The ACT Math section has questions in minutes, so students need fast recall of common formulas and problem types. This cheat sheet brings together the algebra, geometry, trigonometry, statistics, and probability ideas that appear most often. It is designed as a quick reference for review, practice, and last-minute studying before test day.
Core ACT Math skills include solving equations, graphing lines and parabolas, using right triangle relationships, and applying area and volume formulas. Students should know slope-intercept form, the quadratic formula, special right triangles, circle formulas, and basic probability rules. Pacing matters because easier questions usually come earlier, while later questions often combine several concepts in one problem.
Key Facts
- The ACT Math section has questions in minutes, so the average pace is minute per question.
- Slope is , and slope-intercept form is .
- The quadratic formula is for equations in the form .
- The distance between two points is , and the midpoint is .
- For right triangles, the Pythagorean theorem is , and common triples include -- and --.
- Special right triangles have side ratios -- and --.
- Circle formulas include circumference , area , and arc length .
- Probability is , and independent events multiply as .
Vocabulary
- Slope
- Slope measures the steepness of a line and is found using .
- Intercept
- An intercept is a point where a graph crosses an axis, such as the -intercept in .
- Discriminant
- The discriminant is and tells how many real solutions a quadratic equation has.
- Median
- The median is the middle value in an ordered data set, or the average of the two middle values when there is an even number of values.
- Function
- A function is a rule that assigns each input exactly one output, often written as .
- Complementary Probability
- Complementary probability uses to find the chance that an event does not happen.
Common Mistakes to Avoid
- Using the wrong sign in slope, such as , is wrong because the order of subtraction must match in the numerator and denominator.
- Forgetting parentheses in the distance formula, such as writing , is wrong because each coordinate difference must be squared as a whole: .
- Using diameter instead of radius in is wrong because the formula requires the radius, which is half the diameter.
- Assuming every triangle is a right triangle is wrong because applies only to right triangles.
- Spending too long on one hard ACT question is a mistake because every question has the same point value, so guessing and returning later often protects your score.
Practice Questions
- 1 A line passes through and . Find its slope and write the equation in the form .
- 2 Solve using factoring or the quadratic formula.
- 3 A circle has radius . Find its circumference and area in terms of .
- 4 On the ACT, why can it be better to skip a difficult question temporarily instead of spending minutes trying to solve it?
Understanding ACT Math Reference
Fast ACT math work depends on recognizing structure before doing calculations. In algebra, first decide what the problem is asking you to find. Then identify the relationship that controls it.
A linear relationship changes by a constant amount. A quadratic relationship often has a highest or lowest point and can cross the horizontal axis up to two times. Factoring is usually quicker than a general solving method when whole-number factors are visible.
Substitution works well when one equation already isolates a variable. Elimination works well when two equations line up neatly.
Check signs carefully, especially when distributing a negative number or squaring a negative value. These small errors change an otherwise correct solution.
Graphs turn equations into pictures. A line can be understood through its starting height and its rate of rise or fall. A positive slope rises from left to right, while a negative slope falls.
For parabolas, pay attention to whether the graph opens up or down, where its turning point sits, and where it meets the axes. Coordinate geometry questions often connect algebra with geometry. The distance formula comes from the Pythagorean theorem because horizontal and vertical changes form the legs of a right triangle.
A midpoint is simply the average location between two points. In real life, these ideas describe travel routes, ramps, profit graphs, and the position of objects on maps.
Geometry rewards careful reading of units and diagrams. A drawing may not be to scale, so use the stated facts rather than appearances. Area measures the space inside a flat figure.
Volume measures the space inside a solid. Keep track of square units for area and cubic units for volume. In circle problems, decide whether the question concerns distance around the edge, space inside the circle, or only part of the circle.
Trigonometry connects an angle in a right triangle to side lengths. Use sine, cosine, or tangent only after labeling the side across from the angle, the side next to the angle, and the longest side. Special triangles save time because their side patterns are fixed.
Statistics and probability questions test interpretation as much as arithmetic. The mean can be pulled strongly by one extreme value, while the median is the middle value after data are ordered. A graph can be misleading when its scale begins above zero or when categories have unequal widths.
For probability, list the possible outcomes when the sample space is small. Decide whether events affect each other. Drawing cards without replacement changes the second probability because the total number of cards changes.
Test strategy matters throughout. Estimate before calculating, use answer choices to test a result when appropriate, and move on when a method is taking too long.
Mark hard questions for a later pass. A calm check of units, signs, and reasonableness catches many mistakes.