ACT Math Fundamentals

This course covers the core skills represented in the ACT Math curriculum.

Students begin by taking a diagnostic assessment, which will identify their current knowledge frontier and any weaknesses in their foundational knowledge.

Once the student has completed the diagnostic assessment, our sophisticated algorithms will create a custom course that addresses any foundational weaknesses along with a plan for building on their existing strengths to attain the highest possible score.

The course begins with preliminaries, including rational-number arithmetic, exponents, scientific notation, radicals, order of operations, number theory, conditional statements, algebraic expressions, and simplifying linear expressions. These topics ensure that students have the numerical and symbolic fluency needed to work efficiently across the full range of ACT math problems.

Students then study linear equations and inequalities. They solve and model with one-variable equations, graph and interpret two-variable linear equations, solve systems of equations, work with linear inequalities, use interval notation, graph two-variable inequalities, and model constraints with inequalities.

The geometry portion of the course develops students’ understanding of points, lines, angles, transformations, polygons, solids, congruence, similarity, length, area, volume, circles, and radians. Students solve problems involving angle relationships, triangle and polygon properties, surface area, volume, coordinate geometry, inscribed figures, circle theorems, equations of circles, tangent lines, arc length, and sector area.

The trigonometry portion of the course covers right-triangle trigonometry, the unit circle, special trigonometric ratios, general triangles, graphs of trigonometric functions, transformed trigonometric functions, trigonometric identities, double-angle formulas, sum and difference formulas, and inverse trigonometric functions.

Students then move into advanced algebra. They study functions, linear functions, graph transformations, absolute value expressions and functions, polynomials, quadratic equations and functions, nonlinear systems, exponential equations and functions, logarithms, rational expressions and equations, rational functions, radical expressions and functions, complex numbers, sequences and series, conic sections, parametric and polar equations, vectors, and matrices.

The course concludes with problem solving, ratios, percentages, units, statistics, and probability. Students solve problems involving ratios, unit rates, proportional relationships, percentages, unit conversions, sampling, data displays, frequency tables, measures of center and spread, correlation, probability, compound events, combinatorics, and discrete random variables.

Preliminaries

• Add, subtract, multiply, and divide rational numbers.
• Solve problems using the four operations on rational numbers.
• Use additive inverses, absolute value, reciprocals, and fractions of fractions.
• Evaluate expressions involving positive, zero, negative, and fractional exponents.
• Apply the product, quotient, power, power-of-product, and power-of-quotient rules for exponents.
• Write, compare, add, subtract, multiply, and divide numbers in scientific notation.
• Evaluate square roots, cube roots, surds, and radical expressions.
• Simplify, compare, add, subtract, and rationalize radical expressions.
• Convert between radical notation and fractional exponent notation.
• Apply order of operations to expressions involving rational numbers, exponents, radicals, and parentheses.
• Classify natural numbers, integers, rational numbers, irrational numbers, and real numbers.
• Use divisibility rules, prime factorization, greatest common factors, and lowest common multiples.
• Interpret conditional statements and related statements, including converses, inverses, and contrapositives.
• Identify necessary and sufficient conditions.
• Construct, evaluate, simplify, and factor algebraic expressions.
• Collect like terms and apply the distributive law.

Algebra - Linear Equations in One Variable

• Solve one-step, two-step, and multi-step linear equations.
• Solve equations with fractional coefficients, decimal coefficients, variables on both sides, binomials, and rational expressions.
• Solve equations using cross-multiplication.
• Solve two-variable equations for a specified variable.
• Identify equations with one solution, no solution, or infinitely many solutions.
• Solve equations with unknown coefficients and unknown parameters.
• Rearrange many-variable equations and formulas.
• Interpret coefficients and constants in linear expressions.
• Model and solve contextual problems with linear equations.
• Solve consecutive integer, speed-distance-time, work, and mixture problems.

Algebra - Linear Equations in Two Variables

• Use the Cartesian coordinate system.
• Find horizontal and vertical distances between points.
• Identify solutions to two-variable linear equations.
• Graph linear equations using tables and points.
• Work with horizontal and vertical lines.
• Calculate slopes of straight lines.
• Write and interpret equations of lines in slope-intercept, point-slope, and standard form.
• Determine slopes and intercepts from line equations.
• Identify and write equations of parallel and perpendicular lines.
• Model real-world situations with two-variable linear equations.
• Interpret coefficients, unit rates, intercepts, distance-time graphs, and speed-time graphs.
• Solve systems of linear equations by graphing, substitution, and elimination.
• Solve systems with fractional and decimal coefficients.
• Identify systems with one solution, no solution, or infinitely many solutions.
• Determine unknown parameters in systems of equations.
• Calculate intersections of lines.
• Solve larger systems and systems by identifying structure.
• Model contextual problems using systems of linear equations.

Algebra - Linear Inequalities

• Represent inequalities and compound inequalities on number lines.
• Verify solutions of linear inequalities.
• Solve one-step, two-step, and advanced linear inequalities.
• Solve inequalities that require reversing the inequality sign.
• Solve compound inequalities.
• Use interval notation, unbounded intervals, unions, and intersections.
• Optimize sums, differences, products, and quotients under constraints.
• Graph strict and non-strict two-variable linear inequalities.
• Graph systems of linear inequalities.
• Model real-world situations with one-variable and two-variable inequalities.
• Interpret feasible regions and optimal solutions.

Geometry - Points, Lines, & Angles

• Use appropriate units for area and volume.
• Identify points, lines, rays, segments, parallel lines, and perpendicular lines.
• Measure and calculate segment lengths.
• Use midpoint properties and the midpoint formula.
• Apply the segment addition postulate.
• Measure and classify angles.
• Use right, straight, full, null, acute, obtuse, reflex, congruent, and vertical angles.
• Solve problems involving complementary and supplementary angles.
• Use corresponding, alternate, and consecutive angle relationships.
• Apply angle criteria for parallel lines.
• Use angle bisectors, segment bisectors, and perpendicular bisectors.
• Translate, rotate, reflect, dilate, and stretch figures.
• Combine geometric transformations.
• Analyze reflective and rotational symmetry.

Geometry - Polygons & Solids

• Use triangle angle relationships, exterior angles, and the exterior angle theorem.
• Classify triangles by side lengths and angle measures.
• Apply the triangle inequality.
• Use the isosceles triangle theorem.
• Use heights, medians, and centroids of triangles.
• Apply the Pythagorean theorem.
• Use properties of 45-45-90 and 30-60-90 triangles.
• Calculate areas of special right triangles and equilateral triangles.
• Work with diagonals of squares.
• Solve compound right-triangle problems.
• Identify polygons and their parts.
• Calculate interior and exterior angles of polygons.
• Work with congruent polygons, regular polygons, quadrilaterals, parallelograms, rectangles, squares, and trapezoids.
• Identify three-dimensional shapes.
• Count faces, vertices, and edges of polyhedra.
• Apply Euler’s formula.
• Use nets to identify solids and calculate surface areas.

Geometry - Congruence & Similarity

• Apply ASA, AAS, SAS, SSS, and HL congruence criteria.
• Combine congruence criteria to prove or identify triangle congruence.
• Identify corresponding parts of congruent triangles.
• Reason about information needed to prove triangle congruence.
• Determine similarity ratios between similar polygons.
• Use corresponding side lengths and angles in similar polygons.
• Use area relationships in similar polygons.
• Work with scale models and areas between similar shapes.
• Apply AA, SSS, and SAS similarity criteria.
• Combine similarity criteria to identify similar triangles.
• Use corresponding angles and side lengths in similar triangles.
• Apply the midpoint theorem.
• Apply the triangle proportionality theorem.
• Use surface area and volume ratios for similar solids.

Geometry - Length, Area & Volume

• Calculate perimeters of polygons.
• Calculate areas of rectangles, squares, parallelograms, triangles, trapezoids, and composite shapes.
• Calculate areas between shapes.
• Solve border problems.
• Construct functions representing areas and perimeters of two-dimensional shapes.
• Use the midpoint formula and distance formula in the coordinate plane.
• Use the distance formula in three dimensions.
• Calculate perimeters and areas in the coordinate plane.
• Analyze inscribed shapes, including circles and squares, squares and circles, equilateral triangles and circles, and triangles inscribed in rectangles.
• Calculate volumes, surface areas, face diagonals, and space diagonals of cubes.
• Calculate surface areas and volumes of rectangular solids.
• Calculate volumes and surface areas of cylinders, spheres, cones, and pyramids.
• Reason about inscribed spheres and cubes.
• Construct functions representing volumes and surface areas.
• Find volumes and surface areas of composite solids and combined prisms.

Geometry - Circles

• Use circle terminology involving interiors, exteriors, arcs, sectors, and segments.
• Calculate circumference, area, radius, and diameter.
• Solve problems involving arcs, sectors, and segments.
• Use central angles, inscribed angles, and Thales’ theorem.
• Analyze inscribed quadrilaterals.
• Use tangent, secant, chord, and circle theorem relationships.
• Use equal chords, perpendicular bisectors of chords, tangents, and radii.
• Work with equations of circles in the coordinate plane.
• Find centers and radii of circles from equations.
• Complete the square to determine circle properties.
• Find intercepts and intersections involving circles.
• Write equations of tangent lines to circles.
• Convert between degrees and radians.
• Use radians to calculate arc length and sector area.

Trigonometry

• Use right-triangle trigonometric ratios.
• Solve right triangles using sine, cosine, and tangent.
• Apply inverse trigonometric functions to find angle measures.
• Model real-world situations with right-triangle trigonometry.
• Use the unit circle to represent angles in degrees and radians.
• Work with negative, coterminal, reference, quadrantal, and special angles.
• Extend trigonometric ratios to all quadrants.
• Use reciprocal trigonometric ratios.
• Apply the Pythagorean identity.
• Use the law of sines and law of cosines.
• Calculate areas of general triangles.
• Graph sine, cosine, tangent, cotangent, secant, and cosecant functions.
• Transform trigonometric functions using translations, stretches, reflections, and phase shifts.
• Determine amplitude, period, midline, range, extrema, zeros, domains, and asymptotes of transformed trigonometric functions.
• Use trigonometric identities.
• Apply double-angle, sum, and difference formulas.
• Evaluate inverse trigonometric functions.

Advanced Algebra - Functions

• Determine whether a relation is a function.
• Use function notation and evaluate functions.
• Determine domain and range.
• Identify global and local extrema, roots, intercepts, increasing and decreasing intervals, and end behavior.
• Analyze one-to-one functions, inverse functions, and invertible intervals.
• Work with piecewise functions.
• Calculate and interpret average rates of change.
• Construct and analyze linear functions.
• Add, subtract, multiply, divide, and compose functions.
• Transform functions using translations, stretches, compressions, and reflections.
• Determine points, extrema, domains, and ranges of transformed functions.
• Compare functions across representations.

Advanced Algebra - Absolute Value

• Evaluate and simplify absolute value expressions.
• Apply properties of absolute value.
• Solve absolute value equations and inequalities.
• Identify no-solution and extraneous-solution cases.
• Graph and transform absolute value functions.
• Determine domains, ranges, and roots of absolute value functions.
• Solve equations involving absolute value functions and linear functions.

Advanced Algebra - Polynomials

• Identify, classify, simplify, add, subtract, and multiply polynomials.
• Determine degree, leading term, leading coefficient, and constant term.
• Factor polynomials using greatest common factors, differences of squares, perfect square trinomials, grouping, and trinomial factoring.
• Solve quadratic equations by factoring, square-root methods, completing the square, and the quadratic formula.
• Use the discriminant to determine the number and type of solutions.
• Determine unknown parameters in quadratic equations.
• Graph and transform quadratic functions.
• Analyze vertices, axes of symmetry, roots, intercepts, domains, ranges, and extrema.
• Solve systems involving quadratic functions and lines.
• Model vertical motion, revenue, cost, profit, and break-even points with quadratic functions.
• Factor and solve higher-order polynomial equations.
• Apply the factor theorem, remainder theorem, rational roots theorem, and multiplicity of roots.
• Graph polynomial functions using roots, multiplicities, leading coefficients, and end behavior.

Advanced Algebra - Exponential Equations & Functions

• Apply exponent rules to exponential expressions.
• Solve exponential equations.
• Model exponential growth and decay.
• Interpret exponential expressions, growth rates, decay rates, and outputs.
• Compare linear and exponential change.
• Determine whether a model changes linearly or exponentially.
• Graph and transform exponential functions.
• Determine domains, ranges, intercepts, asymptotes, roots, extrema, and end behavior of exponential functions.
• Convert between exponential and logarithmic forms.
• Evaluate common and natural logarithms.
• Apply logarithm laws and the change-of-base formula.
• Solve logarithmic equations.
• Graph and transform logarithmic functions.
• Determine domains, ranges, roots, and asymptotes of logarithmic functions.

Advanced Algebra - Rational and Radical Equations

• Simplify rational expressions using factoring and common factors.
• Add, subtract, multiply, and divide rational expressions.
• Split rational expressions into separate terms.
• Solve rational equations.
• Identify restrictions and check solutions involving denominators.
• Graph reciprocal and rational functions.
• Transform rational functions.
• Determine domains, ranges, asymptotes, inverses, and intersections of rational functions.
• Simplify radical expressions.
• Apply product and quotient rules for radicals.
• Rationalize denominators.
• Solve radical equations.
• Graph and transform square root, cube root, and radical functions.
• Determine domains, ranges, roots, inverses, and intersections of radical functions.

Advanced Algebra - Complex Numbers

• Evaluate square roots of negative numbers.
• Identify real and imaginary parts of complex numbers.
• Add, subtract, multiply, and divide complex numbers.
• Use complex conjugates.
• Solve quadratic equations with imaginary or complex roots.
• Evaluate powers of the imaginary unit.
• Plot and interpret complex numbers in the complex plane.
• Find moduli and arguments of complex numbers.

Advanced Algebra - Sequences & Series

• Identify and evaluate terms of sequences.
• Use sequence notation and function notation.
• Define and evaluate recursive sequences.
• Analyze arithmetic sequences.
• Write explicit and recursive formulas for arithmetic sequences.
• Translate between explicit and recursive formulas.
• Analyze geometric sequences.
• Write explicit and recursive formulas for geometric sequences.
• Use sigma notation.
• Calculate sums of arithmetic series.
• Calculate sums of finite geometric series.
• Write arithmetic and geometric series in sigma notation.
• Model contextual and financial situations with sequences and series.

Advanced Algebra - Conic Sections

• Write and interpret equations of ellipses.
• Determine centers, vertices, co-vertices, axes, and foci of ellipses.
• Graph ellipses from equations.
• Write and interpret equations of hyperbolas.
• Determine centers, vertices, transverse axes, conjugate axes, foci, and asymptotes of hyperbolas.
• Graph hyperbolas from equations.

Advanced Algebra - Parametric & Polar Equations

• Evaluate and graph parametric equations.
• Convert between parametric and rectangular representations in simple cases.
• Interpret motion described parametrically.
• Plot points in polar coordinates.
• Convert between polar and rectangular coordinates.
• Interpret polar-coordinate representations.

Advanced Algebra - Vectors & Matrices

• Represent vectors geometrically.
• Add and subtract vectors.
• Multiply vectors by scalars.
• Work with two-dimensional and three-dimensional Cartesian vectors.
• Calculate vector magnitudes and components.
• Use unit vectors.
• Apply the dot product.
• Interpret angles and projections using vectors.
• Use matrices to organize data.
• Add, subtract, and scale matrices.
• Multiply matrices using the row-column rule.
• Use determinants in two-by-two matrix contexts.

Problem Solving - Ratios & Percentages

• Simplify, compare, and interpret ratios.
• Divide quantities in ratios.
• Work with equivalent ratios.
• Find and interpret unit rates.
• Solve proportional relationship problems.
• Use direct variation and proportional models.
• Solve percent problems involving percent of a quantity, percent change, reverse percentages, discounts, markups, tax, tip, commission, interest, and successive percentage changes.
• Determine units in formulas.
• Convert between units of length, area, volume, time, speed, density, and other compound measures.
• Use unit analysis to guide multi-step solutions.

Data Analysis - Statistics & Probability

• Identify statistical questions, populations, samples, and sampling methods.
• Understand bias, sample size, and reliability.
• Summarize data using measures of centrality and spread.
• Use mean, median, mode, range, quartiles, interquartile range, mean absolute deviation, and standard deviation.
• Represent data using frequency tables, dot plots, box plots, histograms, and other displays.
• Compare distributions using center, spread, shape, skew, and outliers.
• Interpret scatter plots and bivariate data.
• Identify correlation, trends, and regression models.
• Estimate and interpret trend lines.
• Calculate and interpret probabilities of single events.
• Use complements, unions, intersections, and Venn diagrams.
• Solve probability problems involving compound events.
• Use conditional probability and independent events.
• Apply counting principles, permutations, and combinations.
• Work with discrete random variables and probability distributions.