ACT Math: Study Guide & Free Lessons
Short lessons, worked examples, and practice that connects.
Numbers, vectors & matrices
Choose a representation before calculating. Fractions, powers, complex numbers and arrays encode different operations.
Equations with restrictions
An algebraic solution must satisfy both the equation and any stated restrictions. Squaring or dividing can introduce or remove candidates.
Functions as input rules
Function notation names the output produced from an input. Composition applies one rule and then another; an inverse reverses a one-to-one rule.
Read quadratic structure
Standard, factored and vertex forms make different features visible. The requested feature determines which representation is most efficient.
Coordinate geometry connections
Coordinates connect geometric relationships to algebra. Distance uses squared differences; a midpoint averages coordinates; slope compares signed changes.
Geometry with dimensions
Area measures a surface and volume measures three-dimensional space. Dimension and units help distinguish a formula from a similar-looking one.
Right-triangle trigonometry
A trigonometric ratio compares two sides relative to a specified acute angle. The hypotenuse is always opposite the right angle.
Summaries from data
A data summary preserves some information and discards other information. A mean uses every value; a median depends on sorted position.
Count the sample space
A probability uses favorable outcomes divided by all equally likely outcomes in the relevant sample space. Conditioning changes that space.
Rates, scales & mixtures
Rates connect unlike units while ratios compare quantities. A mixture calculation combines the amount of an ingredient, not merely the percentages.
Systems and practical constraints
Two equations describe conditions that must hold at the same time. Their solution is a shared intersection rather than a separate answer to each equation.
Patterns, counting & reasoning
A pattern becomes useful when its rule is stated precisely. Distinguish repeated addition, repeated multiplication, and arrangements with or without order.
Vectors and matrix structure
A vector stores directed components; a matrix stores an ordered rectangular array. Position and operation both matter.
Rational powers and logarithmic inverses
An exponent describes repeated multiplicative structure. A logarithm asks for an exponent, while a negative exponent asks for a reciprocal.
Track points and domains through transformations
A function transformation changes input coordinates and output coordinates in different ways. A point must satisfy the new rule, not merely resemble the old graph.
Polynomial structure and rational restrictions
Coefficients, factors and roots describe the same polynomial from different perspectives. The most useful form depends on the feature being requested.
Read circles, ellipses and hyperbolas
Conic equations encode centers, lengths and directions. Squared distances reveal a center after completing squares. Denominators in standard forms are squared lengths.
Surfaces, solids and composite regions
Surface area measures a boundary; volume measures interior capacity. A closed cylinder includes two end disks as well as a curved surface.
Angles beyond one right triangle
The unit circle extends trigonometric ratios beyond acute angles. Symmetry, periodicity and the identity sin²θ+cos²θ=1 constrain values.
Sampling, response and fair comparisons
A numerical comparison needs a design. Sampling, random assignment and response describe different stages, and each limits the claims we can make.
Check predictions with residuals
A fitted model supplies predictions, not guaranteed observations. A residual is observed minus predicted and can be positive or negative.
Frequency, spread and distribution summaries
A frequency table represents repeated observations. A box plot summarizes ordered positions, while standard deviation measures spread about the mean.
Overlaps, paths and expected value
Compound events require attention to overlap and to which outcomes are being conditioned on. Expected value is a probability-weighted long-run average, not a promised single outcome.
Reverse models and test their structure
A model connects quantities using assumptions. Reversing successive percent changes, combining work rates and rounding capacity each require a different operation.
400 parameterized questions across 96 families, not 400 independent designs
A clear route through the material
Follow the topics in order or use the prerequisites to choose a starting point. Read the worked example before focused practice, then use review cards to revisit the idea.
Continue your preparation
Scope: Enhanced ACT Mathematics foundations