ACT Math study guide · Chapter 46
Question Archetype Index
The same underlying skills reappear across the ACT Math section in a limited number of recognizable forms. This index lists those forms, what each one looks like on the page, the method that solves it efficiently, and the specific error each one is designed to catch. Use it as a reference during practice, not as new material to memorize; every entry points to the chapter that teaches the skill in full.
Translating words into an equation. Looks like a word problem describing a rate, a percentage, a mixture, or a geometric relationship in prose. Define a variable for the unknown and translate each stated relationship into an equation before calculating anything. The common error is solving for an intermediate quantity instead of the one the question actually asks for. Taught in Rates, Percentages, and Proportions.
Reverse average or missing value. Looks like a mean given for a set of values with one value missing. Recover the total by multiplying the mean by the count, then subtract the known values. The common error is treating the mean itself as if it were the total. Taught in Center and Spread and Averages in Context.
Percent change. Looks like a value that increases or decreases by a percentage, sometimes more than once in sequence. Multiply by (1 plus or minus the rate) for each change, in order. The common error is assuming a percent increase and an equal percent decrease cancel out; they do not, because the second change applies to a different base value. Taught in Rates, Percentages, and Proportions.
System intersection. Looks like two equations, or two conditions described in words, that must both be true at once. Choose substitution, elimination, or graphing based on which is fastest for the given coefficients. The common error is solving only one equation and treating that as the full answer. Taught in Linear Equations and Systems.
Function notation and composition. Looks like f(x), g(x), a table of input-output pairs, or a composition such as f(g(x)). Evaluate from the inside out, treating each function strictly as a rule that transforms an input into an output. The common error is treating f(x) as f multiplied by x. Taught in Function Notation and Composition.
Graph transformation or feature. Looks like a shifted, reflected, or scaled version of a familiar graph, or a request for a zero, an extremum, or an asymptote. Track what happened to specific points or features instead of replotting the whole graph. The common error is reversing the direction of a horizontal shift. Taught in Function Transformations and Piecewise Functions.
Quadratic roots and features. Looks like an equation or graph asking for roots, the vertex, or the relationship between coefficients and solutions. Match the given form, standard, factored, or vertex, to the fastest available relationship instead of expanding first by default. The common error is a sign error while factoring or applying the quadratic formula. Taught in Quadratic Equations.
Exponent or radical equivalence. Looks like an expression to rewrite, or an equation to solve, involving exponents or radicals. Apply exponent rules to combine or simplify terms before solving. The common error is adding exponents when the bases should be multiplied together, or the reverse. Taught in Exponents and Radicals and Other Types of Equations.
Matrix operation. Looks like two small matrices to add, multiply, or interpret. Check the dimensions before multiplying, and multiply row by column instead of entry by entry. The common error is multiplying matrices as though they were ordinary numbers. Taught in Vectors and Matrices.
Complex-number simplification. Looks like a power of i, or arithmetic with complex numbers. Reduce a power of i using its remainder when divided by 4, and multiply by the conjugate to divide. The common error is treating i² as 1 instead of −1. Taught in Real and Complex Numbers.
Triangle similarity in disguise. Looks like lengths embedded in a diagram involving shadows, parallel lines, or nested triangles, without the word similar appearing anywhere. Establish which angles are equal first, then set up the ratio between corresponding sides. The common error is pairing sides that do not actually correspond. Taught in Congruence and Similarity.
Right-triangle trigonometry. Looks like an angle and two sides in a right triangle, sometimes with elevation or depression language. Label opposite, adjacent, and hypotenuse relative to the given angle before choosing a ratio. The common error is measuring a side relative to the wrong angle. Taught in Right Triangles and Trigonometry.
Law of Sines or Law of Cosines. Looks like a triangle with no right angle, given some combination of sides and angles. Use the Law of Sines when a side and its opposite angle are both known; use the Law of Cosines when two sides and the included angle, or all three sides, are known. The common error is applying right-triangle trigonometry to a triangle that has no right angle. Taught in Trigonometry Beyond Right Triangles.
Circle arc, sector, or tangent. Looks like a circle diagram with a central angle, a tangent line, or a request for an arc length or a sector area. Convert the central angle to a fraction of 360° first, then apply that same fraction to the circumference or the area. The common error is confusing which fraction applies to length and which applies to area; both use the identical fraction of the full circle. Taught in Circle Theorems and Circles and Conic Sections.
Conic recognition. Looks like a graph or a standard-form equation of a circle, ellipse, parabola, or hyperbola. Identify the center first, then use the signs and the relative sizes of the denominators to determine the type and orientation. The common error is confusing the standard forms for an ellipse and a hyperbola. Taught in Circles and Conic Sections.
Counting and sample spaces. Looks like a question about arrangements, selections, or a multi-stage process, sometimes with a restriction. Decide whether order matters before choosing between a permutation and a combination. The common error is double-counting arrangements that are actually identical. Taught in Probability.
Probability from a table or a restricted subset. Looks like a table, a described sample space, or a condition that restricts which outcomes count. Identify the eligible group first, and use only that group as the denominator. The common error is dividing by the total sample instead of the restricted subgroup the condition describes. Taught in Probability.
Distribution, box plot, or statistical summary. Looks like a graphical distribution, a box plot, or a request to compare center and spread across data sets. Read only what the display actually shows before drawing a conclusion. The common error is inferring a shape or a specific value that the display does not actually support. Taught in Center and Spread.
Unit or dimensional reasoning. Looks like a rate, a geometry problem, or a comparison that mixes units, such as feet and inches or minutes and hours. Convert every quantity to the same unit before combining them, and track the units through each step. The common error is combining quantities in different units without converting first. Taught in Measurement and Unit Conversion.
Testing the answer choices directly. Looks like a question where the algebra to solve directly is more awkward than checking a specific number, or where the answer choices are themselves expressions in the same variable. Full treatment, including when this method is not the fastest choice, is in Backsolving, Plugging In, and Estimating.