Content Covered by the ACT Mathematics Test

Content Covered by the ACT Mathematics Test

In the Mathematics Test, three subscores are based on six content areas:

pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry.

Pre-AlgebraPre-Algebra (23%). Questions in this content area are based on basic operations using whole numbers, decimals, fractions, and integers; place value; square roots and approximations; the concept of exponents; scientific notation; factors; ratio, proportion, and percent; linear equations in one variable; absolute value and ordering numbers by value; elementary counting techniques and simple probability; data collection, representation, and interpretation; and understanding simple descriptive statistics.

Elementary Algebra

Elementary Algebra (17%). Questions in this content area are based on properties of exponents and square roots, evaluation of algebraic expressions through substitution, using variables to express functional relationships, understanding algebraic operations, and the solution of quadratic equations by factoring.

Intermediate Algebra

Intermediate Algebra (15%). Questions in this content area are based on an understanding of the quadratic 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.

Coordinate GeometryCoordinate Geometry (15%). Questions in this content area 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.

Plane GeometryPlane Geometry (23%). Questions in this content area are based on the properties and relations of plane figures, including angles and relations among perpendicular and parallel lines; properties of circles, triangles, rectangles, parallelograms, and trapezoids; transformations; the concept of proof and proof techniques; volume; and applications of geometry to three dimensions.

Trigonometry

Trigonometry (7%). Questions in this content area are based on understanding trigonometric relations in right triangles; values and properties of trigonometric functions; graphing trigonometric functions; modeling using trigonometric functions; use of trigonometric identities; and solving trigonometric equations.

• Read each question carefully to make sure you understand the type of answer required.

• If you use a calculator, be sure it is working on test day and has reliable batteries.

• Solve the problem.• Locate your solution among the answer

choices.• Make sure you answer the question asked.• Make sure your answer is reasonable.• Check your work.

Pre-Algebra

1. Number Problems2. Multiples, Factors, & Primes3. Divisibility and Remainders4. Percentages5. Ratios and Proportions6. Mean, Median, & Mode7. Probability8. Absolute Value9. Exponents and Roots10. Series

Elementary Algebra

1. Substitution2. Simplifying Algebraic Expressions3. Writing Expressions & Equations4. Solving Linear Equations5. Multiplying Binomials6. Inequalities

Intermediate Algebra

1. Solving & Factoring Quadratic Equations2. Solving Systems of Equations3. Relationship between Sides of an Equation4. Functions5. Matrices6. Logarithms

Plane Geometry

1. Angles2. Triangles3. Polygons4. Circles5. Simple 3-D Geometry

Coordinate Geometry

1. Number Lines & Inequalities2. The (x,y) Coordinate Plane3. Distance and Midpoints4. Slope5. Parallel & Perpendicular Lines6. Graphing Equations7. Conic Sections

Trigonometry

1. SOHCAHTOA2. Solving Triangles3. Trigonometric Identities4. Trigonometric Graphs

Whole Numbers

Natural or Counting Numbers

Whole Numbers

Integers

Rational Numbers Irrational Numbers

Real Numbers

Terms and Operations

1. Basic Terms• sum (total)• difference• product• quotient• remainder


2. Symbols of inclusion• parentheses• brackets• braces

(2 + 3) •4 = 20

2 + (3 • 4) = 14


3. Order of Operation• Parentheses• Exponents, radicals• Multiplication/division• Addition/subtraction


4. Factoring and Canceling• 5(2 + 3 + 4) = 5(9) = 45

Factors, Multiples, and Primes

• Factor - A number that divides another number evenly.

• Multiple - A number that is evenly divisible by another number.

• Prime - A number that is only divisible by 1 and itself.

Odd and Even Numbers

• Odd number - A number that is not evenly divisible by 2.

• Even number - A number that is divisible by 2.

1. Even + Even = Even 5. Even • Even = Even 2. Even + Odd = Odd 6. Even • Odd = Even 3. Odd + Even = Odd 7. Odd • Even = Even 4. Odd + Odd = Even 8. Odd • Odd = Odd

Consecutive Integers

Consecutive integers immediately follow one another 1st 2nd 3rd n n + 1 n + 2

Consecutive even integers and consecutive odd integers

1st 2nd 3rd n n + 2 n + 4

Fractions

• Convert Mixed Numbers to Improper Fractions

• Convert Improper Fractions to Mixed Numbers

• Reducing Fractions to Lowest Terms• Common Denominators• Operations of Fractions• Comparing Fractions

Signed Numbers• Absolute Value• Adding Negative Numbers• Subtracting Negative Numbers• Multiplying Negative Numbers• Dividing Negative Numbers

1. Subtracting a negative is the same as adding a positive number.

2. Adding a negative number is the same as subtracting a positive number.

3. Multiplying or dividing an odd number of negative signs will always result in a negative number.

4. Multiplying or dividing an even number of negative signs will always result in a positive number.

Decimals• A decimal is a special

way of writing fractions using a denominator that is a power of 10.

Millionths

Hundred T

housandthsTen T

housandthsT

housandthsH

undredthsTenths

0 . 1 2 3 4 5 6

Decimals

• Converting Fractions to Decimals• Converting Mixed Numbers to Decimals• Converting Improper Fractions to

Decimals• Converting Decimals and Fractions to

Mixed Numbers• Operations of Decimals

Percents

• Converting to and from Percents• Operations of Percents• Percent Story Problems

Mean, Median, and Mode

• Mean - The average of a set data values

• Median– The median of an odd number of data

values is the middle value– The median of an even number of data

values is the average of the middle two• Mode - The value that occurs most

often

Ratios and Proportions

• Ratio - A relationship between two quantities

• Proportion - An equation relating two proportions

1:2 = 2:4 1*4 = 2*2The product of the means equals the product of the extremes.

Exponents

Rules For ExponentsIf a > 0 and b > 0, the following hold true for all real numbers x and y.

yxyx aaa 1.

yxy

x

aa

a 2.

xyyx aa 3.

xxx (ab)ba 4.

x

xx

b

a

b

a 5.

1a 6. 0

x

1a 7. x-

q pq

p

aa 8.

Roots and Radicals

• Square Root

• Cube Root

• Rational Exponents

x b, or x b2

x3 b, or x b3

xn x1

n

Algebraic Operations

• Elements of Algebra– Algebraic Terms

• Term• Coefficient• Variable• Exponent

3x 4 2y

Coefficients

Variables

Exponent

Operations of Algebraic Terms

• Adding and Subtracting• Multiplying and Dividing• Algebraic Fractions• Factoring Algebraic Expressions• Absolute Value in Algebraic Expressions• Radicals in Algebraic Expressions

Algebraic Equations and Inequalities

• Solving Algebraic Formulas• Solving Linear Equations• Solving Simultaneous Equations• Solving Quadratic Equations• Algebraic Inequalities• Exponents in Equations and Inequalities• Rational Equations and Inequalities• Radical Equations

Geometry Notation

P QLine Segment PQ

Vertex A

B

C

Angle BAC <BAC

Angle Properties

0°

270°

90°

180°

Angle Properties

• Right Angle

• Acute Angle

• Obtuse Angle

• Vertical Angle xyw

z

x = zw = y

Angle Properties

• Transversal - A line that intersects parallel line

1 2

5 6

3 4

7 8

– Alternate Interior– Alternate Exterior– Vertical Angles– Same Side Interior Angles

Polygon Properties

• Polygon - A closed figure created by three or more lines.

• Triangle - A polygon with exactly three sides• Quadrilateral - A polygon with exactly four

sides• Pentagon - A polygon with exactly five sides• Hexagon - A polygon with exactly six sides

Polygon Properties

• Regular Polygon - A polygon with equal sides and equal angles.

• The sum of the exterior angles of a polygon is 360°

• The sum of the interior angles of a polygon is 180(n - 2) where n is the number of sides in the polygon

Triangle Properties

• Scalene Triangle - A triangle with no equal sides

• Isosceles Triangle - A triangle with exactly two sides equal, If the two sides are equal than the angles opposite the two sides are equal and vice versa.

• Equilateral Triangle - A triangle with three sides equal, If the three sides are equal than each angle equal 60° and vice versa

Perimeter and Area

Area and Volume

Circle Properties

chord

radius

tangent

secant

diameter

Circle Properties

O

B

CE

D

A

Central Angle - An angle withthe vertex at the center and with sides that are radii.

Inscribed Angle - An angle witha vertex on the circle and withsides that are chords.

Coordinate Geometry

d(P,Q ) (x1 x2)2 (y1 y2)2

midpoint a c

2,b d

2

Coordinate Geometry

m y2 y1x2 x1

y2 y1 m(x2 x1)

y mx b

Slope Equations of Lines

Coordinate Geometry

Parallel lines m1 = m2

Perpendicular lines m1 = -1/m2

Coordinate GeometryTransformationsh(x) = af(x) vertical stretch or shrink

h(x) = f(ax) horizontal stretch or shrink

h(x) = f(x) + k vertical shift

h(x) = f(x+h) horizontal shift

h(x) = -f(x) reflection in the y-axis

h(x) = f(-x) reflection in the x-axis

1. log a (ax) = x for all x 2. a log ax = x for all x > 03. log a (xy) = log a x + log a y4. log a (x / y) = log a x – log a y5. log a xn = n log a x

Common Logarithm: log 10 x = log xNatural Logarithm: log e x = ln x

All the above properties hold.

Logarithms

(1,0)

3

2,-

1

2

2

2,

2

2

1

2,

3

2

(-1,0)

(0,1)

(0,-1)

1

2,

3

2

1

2,

3

2

2

2,-

2

2

2

2,-

2

2

2

2,

2

2

3

2,1

2

3

2,1

2

3

2,-

1

2

Trigonometric Functions

60° 1

3

2

30°

45°

2

1

145°

A

CT

S

2

3,

2

1

Trigonometric Functions

Coordinate Geometry

Quadratic functions f(x) = ax2+bx+cComplete the square f(x) = a(x-h)2+ kDiscriminant b2-4acVertex (h,k)

Quadratic Formula

b b2 4ac

2a

Conic Sections

• Circles (x – h)2 + (y – k)2

• Parabolas (x – h)2 = 4a(y – k) (Up/Down) (y – k)2 = 4a(x – h) (Right/Left)

• Ellipses (x – h)2 /a2 + (y – k)2/ b2 = 1 (y – k)2 /a2 + (x – h)2/ b2 = 1

c 2 = a 2 - b 2

• Hyperbolas (x – h)2 /a2 - (y – k)2/ b2 = 1 (y – k)2 /a2 - (x – h)2/ b2 = 1 c 2 = a 2 + b 2

Circles

Standard Equation of a Circle [Center at (h,k)]

The standard form of the equation of a circlecentered at (h,k) is

222 )()( rkyhx y

Radius rPoint on circle (x,y)

Center (h,k)

x

Parabolas

x

y

F

a

V(h, k)V(h, k)

x

y

F

a

(x – h)2 = 4a(y – k)Vertex (h, k)Focus (h, k + a)a > 0 opens upa < 0 opens down

(y – k)2 = 4a(x – h)Vertex (h, k)Focus (h + a, k)a < 0 opens lefta > 0 opens right

x

y

(h, k) b

a

y

x

ba

(h, k)

(x – h)2

a2 + (y – k)2

b2 = 1 a > b > 0 (x – h)2

b2 + (y – k)2

a2 = 1

Center (h, k) Center (h, k)Major axis 2a Major axis 2aMinor axis 2b Minor axis 2b

Ellipses

a

b

y

x

(h, k)

y

x

ab(h, k)

Hyperbolas

(x – h)2

a2 – (y – k)2

b2 = 1 (y – k)2

a2 + (x – h)2

b2 = 1

Center (h, k) Center (h, k) Transverse axis 2a Transverse axis 2a Conjugate axis 2b Conjugate axis 2b

-

DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document. Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test. You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator. Note: Unless otherwise stated, all of the following should be assumed.

1 Illustrative figures are NOT necessarily drawn to scale.

2. Geometric figures lie in a plane. 3. The word line indicates a straight line. 4. The word average indicates arithmetic mean.

1. Which of the following is equivalent to (x)(x)(x)(x), for all x ?

a. 4x b. x4 c. x + 4 d. 4 + x e. 2x2

2. A rectangle is twice as long as it is wide. If the width of the rectangle is 3 inches, what is the rectangle's area, in square inches?

F. 6 G. 9 H. 12 J. 15 K. 18

3. A vendor has 14 helium balloons for sale: 9 are yellow, 3 are red, and 2 are green. A balloon is selected at random and sold. If the balloon sold is yellow, what is the probability that the next balloon, selected at random, is also yellow?

A. 8

15 B.

9

15 C.

5

14 D.

8

13 E.

9

14

4. 3 x 10 - 4 = ?

F. -30,000 G. -120 H. 0.00003 J. 0.0003 K. 0

5. For all x > 0, simplifies to:

A. x + 3B. x + 4C. 2(x + 3)D. 2(x + 4)E. 2(x + 3)(x + 4)

424142 2

xxx

6. If one leg of a right triangle is 8 inches long, and the other leg is 12 inches long, how many inches long is the triangle's hypotenuse?

F. G. H. J. K. 4

4 13

4 10

2 10

4 5

7. In the standard (x,y) coordinate plane, the graph of (x - 2)2 + (y + 4)2 = 9 is a circle. What is the area enclosed by this circle, expressed in square coordinate units?

A. 3π B. 4π C. 6π D. 9π E. 16π

8. How many solutions are there to the equation x2 - 15 = 0 ?

F. 0G. 1H. 2J. 4K. 15

9. A circle with center (-3,4) is tangent to the x-axis in the standard (x,y) coordinate plane. What is the radius of this circle?

A. 3B. 4C. 5D. 9E. 16

10. In ∆ABC, if <A and <B are acute angles, and sin A = what is the value of cos A ?

F.

G.

H.

J.

K

10

13

69

13

3

13

3

13

3 13

13

3

13

11.What are the values of a and b, if any, where a|b - 2| > 0 ?

A. a < 0 and b < 2

B. a < 0 and b = 2

C. a < 0 and b > 2

D. a > 0 and b < 2

E. There are no such values of a and b.

12. In a shipment of 1,000 light bulbs, of the bulbs were defective. What is the ratio of defective bulbs to nondefective bulbs?

F.

G.

H.

J.

K.

1

40

1

25

1

39

1

40

39

1

40

1

13. Which of the following is divisible by 3 (with no remainder)?

A. 2,725B. 4,210C. 4,482D. 6,203E. 8,105

14. A particle travels 1 x 108 centimeters per second in a straight line for 4 x 10 -6 seconds. How many centimeters has it traveled?

F 2.5 x 102 G. 2.5 x 10 13 H. 4 x 102 J. 4 x 10 -14 K. 4 x 10 -48

15. The triangle below is isosceles with

<M = < N. What is the measure of <N ?

A. 22°

B. 68°

C. 78°

D. 79°

E. 89°

16. In the figure, AB = AC and BC is 10 units long. What is the area, in square inches, of ABC ?

F. 12.5 G. 25 H. J. 50 K. Cannot be determined from the given information

17. Which of the following statements

completely describes the solution set for

3(x - 4) = 3x - 12 ?

F. x = 3 only

G. x = 0 only

H. x = -12 only

J. There are no solutions for x.

K. All real numbers are solutions for x.

18. When graphed in the (x,y) coordinate

plane, at what point do the lines

x + y = 5 and y = 7 intersect?

A. (-2,0)

B. (-2,7)

C. (0,7)

D. (2,5)

E. (5,7)

19. The area of a trapezoid is h/2 (b1 + b2),

where h is the altitude, and b1 and b2 are the lengths of the parallel bases. If a trapezoid has an altitude of 5 inches, an area of 55 square inches, and one base 13 inches long, what is the length, in inches, of its other base?

F. 9.0 G. 16.8 H. 19.4 J. 45.0 K. 97.0

20. If you have gone 4.8 miles in 24 minutes, what was your average speed, in miles per hour?

A. 5.0

B. 10.0

C. 12.0

D. 19.2

E. 50.0

21. If a and b are any real numbers such that 0 < a < 1 < b, which of the following must be true of the value of ab ?

F. 0 < ab < a G. 0 < ab < 1 H. a < ab < 1 J. a < ab < b

K. b < ab

Content Covered by the ACT Mathematics Test

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