ACT^® Mathematics

Real Number Properties and Operations

• Identify properties of the real number system including closure, commutativity, associativity, distributivity, and the identities and inverses for addition and multiplication.
• Apply rules for integer and rational exponents to simplify expressions, including negative exponents, fractional exponents, and the laws of exponents for products, quotients, and powers.
• Calculate absolute value expressions and solve equations and inequalities involving absolute value, interpreting solutions on a number line.
• Apply radical simplification rules to simplify, add, subtract, multiply, and rationalize expressions containing square roots and higher-order radicals.

Complex Numbers

• Define complex numbers in the form a + bi, identify the real and imaginary parts, and state that i is defined as the square root of negative one with i squared equal to negative one.
• Apply operations (addition, subtraction, multiplication, division) to complex numbers, including rationalizing denominators with complex conjugates.

Vectors and Matrices

• Define vectors as quantities with magnitude and direction, identify vector components, and describe how vectors are represented graphically and in component form.
• Apply vector addition, subtraction, and scalar multiplication to solve problems involving displacement, velocity, or force in two dimensions.
• Define matrix dimensions and elements by row and column position, and apply matrix addition, scalar multiplication, and matrix multiplication to calculate results, determining when multiplication is defined.

Linear Expressions, Equations, and Inequalities

• Identify the components of linear expressions and equations, including coefficients, variables, constants, and terms, and describe the standard forms of linear equations.
• Solve one-variable linear equations and inequalities, including those requiring distribution, combining like terms, and handling variables on both sides of the equation.
• Solve systems of two linear equations in two variables using substitution, elimination, and graphing methods, and interpret the solution as the point of intersection.
• Analyze word problems to set up and solve linear equations or systems of equations, identifying variables, writing constraint equations, and interpreting solutions in context.

Quadratic and Polynomial Expressions

• Identify the standard form, vertex form, and factored form of quadratic expressions, and describe how each form reveals different features of the parabola.
• Solve quadratic equations by factoring, completing the square, and applying the quadratic formula, including determining the discriminant to predict the number and nature of solutions.
• Apply polynomial operations (addition, subtraction, multiplication) and factor polynomials using greatest common factor, grouping, and special patterns (difference of squares, perfect square trinomials).
• Analyze the relationship between the factors of a polynomial, its zeros, and the x-intercepts of its graph to determine solutions and sketch polynomial behavior.

Function Concepts and Notation

• Define a function as a relation that assigns exactly one output to each input, identify domain and range, and use function notation f(x) to evaluate functions at given values.
• Determine the domain and range of functions from equations, graphs, and tables, identifying restrictions caused by division by zero, square roots of negatives, and logarithms of non-positives.
• Calculate the composition of two functions f(g(x)) and g(f(x)), and determine the domain of a composite function based on the domains of its component functions.

Types of Functions and Their Graphs

• Identify the characteristics of linear functions (constant rate of change, y-intercept) and write equations in slope-intercept, point-slope, and standard forms.
• Interpret the key features of quadratic function graphs (vertex, axis of symmetry, direction of opening, intercepts) and convert between standard, vertex, and factored forms.
• Identify the characteristics of exponential functions (growth factor, initial value, asymptote) and distinguish exponential growth from exponential decay in equations, tables, and graphs.
• Apply properties of logarithms (product, quotient, power rules) to simplify logarithmic expressions and solve logarithmic and exponential equations.
• Analyze the end behavior, intercepts, and number of turning points of polynomial functions to match equations with their graphs and predict function behavior.

Function Transformations

• Describe the effects of transformations on function graphs, including vertical and horizontal shifts, reflections across the x-axis and y-axis, and vertical and horizontal stretches and compressions.
• Apply transformation rules to write the equation of a function that has been shifted, reflected, or stretched from its parent function, and identify transformations from a given equation.
• Analyze how multiple transformations applied in sequence affect the graph of a parent function, determining the resulting equation and graph features.

Properties of Shapes and Solids

• List the formulas for area, perimeter, and angle measures of triangles, rectangles, parallelograms, trapezoids, and regular polygons, and describe when each formula applies.
• Calculate the area, perimeter, and missing angle measures of two-dimensional shapes, including composite figures formed by combining standard shapes.
• List the formulas for volume and surface area of prisms, cylinders, cones, pyramids, and spheres, and describe the relationship between dimensions and these measurements.
• Calculate the volume and surface area of three-dimensional solids, including problems that require determining missing dimensions from given constraints.

Triangles and the Pythagorean Theorem

• State the Pythagorean theorem, identify the special right triangle ratios (30-60-90 and 45-45-90), and describe the triangle inequality theorem.
• Apply the Pythagorean theorem and special right triangle ratios to calculate unknown side lengths and solve problems involving right triangles in geometric and applied contexts.
• Apply properties of similar triangles (proportional sides, equal angles) and congruent triangles (SSS, SAS, ASA, AAS) to determine unknown measurements and justify geometric relationships.
• Analyze complex geometric figures involving overlapping or embedded triangles to determine unknown measurements using combinations of Pythagorean theorem, similarity, and angle relationships.

Circles

• State the formulas for circumference, area, arc length, and sector area of circles, and define key terms including radius, diameter, chord, tangent, secant, central angle, and inscribed angle.
• Calculate circumference, area, arc length, and sector area, and apply relationships between central angles, inscribed angles, arcs, and chords to solve problems involving circles.

Coordinate Geometry

• State the distance formula, midpoint formula, and slope formula, and describe how slope relates to parallel and perpendicular lines.
• Apply the distance, midpoint, and slope formulas to solve coordinate geometry problems, including finding equations of parallel and perpendicular lines through given points.
• Identify the standard form of a circle equation (x-h)^2 + (y-k)^2 = r^2, determine center and radius from the equation, and write circle equations from given information.
• Analyze coordinate geometry problems that require integrating multiple concepts (distance, slope, midpoint, circle equations) to determine properties of geometric figures placed on the coordinate plane.

Geometric Transformations

• Describe the four basic geometric transformations (translation, rotation, reflection, dilation) and state which preserve congruence and which preserve similarity.
• Apply coordinate rules to perform translations, reflections over the axes and lines y=x, rotations of 90, 180, and 270 degrees, and dilations centered at the origin on given figures.
• Determine the sequence of transformations that maps one figure onto another and evaluate whether two figures are congruent or similar based on the transformations applied.

Measures of Center and Spread

• Define mean, median, mode, range, interquartile range, and standard deviation, and describe when each measure is most appropriate for summarizing a data set.
• Calculate mean, median, mode, and range from data sets presented in lists, frequency tables, and histograms, including weighted averages and missing-value problems.
• Evaluate the effect of outliers, skewness, and data distribution shape on measures of center and spread, determining which summary statistics best represent a given data set.

Data Interpretation and Collection

• Interpret data presented in bar graphs, line graphs, scatter plots, histograms, box plots, pie charts, and two-way frequency tables to extract specific values and identify trends.
• Analyze scatter plots to describe the direction, form, and strength of associations between variables and interpret lines of best fit to make predictions.
• Evaluate data collection methods (random sampling, surveys, observational studies, experiments) and assess how sampling bias and study design affect the validity of conclusions.

Probability and Counting

• Define probability as a ratio of favorable outcomes to total outcomes and describe the addition and multiplication rules for independent and mutually exclusive events.
• Calculate probabilities of simple and compound events, including problems involving conditional probability, complementary events, and geometric probability.
• Apply the fundamental counting principle, permutations, and combinations to determine the number of possible outcomes in arrangement and selection problems.
• Analyze probability problems that require combining counting principles with probability rules to determine the likelihood of complex events in multi-step scenarios.

Rates, Ratios, and Proportions

• Define ratios, rates, unit rates, and proportions, and describe how each is used to express relationships between quantities in applied contexts.
• Solve proportion problems using cross-multiplication and unit rate methods, including scale factor problems, map reading, and unit conversion problems.
• Analyze multi-step rate problems involving distance-rate-time, work rates, density, and unit pricing, setting up and solving proportional equations to determine unknown quantities.

Percent Problems

• Describe the relationship between fractions, decimals, and percents, and convert fluently among these representations for common and uncommon values.
• Solve percent problems including finding the percent of a number, determining what percent one number is of another, percent increase and decrease, and successive percent changes.
• Analyze applied percent problems involving tax, tip, discount, markup, simple and compound interest, and percent change in multi-step real-world scenarios.

Applied Area, Perimeter, and Volume

• Solve applied problems requiring calculation of area, perimeter, or volume in real-world contexts such as flooring, fencing, painting, and container capacity.
• Analyze multi-step geometry word problems that combine area, perimeter, and volume calculations with unit conversion, cost estimation, and material requirements.

Right Triangle Trigonometry

• Define the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) for acute angles in right triangles using opposite, adjacent, and hypotenuse side relationships.
• Apply trigonometric ratios to calculate unknown side lengths and angle measures in right triangles, including problems involving angles of elevation and depression.
• Analyze applied trigonometry problems that require setting up right triangle models for real-world situations involving heights, distances, and indirect measurement.

Unit Circle and Radian Measure

• Define radian measure, describe the relationship between degrees and radians, and identify the coordinates of key points on the unit circle at standard angles.
• Use the unit circle to determine trigonometric function values for angles in all four quadrants, including reference angles and the signs of trig functions by quadrant.

Trigonometric Identities and Equations

• State the Pythagorean identity, reciprocal identities, and quotient identities, and describe how they relate the six trigonometric functions to one another.
• Apply trigonometric identities to simplify expressions, verify identities, and solve basic trigonometric equations for values within specified intervals.
• Evaluate which trigonometric identity or technique is most efficient for simplifying a given expression or solving a given equation, justifying the chosen approach.

Law of Sines and Law of Cosines

• State the law of sines and the law of cosines and describe the types of triangle problems (AAS, ASA, SSA, SAS, SSS) for which each law is applicable.
• Apply the law of sines and the law of cosines to solve for unknown sides and angles in oblique (non-right) triangles, including the ambiguous case of the law of sines.

Graphs of Trigonometric Functions

• Describe the key features of sine and cosine graphs including amplitude, period, phase shift, vertical shift, and midline, and state how these features relate to the equation parameters.
• Determine the amplitude, period, phase shift, and vertical shift from a given trigonometric equation and use these values to sketch or match the corresponding graph.
• Analyze trigonometric graphs to write equations that model periodic phenomena, determining appropriate values for amplitude, period, and phase shift from contextual information.

Interpret mathematical models (equations, graphs, tables) in context, identifying what the variables, coefficients, and constants represent in real-world scenarios.

Use mathematical models to calculate specific values, make predictions, and interpolate or extrapolate data within and beyond the given domain of the model.

Determine which type of mathematical model (linear, quadratic, exponential) best fits a given data set or real-world scenario based on the pattern of change observed.

Evaluate the limitations and appropriateness of a mathematical model by identifying assumptions, assessing the domain of validity, and comparing model predictions to actual data.