## SAT Math Content

The SAT math section tests around 25-30 topics that are broken down into four separate areas of focus. The topics themselves are ones students generally see and recognize from school, but the areas of focus are given strange names by The College Board: Heart of Algebra, Problem Solving and Data Analysis, Passport to Advanced Math, and Additional Topics in Math.

The goal of this page is to help you understand what concepts are tested and make up the areas of focus above in plain English. If you use our score reporting (and why wouldn’t you?), it also uses the same language to help you identify your areas of improvement. Additionally, there are some examples of questions from each topic so that you get the flavor of what you may encounter test day.

One final word on the math section. The SAT tests many math questions based on your competency and fluency. Simply put, competency deals with your ability to solve a relatively straightforward question in a way that is probably familiar from school. Fluency, on the other hand, tests your ability to apply knowledge from school in a somewhat unfamiliar way; it is what we call critical reasoning

Here are two questions that test the same skill—solving an equation. However, one tests competency and one tests fluency.

#### Competency Notice that this question asks you to find something familiar—x—and the most common method of finding it is mechanical: distribute the 3, add 9 to both sides, and then divide by 6.

NOTE: dividing by 3 first is a great strategy as well.

#### Fluency Notice that this question asks you to find something that you are not usually asked to find in school for these types of equations—a binomial. Although you could solve in a similar method as in the question to the left and then substitute x=7 for 2x-3 to get 11, the reasoning aspect comes in by noticing that simply dividing both sides by 3 gives the quantity you are being asked to find.

NOTE: It is not a coincidence that the value of x appears as an answer choice here. The question writers almost always put partial answer choices—ones that appear from correct work on a question, but an incorrect understanding of what you’re asked to find.

### Heart of Algebra

Heart of Algebra is a fancy name for Algebra 1. It makes up around 33% of the SAT (19 out of 58 questions), making it the most heavily tested area of focus on the math section. Heart of Algebra appears on both the No Calculator and Calculator sections.

#### Equation of a Line

Equations of a line are one of the most frequently occurring topics on the test, making up close to 9% of all math questions. Although this is not an exhaustive list, the SAT tests the following equation of a line concepts:

• Finding the slope of a line when you’re given two points, a table of values, or an equation of a line.
• Identifying the graph of a line when given an equation and some information or finding the equation of a line when given a graph
• Finding the equation of a line, the slope of a line, or a point on a line when lines are parallel or perpendicular.
• Identifying a point or a part of a point (an x or y coordinate, for instance) that is on a line.
• Working with lines when written as a linear function. In the question to the right, you are asked to find the equation of the line. Each of the answer choices are in slope-intercept form, that is , y=mx+b, so you should find the slope and the y-intercept.

Since the point (0,3) is given, the y-intercept must be 3; this eliminates choices B and D from contention.

To find the slope, use the slope formula, \frac{y_2 -y_1}{x_2 – x_1} with the given points.

\frac{5-3}{1-0}=2

Alternatively, you can substitute the point (1,5) into each answer and see that only C creates a true equation. In the question to the right, you asked to find the y-coordinate of a point on the line when given function notation. In this case, the x-value is the 12 from g(12).

One way to figure this out is to find the equation of the line from two given points. Because who wants to work with negatives, let’s use (3,11) and (6,20) from the table.

The slope is \frac{20-11}{6-3}=3. You can now find the y-intercept by substituting a point and slope into slope-intercept form:

y=mx+b\Longrightarrow 20=6(3)+b. Solving this gives b=2. We now know the equation of the line: y=3x+2. Substitute in x=12 to find the answer: y=3(12)+2\Longrightarrow y=38, or choice C.

Alternatively, we might recognize that going from x=3 to x=6 increases the function by 9. To get from x=6 to x=12, the function would increase by 9 twice, so 20+2(9)=38.

##### Equations and relationships you should know #### Linear Systems

Linear systems of equations and inequalities A description of the question

#### Parts of a Linear Model

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#### Solving Linear Quantities

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#### Translation

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### Problem Solving and Data Analysis

Problem Solving and Data Analysis is mostly concepts that students learn in a Prealgebra course along with statistics. For some students, certain concepts—like study design or standard deviation—are not part of the school curriculum. Problem Solving and Data Analysis questions make up around 30% of the test (17 out of 58 questions), making it the second most frequent area of focus on the SAT. These questions only appear on the Calculator section, so expect around half of the questions you see on section 4 to come from the topics here.

#### Linear and Nonlinear Modeling

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#### Percent and Probability

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#### Rates and Ratios

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#### Scatterplots

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#### Statistics

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#### Study Design

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#### Using Graphics

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Passport to Advanced Math is apparently what the College Board considers Algebra 2 to be. It makes up a little less than 30% of the test (16 out of 58 questions), making it the third most frequently occurring area of focus. These questions pop up on both the No Calculator and Calculator section of the test. Additionally, these questions test fluency more often than the other areas of focus do.

#### Exponents and Exponentials

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#### Functions

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#### Literal Equations

Literal equations usually look worse than they actually are; if you can solve an algebraic equation, you can rework a literal equation as well. These question types appear as around 2% of the questions, so you should expect to see around one literal equation question per test. The goal of any literal equation is to solve for some variable of combination of variables in a multivariable equation. From the question, you are told the density is the mass divided by the volume; you can write this relationship as D=\frac{m}{v} with the given variables in the question.

Next, you are told to express the volume in terms of its density and mass—this is a fancy way of saying solve for v. There are a couple ways to solve this, but

1. Multiply both sides of the equation by v to get it out of the denominator: vD=m.
2. Solve for v by dividing both sides of the equation by D: v=\frac{m}{D}, which is choice B. This question is a harder literal equation question. First, if you are looking for something equivalent to a you will want to solve for/isolate the a.

1. Move the ac to the left side of the equation by subtracting it from both sides: abc+ab+bc-ac=0.
2. Since you want to solve for a, get everything that does not have an a to the right side of the equal sign by subtracting bc from both sides: abc+ab-ac=-bc.
3. Now, the tricky part. In order to isolate the a, you need to factor it out of the left side of the equation: a(bc+b-c)=-bc.
4. Finally, to isolate the a, divide both sides of the equal sign by bc+b-c to arrive at the answer: a=\frac{-bc}{bc+b-c}. This is choice D.

#### Operations with Polynomials

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#### Rational Functions

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This area of focus basically amounts to et cetera. It makes up about 10% of the math section (6 out of 58 questions), making it the least frequently occurring area of focus. There are 3 Additional Topics in Math questions on both the No Calculator and Calculator sections of the test.

#### Complex Numbers

Complex number questions are very rarely tested on the SAT, but they do occasionally appear. You should know how to do basic operations with complex numbers: adding, subtracting, multiplying, and dividing. Additionally, you should understand how powers of i work. This is one of the more basic complex number questions we will see. Simply combine like terms after you’ve distributed the negative sign.

5+7i-2+3i\Longrightarrow 3+10i Here, we need to remember our powers of i or derive them from i=\sqrt{-1}.

if i=\sqrt{-1}, i^2=-1 and i^4 = (i^2)^2 \Longrightarrow i^4=(-1)^2 =1.

Thus, we have 32(1)-16(-1)+8\Longrightarrow 32+16+8=56

##### Equations and relationships you should know #### Equations of a Circle

Equations of a circle tend to appear as medium or difficult questions on both the no calculator and calculator sections; they are somewhat rare (about 1.5% of all questions), but can be a somewhat easy point to get test day if you know a couple of formulas. The SAT wants students to be able to identify the center and radius of a circle in center-radius form, complete the square to turn standard form into center-radius form, and understand relationships among points on the circle. This question is a relatively straightfoward application of center-radius form of a circle, (x-h)^2+(y-k)^2=r^2, where (h,k) is the center of the circle and r is the radius.

Since you are told the center and radius, all you need to do is substitute them into the equation.

(x-2)^2+(y-(-1))^2=(3)^2 to end up with (x-2)^2+(y+1)^2=9, or answer choice B. This question is one of the more difficult questions involving equations of a circle  you will see on the SAT. You are given an equation in standard form, Ax^2+Bx+Ay^2+Cy+D=0 and you need to complete the square to massage the equation into center-radius form. If you are unsure how to complete the square, check out our video on it.

1. Since your constant is already by itself on one side of the equal sign, you do not need to move it. Also, since A=1, you do not have to divide by anything.
2. Divide the B term, 16, by two and add the square to both sides: x^2 +16x+(8)^2+y^2-12y=-21+(8^2).
3. Divide the C term, -12, by two and add the square to both sides: x^2 +16x+(8)^2+y^2-12y+(-6)^2=-21+(8^2)+(-6)^2.
4. Factor x^2 +16x+(8)^2 and y^2-12y+(-6)^2 into perfect squares: (x+8)^2+(y-6)^2=-21+64+36.
5. Use this to find the center: (-8, 6), or choice B.

Note, if you needed to find the radius, it would be the square root of the constant, or \sqrt{79}.

##### Equations and relationships you should know #### Geometry

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#### Trigonometry

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