top 75 math question types
With the latest MAK, I was able to put together a top 75 list for ACT Math. We still don't have a ton of material (the online tests, the other ones in the red book, the two MAKs), but it's enough to get a sense. For example, these ones made the list for the first time: Binomial expansion, Hyperbola, Irrational, Find inverse function.
1. Fractions and Decimals - All four operations. Mixed numbers.
2. Area / Perimeter of basic shapes - Triangles, rectangles, circles.
3. Probability - Know the basic part:whole versions. There is usually a harder one also (like one with two events).
4. Ratio - Part:part, part:whole.
5. Linear Equations / Slope - Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
6. Quadratic skills - Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
7. Plug In - You have a value or a point that needs to be plugged into an equation or function. Distinct from “Plug in answers” (using the choices).
8. Average - Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
9. Factoring - Mostly the basics. Almost never involves a leading coefficient.
10. Solving Equations - Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
11. Functions - Function notation (f(x), g(x), etc.) and basic evaluation. Show up across many question types.
12. Conversion - Watch out for ones that involve powers. For example, they might give you the conversion for yards to miles, but then ask about square miles.
13. Angle chasing - 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
14. Exponents - All operations. Fractional and negative exponents are very common too (see below).
15. MPH - The concept of speed in miles per hour is very common (sometimes combined with other conversion).
16. Percents - Know all basic variations. More advanced ones are common also.
17. Radicals - Basic operations. Translate to fractional exponents.
18. SOHCAHTOA - Every variation of right triangle trig, including word problems.
19. Arithmetic sequence - Usually asks you to find a specific term, sometimes asks you to find the formula.
20. Fractional Exponents - Rewrite radicals as fractional exponents and vice versa.
21. Graph translations - Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
22. Imaginary numbers - Powers of i. What is i^2? The complex plane.
23. Negatives - Be comfortable with all operations.
24. Probability, two events - If there’s a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
25. Similar triangles - Relate the sides with a proportion.
26. Number Properties - Properties of integers: even/odd, divisibility, factors, multiples, primes.
27. Negative exponents - Know what they do and how to combine them with other exponents.
28. Picking Numbers - You never have to use this but it will be a useful option on every test.
29. Plug in answers - Like picking numbers, it’s not required but it’s often helpful.
30. Shaded area - The classic one has a square with a circle inside.
31. Triangle opposite side rule - There is a relationship between an angle and the side across from that angle.
32. Weird shape area - It’s an unusual shape but you can use rectangles and triangles to find the area.
33. Periodic function graph - Reading or interpreting graphs of sin/cos or other periodic functions. Watch for amplitude and period.
34. Weighted average - Average where each value is multiplied by its weight or frequency before summing.
35. Average sum trick - 5 tests, average is 80. After the 6th test, the average is 82. What was the 6th test score?
36. Composite function - As in g(f(x)).
37. Expected value - There is a 0.3 chance of winning $100 in Game A and 0.2 chance of winning $200 in Game B, which is unrelated to Game A. If you place bets on both games, what is the expected value of your bets?
38. Logarithms - Rewrite in exponential form. Basic operations.
39. Matrices - Adding, subtracting, multiplying. Knowing when products are possible.
40. Median - Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
41. Special right triangles - 30:60:90, 45:45:90.
42. System of Equations - Elimination. Substitution. Word problems.
43. Use the radius - A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
44. Venn - There are 30 kids. 18 are in Algebra. 20 are in French. How many are in both?
45. Circle equations - (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
46. Conjugates - Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
47. Difference of two squares - (x + y)(x - y) = x^2 - y^2
48. Given points, find equation - You’re given two ordered pairs and must find the linear equation.
49. LCM - Straight up. In word problems. In algebraic fractions.
50. Law of Cosines - They almost always give you the formula. Then you just have to plug things in.
51. Linear inequality - Be comfortable solving algebraic inequalities. Graphs appear sometimes also.
52. Midpoint - Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
53. Multistep conversion - For example, they might give you a mph and a cost/gallon and then ask for the total cost.
54. Pythagorean Theorem - Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
55. Remainders - Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
56. Amplitude - The height of a periodic function from midline to peak. Usually extracted from an equation or graph.
57. Arc length - Length of a portion of a circle’s circumference. Usually involves setting up a proportion with the central angle.
58. Binomial expansion - Expand expressions like (a+b)^n. May use Pascal’s Triangle or the binomial theorem.
59. Circle wedge - Sector area or arc length using the central angle / 360 (or radians) proportion.
60. Hyperbola - Recognize the standard form of a hyperbola equation and its key features. You often can solve by plugging in values or looking at the graph.
61. Identify function - “Which equation matches this graph?” — most often an exponential or shifted parabola.
62. Irrational - Recognize or reason about irrational numbers (e.g., non-terminating non-repeating decimals, square roots of non-perfect squares).
63. Polynomial remainder theorem - If you know that (x - 5) is a factor of a polynomial, you can plug in 5 and set it equal to 0 in order to solve for a constant.
64. Absolute Value Equation - Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
65. Algebra LCD - Find the lowest common denominator, then combine the numerators.
66. Domain - Usually you can think of it as “possible x values”.
67. FOIL - This has to be automatic.
68. Find inverse function - Switch y and x, then isolate y.
69. Mixed Numbers - All four operations. Often combined with word problems.
70. Permutation - You have 5 plants and 3 spots. How many ways can you arrange them?
71. Prime numbers - Usually combined with something else, like basic probability.
72. Scientific notation - Go back and forth between standard and scientific notation. All four operations.
73. Time - Hours to minutes, minutes to seconds, time elapsed.
74. Value/frequency table - Find the median and mean in this format.
75. Congruent - Know that congruent shapes must have the same angles and the same size (not simply proportional).
Notable Omissions: Geometric sequence, Inverse trig, Law of Sines, Ellipses. These all appeared once in my sample, but they’ve appeared on previous Top 75 lists. There's also a very long tail of topics that have appeared once or twice so far but might be more common going forward.