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ACT Test Prep Math

Prepare for the ACT Math Section with helpful advice such as using scratch paper, mastering formulas, and time-saving strategies in this comprehensive guide.

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ACT Test Prep Math

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  1. ACT Test PrepMath

  2. Before we start • Get a good night’s rest. Eat what you always eat for breakfast. • Use the test booklet for scratch paper. You can’t bring your own. • Remember your formulas. You will not get them on the test. • Turn word problems into equations or equations into word problems -- whichever is easiest for you! • You can use a calculator. • Don’t be afraid! Self-doubt lowers scores. • Hard questions vs. easy questions • Must answer all easy questions • Go back and guess on hard ones if you run out of time • One minute per question • Faster on easy questions • Skip questions that take too much time • Guess if you run out of time

  3. 60 questions in 60 minutes Scores reported: Total Mathematics Test score based on all 60 questions. Pre-Alegebra/Elementary Algebra Subscore Intermediate Algebra/Coordinate Geometry Subscore Plane Geometry/Trigonometry Subscore Source: The Real ACT Prep Guide. ACT. 2nd Ed.

  4. Math Section of the ACT 60 Questions in 60 Minutes Goal: Answer 70% correctly (42 out of 60) This means you need a strategy to confidently answer 42 questions correctly in 60 minutes.

  5. Math Section Content Pre-algebra Elementary algebra Intermediate algebra Coordinate geometry Plane geometry Trigonometry Miscellaneous topics Math test-taking strategy

  6. Math Vocabulary area of a circle chord circumference collinear complex number congruent consecutive diagonal directly proportional endpoints function y = R (x) hypotenuse integer intersect irrational number least common denominator logarithm matrix mean median obtuse perimeter perpendicular pi polygon prime number quadrant quadratic equation quadrilateral quotient radian radii radius rational number real number slope standard coordinate plane transversal trapezoid vertex x-intercept y-intercept

  7. √ 3 Math Vocabulary area of a circle—A = π r2 chord—a line drawn from the vertex of a polygon to another non adjacent vertex of the polygon circumference—the perimeter of a circle = 2 π r collinear—passing through or lying on the same straight line complex number—is an expression of the form a+bi, where a & b are real numbers and i2 = -1 congruent—corresponding; equal in length or measure consecutive—uninterrupted sequence diagonal—a line segment joining two nonadjacent vertices of a polygon or solid (polyhedron) directly proportional—increasing or decreasing with the same ratio endpoints—what defines the beginning and end-of-line segment Function y = R (x)—a set of number pairs related by a certain rule so that for every number to which the rule may be applied, there is exactly one resulting number hypotenuse—the longest side of a right-angle triangle, which is always the side opposite the right angle integer—a member of the set ..., -2, -1, 0, 1, 2, … intersect—to share a common point irrational number—cannot be expressed as a ratio of integers, eg., , π, etc. least common denominator—the smallest number (other than 0) that is a multiple of a set of denominators (for example, the LCD of ¼ and ⅓ is 12) logarithm—log a x means ay = x matrix—rows and columns of elements arranged in a rectangle mean—average; found by adding all the terms in a set and dividing by the number of terms median—the middle value in a set of ordered numbers obtuse—an angel that is larger than 90°

  8. Math Vocabulary (continued) perimeter—the distance from one point around the figure to the same point perpendicular—lines that intersect and form 90-degree angles pi— = 3.14 … polygon—a closed, plane geometric figure whose sides are line segments prime number—a positive integer that can only be evenly divided by 1 and itself quadrant—any one of the four sectors of a rectangular coordinate system, which is formed by two perpendicular number lines that intersect at the origins of both number lines quadratic equation—Ax2 + bx + C = D, A ≠ 0 quadrilateral—a four sided polygon quotient—the result of division radian—a unit of angle measure within a circle radii—the plural form of radius radius—a line segment with endpoints at the center of the circle and on the perimeter of the circle, equal to one-half the length of the diameter rational number—r can be expressed as r = where m & n are integers and n ≠ 0 real number—all numbers except complex numbers slope—m = standard coordinate plane—a plane that is formed by a horizontal x-axis and a vertical y-axis that meet at point (0,0) (also known as the Cartesian Coordinate Plane) transversal—a line that cuts through two or more lines trapezoid—a quadrilateral (a figure with four sides) with only two parallel lines vertex—a point of an angle or polygon where two or more lines meet x-intercept—the point where a line on a graph crosses the x-axis y-intercept—the point where a line on a graph crosses the y-axis m n y2 – y1 x2– x1

  9. Pre-Algebra • Operations using whole numbers, fractions, and decimals. • PEMDAS • 2x3= ? • 4/2 x 6/2= ? • 1/5 x .5 = ? • 4/.5 = ? • Numbers raised to powers and square roots. • 22 • 4.5 • Simple linear equations with one variable. • 3x+7=16. Solve for X. • Simple probability and counting the number of ways something can happen. • On a six sided die, what are the chances of rolling a five?

  10. Pre-Algebra • Ratio, proportion, and percent. • 3 is what percent of 6? What is 50% of 6? • Absolute value. • What is the absolute value of -3? • |-3| = ? • Ordering numbers from least to greatest. • Reading information from charts and graphs. • Simple stats • Mean: add all terms together and divide by number of terms. • Median: order terms from lowest to highest. Eliminate high and low terms till you’ve reached the middle. If two terms are left, take the mean. • Mode: most frequent term.

  11. Pre-Algebra – Word Problems Converting a word problem into an equation: If a discount of 20% off the retail price of a desk saves Mark $45, how much did Mark pay for the desk?

  12. Pre-Algebra If a discount of 20% off the retail price of a desk saves Mark $45, how much did Mark pay for the desk? Amount Paid (Sales Price) = Retail Price – Discount Discount = 20% × Retail Price $45 = 20% × Retail Price Retail Price = $45/.2 = $225 Sales Price = $225 − $45 = $180

  13. Pre-Algebra A lawn mower is on sale for $1600. This is 20% off the regular price. How much is the regular price?

  14. Pre-Algebra A lawn mower is on sale for $1600 which is 20% off the regular price. How much is the regular price? Sales Price = Regular Price – Discount Discount = 0.20 × Retail Price Sales Price = Regular Price – 0.20 × Retail Price $1600 = 0.80 × Regular Price Regular Price = $1600 / 0.8 = $2000

  15. Pre-Algebra If 45 is 120% of a number, what is 80% of the same number?

  16. Practice Questions

  17. Practice Questions 4. Marlon is bowling in a tournament and has the highest average after 5 games, with scores of 210, 225, 254, 231, and 280. In order to maintain this exact average, what must be Marlon’s score for his 6th game? F. 200 G. 210 H. 231 J. 240 245 5. Joelle earns her regular pay of $7.50 per hour for up to 40 hours of work in a week. For each hour over 40 hours of work in a week, Joelle is paid 1 times her regular pay. How much does Joelle earn for a week in which she works 42 hours? A. $126.00 B. $315.00 C. $322.50 D. $378.00 E. $472.50 6. Which of the following mathematical expressions is equivalent to the verbal expression “A number, x, squared is 39 more than the product of 10 and x” ? F. 2x = 390 + 10x G. 2x = 39x + 10x H. x2= 390 − 10x J. x2= 390 + x10 K. x2=390 + 10x

  18. Practice Questions

  19. Pre-Algebra If 45 is 120% of a number, what is 80% of the same number? 45 = 1.2 (X) X = 45/1.2 = 37.5 Y = 0.8 (37.5) = 30

  20. Elementary algebra • Substituting the value of a variable in an expression. • Add like terms. Separate different terms. • 2x+2x+7y=15. • Y=2. Solve for X. • Performing basic operations on polynomials and factoring polynomials. • FOIL • (x-3)(x+7) = ? • x2+8x+12=0. Solve for X. • Factor x2-11+30. • Solving linear inequalities with one variable. • X+7<12. What do we know about x? • X+6>19 and x-8<6. What do we know about x?

  21. Elementary Algebra – Substitution, 2 Equations, 2 Unknowns If a – b = 14, and 2a + b = 46, then b = ? a = 14 + b; substitute 2(14 + b) + b = 46 28 + 2b + b = 46 3b = 18 b = 6, a = 20

  22. Elementary Algebra a c + = (a + c) / b + = (ad + bc) / bd b b a c b d 3x3 + 9x2 – 27x = 0; 3x (x2 + 3x – 9) = 0 (x+2)2 = (x+2)(x+2) (x/y)2 = x2/y2 X0 = 1

  23. Intermediate algebra • Quadratic Formula • When you can’t factor a polynomial cleanly. You can always use the quadratic formula • In x2+7x+15=0, what is a, b, and c?

  24. Intermediate algebra Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf \

  25. Intermediate algebra Source: http://www.mathsisfun.com/algebra/matrix-multiplying.html • What are the dimensions of a matrix? • Up and over. • Multiplying Matrices • Scalar multiplication • A number times everything inside the matrix.

  26. Intermediate algebra Source: http://www.mathsisfun.com/algebra/matrix-multiplying.html • Multiplying a matrix by another matrix • 2x3 * 3x2. • Can we do it? • What will the final matrix look like?

  27. Intermediate Algebra – Quadratics x2 + 3x – 4 = y x2+ 3x – 4 = 0 Factoring: (x – 1) (x + 4) = 0 X = 1, -4 For ax2 + bx + c = 0, the value of x is given by: X= (-3 + (32 – 4*1*-4).5)/2 = 1 X= (-3 - (32 – 4*1*-4).5)/2 = -4 Quadratic Formula

  28. Intermediate Algebra – Factoring Polynomials, Solve for x x2 - 2x - 15 = 0 (x - 5) (x + 3) = 0 x = 5, -3

  29. Intermediate Algebra – Factoring Polynomials Example 2 Example 1 x3 + 3x2 + 2x + 6 (x3 + 3x2) + (2x + 6) x2(x + 3) + 2(x + 3) (x + 3) (x2 + 2) x3 + 3x2 + 2x + 6 / (x + 3) ((x3 + 3x2) + (2x + 6)) / (x+3) (x2(x + 3) + 2(x + 3)) / (x+3) ((x + 3) (x2 + 2)) / (x+3) x2 + 2

  30. Intermediate Algebra – Exponents x3 * x2 = x5 x9 / x2 = x7 (x2)5 = x10 1/x4 = x-4 x2 * x.5 = ? x4 / x8 = ? (x.5)2 = ? 1/x-z = ? x2 * x.5 = x2.5 x4 / x8 = x-4 (x.5)2 = x 1/x-z = xz

  31. Intermediate Algebra – Imaginary Numbers

  32. Coordinate geometry Graphs of lines, curves, points, polynomials, circles in an (x,y) plane. Relationship between equations and graphs, slope, parallel and perpendicular lines, distance, midpoints, transformations, and conics. It’s coordinate, so draw it on the graph!

  33. Coordinate geometry Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf • Lines • Aline goes through points A(2, 3) and B(4, 5). You should be able to find the following: • Parallel lines have the same slope. Perpendicular lines have inverted slopes.

  34. Coordinate Geometry – Coordinates Equation of a Line y = mx + b, equation of a linear (straight) line m = slope of the line = change in Y / change in X b = y intercept If m is negative, the line is going down and if positive the line is going up (left to right). What is the equation for the line between points, (1, -2) & (6, 8)? m = change in y values / change in x values = (y1 – y2) / (x1– x2) m = [8- (-2)] / (6 - 1) = 10/5 = 2 b = y – mx; b = 8 – (2) × (6) = 8 – 12 = -4 y = 2x -4

  35. Coordinate Geometry – Coordinates What is the distance between these points (-1, 2) and (6, 8)?

  36. Coordinate Geometry – Coordinates What is the distance between these (-1, 2) and (6, 8)? * 6, 8 c b 6 a * -1, 2 7

  37. Plane geometry • Relations and properties of shapes (triangles, rectangles, parallelograms, trapezoids, and circles), angles, parallel lines, and perpendicular lines. • What happens when you move or change these shapes? • Translations, rotations, reflections • Proofs • Justification, logic. • Three-dimensional geometry • Measurements: perimeter, area, and volume.

  38. Plane geometry Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf Circles

  39. Plane geometry Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf Lines in a plane What do we know about a and b in both of these cases?

  40. Plane geometry Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf Other shape areas and perimeters. If an angle is greater than 90, it is obtuse. If an angle is less than 90, it is acute. If an angle is 90, it is a right angle. TRIANGLE: SUM OF ALL ANGLES = 180 SQUARE AND RECTANGLE: SUM OF ALL ANGLES = 360

  41. Plane geometry Source: http://www.erikthered.com/tutor/act-facts-and-formulas.pdf • Right Triangles • How do you find the length of a side in a right triangle? Pythagorean Theorem. • Other Triangles: Equilateral (all three sides are equal), Isosceles (two equal sides), and Similar (corresponding angles are equal and sides are in proportion).

  42. Plane Geometry • Lines and Angles • Triangles • Circles • Squares and Rectangles • Multiple Figures

  43. Plane Geometry: Lines c abc + cbd = 1800 a d b a b d c Opposite (vertical) angles are congruent (equal) All angles combined = 3600 Transversal line thru two parallel lines creates equal opposite angles.

  44. Plane Geometry: Triangles

  45. Plane Geometry Area of a triangle = ½ (base * height) The sum of the three angles = 1800 Area of a trapezoid = ½ (a +b)*(height) where a and b are the lengths of the parallel sides Diameter = 2 * radius of a circle Volume of cylinder = area of circle * height a b r h

  46. Plane Geometry Example What is the area of the square if the radius equals 5? L L r Diameter = 2 x r The diameter = 1 side of the square Area = L x L Diameter = 10 (same as a length of a side), Area = 100

  47. Plane Geometry Parallelogram Area = Base x Height h b Note a rectangle is a parallelogram. The sum of the angles = 3600

  48. Plane Geometry Circles

  49. Plane Geometry Circles What is the equation of these circles? (x-1)2 + y2 = 1 (x-3)2 + (y-1)2 = 4

  50. Plane Geometry Terms Congruent = equal lengths Co-linear = on same line abc = the angle of b in the triangle abc Acute = less than 90 degrees (A cute little angle) Obtuse = greater than 90 degrees

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