx + b), where m, the coefficient of x, is the slope of the line. The easiest way to get the slope is to just rework the equation into the slope-intercept form: 5x + 10y = 7 10y = –5x + 7 y= -1 x+ 7 2 10 So the slope is - 1 , which is Answer B. 2
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Test #3—Answers and Explanations
70. A. The answer is the graph that represents these two equations. The easiest way to determine that is to convert these two equations to the slope-intercept form, y = mx + b. –x + y = 3 → y = x + 3 –4x + 2y = 6 → y = 2x + 3 The lines in the answer both have a positive slope and a y-intercept of +3. Lines with a positive slope travel upward from left to right; the y-intercept is the point on the y-axis that the line passes through. All four answers show graphs with correct y-intercepts. However, only Graph A shows two lines with a positive slope, so we can identify it as the only possible answer without going any further. 71. D. The trick to simplifying this expression is to correctly distribute –2x to both terms within the parentheses: 3x – 2x(x – 2) + 2x2 = 3x + (–2x)(x) + (–2x)(–2) + 2x2 = 3x – 2x2 + 4x + 2x2 = 7x 72. A. The formula for solving this type of time problem is: distance = time rate So, to drive 200 miles at 60 mph takes: 200 miles = 10 = 3 1 hours 3 3 60 mph To drive 200 miles at 75 mph takes: 200 miles = 8 = 2 2 hours 3 3 75 mph The difference in time is 3 1 - 2 2 = 2 of an hour, which equals 40 minutes. 3 3 3
Mathematical Reasoning (MR) 73. C. Though Answers A, B, and D all mention factors that could affect Paula’s plans to graduate in 4 years, Answer C identifies the specific information needed to formulate her initial class schedule. For instance, if each class is worth 5 units, Paula only needs to take 9 classes a year to earn 45 units. However, if each class is worth only 3 units, Paula needs to take 15 classes to earn the same number of units. 74. A. An easy way to solve this kind of problem is to consider a specific case of this situation, and then substitute the numbers into the expressions to see which expression yields the desired total. If each man gets only 1 fish, together the men have 3 fish, plus one they gave to the cat, for an initial total of 4 fish. So with x = 1, which of these expressions is worth 4? Only the expression in Answer A: 3x +1 = 3(1) + 1 = 3 + 1 = 4 75. D. Positive integers are whole numbers, like 3 or 7. If division by a particular number results in a quotient that is a whole number, the number is said to be divisible by that divisor. The symbolic information presented in the box gives the same information as Answer D. Answer A might be true, but we do not have enough information to be sure: Many numbers that are divisible by both 2 and 3 are not multiples of 4, such as the number 6. 76. C. The expression for this answer rounds each item to the closest $10. Answers B and C round the computer printer to the closest $100, which introduces a significant error of about $46. Answer A incorrectly rounds the printer to $200, introducing an even bigger error.
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Part II: Full-length Practice Tests
77. B. The decimal equivalent for 1 is a repeating decimal, .3, which is significantly larger than .3. By rounding 3 off so early in his computation, and then rounding off the other addends in the same way, Jones introduces a significant error that produces an incorrect sum. This is a common problem when converting fractions to decimals to do computations. 78. B. To determine the number of people who did not send back responses, add the number of responses reported from each category and subtract the total from 500: 130 + 140 + 20 + 20 + 10 = 320 500 – 320 = 180 79. A. For each of the x/y pairs, multiplying the x by 2, then adding 3, gives the corresponding y value. The other answer choices might be true for a single x/y pair, but not for all pairs. 80. B. In this equation, x represents the number of dimes. When there are 13 coins, 13 – x represents the number of nickels. The value of one dime multiplied by the number of dimes is .10x. This gives the total value of the dimes in the combination. Since the value of one dime does not change, this part of the equation should not be changed.
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Arithmetic/Statistics and Probability Glossary of Terms ABSOLUTE VALUE: The distance a number is from zero on the number line. Considering + and − as directions on the number line, absolute refers to distance, not direction. For example, the absolute value of −3 is 3. Also written as |−3| = 3. The absolute value of + 3 is also 3. ADDITIVE INVERSE: The opposite (negative) of a number. Any number plus its additive inverse equals 0. For example, 3 + (−3) = 0. ASSOCIATIVE PROPERTY: A property stating that the grouping of elements does not make any difference in the outcome. This is only true for multiplication and addition. For example, (2 + 3) + 4 = 2 + (3 + 4). BAR GRAPH: A graph using horizontal or vertical bars to display the data. The longer the bar, the greater the quantity. BRACES: Grouping symbols used after the use of brackets. Also used to represent a set. { } BRACKETS: Grouping symbols used after the use of parentheses. [ ] CANCELING: In multiplication of fractions, dividing the same number into both a numerator and a denominator. CIRCLE GRAPH (or pie chart): Displaying data on a circular graph by dividing the circle into sections. COMBINATIONS: The total number of independent possible choices. COMMON DENOMINATOR: A number that can be divided evenly by all denominators in the problem. For example, the common denominator of 1 and 1 is 6. 2 3 COMMON FACTORS: Factors that are the same for two or more numbers. For example, 3 is a common factor of 6 and 9. COMMON MULTIPLES: Multiples that are the same for two or more numbers. For example, 10 is a common multiple of 2 and 5. COMMUTATIVE PROPERTY: A property stating that the order of elements does not make any difference in the outcome. This is only true for multiplication and addition. For example, 2 + 3 = 3 + 2. COMPLEX FRACTION: A fraction having a fraction in the numerator and/or denominator. COMPOSITE NUMBER: A number divisible by more than just 1 and itself. {4, 6, 8, 9, . . .}. 0 and 1 are not composite numbers. CORRELATION: Comparing the relationship of two pairs of data. CUBE: The result when a number is multiplied by itself twice. For example, 8 is a cube because 2 × 2 × 2 = 8. CUBE ROOT: The number that is multiplied by itself twice to get the resulting cubed number. For example, 5 is the cube root of 125 because 5 × 5 × 5 = 125. Its symbol is 3 . 3 125 = 5 . DECIMAL FRACTION: A fraction with a denominator of 10, 100, 1000, or any multiple of 10, written using a decimal point (for example, .3, .275). DECIMAL POINT: A point used to distinguish decimal fractions from whole numbers. DECREASED BY: To make a quantity smaller by a certain value. DENOMINATOR: The bottom symbol or number in a fraction.
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CliffsTestPrep California High School Exit Exam: Math
DEPENDENT EVENT: When the outcome of one event has a bearing or effect on the outcome of another event. DIFFERENCE: The result of subtraction. DISTRIBUTIVE PROPERTY: The process of distributing a number on the outside of a set of parentheses to each number on the inside, for example, 2(3 + 4) = 2(3) + 2(4). This can also be written algebraically as a(b + c) = ab + ac. EVEN NUMBER: An integer (positive whole numbers, zero, and negative whole numbers) divisible by 2 (which means with no remainder). EXPANDED NOTATION: Pointing out the place value of a digit by writing the number as the digit multiplied by its place value. For example, 342 = (3 × 102) + (4 × 101) + (2 × 100). EXPONENT: A small number placed above and to the right of a number. It expresses the power to which the quantity is to be raised or lowered. For example, in the number 32, 2 is the exponent. FACTOR (noun): A number or symbol that divides evenly into a larger number. For example, 6 is a factor of 24. FACTOR (verb): To find two or more quantities whose product equals an original quantity. For example, 15 can be factored into 3 × 5. FRACTION: A symbol expressing part of a whole. It consists of a numerator and a denominator (for example, 3 , 9 ). 5 4 GREATEST COMMON FACTOR: The largest factor common to two or more numbers (that is, the largest number that divides into two or more numbers evenly). For example, 6 is the greatest common factor of 18 and 24. HUNDREDTH: The second decimal place to the right of the decimal point. For example, .08 is eight hundredths. IDENTITY ELEMENT FOR ADDITION: 0. Any number added to 0 gives the original number. For example, 2 + 0 = 2. IDENTITY ELEMENT FOR MULTIPLICATION: 1. Any number multiplied by 1 gives the original number. For example, 3 × 1 = 3. IMPROPER FRACTION: A fraction in which the numerator is greater than the denominator. For example, 3 . 2 INDEPENDENT EVENT: When the outcome of one event has no bearing or effect on the outcome of another event. INTEGER: A whole number, either positive, negative, or zero. {. . .−3, −2, −1, 0, 1, 2, 3 . . .} INVERT: To turn upside down, as in “invert 2 ” = 3 . 3 2 IRRATIONAL NUMBER: A number that is not rational (that is, it cannot be written as a fraction xy , with x as an integer and y as a natural number). For example, 3 or π. LEAST COMMON MULTIPLE: The smallest multiple that is common to two or more numbers. For example, 6 is the least common multiple of 2 and 3. LINE GRAPH: Graphing on an x-y graph by placing points on the graph and connecting them to show relationships in the data. LOWEST COMMON DENOMINATOR: The smallest number that can be divided evenly by all denominators in the problem. For example, in the problem 2 + 1 , the lowest common denominator is 12. 3 4 MEAN (arithmetic): The average of a number of items in a group (found by totaling the items and dividing by the number of items). MEDIAN: The middle item in an ordered group. If the group has an even number of items, the median is the average of the two middle items. The items in the group have to be in placed in consecutive order.
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Arithmetic/Statistics and Probability Glossary of Terms
MIXED NUMBER: A number containing both a whole number and a fraction (for example, 5 1 ). 2 MODE: The number appearing most frequently in a group. MULTIPLES: Numbers found by multiplying a number by 2, by 3, by 4, and so on. MULTIPLICATIVE INVERSE: The reciprocal of a number. Any number multiplied by its multiplicative inverse equals 1 (for example, 1 # 3 = 1). 3 NATURAL NUMBER: A counting number. {1, 2, 3, 4, . . .} NEGATIVE CORRELATION: In a scatterplot, when one set of data increases while another decreases. NEGATIVE NUMBER: A number less than 0. NUMBER LINE: A visual representation of the positive and negative numbers and zero. The line can be thought of as an infinitely long ruler with negative numbers to the left of zero and positive numbers to the right of zero. NUMBER SERIES: A sequence of numbers with some pattern. One number follows another in some defined manner. NUMERATOR: The top symbol or number in a fraction. ODD NUMBER: An integer that is not divisible by 2. OPERATION: Multiplication, addition, subtraction, or division. ORDER OF OPERATIONS: The priority given to an operation relative to other operations. For example, multiplication takes precedence (is performed before) addition. PARENTHESES: Grouping symbols. ( ) PERCENT OR PERCENTAGE: A common fraction with 100 as its denominator (for example, 37% is 37 ). 100 PIE CHART (or circle graph): Displaying data on a circular graph by dividing the circle into sections. PLACE VALUE: The value given to a digit by the position of that digit in a number. For example, in the number 37, 3 is in the tens place (meaning 3 tens). POSITIVE CORRELATION: In a scatterplot, when one set of data increases while another also increases. POSITIVE NUMBER: A number greater than 0. POWER: A product of equal factors. 4 × 4 × 4 = 43, read “four to the third power” or “the third power of four.” Power and exponent are sometimes used interchangeably. PRIME NUMBER: A number that can be divided by only itself and one. A number that has exactly two factors (one and itself), for example, 2, 3, 5, 7, and so on. 0 and 1 are not prime numbers. PROBABILITY: The numerical measure of the chance of an outcome or event occurring. PRODUCT: The result of multiplication. PROPER FRACTION: A fraction in which the numerator is less than the denominator (for example, 2 ). 3 5 PROPORTION: An expression written as two equal ratios (for example, 5 is to 4 as 10 is to 8, or = 10 ). 4 8 QUOTIENT: The result of division. RANDOM: When all possible choices have an equal probability of being selected.
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CliffsTestPrep California High School Exit Exam: Math
RANGE: The difference between the largest and the smallest number in a set of numbers. RATIO: A comparison between two numbers or symbols. It can be written x:y, xy , or x is to y. For example, 1:2 is the same as 1 , which is the same as 1 is to 2. 2 RATIONAL NUMBER: An integer or fraction such as 7 , 9 , or 5 . Any number that can be written as a fraction 2 8 4 1 3 (where x is an integer and y is a natural number). REAL NUMBER: Any rational or irrational number. RECIPROCAL: The multiplicative inverse of a number. For example, 2 is the reciprocal of 3 . 3 2 2 1 REDUCING: Changing a fraction into its lowest terms. For example, can be reduced to . 4 2 ROUNDING OFF: Changing a number to a nearest place value as specified; it is a method of approximating. For example, 56 rounded off to the nearest ten is 60. SCATTERPLOT: Plotting points on an x-y graph to find a correlation between data. SCIENTIFIC NOTATION: A number equal to or greater than 1 and less than 10 that is multiplied by a power of 10. It is used for writing very large or very small numbers (for example, 2.5 × 104). SIMPLE INTEREST: Interest calculated by multiplying an amount of money by an interest rate and an amount of time. SQUARE: The result when a number is multiplied by itself. For example, 16 is a square number because 4 × 4 = 16. SQUARE ROOT: The number that is multiplied by itself to get the resulting square number. For example, 5 is the square root of 25 because 5 × 5 = 25. Its symbol is . 25 = 5 . SUM: The result of addition. TENTH: The first decimal place to the right of the decimal point. For example, .7 is seven tenths. WEIGHTED MEAN: The mean of a set of numbers that has been weighted (that is, multiplied by the relative importance or frequency of occurrence of the numbers). WHOLE NUMBERS: 0, 1, 2, 3, and so on. ZERO CORRELATION: In a scatterplot, when no apparent pattern or relationship exists between data.
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Algebra Glossary of Terms ABSCISSA: The distance along the horizontal axis in a coordinate graph. ABSOLUTE VALUE: The numerical value when direction or sign is not considered. The symbol for absolute value is | |. ALGEBRA: Arithmetic operations using letters and/or symbols in place of numbers. ALGEBRAIC FRACTIONS: Fractions using a variable in the numerator and/or denominator. ASCENDING ORDER: Basically, when working with polynomials, the power of a term increases for each succeeding term (for example, x2 + x3 + x4). BINOMIAL: An algebraic expression consisting of two terms (for example, x − 3 or 2x + 3y). CARTESIAN COORDINATES: A system of assigning ordered number pairs to points on a plane. COEFFICIENT: The number in front of a variable. For example, in 9x, 9 is the coefficient. COORDINATE AXES: Two perpendicular number lines used in a coordinate graph. COORDINATE GRAPH: Two perpendicular number lines, the x-axis and the y-axis, creating a plane on which each point is assigned a pair of numbers. These numbers are referred to as ordered pairs. CUBE: The result when a number is multiplied by itself twice. This is designated by the exponent 3, as in x3. For example, 2 × 2 × 2 = 23, and y × y × y = y3. CUBE ROOT: The number that is multiplied by itself twice to get the resulting cubed number. For example, 5 is the cube root of 125 because 5 × 5 × 5 = 125. Its symbol is 3 . 3 125 = 5 . DENOMINATOR: Everything below the fraction bar in a fraction. DESCENDING ORDER: Basically, when working with polynomials, the power of a term decreases for each succeeding term. For example, x4 + x 3 + x2. EQUATION: A balanced relationship between numbers and/or symbols, that is, a mathematical sentence. EQUIVALENT EXPRESSIONS: Numerical expressions that have the same value, that is, variable expressions (expressions containing variables) that have the same values for every value of the variable. EVALUATE: To determine the value or numerical amount. EXPONENT: A numeral or symbol used to indicate the power of a number. EXPRESSION: A number, a variable, or a combination of numbers, variables, and symbols (for example, 3x2, 2x + 3y). EXTREMES: Outer terms. For example, in the multiplication problem (x + 3)(3x + 2), the extremes would be x and 2. FACTOR: To find two or more quantities whose product equals the original quantity. For example, factoring 4x + 2 gives 2(2x + 1). A number of forms of factoring exist in Algebra. FINITE: Countable; having a definite ending. F.O.I.L. METHOD: A method of multiplying binomials in which first terms, outside terms, inside terms, and last terms are multiplied. IMAGINARY NUMBERS: Square roots of negative numbers. The imaginary unit is i.
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CliffsTestPrep California High School Exit Exam: Math
INEQUALITY: A statement in which the relationships are not equal; the opposite of an equation. For example, 3x + 2 > 4 is an inequality. INFINITE: Uncountable; continues forever. LINEAR EQUATION: An equation whose solution set forms a straight line when plotted on a coordinate graph. LITERAL EQUATION: An equation having mostly variables. MEANS: Inner terms. For example, in the multiplication problem (x + 3)(3x + 2), the means are 3x and 3. MONOMIAL: An algebraic expression consisting of only one term (for example, x2, 3x, or 2xy2). NONLINEAR EQUATION: An equation whose solution set does not form a straight line when plotted on a coordinate graph. NUMBER LINE: A graphic representation of integers and real numbers. The point on this line associated with each number is called the graph of the number. NUMERATOR: Everything above the fraction bar in a fraction. ORDERED PAIR: Any pair of elements (x, y) having a first element x and a second element y. This is used to identify or plot points on a coordinate graph. ORDINATE: The distance along the vertical axis on a coordinate graph. ORIGIN: The point of intersection of the two number lines on a coordinate graph. This is represented by the coordinates (0,0). POLYNOMIAL: An algebraic expression consisting of two or more terms (for example, 2x + 3, 3xy + 2x + 4, or x2 + 5x − 1). PROPORTION: Two ratios equal to each other. For example, a is to c as b is to d. QUADRANTS: The four quarters or divisions of a coordinate graph. The quadrants are numbered I, II, III, and IV; starting in the upper right quarter and moving counterclockwise. QUADRATIC EQUATION: An equation that can be written in the form Ax2 + Bx + C = 0. RADICAL SIGN: The symbol used to designate a square root. RATIO: A method of comparing two or more numbers. For example, a:b. It is often written as a fraction. REAL NUMBERS: The set consisting of all rational and irrational numbers. SIMPLIFY: To combine several (or many) terms into fewer terms. SLOPE OF A LINE: On a graph, the ratio of the horizontal change in a line, calculated as the rise over the run. The change in y over the change in x. Slope = rise run . SOLUTION SET (or solution): All the answers that satisfy an equation. SQUARE: The result when a number is multiplied by itself. It is designated by the exponent 2 (for example, x2). SQUARE ROOT: The number that is multiplied by itself to get the resulting squared number. For example, 5 is the square root of 25 because 5 × 5 = 25. Its symbol is . 25 = 5 . SYSTEM OF EQUATIONS: Simultaneous equations. Two equations that when plotted on a coordinate graph either intersect (giving an ordered pair that is a solution) or are parallel (having no solution).
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Algebra Glossary of Terms
TERM: A numerical or literal expression with its own sign. TRINOMIAL: An algebraic expression consisting of three terms (for example, x2 + 2x + 1 or 3x + 2y − 4). UNKNOWN: A letter or symbol whose value is not known. VALUE: A numerical amount. VARIABLE: A symbol used to stand for a number. X-AXIS: The horizontal axis in a coordinate graph. X-COORDINATE: The first number in an ordered pair. It refers to the distance on the x-axis (abscissa). X-INTERCEPT: The value of x in an ordered pair where the graph of the line crosses the x-axis. In an ordered pair, if y is 0, x is the x-intercept. For example, (3,0) indicates an x-intercept of 3. Y-AXIS: The vertical axis in a coordinate graph. Y-COORDINATE: The second number in an ordered pair. It refers to the distance on the y-axis (ordinate). Y-INTERCEPT: The value of y in an ordered pair where the graph of the line crosses the y-axis. In an ordered pair, if x is 0, y is the y-intercept. For example, (0,3) indicates a y-intercept of 3.
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Measurement and Geometry Glossary of Terms ALTITUDE OF A TRIANGLE (height of a triangle): The perpendicular line segment from a vertex (point) to the opposite side (or an extension of the opposite side). ARC OF A CIRCLE: A connected portion of a circle. AREA: A measure of the interior of a flat (two-dimensional) figure. It is expressed in square units such as square inches (in2) or square centimeters (cm2) or in special units such as acres. BASE OF A TRIANGLE: Any side of a triangle. BASES OF A PARALLELOGRAM: Each pair of parallel sides in a parallelogram can serve as bases. BASES OF A TRAPEZOID: The parallel sides of a trapezoid. CENTER OF A CIRCLE: The (fixed) interior point that is equidistant from all points on a circle. CIRCLE: A flat figure with all its points equidistant from a fixed point (the center). CIRCUMFERENCE: The distance around a circle. CONGRUENT TRIANGLES (or figures): Triangles (or figures) that have exactly the same size and shape. COORDINATES OF A POINT: The ordered pair of numbers assigned to a point in a plane. COORDINATE PLANE: The x-axis, the y-axis, and all the points in the plane they determine. CORRESPONDING PARTS OF TRIANGLES: The parts of two (usually) congruent or similar triangles that are in the same relative position. CUBE: A six-sided solid where all sides are equal squares, and all edges are equal. CYLINDER: A prism-like solid whose bases are circles. DECAGON: A 10-sided polygon. DEGREE MEASURE OF A SEMICIRCLE: 180°. DIAGONAL OF A POLYGON: Any line segment that joins two nonconsecutive vertices of a polygon. DIAMETER OF A CIRCLE: A line segment that passes through the center of a circle, starting and ending on the circle. EXTREMES OF A PROPORTION: When a proportion is written in the form a:b = c:d, a and d are referred to as extremes of the proportion. GRAPH OF AN ORDERED PAIR: The point (in a coordinate plane) associated with an ordered pair of real numbers. HEIGHT OF A PARALLELOGRAM (or trapezoid): Any perpendicular segment connecting two bases of a parallelogram (or trapezoid). HEIGHT OF A TRIANGLE (altitude of a triangle): The perpendicular line segment from a vertex (point) to the opposite side (or an extension of the opposite side) of a triangle. HEPTAGON: A seven-sided polygon. HEXAGON: A six-sided polygon.
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CliffsTestPrep California High School Exit Exam: Math
HYPOTENUSE: The side opposite the right angle in a right triangle. INTERSECTING LINES: Two or more lines that meet at a single point. LEGS OF A RIGHT TRIANGLE: The two sides other than the hypotenuse in a triangle. LEGS OF A TRAPEZOID: The nonparallel sides of a trapezoid. LINEAR EQUATION: An equation whose graph is a straight line. MEANS OF A PROPORTION: When a proportion is written in the form a:b = c:d, b and c are referred to as means of the proportion. MIDPOINT OF A LINE SEGMENT: The point on a line segment that is equidistant from the endpoints; the halfway point. NONAGON: A nine-sided polygon. OCTAGAN: An eight-sided polygon. ORDERED PAIR: A pair of numbers whose order is important; these are used to locate points in a plane. ORIGIN: In two dimensions, the point (0,0); it is the intersection of the x-axis and the y-axis. PARALLEL LINES: Two lines that lie in the same plane and never intersect. PARALLELOGRAM: Any quadrilateral with both pairs of opposite sides parallel. PENTAGON: A five-sided polygon. PERIMETER: The distance around a figure. PERPENDICULAR LINES: Lines that intersect and form right angles. POINT-SLOPE FORM OF THE EQUATION OF A LINE: The form y − y1 = m (x − x1), where m is the slope of a line, and (x, y) (x1, y1) are specific points on that line. POLYGON: A plane closed figure with three or more sides. PROPORTION: An equation stating that two ratios are equal. PYTHAGOREAN THEOREM: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle: a2 + b2 = c2 where c is the longest side (the hypotenuse). PYTHAGOREAN TRIPLE: Three positive integers (a, b, c) that satisfy the equation a2 + b2 = c2 (for example, 3, 4, 5 or 5, 12, 13). QUADRANTS: The four regions of a coordinate plane separated by the x-axis and the y-axis. QUADRILATERAL: A four-sided polygon. RADIUS OF A CIRCLE: A line segment with the center of a circle and a point on that circle as endpoints (plural: radii). RATIO OF TWO NUMBERS a AND b: The fraction a/b, usually expressed in simplest form; also denoted a:b. RECTANGLE: A quadrilateral in which all the angles are right angles. Opposite sides are parallel and equal. REFLECTION: Transforming each point in a plane by mapping it to its mirror image across a line. When the x-coordinate of each point is changed to its opposite sign, the image is reflected across the y-axis. When the y-coordinate of each point is changed to its opposite sign, the image is reflected across the x-axis.
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Measurement and Geometry Glossary of Terms
RHOMBUS: A quadrilateral with all four sides equal. RIGHT ANGLE: A 90° angle. RIGHT CIRCULAR CYLINDER: A cylinder with the property that the segment joining the centers of the circular bases is perpendicular to the planes of the bases. SECTOR OF A CIRCLE: A region bounded by two radii and an arc of the circle intercepted by (an angle formed by) those two radii. SEMICIRCLE: An arc whose endpoints are the endpoints of a diameter of the circle. A semicircle measures 180°. SIMILAR POLYGONS: Polygons with the same shape; all their corresponding angles have the same measure. SLOPE-INTERCEPT FORM OF A LINE: The form y = mx + b where m is the slope of the line, and b is the y-intercept. SLOPE OF A LINE: On a graph, the ratio of the horizontal change in a line, calculated as the rise over the run. The change in y over the change in x. Slope = rise run . SQUARE: A quadrilateral in which all the angles are right angles, and all the sides are equal. STANDARD FORM FOR THE EQUATION OF A LINE: The form Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero. TRANSLATION: Transforming each point in a plane by mapping it to a different location by sliding the image a specific distance and direction. TRAPEZOID: A quadrilateral with only one pair of opposite sides parallel. TRIANGLE: A three-sided plane figure (polygon). VERTEX: An endpoint of a side of a polygon. VOLUME: The measure of the interior of a solid; the number of unit cubes necessary to fill the interior of such a solid. X-AXIS: A horizontal line used to help locate points in a plane, also called the abscissa. X-COORDINATE: The first term of an ordered pair; it appears to the left of the comma and indicates movement along the x-axis. X-INTERCEPT: The point where a line crosses or intersects the x-axis. Y-AXIS: A vertical line used to help locate points in a plane, also called the ordinate. Y-COORDINATE: The second term of an ordered pair; it appears to the right of the comma and indicates movement along the y-axis. Y-INTERCEPT: The point where a line crosses or intersects the y-axis.
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Final Preparations Final Preparation and Sources Finishing Touches 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Make sure that you are familiar with the areas covered on the test. Spend the last week of preparation on a general review of key concepts and strengthening weak areas. Don’t cram the night before the exam. It is a waste of time! Start off crisply, working the questions you know first, then going back and trying to answer the others. Try to eliminate one or more choices before you guess, but make sure that you fill in all the answers. There is no penalty for guessing! Underline key words in questions. Write out important information, and make notations on diagrams. Take advantage of being permitted to write in the test booklet. Make sure that you answer what is being asked and that your answer is reasonable. Cross out incorrect choices immediately: This can keep you from reconsidering a choice that you have already eliminated. Don’t get stuck on any one question. They are all of equal value. The key to getting a good score on CAHSEE Math is reviewing properly, practicing, and getting the questions right that you can and should get right. A careful review of Parts I and II of this book helps you focus during the final week before the exam.
Sources If you need additional review or practice, the following books can be very helpful: CliffsQuickReview Algebra I CliffsQuickReview Basic Math and Pre-Algebra Cliffs Math Review for Standardized Tests
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