Improving College Admission Test Scores - Miss Noble's Class Website

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ACT MATHEMATICSImproving College Admission Test Scores

Contributing WritersMarie HaisanL. RamadeenMatthew MiktusDavid HoffmanACT is a registered trademark of ACT Inc.Copyright2004 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

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 are1.2.3.4.5.6.Pre-Algebra. 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 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—they number (6)—is called the ordinate.CORRESPONDING ANGLESTwo angles formed by a line (the transversal) thatcuts two parallel lines. The angles, one exterior andone interior, are on the same side of the transversal.CORRESPONDING SIDESSides of similar figures that are proportional.COSINEThe cosine of an acute angle of a triangle is the ratioof the length of the side adjacent to the angle of thehypotenuse.CUBEA rectangular prism whose six faces are squares.CUBE of a numberThe third power of a number; e.g., the cube of 2,written 23, is 2 2 2 or 8.CUBICOf the third degree; cubic equation; e.g.,2x3 3x2 4 0CYLINDERA space figure that has two circular bases that are thesame size and are in parallel planes. It has onecurved face.DECAGONA polygon that has 10 sides.DECIMALAny number written in decimal notation (a decimalpoint followed by one or more digits). Decimalpoints followed by one digit are tenths: 0.8 is read “8viiiDEGREESA unit of measure of angles or temperatures; e.g.,there are 90 degrees in a right angle; today’stemperature is 48 degrees.DENOMINATORThe term below the line in a fraction; e.g., thedenominator of2is 3.3DEPENDENT EQUATIONSA system of equations in which every set of valuesthat satisfies one of the equations satisfies them all;e.g.,5x 8y 1010x 16y 20DEPENDENT VARIABLESA variable whose values are considered to bedetermined by the values of another variable; y 2x 3; if x 4 then y 11, but if x 1 then y 5.DESCENDING ORDERFrom highest to lowest; the algebraic expression 5x4 x3 – 2x2 3x – 1 is arranged in descending order ofpowers of x.DIAGONALThe line segment joining two non-adjacent verticesin a quadrilateral.DIAMETEROf a circle is a straight line passing through thecenter of the circle and terminating at two points onthe circumference.DIFFERENCEThe result of subtracting one quantity from another;320 is the difference between 354 and 34.DIRECTProof: Uses an argument that makes direct use ofthe hypotheses and arrives at a conclusion.Variation:A relationship determined by theequation y kx, where k is a constant.

DISTANCEThe length of the line joining two points or thelength of a perpendicular line joining two lines.Distance may be expressed in inches, feet, yards,miles, etc.DISTRIBUTIVE LAWFor any numbers replacing a, b, and c,a (b c ) ab ac2(3 5) 2(3) 2(5)2(8) 6 1016 16DIVIDENDA quantity being divided in a division problem; e.g.,30 5 6 (30 is the dividend).DIVISIBLEThe ability to be evenly divided by a number; e.g.,10 is divisible by 2 because 10 2 5.DIVISORThe quantity by which the dividend is being divided;e.g., 30 5 6 (5 is the divisor).22 2 1 4 2 1 7EVEN NUMBERAn integer that is divisible by 2. All even numberscan be written in the form 2n, where n is any integer.EXCLUSIONThe act of leaving something out; e.g., write the setof all even numbers between 1 and 11. The solutionset is {2, 4, 6, 8, 10}; the odd numbers from 1 to 11are excluded from the solution set.EXPONENTA number placed at the right of and above a symbol.The number indicates how many times this symbol isused as a factor; e.g., in x3, 3 is the exponentindicating that x is used as a factor three times. x3 (x)(x)(x).EXTERIOR ANGLEOf a triangle is an angle formed by the one side of atriangle and the extension of the adjacent side.DOMAINThe defined set of values the independent variable isassigned; e.g., in y x 5, x is the independentvariable. If x {0, 1} is the domain, then y {5, 6}.FACTORIALFor a positive integer n, the product of all thepositive integers less than or equal to n. Factorial nis written n!1! 12! (1)(2)3! (1)(2)(3)EQUATIONA statement of equality between two expressions;e.g., 3 x 8. The left-hand member 3 x isequivalent to the right-hand member 8.Literal equation: An equation containing variablesas its terms.Fractional equation: An equation with at least oneterm being a fraction.Radical equation: An equation with at least oneterm being a square root.FACTORINGThe process of finding factors of a product. Types:(a) greatest common factor2x2 2xy 2x(x y)(b) difference between 2 squaresx2 – 25 (x – 5)(x 5)(c) factoring a trinomialx2 6x 5 (x 1)(x 5)(d) factoring completely5x2 – 5 5(x 2 –1) 5(x – 1)(x 1)EQUILATERALAll sides are the same measure; e.g., an equilateraltriangle contains three equal sides.FACTORSAny of a group of numbers that are multipliedtogether yielding the original given number; e.g., thepositive factors of 12 are:2 and 6 (2 6 12)3 and 4 (3 4 12)1 and 12 (1 12 12)EQUIVALENTEquations: Equations that have the same solutionset; e.g., the equation x 6 10 and 4x 16 areequivalent because 4 is the only solution for both.Expressions: Expressions that represent the samevalue for any variable involved; e.g., 3x 3y and 3(x y).EVALUATETo find the value of; e.g., to evaluate 3 2 4 meansto compute the result, which is 10; to evaluate x2 x 1 for x 2 means to replace x with 2; e.g.,FORMULAA special relationship between quantities expressedin symbolic form, an equation; e.g., area of arectangle is length times width. The formula is A lw.ix

FRACTIONSA fraction is part of a whole. It is written A . B isBthe denominator and tells how many parts the wholewas divided into. A is the numerator and tells thenumber of equal parts used; e.g., in 3 the whole is4divided into 4 parts with 3 of the 4 being used.GREATEST COMMON FACTOR (GCF)The greatest integer that is a factor of both integersbeing considered; e.g., the GCF of 5 and 20 is 5.HEXAGONA polygon that has six sides.HORIZONTALParallel to level ground.HUNDREDTHSA decimal point followed by two digits; e.g., .27 is27 hundredths and .09 is 9 hundredths. See decimalHYPOTENUSEThe side opposite the right angle in a right triangle.It is the longest side of the triangle.IDENTITYA statement of equality; any quantity is equal toitself; e.g.,4 4AB ABx 6 x 6INEQUALITYA statement that one quantity is less than (or greaterthan) another; not equal to ( ); e.g.,A B A is less than BA B A is greater than BA B A is not equal to BINSCRIBED ANGLEAn angle whose sides are chords of a circle andwhose vertex is a point on the circumference.INSCRIBED CIRCLEA circle within a polygon, the circle being tangent toevery side of the polygon.INTEGERAny of the counting numbers, their additive inverses,and 0; e.g.,{ -4, -3, -2, -1, 0, 1, 2, 3, 4, }INTERCEPTTo pass through a point on a line; x-intercept is thepoint on the x-axis where a line intersects it; yintercept is the point on the y-axis where a lineintersects it.INTERSECTIONof two lines: is the point where they meet.of two sets: consists of all the members that belongto both sets. The symbol used is “ ”; e.g.,Set A {2, 4, 6}Set B {2, 3, 4}A B {2, 4}Additive identity (0): a number that can be addedto any quantity without changing the value of thequantity.Multiplicative identity (1): a number that can bemultiplied times any quantity without changing thevalue of the quantity.INVERSESee additive inverse, multiplicative inverseVariation: When the product of two variables isconstant, one of them is said to vary inversely as theIMPROPER FRACTIONA fraction whose numerator is equal to or greater3 16 5than its denominator; e.g., ,, .3 7 4as x or x to vary inversely as y.INCONSISTENT EQUATIONSEquations that have no common solution set.Graphically they appear as parallel lines, since therewould be no intersecting point; e.g.,x y 8x y 4INDEPENDENT VARIABLEA variable considered free to assume any one of agiven set of values; e.g., in y 3x, x can be anyinteger, and y is the dependent variable.xother. If y cor xy c, y is said to vary inverselyxIRRATIONAL NUMBERAny real number that is not the quotient of twointegers; e.g.,2,7π .ISOSCELES TRAPEZOIDA trapezoid whose non-parallel sides are equal.ISOSCELES TRIANGLEA triangle with two equal sides.LEGSThe sides of a right triangle adjacent to the rightangle are called legs.

LIKE TERMSTerms whose variables (letters) are the same;e.g., 3x and 12x.LINE SEGMENTA part of a line that consists of two points on the line,called endpoints, and all the points between them.LINEAR EQUATIONAn equation of the first degree. The graph of a linearequation in two variables is a straight line.LITERAL EQUATIONAn equation containing variables as its terms.LOCUSThe set of all points, and only those points, thatsatisfy a given condition.LOWEST COMMON DENOMINATOR (LCD)The smallest natural number into which each of thedenominators of a given set of fractions divide13 2is 12.exactly, e.g., the LCD for , , and64 3MAJOR ARCA major arc is an arc that is larger than a semi-circle;the larger arc formed by an inscribed or central anglein a circle.MAXIMUMThe greatest value of an item; e.g., the maximumvalue of the sine of an angle is 1.e.g., 8 1 1, therefore 1 is the multiplicative88inverse of 8 or 8 is the multiplicative inverse of 1 .8NETClear of all charges, cost, loss; e.g., net salary issalary after all deductions have been subtracted fromthe gross salary.NUMERATORThe expression above the line in a fraction. In thefraction 3 , 3 is the numerator.4OBTUSEObtuse angle is an angle greater than 90 and smallerthan 180 . Obtuse triangle is a triangle, one ofwhose angles is obtuse.OCTAGONA polygon that has eight sides.ODDAn odd number is a number that is not evenlydivisible by 2; e.g., 1, 3, 5, 7, 9, OPEN SENTENCEA sentence or equation that is neither true nor false;e.g., x 3 7. If x 4, the sentence is true; for allother values of x the sentence is false.MINIMUMThe lowest value of an item.ORIGINThe point on a line graph corresponding to zero. Thepoint of intersection of the x-axis and y-axis. Thecoordinates of the origin are (0, 0).MINOR ARCAn arc that is smaller than a semi-circle; the smallerarc formed by an inscribed or central angle of acircle.ORDER OF OPERATIONSIn performing a series of operations, multiplicationand division are performed before addition andsubtraction in order from left to right.MONOMIALAn algebraic expression consisting of a single term;e.g., 8x2, 5xy.ORDERED PAIRAn 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.MULTIPLEA number that is the product of a given integer andanother integer; e.g., 12 is a multiple of 2, 3, 4, 6 or12.MULTIPLICATIVE INVERSEWhen the product of two numbers is 1, one is calledthe reciprocal or multiplicative inverse of the other;PARALLELEverywhere equally distant; parallel lines are twolines that never meet no matter how far they areextended. The symbol is .PARALLELOGRAMA polygon with four sides and two pairs of parallelsides.xi

PENTAGONA polygon that has five sides.PERCENT(AGE)Hundredths (symbol %); e.g., 5% of a quantity is5 of it.100PERFECT SQUAREA perfect square is the exact square of anothernumber; e.g., 4 is the perfect square of 2, since 2 2 4.PERIMETERThe sum of the lengths of the side of a polygon; thedistance around an area.PERPENDICULARPerpendicular lines are lines that meet and form rightangles (symbol ).PiThe name of the Greek letter that corresponds to theletter P (symbol π). It represents the ratio of thecircumference of a circle to its diameter. The122, 3 , or 3.14.equivalent value assigned is77POINTAn undefined element of geometry; it has positionbut no non-zero dimensions.POLYGONA plane figure consisting of a certain number ofsides. If the sides are equal, then the figure isreferred to as regular. Examples are: triangle (3sided); quadrilateral (4-sided); pentagon (5-sided);hexagon (6-sided); heptagon (7-sided); octagon (8sided); nonagon (9-sided); decagon (10-sided);dodecagon (12-sided); n-gon (n-sided).POLYNOMIALA special kind of algebraic expression usually usedto describe expressions containing more than threeterms: one term monomial; two terms binomial;three terms trinomial; four or more polynomial.POSITIVEHaving a value greater than zero.POWERSee exponentPRIME FACTORA factor that is a prime number; e.g., 2, 3, and 5 arethe prime factors of 30.xiiPRIME NUMBERA natural number greater than 1 that can only bedivided by itself and 1. A prime number has nofactors other than itself and 1; e.g.,2 2 13 3 15 5 1PRINCIPAL SQUARE ROOTThe positive square root of a number; e.g., theprincipal square root of 100 is 10.PROBABILITYThe likelihood of something happening.PRODUCTThe answer to a multiplication problem; e.g., theproduct of 8 and 5 is 40.PROOFThe logical argument that establishes the truth of astatement.PROPER FRACTIONA fraction whose numerator is smaller than itsdenominator; e.g.,1 3 7, ,.2 4 11PROPORTIONThe equality of two ratios. Four numbers A, B, C,and D are in proportion when the ratio of the firstpair A:B equals the ratio of the second pair C:D.A CUsually written as . A and D are the extremesB Dand B and C are the means.PYTHAGOREAN THEOREMThe sum of the squares of the lengths of the legs of aright triangle is equal to the square of the length ofthe hypotenuse. (Given sides a and b of a righttriangle with hypotenuse c, then a2 b2 c2.)PYTHAGOREAN TRIPLESAny set of numbers that satisfies the PythagoreanTheorem a2 b2 c2; e.g., 3, 4, 5; 5, 12, 13; and 7,24, 25 are Pythagorean triples.QUADRANTIn the coordinate system, one of the four areasformed by the intersection of the x-axis and the yaxis.QUADRATICOf the second degree; a quadratic equation is apolynomial equation of the second degree; e.g.,x2 3x 5 0

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 . classified in the Mathematics Test (see chart, page v). The 60 test questions reflect an appropriate balance of content and skills (low, middle, and high difficulty) and range of performance.

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