ACT MATHEMATICSImproving College Admission Test Scores

Contributing WritersMarie HaisanL. RamadeenMatthew MiktusDavid HoffmanACT is a registered trademark of ACT Inc.Copyright 2004 by Instructivision, Inc., revised 2006, 2009, 2011, 2014ISBN 973-156749-774-8Printed in Canada.All rights reserved. No part of the material protected by this copyright may be reproduced inany form or by any means, for commercial or educational use, without permission in writingfrom the copyright owner. Requests for permission to make copies of any part of the workshould be mailed to Copyright Permissions, Instructivision, Inc., P.O. Box 2004, Pine Brook,NJ 07058.Instructivision, Inc., P.O. Box 2004, Pine Brook, NJ 07058Telephone 973-575-9992 or 888-551-5144; fax 973-575-9134, website: www.instructivision.comii

TABLE OF CONTENTSIntroductionivGlossary of TermsviSummary of Formulas, Properties, and LawsxviPractice Test A1Practice Test B16Practice Test C33Pre AlgebraSkill Builder One51Skill Builder Two57Skill Builder Three65Elementary AlgebraSkill Builder Four71Skill Builder Five77Skill Builder Six84Intermediate AlgebraSkill Builder Seven88Skill Builder Eight97Coordinate GeometrySkill Builder Nine105Skill Builder Ten112Plane GeometrySkill Builder Eleven123Skill Builder Twelve133Skill Builder Thirteen145TrigonometrySkill Builder FourteenAnswer Forms158165iii

INTRODUCTIONThe American College Testing Program(ACT) is a comprehensive system of datacollection, processing, and reporting designed toassist students in the transition from high school tocollege. The academic tests in English, mathematics, reading, and science reasoning emphasizereasoning and problem-solving skills. The testitems represent scholastic tasks required toperform college level work.ACT questions are designed to measure awide range of abilities and knowledge.Consequently, some of the items are difficult whileothers are fairly easy. A background of strongacademic courses combined with a worthwhilereview will enable you to meet this challengesuccessfully.The Mathematics TestThe Mathematics Test is a 60-question, 60minute examination that measures mathematicsreasoning abilities. The test focuses on thesolution of practical quantitative problems that areencountered in high school and some collegecourses. The test uses a work-sample approachthat measures mathematical skills in the context ofsimple and realistic situations. Each of themultiple-choice questions has five alternativeresponses. Examine the choices, and select thecorrect response.Three subscores based on six content areas areclassified in the Mathematics Test (see chart, pagev). The 60 test questions reflect an appropriatebalance of content and skills (low, middle, andhigh difficulty) and range of performance.Because there is no penalty for guessing, answerevery question. There are no trick questions; Insome problems, you may have to go through anumber of steps in order to find the correct answer.In order to perform efficiently and accuratelythroughout the examination, you must understandand apply fundamental mathematical concepts.Spending too much time on any one item isunwise. On the average, spend about one minuteon each question. Any remaining time should bespent in completing unanswered questions orreviewing previous work.How to Use the Mathematics WorkbookThis workbook consists of the introduction, aglossary of terms, formulas, three practice tests,skill builders, and additional questions for review.ivGlossary: The glossary defines commonly usedmathematical expressions and many special andtechnical words.Formulas: Formulas that are commonly applied tomathematical problems are listed in a separatesection. This section can be used as a convenientreference for formulas relating to geometric shapesand algebraic functions.Practice Tests: There are three full-length practicetests. Under actual testing conditions, you areallowed 60 minutes for the entire test. Theinstructions should be followed carefully.Skill Builders: The skill builders describe andillustrate each of the content areas in theMathematics Test. The skill builders are dividedinto sections, each of which relates to one of theprincipal categories covered in the test. Each skillbuilder consists of a series of examples, orientationexercises, practice exercises, and a practice test.The answers to the sample tests and the skillbuilder exercises and practice tests are not foundin the Student Workbook. They are included in theTeacher Manual.How the ACT is ScoredThe “raw” score of 1 point for each correctanswer will be converted to a “scale” score. Thescale on which ACT academic test scores arereported is 1-36, with a mean (or average) of 18,based on a nationally representative sample ofOctober-tested 12th grade students who plan toenter two-year or four-year colleges oruniversities. The scale for each subscore is 1-18,with a mean of 9. A guidance counselor will beglad to answer questions regarding the scoringprocess and the score reports.Math Strategies1. Answer all questions. First do those problemswith which you are most familiar and whichseem the easiest to solve, and then answerthose you find more difficult.2. Practice pacing yourself. Try to solve most ofthe problems in less than one minute each.3. Pay close attention to the information in eachproblem. Use the information that is importantin solving the problem.4. If you are making an educated guess, try toeliminate any choices that seem unreasonable.

5. If the item asks for an equation, check to see ifyour equation can be transformed into one ofthe choices.6. Always work in similar units of measure.7. Sketch a diagram for reference when feasible.8. Sometimes there is more than one way tosolve a problem. Use the method that is mostcomfortable for you.9. Use your estimation skills to make educatedguesses.10. Check your work.Items are classified according to six contentareas. The categories and the approximateproportion of the test devoted to each are1. Items in this category are basedon operations with whole numbers, decimals,fractions, and integers. They also may requirethe solution of linear equations in onevariable.Elementary Algebra. Items in this categoryare based on operations with algebraicexpressions. The most advanced topic in thiscategory is the solution of quadratic equationsby factoring.Intermediate Algebra. Items in this categoryare based on an understanding of the quadraticformula, rational and radical expressions,absolute value equations and inequalities,sequences and patterns, systems of equations,quadratic inequalities, functions, modeling,matrices, roots of polynomials, and complexnumbers.Coordinate Geometry. Items in this categoryare based on graphing and the relationsbetween equations and graphs, includingpoints, lines, polynomials, circles, and othercurves; graphing inequalities; slope; paralleland perpendicular lines; distance; midpoints;and conics.Plane Geometry. Items in this category arebased on the properties and relations of planefigures.Trigonometry. Items in this category are basedon right triangle trigonometry, graphs of thetrigonometric functions, and basic trigonometric identities.ACT Assessment Mathematics Test60 items, 60 minutesProportion NumberContent Areaof Testof ItemsPre-Algebra/Elementary Algebra.4024Intermediate Algebra/Coordinate Geometry.3018Plane Geometry/Trigonometry.3018Total1.0060Scores reported:Pre-Algebra/Elementary Algebra (24 items)Intermediate Algebra/Coordinate Geometry(18 items)Plane Geometry/Trigonometry (18 items)Total possible maximum raw test score (60items) is 60. Because the formula forcalculating the final score varies slightly eachyear, we have not included this informationhere.v

GLOSSARY OF TERMSABSCISSAAn ordered pair (x, y) specifying the distance ofpoints from two perpendicular number lines (x and yaxis). E.g., in (4, 6) the first number—the x number(4)—is called the abscissa. The second number—the y number (6)—is called the ordinate.ABSOLUTE VALUEThe absolute value of a number x, written x , is thenumber without its sign; e.g., 8 8, 0 0, or -4 4. On a number line it can be interpreted as thedistance from zero, regardless of direction.ACUTE ANGLEAn angle whose measure is less than 90 degrees.ACUTE TRIANGLEA triangle whose three angles each measure less than90 degrees.ADDITIVE INVERSEThe additive inverse of a number a is the number -afor which a (-a) 0. You can think of the additiveinverse of a number as its opposite; e.g., the additiveinverse of -5 is 5 because (-5) ( 5) 0.ADJACENT ANGLESTwo angles having a common vertex and a commonside between them.ALGORYTHMA finite set of instructions having the followingcharacteristics:- Precision. The steps are precisely stated.- Uniqueness. The intermediate results of each stepof execution are uniquely defined and depend onlyon the inputs and the results of the preceding steps.- Finiteness. The algorithm stops after finitely manyinstructions have been executed.- Input. The algorithm receives input.- Output. The algorithm produces output.- Generality. The algorithm applies to a set of inputs.ALTERNATE INTERIOR ANGLESTwo angles formed by a line (the transversal) thatcuts two parallel lines. The angles are interior angleson opposite sides of the transversal and do not havethe same vertex.ALTITUDE of a triangleA line segment drawn from a vertex pointperpendicular to the opposite side (base); the lengthis referred to as the height of the triangle. In a righttriangle, the altitude is one of the legs. In an obtusevitriangle, the altitude meets the base at a point on itsextension.ANGLEA figure formed by two rays that have the sameendpoint. The rays are the sides of the angle. Theendpoint of each ray is called the vertex.ARCA segment or piece of a curve.AREAThe measure of a surface; e.g., number of squareunits contained within a region. Area of a rectangle length times width.ASSOCIATIONA special grouping of numbers to make computationeasier; e.g., 245 (5 2) 245 10 2,450 insteadof (245 5) 2 1,225 2 2,450.ASSOCIATIVE LAWof addition: The way numbers are grouped does notaffect the sum; e.g.,a (b c) (a b) c5 (6 3) (5 6) 35 9 11 314 14of multiplication: The way numbers aregrouped does not affect the product; e.g.,a (bc) (ab)c3( 4 5) (3 4)53( 20) (12)560 60AVERAGEThe average of a group of numbers is found byadding all the quantities being averaged and thendividing by the number of quantities being averaged;e.g., 60, 70, 80, and 90.Average 60 70 80 90 300 7544AXES GRAPHINGTwo perpendicular lines used as a reference forordered pairs.Vertical AxisHorizontal Axis

BASE of a powerThe number to which an exponent is attached. In theexpression x3, x is the base, 3 is the exponent.BASE of a triangleThe side of a triangle to which the altitude is drawn.BASE ANGLES of a triangleThe two angles that have the base of the triangle as acommon side.BINOMIALAn algebraic expression consisting of two terms: 3x 5y is a binomial.BISECTTo divide in half.Bisect an angle: to draw a line through the vertexdividing the angle into two equal angles.Bisect a line segment: to divide the line into twoequal line segments.CENTER of a circleThe fixed point in a plane about which a curve isequally distant. The center of a circle is the pointfrom which every point on the circumference isequidistant.CENTRAL ANGLEIn a circle, an angle whose vertex is the center andwhose sides are radii.CHORDA chord of a circle is a line segment joining any twopoints on the circle.CIRCLEThe set of points in a plane at a given distance (theradius) from a fixed point in the plane (called thecenter).CIRCUMFERENCEThe distance around a circle.CIRCUMSCRIBEDTo draw a line around a figure; e.g., a circlecircumscribed around a triangle is a circle that passesthrough each vertex of the triangle.COEFFICIENTA coefficient is the number before the letters in analgebraic term, in 3xyz, 3 is the coefficient.COMBINATIONThe arrangement of a number of objects into groups;e.g., A, B, and C into groups AB, AC, and BC.COMMON DENOMINATORA common denominator is a common multiple of thedenominators of the fractions.A common11andis 6 because 1 3 anddenominator for32 621 2 .3 6COMMUTATIVE LAWof addition: The order of the numbers does notaffect the sum; e.g.,a b b a8 3 3 811 11of multiplication: The order of the numbers doesnot affect the product; e.g.,ab ba(6)(8) (8)(6)48 48COMPLEMENTARY ANGLESTwo angles whose sum is a right angle (90 ).COMPOSITE NUMBERA composite number is a natural number that can bedivided by 1 or by some number other than itself. Acomposite number has factors other than itself and 1;e.g.,4 (4)(1) and (2)(2)6 (6)(1) and (3)(2)CONEA space figure with one flat face (known as a base)that is a circle and with one other face that is curved.CONGRUENTtriangles: two triangles that can be made to coincide(symbol ).lines: lines that are the same length.angles: angles that have the same measure indegrees.CONSECUTIVE INTEGERSNumbers that follow in order; e.g., 1, 2, 3, 4, 5, 6,etc. Even consecutive integers 2, 4, 6, 8, Oddconsecutive integers 1, 3, 5, 7, CONSECUTIVE INTERIOR ANGLESTwo angles of a polygon with a common side.vii

CONSTANTA symbol representing a single number during aparticular discussion; e.g., x2 x 5 has 5 as theconstant that does not vary in value.tenths.” Decimal points followed by two digits arehundredths: 0.05 is read “5 hundredths.” Decimalpoints followed by three digits are thousandths:0.123 is read “123 thousandths.”CONVERSIONTo change the units of an expression; e.g., convert 2hours and 3 minutes to 123 minutes.DEGREEof a term: with one variable is the exponent of thevariable; e.g., the term 2x4 is of the fourth degree.of an equation: with one variable is the value of thehighest exponent; e.g., 3x3 5x2 4x 2 0 is athird degree equation.COORDINATES OF A POINTAn ordered pair (x, y) specifying the distance ofp

scale on which ACT academic test scores are reported is 1-36, with a mean (or average) of 18, based on a nationally representative sample of October-tested 12th grade students who plan to enter two-year or four-year colleges or universities. The scale for each subscore is 1-18, with a mean of 9. A guidance counselor will be