Math Study Guide for the SAT Exam

General Information

In 2024, the College Board began administering a digital form of the SAT. In addition to the technical changes, there were some procedural updates involved, including test timing, numbers of questions, and calculator use in the math section. Here is a synopsis of some of the most important current SAT guidelines.

Formula Reference Sheet

Throughout the entire SAT Math section, you will have access to the same formula reference sheet used with the previous version of the SAT (shown below). It is important to know when and how to use these and to know formulas that are not on the reference sheet.

$1 Math Reference Sheet.png$

Retrieved from: https://sat.blog.targettestprep.com/sat-math-reference-sheet/#The-SAT-Math-Reference-Sheet

Note: This guide will review all of these concepts in greater detail. While you don’t need to memorize all the formulas on this reference sheet, being familiar with them will save you time on the test.

Timing and Question Numbers

The digital SAT Math section covers basically the same material as the previous SAT edition, but there are a few changes in the presentation. Note the following:

The biggest change is that a calculator is allowed for all of the math questions on the SAT. It will be your job to determine whether it is advantageous to use the calculator based on the content of each question. You can find calculator information on the SAT website.
There are fewer questions, with only 44 in total.
The questions are divided into two 35-minute modules with 22 questions in each.
- There are two unscored questions in each module, but you will not know which ones they are.
- The second module is adaptive, meaning the questions you receive will be based on your performance on the first module.
- The timing will allow you about one and a half minutes per question.
The word problems are shorter.
There is only one question about each image or passage given.

Content Coverage

The digital SAT Math section assesses skills in the following areas:

algebra—These questions include linear equations and functions, systems of equations, and linear inequalities.
advanced math—Concepts tested include nonlinear functions and equations, systems of equations in two variables, and equivalent expressions. Complex numbers are no longer tested.
problem-solving and data analysis—Questions in these areas will involve ratios, rates, proportions, percentages, data distribution and measures, two-variable data, probability, statistical inferences, margin of error, and statistical claim evaluation.
geometry and trigonometry—Questions will include the concepts of area, volume, lines, angles, triangles, circles, and trigonometry.

Three-fourths of the digital SAT Math questions are typical multiple-choice questions, but you’ll need to enter the answer for the other 25%. These student-produced response (SPR) questions are dispersed throughout the test.

Algebra Basics

One of the four types of questions on the SAT Math section involves algebra concepts. These questions test your understanding of, and ability to use, linear equations, systems of these equations, and inequalities. Just what do you need to know?

First, you need to be familiar with the terms used in algebra so they don’t trip you up or cause you to spend extra time dissecting a question. Be sure to have a complete understanding of all of these concepts and anything else you encounter while studying for this type of question.

Variable vs. Constant

A variable, often designated with a letter (e.g., $x$, $y$, $z$), is used to represent an unknown quantity that may change its value over the course of a problem.

A constant, on the other hand, is a value that does not change, regardless of the circumstances.

Consider the following situation: John is paid a weekly base wage of $\$15$. In addition to this base wage, he earns $\$0.25$ for every magazine he sells. You need to write an equation representing John’s total wage.

In this case, the base wage John earns is a constant, so regardless of the number of magazines he sells, he will earn $\$15$. If he sells zero magazines, he will earn $\$15$. If he sells one magazine, he will earn $\$15 + \$0.25$, or $\$15.25$. We can use a variable to represent the unknown number of magazines he sells and combine this with his base wage to express his total wage. Let $m$ represent the unknown number of magazines sold, and let $f(m)$ be his total wage:

\[f(m) = 0.25 \cdot m + 15\]

Because he earns $\$0.25$ per magazine sold, when we find the product of this with the number of magazines sold, $m$, we are finding the total amount of money earned from the magazines he has sold. By combining this with his base wage, we calculate his total wage.

Note: There are two common ways to represent multiplication: $a \cdot b$ and $a \times b$. You may see both on the test, so it’s important to recognize both.

Exponents and Radicals

Exponents and radicals can seem complex at first. However, once you understand them, they are logical and can even make doing higher level math easier.

When you first learned multiplication, you were probably told to think of it as repeated addition, for example:

\[a \times 3 = a + a + a\]

Exponents, also known as powers, can first be thought of as repeated multiplication:

\[a^3 = a \times a \times a\]

With $a^3$, $a$ is the base and $3$ is the exponent.

Radicals can be thought of as the opposite of exponents. They are also known as roots.

When trying to solve $\sqrt[3]{b}$, we are looking for a number, $a$, such that $a^3 = b$.

The $\sqrt{\text{ }}$ is the radical symbol, the number under the symbol is the radicand, and the smaller number above the symbol is the index. If there is no smaller number, it is assumed to be $2$, meaning a square root.

Rules for Exponents and Radicals

Exponents and radicals work in specific ways according to specific rules. These rules build off of each other. So, for instance, for any numbers $a$, $b$, and $c$, these rules apply:

Rule 1—$a^b \times a^c\;= \;a^{b+c}$

We define it this way because this is what we’d expect for positive integers:

\[a^2 \times a^3 = (a \times a) \times (a \times a \times a) = (a \times a \times a \times a \times a) = a^5\]

Now what should a negative exponent mean? Using the first rule, we get this:

\[a = a^{2 - 1} = (a \times a) \times a^{-1}\]

We can divide both sides by $a$ (i.e., cancel out an $a$), and we have:

\[1 = a \times a^{-1}\]

We’ll divide both sides by $a$ again to find:

\[a^{-1} = \frac{1}{a}\]

We can raise this to any power, $b$, and use the first rule again to find the second rule:

Rule 2—$a^{-b} =\frac{1}{a^b}$

What this is stating is that any number with a negative power is the reciprocal (or multiplicative inverse) of that number.

This rule also shows us what we get when we raise to the power of zero:

\[a^0 = a^{1-1} = a \times a^{-1} = \frac{a}{a} = 1\]

Rule 3—$a^0 = 1$

Any integer raised to the power of zero equals one.

Now we know how exponents work for any integer. To go further, we need to think about integers again. Let’s raise a number to two powers:

\[(a^3)^2 = (a \times a \times a) \times (a \times a \times a) = a^{3 + 3} = a^{3 \times 2}\]

We, therefore, write the next rule:

Rule 4—$(a^b)^c = a^{b \, \times \, c}$

What about fractional exponents? We use the rule we just found:

\[a = a^1 = a^{\frac{b}{b}} =(a^b)^\frac{1}{b}\]

So, raising to a fractional power “undoes” raising to a whole number power. For example:

$a^\frac{1}{2}$ is the square root of $a$, or $\sqrt{a}$.

Note that just knowing the square of a number is not enough to know what that number is. For example, if we know that $x^2 = 25$, then it is possible that $x = 5$ or $x = -5$:

\[5^2 = 5 \times 5 = 25\] \[(-5)^2 = -5 \times -5 = (-1 \times 5) \times (-1 \times 5) = (-1 \times -1) \times (5 \times 5) = 25\]

So, in general, square roots (and any other even number roots) can be either positive or negative. Keeping that in mind, we have our next rule:

Rule 5—$a^{\frac{b}{c}} = \sqrt[c]{a^b}$

$\sqrt[c]{a^b}$ means “the $c$ th root of $a^b$.”

There’s one more rule to know:

Rule 6—$(a \times b)^c = (a^c)(b^c)$

When shown in full, it looks like this:

\[(a \times b)^2 = (a \times b) \times (a \times b)=(a \times a)(b \times b) = (a^2)(b^2)\]

Equation vs. Expression

The fundamental difference between an equation and an expression is that an equation contains two expressions that are equal to each other. This equality is represented with the equal sign ($=$).

Expressions cannot be solved. Instead, you will be asked to “simplify” or “reduce” them. To simplify an expression is to apply algebraic rules in order to find a simpler expression to work with. For example, we can write $4x$ as $x + x + x + x$, but it is often more convenient to simplify the second expression to the first. Consider the following expression:

\[4x - 2x + 3x^2 + 5x^2\]

The terms containing like terms (the same variable type) can be combined, and the expression can be simplified:

\[4x - 2x = 2x\]

and

\[3x^2 + 5x^2 = 8x^2\]

So, the expression can be rewritten as:

\[2x + 8x^2\]

Remember that only like terms can be combined. So, in this example, $2x$ and $8x^2$ cannot be combined because the variable is to a different degree.

Evaluating Expressions

To interpret, or evaluate, an expression is to determine its numerical value. Sometimes these expressions will include variables, but to evaluate the value of the expression, the value of each of the variables must also be known. The first step in evaluating expressions is often simplifying. Consider this example:

\[-3(x +y - 3) + 2x - 4y\]

Before we can simplify this expression, we must first distribute the $-3$ to the terms inside the parentheses (this is known as applying the distributive property). After that is done, we can combine like terms:

\[-3x - 3y + 9 + 2x - 4y = -x - 7y +9\]

The expression on the right is easier to work with than the expression on the left.

Note: The final expression cannot be simplified any further as there are no more like terms.

Solving Equations

To solve an equation is to find the value or values of a variable that make the equation true. An equation shows that two expressions are equal and the goal is to determine the value of the variable that balances both sides.

Solving often involves isolating the variable, which means using inverse operations to undo addition, subtraction, multiplication, or division. For instance, suppose you have this equation

\[3x - 7 = 2x + 5\]

The first step is to get all variable terms on one side of the equation. To do this, we can subtract $2x$ from both sides:

\[3x - 7 - 2x = 2x + 5 -2x\] \[x - 7 = 5\]

Next, isolate the variable by adding $7$ to both sides:

\[x - 7 + 7 = 5 +7\] \[x = 12\]

The solution is $x = 12$. If we substitute this value back into the original equation, we will see that it makes it true. Just like simplifying expressions, solving equations follows a structured process. You move terms strategically, combine like terms when possible, and always perform the same operation on both sides of the equation.

Note: Always check your solution by substituting it back into the original equation to confirm that both sides are equal.

The Order of Operations

There is a vital concept to know when working with expressions and equations, and that is the order of operations. Operations include addition, subtraction, multiplication, division, and evaluating exponents. Knowing this is going to be especially important as the expressions and equations you work with get more complicated.The easiest way to remember the order of operations is with the acronym PEMDAS:

Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division (MD), left to right.
Then do all addition and subtraction (AS), left to right.
A good way to remember: Please Excuse My Dear Aunt Sally.

Doing operations in this order is important, because if you do them in a different order, you will get a different, wrong answer. Say you have this expression:

\[x(5 + 3) + 4^2 \div 2 - 2x\]

The first step in PEMDAS is parentheses, so we have:

\[x(8) + 4^2 \div 2 - 2x\]

The next step is exponents, which gives us:

\[x(8) + 16 \div 2 - 2x\]

Now we do multiplication and division:

\[8x + 8 - 2x\]

Finally, we end with addition and subtraction:

\[6x +8\]

That is the correct simplified version of the expression we were given.

Polynomials

Polynomials are expressions that can be written as sums of powers of variables. For example, these are all polynomials:

\[x^2 + x - 2\] \[x^5 + x^4 - 7x^ 3 + x - 4\] \[x + 1\]

The powers in polynomials can be any fixed numbers but not variables, and polynomials can have any number of terms. In general, we write a polynomial like this:

\[a_{n} x^{n} + a_{n-1}x^{n-1}+…a_{2} x^{2} + a_{1}x+a_{0}\]

where $n$ is any positive integer and $a_1$, $a_2$, and all the rest of the values up to $a_n$ represent any constant number (possibly zero). This is the most straightforward way to write a polynomial, but it isn’t the only way, as we will see later.

There are special types of polynomials known as binomials, which have two terms, and trinomials, which have three terms.

The FOIL Method

When two polynomials are multiplied by each other, the quickest way to complete the operation is with the FOIL method. FOIL stands for “first, outer, inner, last.” Like PEMDAS, it is telling you the order in which to do the operations. Let’s see how it works with this example:

\[(x^2+3)(x+4)\]

The “first” refers to the first terms of each polynomial, so we multiply those by each other:

\[x^2 \times x= x^3\]

Next, we multiply the “outer” terms:

\[x^2 \times4 = 4x^2\]

Now, we multiply the “inner” terms:

\[3 \times x = 3x\]

And, finally, we multiply the “last” terms:

\[3 \times 4 = 12\]

Putting it altogether, we have:

\[x^3 + 4x^2 + 3x + 12\]

Note: The FOIL method works when multiplying two polynomials with two terms each, but a similar distribution method works for larger polynomials. It is important to pay attention to the signs between the terms.