# ACT vs SAT Math – 6 Types of Problems Unique to the ACT

While the ACT Math Section covers more advanced math topics than the SAT does and requires a fast working pace, its arguable a more straightforward test overall. SAT Math questions can be convoluted and difficult to decipher. For that reason alone, there are many that prefer the ACT over the SAT.

Having said this, I find that a number of students can learn the unique language and style of the SAT questions and master them. That requires at least a couple months of studying. Once they do figure out the test, they can earn a great score.

## ACT Math vs SAT Math – Which One is Best For You?

Which should you pick? Take a full length practice test from both test makers and see where you stand. In addition, take a look at the advanced ACT math topics below (that aren’t found on the SAT) and see if you are fluent in them. If you are, the ACT might be a good test for you.

*note: some of these question types are fairly rare, but a selection of these topics is likely to show up on any given test.

## 1 – Logarithms

The ACT Math Section will usually feature one or two logarithm problems. A logarithm is the power to which a number must be raised in order to produce some other number. Really, they are all about thinking about exponents in a different way.

There are two formal definitions of logarithms that can help you answer a majority of the ACT log problems:

\textbf {If } y=log_b{x}\textbf { then } b^y = x

\textbf {If } b^y = x\textbf{ then } y=log_b{x}

Using these formulas, you can solve a number of sample problems on ACT Practice Tests: #29 on Section 2 of ACT Test Form 3MC, #49 on Section 2 of ACT Test Form 4MC (you’ll need to set the log equation equal to y), and #56 on Section 2 of ACT Test Form 5MC (you’ll need to first use your exponent rules and again think of the equation as being set to y). There are many other examples that can be found on the authentic ACT practice tests.

In addition to the formulas above, you’ll want to memorize these:

log_b{xy} =log_b{x} + log_b{y}

log_b{x/y} =log_b{x} - log_b{y}

log_b{x^y} =y log_b{x}

log_b{x} =log{x}/log{b}

This is only a primer on the subject of Logs. I recommend Khan Academy or MathisFun for a course on the subject.

## 2 – Matrix Problems

At heart, a matrix is simply an array of numbers that are organized in a consistent way. You’ll want to understand the steps you need to take to multiply, divide, add, or subtract multiple matrices. Matrices can appear intimidating at first, but they are some of the easier problems to do once you memorize the rules:

How to Describe a Matrix: To show how many rows and columns a matrix has, we usually write rows x columns. Here is a an example of a 2X3 matrix (2 rows by 3 columns).

\begin{matrix}5&7&23\\1&9&4\\\end{matrix}

Subtracting Matrices: To subtract one matrix from the other, you subtract the numbers in matching positions.

Multiplying or Dividing Matrices are a little trickier. You can find a great lesson on the subject here.

If you are multiplying a matrix by one number (scalar multiplication): You multiply that number by each of the numbers in the matrix and write the results in a matrix that has exactly the same number of rows and columns as it did before.

If you are multiplying matrices: The number of columns of the 1st matrix must equal the number of rows of the second matrix. (The ACT loves to test that you know this). The result will have the same number of rows as the first matrix, and the same number of columns and the second matrix.

Dividing Matrices: You’ll learn that we don’t actually divide matrices. Instead, we multiply by an inverse. Read about that here.

Here are some sample problems on the ACT that test your Matrix chops: #23 on Section 2 of Form 3MC, #42 on Section 2 of Form 2MC, #13 on Section 2 of Form 1MC.

## 3 – Trigonometry for Non-Right Triangles

The SAT tests almost entirely on right triangles. The ACT does too, but you’ll need to use trigonometry on non-right triangles (The SAT doesn’t test this). You’ll need to memorize the Law of Sines (or Sine Rule) and the Law of Cosines (Cosine Rule). Note: The ACT will often give you these formulas in the problem, but you’ll want to be comfortable with them beforehand.

Law of Sines: \frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}

Law of Cosines: c^2=a^2+b^2-2ab\ \cos{C} and two alternatives:

b^2=a^2+b^2-2ac\ \cos{B}

a^2=b^2+c^2-2bc\ \cos{A}

Problems that test your abilities with Sine and Cosine Rules: #35 on Section 2 of Test Form 2MC.

## 4 – Interpreting the Graphs of Trigonometric Functions

The ACT Match section will not ask you to graph a trig function, you do need to recognize what sine, cosine, and tangent graphs look like. You will sometimes be asked to find the period or the amplitude of a sine or cosine graph and you’ll want to be prepared.

You can answer most of these question with a minimal understanding of how trig graphs work. You’ll often these question types in the last 20 questions of the math section, but they can be easy points once you have the concepts down.

You’ll also want to be comfortable with thinking in radians.

The Period goes from one peak to the next (or from any point to the next matching point).

The Amplitude is the height from the center line to the peak (or to the trough).

For the equation y = A sin(B(x + C)) + D, the amplitude is A, period is 2π/B, the phase shift is C (positive is to the left), and the vertical shift is D.

Problems that test your abilities on finding the period or amplitude of a trig function: #53 on Section 2 of Form 3MC, #53 on Section 2 of Form 4MC.

## 5 – Arithmetic and Geometric Sequences

There are two types of sequences, arithmetic and geometric. These topics are fairly rare on the ACT, but they do come up. To prepare for them, you’ll want to memorize the formulas that you’ll need.

Arithmetic sequences have the common difference d. That difference between adjacent terms is always the same. The expression for the nth term of an arithmetic sequence is:

a_n=a_1+(n-1)d where a_1 is the first term and a_n is the last term to be summed. n is the number of terms.

The sum of n consecutive terms of an arithmetic sequence is s_n=\frac{n(a_1+a_n)}{2}

Geometric sequences have the common ratio r, i.e., the ratio of any term to the previous term is always the same. The expression for the nth term of a geometric sequence is

a_n=a_1 r^n-1

The same of n consecutive terms of a geometric sequence is s_n=\frac{a_1(1-r^n)}{1-r} where a_1 is the first term to be summed and n is the number of terms to be summed.

Sample sequence problems from ACT practice tests: #32 of Section 2 on Form 4MC (geometric sequence), #37 of Section 2 on Form 2MC (arithmetic sequence).

## 6 – Factorials

Factorials do come up on the ACT Math Section. The factorial function (symbol !) means that you need to multiply all whole numbers from our chosen number down to 1. The formal definition is n! = n(n-1)(n-2)(n-2)….1. It’s also useful to know that 0!=1.

For example: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

actorials are used in the context of combinations and permutations. You’ll want to study both those concepts.

Calculator Tip: The factorial key can be found on your graphing calculator. Its often found through math-pro-!. You can also solve permutations and combinations through your calculator. It’s best to learn how to do these problems by hand first.

Sample factorial problems from ACT practice tests: #57 on Section 2 of Form 16MC3, #40 on Section 2 of Form 16MC1.

## Comparing the ACT Math Section to the SAT Math Section Overall

• The ACT covers more advanced math topics overall
• The ACT requires a faster working pace than the SAT
• The ACT question are more straightforward overall
• The ACT Math Section lets you use a calculator on ALL the questions
• The ACT might be better suited for those who have already taken Trigonometry in High School (and excelled)

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