ACT Math: Concepts, Skills, and Problem-Solving Methods

Rational number

A number that can be written as a/b, where a and b are integers and b ≠ 0.

Irrational number

A real number that cannot be written as a fraction of integers, such as √2.

Absolute value

The distance of a number from 0 on the number line; it is never negative.

Percent

Per 100; to convert a percent to a decimal, divide by 100.

Ratio

A comparison of two quantities, often written as a:b or a/b.

Proportion

An equation stating that two ratios are equal.

Order of operations

The rule for evaluating expressions: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right.

Distributive property

A rule that removes parentheses by multiplying across a sum or difference: a(b + c) = ab + ac.

Negative exponent

An exponent that means reciprocal: a^(-n) = 1/a^n for nonzero a.

Scientific notation

A way to write very large or very small numbers as a × 10^n, where 1 ≤ a < 10 and n is an integer.

Inequality

A comparison using symbols like

Function

A rule that takes an input and produces exactly one output.

Domain

The set of all allowed inputs for a function.

Slope

The change in y for each 1-unit change in x; m = (y2 - y1)/(x2 - x1).

y-intercept

The value of y when x = 0; in y = mx + b, it is b.

Parallel lines

Lines that have the same slope.

Perpendicular lines

Non-vertical lines whose slopes are negative reciprocals; a vertical line is perpendicular to a horizontal line.

System of linear equations

Two or more equations that must be true at the same time; the solution is the point where their graphs intersect.

Polynomial

An expression made of terms with variables raised to nonnegative integer exponents.

Factoring

Rewriting an expression as a product; it is the reverse of distributing.

Difference of squares

A factoring pattern: a^2 - b^2 = (a - b)(a + b).

Perfect square trinomial

A trinomial that factors as a square: a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2.

Zero-product property

If a product equals 0, then at least one factor must equal 0.

Quadratic formula

A formula for solving ax^2 + bx + c = 0: x = (-b ± √(b^2 - 4ac)) / (2a).

Discriminant

The expression b^2 - 4ac in the quadratic formula; it tells how many real solutions a quadratic has.

Vertex form

A quadratic written as y = a(x - h)^2 + k, where (h, k) is the vertex.

Rational expression

A fraction containing polynomials; it can be simplified by canceling common factors, not terms.

Vertical angles

Opposite angles formed by two intersecting lines; they are equal.

Supplementary angles

Two angles whose measures add up to 180°.

Pythagorean theorem

In a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse.

45°-45°-90° triangle

A special right triangle with equal legs and hypotenuse equal to leg × √2.

30°-60°-90° triangle

A special right triangle with side ratio 1 : √3 : 2; the hypotenuse is twice the short leg.

Similar figures

Figures that have the same shape, with equal corresponding angles and proportional side lengths.

Scale factor

The ratio comparing corresponding lengths in similar figures; lengths scale by k, areas by k^2, and volumes by k^3.

Distance formula

A formula for the distance between two points: d = √((x2 - x1)^2 + (y2 - y1)^2).

Midpoint formula

A formula for the point halfway between two points: M = ((x1 + x2)/2, (y1 + y2)/2).

Slope-intercept form

The equation of a line written as y = mx + b, where m is slope and b is the y-intercept.

Standard equation of a circle

The equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.

Arc length

The length of part of a circle’s circumference: (θ/360) × 2πr.

Sector area

The area of a slice of a circle: (θ/360) × πr^2.

Volume of a cylinder

The amount of space inside a cylinder: V = πr^2h.

Sine

In a right triangle, sin(θ) = opposite/hypotenuse.

Cosine

In a right triangle, cos(θ) = adjacent/hypotenuse.

Tangent

In a right triangle, tan(θ) = opposite/adjacent.

Angle of elevation

The angle measured upward from a horizontal line of sight to an object above.

Mean

The average of a data set: the sum of the values divided by the number of values.

Median

The middle value of a sorted data set, or the average of the two middle values if there is an even number of values.

Probability

The ratio of favorable outcomes to total equally likely outcomes; probabilities range from 0 to 1.

Permutation

A counting arrangement in which order matters.

Combination

A counting selection in which order does not matter.