This is an analysis of the most common math questions on the last 10 released ACTs (through April 2024).

*These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.*

**Fractions and Decimals**– All four operations. Mixed numbers.**Exponents**– All operations. Fractional and negative exponents are very common too (see below).**Probability**– Know the basic part:whole versions. There is usually a harder one also (like one with two events).**Negatives**– Be comfortable with all operations.**Average**– Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).**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.**Quadratic skills**– Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.**Ratio**– Part:part, part:whole.**Area/Perimeter of basic shapes**– Triangles, rectangles, circles.**Percents**– Know all basic variations. More advanced ones are common also.**Absolute Value**– Sometimes basic arithmetic, sometimes an algebraic equation or inequality.**Picking Numbers**– You never have to use this but it will be a useful option on every test.**Plug in answers**– Like picking numbers, it’s not required but it’s often helpful.**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.**Radicals**– Basic operations. Translate to fractional exponents.

*(> 80% chance)*

**FOIL**– This has to be automatic.**SOHCAHTOA**– Every variation of right triangle trig, including word problems.**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?**Average sum trick**– 5 tests, average is 80. After the 6th test, the average is 82. What was the 6th test score?**MPH**– The concept of speed in miles per hour is very common (sometimes combined with other conversion).**System of Equations**– Elimination. Substitution. Word problems.**Composite function**– As in g(f(x)).**Order of operations**– Sometimes directly tested, other times part of a harder question.**Pythagorean Theorem**– Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).**Time**– Hours to minutes, minutes to seconds, time elapsed.**Angle chasing**– 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.**LCM**– Straight up. In word problems. In algebraic fractions.**Imaginary numbers**– Powers of i. What is i^2? The complex plane.**Negative exponents**– Know what they do and how to combine them with other exponents.**Very Likely***(>50% chance)***Factoring**– Mostly the basics. Almost never involves a leading coefficient.**Fractional Exponents**– Rewrite radicals as fractional exponents and vice versa.**Logarithms**– Rewrite in exponential form. Basic operations.**Mixed Numbers**– all four operations. Often combined with word problems.**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?”**Matrices**– Adding, subtracting, multiplying. Knowing when products are possible.**Venn**– There are 30 kids. 18 are in Algebra. 20 are in French. How many are in both?**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?**Algebra LCD**– Find the lowest common denominator, then combine the numerators.**Geometric sequence**– You usually just need to find a subsequent term (not the formula).**Change the base**– If 9^x = 27^5, what is x?**Given points, find equation**– You’re given two ordered pairs and must find the linear equation.**Worth Knowing***(> 25% chance)***Apply formula**– they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.**Domain**– Usually you can think of it as “possible x values”.**Given sine, find cosine**– They give you one trig ratio and ask you to find another. As in, “If the sine of x is 4/5, what is the cosine?”**Law of Cosines**– They almost always give you the formula. Then you just have to plug things in.**Midpoint**– Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.**Scientific notation**– Go back and forth between standard and scientific notation. All four operations.**Shaded area**– The classic one has a square with a circle inside.**Similar triangles**– Relate the sides with a proportion.**Weird shape area**– It’s an unusual shape but you can use rectangles and triangles to find the area.**Conjugates**– Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.**Difference of two squares**– (x + y)(x – y) = x^2 – y^2**Graph translations**– Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 – 5 right away and know exactly what to do.**Multistep conversion**– For example, they might give you a mph and a cost/gallon and then ask for the total cost.**Parallelogram**– Know that adjacent angles add to 180. Area formula.**Prime numbers**– Usually combined with something else, like basic probability.**Probability with “not”**– 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?**Undefined**– You can’t have 0 in the denominator.**Special right triangles**– 30:60:90, 45:45:90.**Amplitude**– If y = 5 sin(x) + 2, what is the amplitude?**Arithmetic sequence**– Usually asks you to find a specific term, sometimes asks you to find the formula.**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?**Volume of a prism**– Know that the volume = area of something x height. Sometimes the base will be a weird shape.**Circle equations**– (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.**Compare numbers**– Be able to order square roots, decimals, and fractions.**Find inverse function**– Switch y and x, then isolate y.**Permutation**– You have 5 plants and 3 spots. How many ways can you arrange them?**Line of best fit**– They’ll sometimes ask you to find the predicted value, or the difference between the predicted and actual values.**Linear inequality**– Be comfortable solving algebraic inequalities. Graphs appear sometimes also.**Triangle opposite side rule**– There is a relationship between an angle and the side across from that angle.**Inverse trig**– Use right triangle ratios to find angles.**Toy Soldier (volume)**– What happens to the height of the water when you drop an object in the bucket?**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.**Periodic function graph**– The basics of sine and cosine graphs (shifts, amplitude, period).**Value/frequency table –**Find the median and mean in this format.

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