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Act Mathematics ACT MATHEMATICS Improving College Admission Test Scores Contributing Writers Marie Haisan L. Ramadeen Matthew Miktus David Hoffman ACT is a registered trademark of ACT Inc. Copyright 2004 by Instructivision, Inc., revised 2006, 2009, 2011, 2014 ISBN 973-156749-774-8 Printed in Canada. All rights reserved. No part of the material protected by this copyright may be reproduced in any form or by any means, for commercial or educational use, without permission in writing from the copyright owner. Requests for permission to make copies of any part of the work should be mailed to Copyright Permissions, Instructivision, Inc., P.O. Box 2004, Pine Brook, NJ 07058. Instructivision, Inc., P.O. Box 2004, Pine Brook, NJ 07058 Telephone 973-575-9992 or 888-551-5144; fax 973-575-9134, website: www.instructivision.com ii TABLE OF CONTENTS Introduction iv Glossary of Terms vi Summary of Formulas, Properties, and Laws xvi Practice Test A 1 Practice Test B 16 Practice Test C 33 Pre Algebra Skill Builder One 51 Skill Builder Two 57 Skill Builder Three 65 Elementary Algebra Skill Builder Four 71 Skill Builder Five 77 Skill Builder Six 84 Intermediate Algebra Skill Builder Seven 88 Skill Builder Eight 97 Coordinate Geometry Skill Builder Nine 105 Skill Builder Ten 112 Plane Geometry Skill Builder Eleven 123 Skill Builder Twelve 133 Skill Builder Thirteen 145 Trigonometry Skill Builder Fourteen 158 Answer Forms 165 iii INTRODUCTION The American College Testing Program Glossary: The glossary defines commonly used (ACT) is a comprehensive system of data mathematical expressions and many special and collection, processing, and reporting designed to technical words. assist students in the transition from high school to college. The academic tests in English, mathe- Formulas: Formulas that are commonly applied to matics, reading, and science reasoning emphasize mathematical problems are listed in a separate reasoning and problem-solving skills. The test section. This section can be used as a convenient items represent scholastic tasks required to reference for formulas relating to geometric shapes perform college level work. and algebraic functions. ACT questions are designed to measure a wide range of abilities and knowledge. Practice Tests: There are three full-length practice Consequently, some of the items are difficult while tests. Under actual testing conditions, you are others are fairly easy. A background of strong allowed 60 minutes for the entire test. The academic courses combined with a worthwhile instructions should be followed carefully. review will enable you to meet this challenge Skill Builders: The skill builders describe and successfully. illustrate each of the content areas in the Mathematics Test. The skill builders are divided The Mathematics Test into sections, each of which relates to one of the The Mathematics Test is a 60-question, 60- principal categories covered in the test. Each skill minute examination that measures mathematics builder consists of a series of examples, orientation reasoning abilities. The test focuses on the exercises, practice exercises, and a practice test. solution of practical quantitative problems that are encountered in high school and some college The answers to the sample tests and the skill courses. The test uses a work-sample approach builder exercises and practice tests are not found that measures mathematical skills in the context of in the Student Workbook. They are included in the simple and realistic situations. Each of the Teacher Manual. multiple-choice questions has five alternative responses. Examine the choices, and select the How the ACT is Scored correct response. The “raw” score of 1 point for each correct Three subscores based on six content areas are answer will be converted to a “scale” score. The classified in the Mathematics Test (see chart, page scale on which ACT academic test scores are v). The 60 test questions reflect an appropriate reported is 1-36, with a mean (or average) of 18, balance of content and skills (low, middle, and based on a nationally representative sample of high difficulty) and range of performance. October-tested 12th grade students who plan to Because there is no penalty for guessing, answer enter two-year or four-year colleges or every question. There are no trick questions; In universities. The scale for each subscore is 1-18, some problems, you may have to go through a with a mean of 9. A guidance counselor will be number of steps in order to find the correct answer. glad to answer questions regarding the scoring In order to perform efficiently and accurately process and the score reports. throughout the examination, you must understand and apply fundamental mathematical concepts. Math Strategies Spending too much time on any one item is 1. Answer all questions. First do those problems unwise. On the average, spend about one minute with which you are most familiar and which on each question. Any remaining time should be seem the easiest to solve, and then answer spent in completing unanswered questions or those you find more difficult. reviewing previous work. 2. Practice pacing yourself. Try to solve most of the problems in less than one minute each. How to Use the Mathematics Workbook 3. Pay close attention to the information in each This workbook consists of the introduction, a problem. Use the information that is important glossary of terms, formulas, three practice tests, in solving the problem. skill builders, and additional questions for review. 4. If you are making an educated guess, try to eliminate any choices that seem unreasonable. iv 5. If the item asks for an equation, check to see if ACT Assessment Mathematics Test your equation can be transformed into one of 60 items, 60 minutes the choices. _____________________________________ 6. Always work in similar units of measure. Proportion Number 7. Sketch a diagram for reference when feasible. Content Area of Test of Items 8. Sometimes there is more than one way to solve a problem. Use the method that is most Pre-Algebra/ comfortable for you. Elementary Algebra .40 24 9. Use your estimation skills to make educated Intermediate Algebra/ guesses. Coordinate Geometry .30 18 10. Check your work. Plane Geometry/ Trigonometry .30 18 Items are classified according to six content _____________________________________ areas. The categories and the approximate Total 1.00 60 proportion of the test devoted to each are 1. Pre-Algebra. Items in this category are based Scores reported: on operations with whole numbers, decimals, fractions, and integers. They also may require Pre-Algebra/Elementary Algebra (24 items) the solution of linear equations in one Intermediate Algebra/Coordinate Geometry variable. (18 items) 2. Elementary Algebra. Items in this category Plane Geometry/Trigonometry (18 items) are based on operations with algebraic expressions. The most advanced topic in this Total possible maximum raw test score (60 category is the solution of quadratic equations items) is 60. Because the formula for by factoring. calculating the final score varies slightly each 3. Intermediate Algebra. Items in this category year, we have not included this information are based on an understanding of the quadratic here. formula, rational and radical expressions, absolute value equations and inequalities, sequences and patterns, systems of equations, quadratic inequalities, functions, modeling, matrices, roots of polynomials, and complex numbers. 4. Coordinate Geometry. Items in this category are based on graphing and the relations between equations and graphs, including points, lines, polynomials, circles, and other curves; graphing inequalities; slope; parallel and perpendicular lines; distance; midpoints; and conics. 5. Plane Geometry. Items in this category are based on the properties and relations of plane figures. 6. Trigonometry. Items in this category are based on right triangle trigonometry, graphs of the trigonometric functions, and basic trigono- metric identities. v GLOSSARY OF TERMS ABSCISSA triangle, the altitude meets the base at a point on its An ordered pair (x, y) specifying the distance of extension. points from two perpendicular number lines (x and y- axis). E.g., in (4, 6) the first number—the x number ANGLE (4)—is called the abscissa. The second number— A figure formed by two rays that have the same the y number (6)—is called the ordinate. endpoint. The rays are the sides of the angle. The endpoint of each ray is called the vertex. ABSOLUTE VALUE The absolute value of a number x, written |x|, is the ARC number without its sign; e.g., |+8| = 8, |0| = 0, or |-4| A segment or piece of a curve. = 4. On a number line it can be interpreted as the distance from zero, regardless of direction. AREA The measure of a surface; e.g., number of square ACUTE ANGLE units contained within a region. Area of a rectangle An angle whose measure is less than 90 degrees. = length times width. ACUTE TRIANGLE ASSOCIATION A triangle whose three angles each measure less than A special grouping of numbers to make computation 90 degrees. easier; e.g., 245 × (5 × 2) = 245 × 10 = 2,450 instead of (245 × 5) × 2 = 1,225 × 2 = 2,450. ADDITIVE INVERSE The additive inverse of a number a is the number -a ASSOCIATIVE LAW for which a + (-a) = 0. You can think of the additive of addition: The way numbers are grouped does not inverse of a number as its opposite; e.g., the additive affect the sum; e.g., inverse of -5 is +5 because (-5) + (+5) = 0. a + (b + c) = (a + b) + c 5 + (6 + 3) = (5 + 6) + 3 ADJACENT ANGLES Two angles having a common vertex and a common 5 + 9 =11 + 3 side between them. 14 =14 of multiplication: The way numbers are ALGORYTHM grouped does not affect the product; e.g., A finite set of instructions having the following a(bc) = (ab)c characteristics: - Precision. The steps are precisely stated.
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