Model Comparison: Math
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Mastery of Digital SAT Mathematics
Mathematics on the SAT is not just about memorizing formulas; it is about fluency in the language of numbers, shapes, and patterns. The test is designed to measure your ability to apply mathematical concepts to real-world situations, interpret data, and solve abstract problems efficiently. With the transition to the Digital SAT, the format has evolved—most notably with the inclusion of the built-in Desmos calculator for all math questions—but the core underlying concepts remain consistent.
These notes are structured around the four primary content domains of the SAT: Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry. Rather than providing a list of shortcuts, we will explore these topics from the ground up, ensuring you understand the "why" behind the operations. This deep understanding is what allows you to adapt to novel question types and achieve a high score.
Algebra: Linear Relationships and Systems
The Algebra domain, often referred to as the "Heart of Algebra," focuses on the mastery of linear equations and functions. This is the bedrock of the SAT math section. If you can fluently speak the language of lines—understanding how things grow at a constant rate—you will be equipped to handle a significant portion of the exam.
Linear Equations in One Variable
A linear equation is a statement that two expressions are equal, where the variable is raised only to the first power. Fundamentally, solving a linear equation is about isolation: manipulating the equation to get the variable by itself. This process relies on inverse operations—undoing what has been done to the variable.
Consider the structure ax + b = c. Here, x is the unknown, while a, b, and c are constants. To solve for x, you must first remove the constant term b by subtraction (or addition if b is negative), and then remove the coefficient a by division (or multiplication). While this seems elementary, the SAT complicates it by embedding these equations in word problems or asking for the conditions under which an equation has specific types of solutions.
Types of Solutions:
One Solution: This happens when the variable terms are different on both sides (e.g., 2x = 3x + 5). There is exactly one value for x that makes the statement true.
No Solution: This occurs when the variable terms are identical on both sides, but the constants are different (e.g., 2x + 5 = 2x + 10). If you subtract 2x from both sides, you are left with 5 = 10, which is false. This represents parallel lines that never intersect.
Infinite Solutions: This happens when both the variable terms and the constants are identical (e.g., 2x + 5 = 2(x + 2.5)). The equation simplifies to 5 = 5, which is always true. This represents two lines that are actually the same line.
Linear Functions and Slope-Intercept Form
A function defines a relationship where every input has exactly one output. Linear functions describe a relationship with a constant rate of change. The most important tool in your arsenal here is the slope-intercept form:
y = mx + b
In this equation:
m represents the slope or the rate of change. It tells you how much y changes for every one-unit increase in x. In word problems, look for keywords like "per," "every," or "rate."
b represents the y-intercept. This is the starting value of y when x = 0. In context, this is often the "flat fee," "initial amount," or "starting height."
Understanding Slope:
Slope is the ratio of vertical change to horizontal change (\frac{\text{rise}}{\text{run}}). If you have two points (x1, y1) and (x2, y2), the formula for slope is:
m = \frac{y2 - y1}{x2 - x1}
If the slope is positive, the line rises from left to right. If it is negative, it falls. A slope of zero indicates a horizontal line (y = b), while an undefined slope indicates a vertical line (x = a).
Systems of Linear Equations
A system of equations is a set of two or more equations with the same variables. The solution to a system is the point (or points) where the lines intersect. Graphically, this is the coordinate pair (x, y) that both lines share. There are two primary algebraic methods to solve these:
Substitution: This is best used when one variable is already isolated or easy to isolate. You solve one equation for x and plug that expression into the second equation.
Elimination: This is best used when variables are aligned in columns. You add or subtract the equations to cancel out one variable.
For example, consider the system:
2x + y = 10
3x - y = 5
Here, elimination is ideal. Adding the two equations directly cancels y:
(2x + 3x) + (y - y) = 10 + 5
5x = 15
x = 3
Substituting x = 3 back into the first equation yields 2(3) + y = 10, so 6 + y = 10, and y = 4. The solution is (3, 4).
Exam Focus
Typical Question Patterns: You will frequently encounter word problems asking you to "interpret the meaning of the constant" in a linear model (answer: it's the starting value) or "interpret the coefficient" (answer: it's the rate of change).
The "No Solution" Trick: A very common question asks for the value of a constant k that results in a system having no solution. Remember: for a system to have no solution, the lines must be parallel. This means their slopes must be equal, but their y-intercepts must be different.
Common Mistakes: Students often flip the slope definition, calculating \frac{\text{run}}{\text{rise}} instead of \frac{\text{rise}}{\text{run}}. Another error is neglecting to distribute a negative sign when subtracting equations during elimination.
Advanced Math: Non-Linear Functions
The "Advanced Math" domain requires you to step away from straight lines and deal with curves. This includes quadratic functions, exponential functions, polynomials, and radicals. The key skill here is structure recognition—seeing a complicated equation and recognizing it as a specific type of function with predictable behavior.
Quadratics and Parabolas
A quadratic function is a polynomial of degree two. Its graph is a U-shaped curve called a parabola. The standard form of a quadratic equation is:
y = ax^2 + bx + c
The sign of a determines the direction: if a > 0, the parabola opens upward (minimum point); if a < 0, it opens downward (maximum point).
To master quadratics, you must understand the three different forms and what they reveal:
Standard Form (y = ax^2 + bx + c): Good for finding the y-intercept, which is c. It is also necessary for using the Quadratic Formula.
Vertex Form (y = a(x - h)^2 + k): This is the most powerful form for graphing. The vertex (the turning point) is at (h, k). If the SAT asks for the minimum or maximum value of a function, you usually want to convert to this form or find the vertex.
Factored (Intercept) Form (y = a(x - r1)(x - r2)): This reveals the x-intercepts (roots or zeros) of the function, which are at x = r1 and x = r2.
Solving Quadratics:
To find the solutions (roots) of ax^2 + bx + c = 0, you can factor the expression if possible. If not, use the Quadratic Formula:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The Discriminant:
The expression inside the square root, b^2 - 4ac, is called the discriminant. It predicts the number of real solutions without needing to solve the whole equation:
If b^2 - 4ac > 0, there are two distinct real solutions (two x-intercepts).
If b^2 - 4ac = 0, there is exactly one real solution (the vertex touches the x-axis).
If b^2 - 4ac < 0, there are no real solutions (the parabola never touches the x-axis).
Exponential Functions
While linear functions change by a constant amount (addition), exponential functions change by a constant percentage or factor (multiplication). The general form is:
y = a(1 \pm r)^t
Or simply y = ab^x.
a is the initial amount (when t=0).
r is the rate of growth (plus) or decay (minus) expressed as a decimal.
t is the time interval.
For example, if a population of 1000 bacteria grows by 5\% every hour, the equation is y = 1000(1 + 0.05)^t or y = 1000(1.05)^t. A common mistake is treating a 5\% decrease as multiplying by 0.05. A 5\% decrease means you retain 95\% of the original, so the base would be (1 - 0.05) = 0.95.
Operations with Polynomials and Rational Expressions
You will also encounter problems involving adding, subtracting, multiplying, and dividing polynomials. The key here is grouping like terms. For multiplication, remember the distributive property (often taught as FOIL for binomials).
Rational Expressions:
These are fractions where the numerator and denominator are polynomials, like \frac{x^2 - 1}{x + 1}. Simplifying these often requires factoring. In this example, x^2 - 1 is a difference of squares:
x^2 - 1 = (x - 1)(x + 1)
So the expression becomes \frac{(x - 1)(x + 1)}{x + 1}. You can cancel the (x + 1) terms, leaving x - 1, provided x \neq -1 (since division by zero is undefined).
Exam Focus
Typical Question Patterns: You will likely be asked to find the vertex of a parabola. If given in standard form y = ax^2 + bx + c, the x-coordinate of the vertex is found at x = -\frac{b}{2a}. Plug this value back into the equation to find the y-coordinate.
Equivalent Expressions: Questions often ask "Which of the following is equivalent to…?" usually involving exponential properties or factoring. For example, knowing that x^a \cdot x^b = x^{a+b} and (x^a)^b = x^{ab} is essential.
Common Mistakes: Students often forget the "middle term" when squaring a binomial. Remember that (x + 3)^2 is NOT x^2 + 9. It is (x + 3)(x + 3) = x^2 + 6x + 9.
Problem Solving and Data Analysis
This domain tests your quantitative literacy. Can you use math to make sense of the real world? Topics here include percentages, ratios, probability, and statistics. These questions are often wordy, so reading comprehension is as important as calculation.
Ratios, Rates, and Proportions
A ratio compares two quantities. A proportion states that two ratios are equal. The most reliable way to solve proportion problems is cross-multiplication or setting up a fraction.
\frac{\text{part A}}{\text{part B}} = \frac{\text{part C}}{\text{part D}}
Unit Conversion:
Dimensional analysis is a foolproof method for converting units. Treat units like algebraic variables that can cancel out. If you need to convert 60 miles per hour to feet per second:
\frac{60 \text{ miles}}{1 \text{ hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}
Notice how "miles" cancels (top and bottom) and "hours" cancels, leaving \frac{\text{feet}}{\text{seconds}}. Multiply the numerators and divide by the denominators.
Percentages
Percentages are fundamentally ratios out of 100. The key to SAT percentage problems is careful translation of English to Math.
"What is 20\% of 80?" translates to x = 0.20 \times 80.
"50 is 25\% of what number?" translates to 50 = 0.25 \times x.
Percent Change:
To calculate the percentage increase or decrease:
\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100
If the result is positive, it is an increase. If negative, it is a decrease.
Statistics: Center and Spread
Statistics questions generally ask you to analyze a dataset to find the "center" or the "spread."
Measures of Center:
Mean (Average): The sum of all data points divided by the number of points.
Median: The middle value when data is ordered from least to greatest. If there is an even number of data points, the median is the average of the two middle numbers.
Mode: The value that appears most frequently.
Measures of Spread:
Range: The difference between the maximum and minimum values (\text{Max} - \text{Min}).
Standard Deviation: A measure of how much the data varies from the mean. You generally do not need to calculate this by hand on the SAT, but you must understand the concept conceptually. A data set with values clustered tightly around the mean (e.g., 4, 5, 5, 6) has a low standard deviation. A data set with values far apart (e.g., 0, 5, 5, 10) has a high standard deviation.
The Effect of Outliers:
An outlier is a data point significantly higher or lower than the rest. Outliers pull the mean toward them. For example, if Bill Gates walks into a bar, the mean income in the bar skyrockets, but the median income stays roughly the same. Therefore, the median is a better measure of center for skewed data or data with outliers.
Exam Focus
Typical Question Patterns: You will see scatterplots and be asked to estimate the line of best fit. You may also be asked to predict a value based on that line. Always calculate the prediction based on the line, not the actual data points.
Probability from Tables: Two-way frequency tables are common. Read the question carefully to determine the denominator. If it asks "Given that a person is left-handed, what is the probability they are female?", the denominator is only the total number of left-handed people, not the total population.
Common Mistakes: Confusing "percent of" with "percent change." If a price goes from 100 to 80, it is 80\% of the original price, but a 20\% decrease.
Geometry and Trigonometry
This domain covers shapes, sizes, and relative positions of figures. While it makes up a smaller portion of the exam than Algebra, the concepts are rigid and rule-based, making them easy to master if you memorize the properties.
Lines, Angles, and Triangles
Parallel Lines:
When two parallel lines are cut by a transversal (a third line), specific angle relationships form. Vertical angles are equal. Alternate interior angles are equal. Corresponding angles are equal. If you see parallel lines, look for the "Z" shape or "F" shape to identify equal angles.
Triangles:
The sum of interior angles in any triangle is always 180^\circ. This simple fact solves many complex problems. The side lengths are related to the angles opposite them: the largest angle opens up to the longest side.
Right Triangles:
The Pythagorean Theorem applies to all right triangles:
a^2 + b^2 = c^2
where c is the hypotenuse (the longest side). You should memorize the common "Pythagorean Triples"—sets of integers that satisfy this theorem—to save time. The most common are (3, 4, 5), (5, 12, 13), and multiples of these (like 6, 8, 10).
Special Right Triangles:
There are two specific triangles that appear constantly. You do not need the Pythagorean theorem for these if you know the ratios:
30-60-90 Triangle: The sides are in the ratio x : x\sqrt{3} : 2x. The hypotenuse is twice the shortest leg.
45-45-90 Triangle (Isosceles Right): The sides are in the ratio x : x : x\sqrt{2}.
Circles
Circles on the SAT are tested in two ways: geometry (angles/arcs) and algebra (equations).
Circle Equations:
The standard equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
(h, k) is the center of the circle.
r is the radius.
Sometimes the SAT gives you a "messy" equation like x^2 + y^2 + 6x - 4y = 12. To find the center and radius, you must complete the square for both x and y. This involves taking half of the linear coefficient, squaring it, and adding it to both sides.
Arc Length and Sector Area:
These are just fractional parts of the circumference and area. If an arc subtends a central angle of \theta degrees:
\text{Arc Length} = \frac{\theta}{360} \times 2\pi r
\text{Sector Area} = \frac{\theta}{360} \times \pi r^2
If the angle is in radians, the formulas are simpler (Arc Length = r\theta), but be careful to check which unit is being used.
Trigonometry
Trigonometry deals with the ratios of sides in right triangles. Remember the mnemonic SOH CAH TOA:
Sine = \frac{\text{Opposite}}{\text{Hypotenuse}}
Cosine = \frac{\text{Adjacent}}{\text{Hypotenuse}}
Tangent = \frac{\text{Opposite}}{\text{Adjacent}}
Complementary Angles:
One critical identity to know is that sine and cosine are "co-functions." In a right triangle, the two non-90-degree angles add up to 90^\circ. Because of this geometry:
\sin(x^\circ) = \cos(90 - x)^\circ
If the SAT asks for \sin(20^\circ) and you know \cos(70^\circ), they are the same value.
Radians vs. Degrees:
To convert degrees to radians, multiply by \frac{\pi}{180}. To convert radians to degrees, multiply by \frac{180}{\pi}. Familiarize yourself with the unit circle basics: 360^\circ = 2\pi radians, 180^\circ = \pi radians, and 90^\circ = \frac{\pi}{2} radians.
Exam Focus
Typical Question Patterns: You will frequently see a circle equation that needs completing the square to identify the radius. Don't forget to square the radius value in the final equation (r^2, not r).
Similarity: Problems often feature a small triangle nested inside a larger one. These triangles are similar, meaning their angles are identical and their side lengths are proportional. If the large triangle's side is 2 times the small triangle's side, then all corresponding lengths are scaled by 2.
Common Mistakes: Calculator mode errors. Ensure your calculator (or the Desmos tool) is in the correct mode (Degrees or Radians) before calculating sine, cosine, or tangent. The default on many devices is Radians, but many geometry problems use Degrees.
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Curriculum & Assessment Framework (College Board SAT)
These notes follow the College Board SAT (Digital SAT) Math framework as published in the SAT Suite assessment specifications and student materials. The official content is organized into four tested content domains:
Algebra (linear equations/inequalities, linear functions, systems)
Advanced Math (nonlinear relationships, quadratics, polynomials, rational expressions, radicals)
Problem-Solving and Data Analysis (ratios, rates, proportional relationships, units, percentages, data interpretation, probability, statistics)
Geometry and Trigonometry (area/volume, lines/angles, circles, right-triangle trig)
Across these domains, the SAT emphasizes skills you repeatedly use in college-level work: modeling (turning a situation into math), fluency (accurate algebra), interpreting graphs/tables, and reasoning (checking whether an answer makes sense). The Digital SAT Math section is split into two timed modules, uses a mix of multiple-choice and student-produced response questions, and provides an on-screen graphing calculator (Desmos) plus a small reference of common geometry formulas.
What follows teaches the full “Math” unit from the ground up—concepts first, then worked examples, plus the most common ways SAT questions are framed.
Algebra Foundations: Expressions, Equivalent Forms, and Rearranging
Algebra starts with the idea that a symbol like x can represent an unknown number or a changing quantity. The SAT uses algebra as a language for describing relationships—between costs and time, distance and speed, inputs and outputs, and more.
Variables, constants, coefficients, and terms
A variable is a symbol that can change (like x). A constant is a fixed value (like 7). A coefficient is the number multiplying a variable (in 5x, the coefficient is 5). A term is a piece separated by addition/subtraction (in 3x - 2y + 9, the terms are 3x, -2y, and 9).
Why this matters: SAT questions often hinge on whether you recognize what is “attached” to a variable and how changing one quantity affects another.
Like terms and simplifying
Like terms have the same variable part raised to the same power, such as 3x and -10x. You can combine like terms because they represent the same kind of quantity.
For example, 3x + 2x means “three x’s plus two x’s,” which is 5x.
A common mistake is combining unlike terms, such as turning x + x^2 into 2x^2. These are different kinds of quantities.
The distributive property (expanding and factoring)
The distributive property connects multiplication and addition:
a(b + c) = ab + ac
This matters because SAT problems frequently require you to rewrite an expression into a different but equivalent form—either to solve an equation or to identify a key feature like a slope or intercept.
Expanding: turning a product into a sum, like 2(x + 5) into 2x + 10.
Factoring: turning a sum into a product, like 6x + 15 into 3(2x + 5).
Factoring is especially useful because it reveals structure—like common factors, zeros, or constraints.
Fractions in algebra: restrictions and structure
Algebraic fractions behave like numerical fractions, but you must track when a denominator is zero.
If you see:
\frac{x + 1}{x - 3}
then x cannot be 3 because division by zero is undefined.
SAT often tests whether you understand that solving an equation may produce an “answer” that must be rejected because it violates a restriction.
Worked example: simplify an expression
Simplify:
3(2x - 5) - 2(x + 1)
Step 1: Distribute.
3(2x - 5) = 6x - 15
-2(x + 1) = -2x - 2
Step 2: Combine like terms.
6x - 15 - 2x - 2 = 4x - 17
So the simplified form is:
4x - 17
Worked example: factor out the greatest common factor
Factor:
12x + 18
Step 1: Find the greatest common factor of the coefficients. The GCF of 12 and 18 is 6.
Step 2: Factor out 6.
12x + 18 = 6(2x + 3)
Factoring like this is often a first step before solving, simplifying a rational expression, or spotting a pattern.
Exam Focus
Typical question patterns:
Rewrite an expression in an equivalent form (expand, factor, or combine) to reveal a feature.
Identify which expression is equivalent to a given one.
Use algebraic structure to match a graph or interpret a real-world model.
Common mistakes:
Combining unlike terms (such as treating x and x^2 as compatible).
Distributing incorrectly across subtraction (missing negative signs).
Canceling terms across addition (you can cancel factors, not terms separated by + or -).
Linear Equations and Inequalities: Solving, Interpreting, and Modeling
A linear equation is an equation where the variable appears only to the first power (no squares, no products of variables). Linear relationships are central on the SAT because they model constant-rate change—hourly pay, constant speed, steady growth, and proportional adjustments.
Solving one-variable linear equations
Solving means isolating the variable using inverse operations—undoing addition/subtraction, then undoing multiplication/division.
Example structure:
ax + b = c
You subtract b from both sides, then divide by a.
Why it matters: Many SAT questions are “hidden linear equations” inside words. If you’re fluent, you can spend brainpower on interpretation instead of algebra mechanics.
Linear equations with variables on both sides
When variables appear on both sides, your goal is still to collect variable terms on one side and constants on the other.
A frequent pitfall: distributing incorrectly or losing a negative when moving terms.
Literal equations (solving for a variable)
A literal equation includes multiple variables, and you solve for one in terms of the others. This is a major SAT skill because it shows you understand equations as relationships, not just “find x.”
Example structure:
A = \pi r^2
Solve for r by dividing by \pi and taking a square root.
Inequalities
An inequality compares values using symbols like <, \le, >, \ge.
Solving an inequality is like solving an equation, with one crucial rule:
When you multiply or divide both sides by a negative number, you must reverse the inequality sign.
For instance, from:
-2x > 6
dividing by -2 gives:
x < -3
SAT also uses compound inequalities (like 2 < x \le 7) and inequality word problems.
Worked example: linear equation with distribution
Solve:
4(2x - 1) = 3x + 10
Step 1: Distribute.
8x - 4 = 3x + 10
Step 2: Move variable terms to one side.
8x - 3x = 10 + 4
5x = 14
Step 3: Divide.
x = \frac{14}{5}
Worked example: inequality with a negative coefficient
Solve:
7 - 3x \le 19
Step 1: Subtract 7 from both sides.
-3x \le 12
Step 2: Divide by -3 and reverse the inequality.
x \ge -4
Exam Focus
Typical question patterns:
Solve for x (including variables on both sides and distribution).
Translate a word problem into an equation/inequality and solve.
Rearrange a formula to solve for a specified variable.
Common mistakes:
Forgetting to reverse the inequality sign when dividing by a negative.
Dropping parentheses or sign errors during distribution.
Treating an inequality solution as a single number instead of a range.
Linear Functions and Graphs: Slope, Intercepts, and Rate of Change
A function is a rule that assigns each input exactly one output. On the SAT, linear functions are the workhorse model for constant change.
A linear function can be written as:
y = mx + b
where:
m is the slope (rate of change)
b is the y-intercept (value when x = 0)
Slope as “change in output per change in input”
Slope measures how much y changes when x increases by 1. More generally:
m = \frac{\Delta y}{\Delta x}
Given two points \left(x1, y1\right) and \left(x2, y2\right), the slope is:
m = \frac{y2 - y1}{x2 - x1}
Why it matters: SAT word problems often hide slope as a unit rate—dollars per month, miles per hour, points per game.
Common misconception: slope is not “rise over run” as two separate numbers you memorize; it is a single ratio describing a rate.
Intercepts and what they mean
The y-intercept occurs where x = 0.
The x-intercept occurs where y = 0.
Intercepts are usually the most interpretable features in context. For example, in a cost model, the y-intercept might be an initial fee.
Forms of a line and when to use them
You’ll see multiple equivalent forms:
Form | Equation | When it’s useful |
|---|---|---|
Slope-intercept | y = mx + b | Reading slope and y-intercept quickly |
Point-slope | y - y1 = m(x - x1) | Writing an equation from a point and slope |
Standard | Ax + By = C | Working with systems; integer coefficients |
A key SAT skill is recognizing that these forms describe the same object.
Parallel and perpendicular lines
Parallel lines have the same slope.
Perpendicular lines have slopes that multiply to -1 (negative reciprocals), when slopes are defined.
If one line has slope m, a perpendicular line has slope:
-\frac{1}{m}
Worked example: equation of a line from two points
Find the equation of the line through \left(2, 5\right) and \left(6, -3\right).
Step 1: Find the slope.
m = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2
Step 2: Use slope-intercept form y = mx + b and plug in a point.
Use \left(2, 5\right):
5 = -2(2) + b
5 = -4 + b
b = 9
So the equation is:
y = -2x + 9
Worked example: interpret slope and intercept in context
A gym charges a signup fee plus a monthly fee. The total cost after m months is modeled by:
C = 35 + 20m
Here, the slope is 20, meaning 20 dollars per month. The intercept is 35, meaning an initial 35 dollar fee at month 0.
Exam Focus
Typical question patterns:
Identify slope/intercepts from an equation, table, or graph.
Build a linear model from two points or a context.
Compare two linear models and interpret differences.
Common mistakes:
Swapping x and y differences in the slope formula.
Confusing slope with intercept (especially in word problems).
Forgetting that slope is a rate with units (dollars per month, not just “20”).
Systems of Linear Equations and Inequalities
A system is a set of equations (or inequalities) that must be true at the same time. On the SAT, systems model situations with two constraints—like mixtures, ticket sales, or intersecting lines.
What a solution means
For a system of two linear equations, a solution is an ordered pair \left(x, y\right) that makes both equations true. Graphically, it’s the intersection point.
Systems can have:
One solution (lines intersect once)
No solution (parallel distinct lines)
Infinitely many solutions (the same line written two ways)
Understanding these cases helps you interpret whether a real-world situation has a unique outcome.
Solving methods
Substitution: solve one equation for one variable, substitute into the other.
Elimination: add/subtract equations to cancel a variable.
Graphing: interpret intersection points (often supported by the graphing calculator).
SAT often rewards elimination because it’s fast and reliable with integer coefficients.
Systems of inequalities
A system of inequalities describes a region of points that satisfy all constraints. SAT questions may ask which graph matches the system or which point satisfies it.
Worked example: elimination
Solve:
2x + y = 11
3x - y = 4
Step 1: Add the equations to eliminate y.
(2x + y) + (3x - y) = 11 + 4
5x = 15
x = 3
Step 2: Substitute into one equation.
2(3) + y = 11
6 + y = 11
y = 5
So the solution is:
\left(3, 5\right)
Worked example: identifying no-solution quickly
Consider:
2x + 4y = 8
x + 2y = 6
Multiply the second equation by 2:
2x + 4y = 12
Now the left sides match but constants differ (8 vs 12), so the system is inconsistent: no solution.
Exam Focus
Typical question patterns:
Solve a system and interpret the intersection as a real-world meaning.
Decide whether a system has one, none, or infinitely many solutions.
Match a system of inequalities to a shaded graph region.
Common mistakes:
Sign errors when adding/subtracting equations.
Assuming a system always has a solution.
Mixing up “same slope” with “same line” (same line requires proportional constants too).
Ratios, Rates, Proportions, Percent, and Units (Problem-Solving and Data Analysis)
This domain is about quantitative reasoning—using numbers to describe comparisons and making sure units and definitions match the context.
Ratios and proportions
A ratio compares quantities, like 3:2 or \frac{3}{2}. A proportion states two ratios are equal:
\frac{a}{b} = \frac{c}{d}
Why it matters: Many real SAT contexts—recipes, map scales, similar figures, currency conversion—are proportional.
A key idea is scale factor: if everything is multiplied by the same factor, the relationships stay consistent.
Rates and unit rates
A rate compares quantities with different units, like miles per hour. A unit rate is “per 1,” like \frac{60\text{ miles}}{1\text{ hour}}.
SAT questions often become straightforward when you identify the unit rate and multiply.
Percent and percent change
A percent is “per 100.” Converting:
p\% = \frac{p}{100}
Percent change is based on the original amount:
\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}}
Common mistake: dividing by the new value instead of the original.
Units and dimensional analysis
Units are not decoration—they guide the math. If a rate is \frac{\text{dollars}}{\text{hour}} and you multiply by hours, hours cancel and you get dollars.
This is a powerful error-checking tool on the SAT.
Worked example: proportion (scale)
A map scale is 1 inch represents 5 miles. Two towns are 3.6 inches apart on the map. What is the actual distance?
Set up the proportion:
\frac{1}{5} = \frac{3.6}{d}
Cross-multiply:
d = 3.6 \cdot 5 = 18
So the towns are 18 miles apart.
Worked example: percent change
A jacket’s original price is 80 dollars. It’s discounted to 60 dollars. What is the percent decrease?
Percent decrease:
\frac{60 - 80}{80} = \frac{-20}{80} = -0.25
Convert to percent: -25\%, so the price decreased by 25\%.
Exam Focus
Typical question patterns:
Multi-step word problems involving unit rates, conversions, or percent change.
Set up and solve a proportion from a context (scale, mixture, density).
Interpret “per” language to build an equation.
Common mistakes:
Using the wrong reference value for percent change (should be original).
Dropping or mismatching units (minutes vs hours, cents vs dollars).
Treating ratios like differences (confusing 3:2 with 3 - 2).
Data Interpretation, Statistics, and Probability (Problem-Solving and Data Analysis)
The SAT tests your ability to read data displays and reason statistically—skills used constantly in science and social science.
Reading tables and graphs
SAT graphs may be linear, nonlinear, or categorical (bar charts). The key is to slow down and identify:
What each axis represents (including units)
The scale (especially if it’s not starting at zero)
Whether values are exact or estimated
A common error is answering from the “shape” of a graph without reading the labels.
Measures of center: mean and median
Mean is the average:
\text{mean} = \frac{\text{sum of values}}{\text{number of values}}
Median is the middle value when ordered (or the average of the two middle values if there are an even number of values).
Why it matters: The SAT often asks which measure changes (or doesn’t) when you alter the data.
If you add the same constant k to every value, both mean and median increase by k. If you multiply every value by k, both mean and median get multiplied by k.
Spread: range and (conceptually) variability
Range is:
\text{range} = \text{max} - \text{min}
The SAT may also use language like “more variable” or “less spread out” without requiring advanced formulas.
Weighted averages
A weighted average accounts for different “amounts” of each value. If scores x1, x2, \dots have weights w1, w2, \dots then:
\text{weighted mean} = \frac{w1x1 + w2x2 + \cdots}{w1 + w2 + \cdots}
This shows up in grade calculations, mixture prices, and combined rates.
Probability basics
Probability is:
P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{number of total equally likely outcomes}}
For independent events (like two separate coin flips):
P(A \text{ and } B) = P(A)P(B)
A frequent SAT pitfall is assuming events are independent when they’re not (like drawing cards without replacement).
Worked example: mean change
The mean of 6 numbers is 10. The sum is:
6 \cdot 10 = 60
If one number increases by 12, the sum becomes 72, so the new mean is:
\frac{72}{6} = 12
The mean increased by 2. Notice the pattern: adding 12 to one value in a set of 6 increases the mean by \frac{12}{6} = 2.
Worked example: probability without replacement
A bag has 3 red marbles and 2 blue marbles. Two marbles are drawn without replacement. Probability both are red:
First draw red:
\frac{3}{5}
Then there are 2 red left out of 4 total:
\frac{2}{4}
Multiply:
\frac{3}{5} \cdot \frac{2}{4} = \frac{6}{20} = \frac{3}{10}
Exam Focus
Typical question patterns:
Compute or compare mean/median from a table or described data.
Interpret a graph and compute a rate of change or difference.
Basic probability, including “without replacement” and multi-step scenarios.
Common mistakes:
Forgetting to order data when finding the median.
Treating “without replacement” as independent.
Missing axis scale details (especially when the vertical axis is truncated).
Advanced Math: Exponents, Radicals, and Rational Expressions
“Advanced Math” on the SAT isn’t about obscure tricks—it’s about handling expressions and equations where the relationship is not purely linear. These tools let you work with growth, area/volume relationships, and inverse variation.
Exponent rules (structure, not memorization)
Exponents represent repeated multiplication. The rules come from that definition.
Key rules:
a^m \cdot a^n = a^{m+n}
\frac{a^m}{a^n} = a^{m-n}
(a^m)^n = a^{mn}
a^{-n} = \frac{1}{a^n}
Also:
a^{\frac{1}{n}} = \sqrt[n]{a}
A common misconception is thinking \left(a + b\right)^2 = a^2 + b^2. The correct expansion is:
(a + b)^2 = a^2 + 2ab + b^2
Radicals and simplifying
A radical like \sqrt{50} can be simplified by factoring out perfect squares:
\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
SAT also expects comfort with rational exponents and basic radical equations.
Rational expressions (fractions with polynomials)
A rational expression is a ratio of polynomials, like:
\frac{x^2 - 1}{x - 1}
You simplify by factoring and canceling common factors (not terms).
For example:
x^2 - 1 = (x - 1)(x + 1)
So:
\frac{x^2 - 1}{x - 1} = x + 1
but only for x \ne 1 because the original expression is undefined at x = 1. SAT sometimes tests this subtlety.
Worked example: exponent manipulation
Simplify:
\frac{2^5 \cdot 2^3}{2^4}
Combine exponents in the numerator:
2^{5+3} = 2^8
Then subtract exponents when dividing:
\frac{2^8}{2^4} = 2^{8-4} = 2^4 = 16
Worked example: simplify a rational expression
Simplify:
\frac{x^2 - 9}{x + 3}
Factor the numerator (difference of squares):
x^2 - 9 = (x - 3)(x + 3)
Cancel the common factor x + 3:
\frac{(x - 3)(x + 3)}{x + 3} = x - 3
Restriction: x \ne -3.
Exam Focus
Typical question patterns:
Simplify expressions using exponent rules or factoring.
Solve equations involving radicals or rational expressions.
Identify restrictions on variables due to denominators or even roots.
Common mistakes:
Canceling terms instead of factors.
Ignoring domain restrictions (like denominator not equal to 0).
Misapplying exponent rules (especially negatives and fractions).
Advanced Math: Polynomials, Factoring, and Quadratic Equations
Quadratics and polynomials model situations where change isn’t constant—projectile motion, area relationships, optimization, and curved graphs.
Polynomials and degree
A polynomial is a sum of terms like ax^n where n is a nonnegative integer. The degree is the highest exponent.
Linear: degree 1
Quadratic: degree 2
SAT focuses heavily on quadratics.
Factoring quadratics
Factoring is rewriting a quadratic as a product, typically:
x^2 + bx + c = (x + p)(x + q)
where p + q = b and pq = c.
Why it matters: Factoring reveals the zeros (x-intercepts) of a quadratic, because a product is zero when any factor is zero.
The quadratic formula
When factoring is difficult or impossible with integers, use the quadratic formula. For:
ax^2 + bx + c = 0
solutions are:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The quantity under the square root is the discriminant:
b^2 - 4ac
If it’s positive: two real solutions
If it’s zero: one real solution (a double root)
If it’s negative: no real solutions
The SAT may ask how many solutions exist without computing them.
Completing the square (vertex form)
Another way to rewrite a quadratic is vertex form:
y = a(x - h)^2 + k
The vertex is \left(h, k\right). This form makes maximum/minimum values and shifts obvious.
Completing the square is the process of rewriting:
x^2 + bx
as:
\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2
Worked example: solving by factoring
Solve:
x^2 - 5x + 6 = 0
Find two numbers that multiply to 6 and add to -5: -2 and -3.
x^2 - 5x + 6 = (x - 2)(x - 3)
Set each factor to zero:
x - 2 = 0
x - 3 = 0
So:
x = 2
x = 3
Worked example: solve with the quadratic formula
Solve:
2x^2 + 3x - 2 = 0
Here a = 2, b = 3, c = -2.
x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}
Compute discriminant:
3^2 - 4(2)(-2) = 9 + 16 = 25
So:
x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}
Two solutions:
x = \frac{-3 + 5}{4} = \frac{1}{2}
x = \frac{-3 - 5}{4} = -2
Exam Focus
Typical question patterns:
Solve quadratic equations (factoring, quadratic formula, or graph interpretation).
Identify the number of solutions using the discriminant.
Interpret vertex form in context (maximum/minimum).
Common mistakes:
Sign errors when factoring or using -b in the quadratic formula.
Forgetting to set the equation equal to 0 before applying methods.
Misinterpreting the vertex (confusing h with -h in x - h).
Functions Beyond Lines: Interpretation, Notation, and Nonlinear Models
SAT function questions test whether you can treat a function as a machine: input goes in, output comes out, and the rule is consistent.
Function notation
If f(x) is a function, then f(3) means “the output when the input is 3.” It does not mean f \cdot x.
This confusion is common and can derail otherwise easy problems.
Domain and constraints
The domain is the set of allowed inputs. Context and algebra both can restrict the domain:
A denominator cannot be zero.
An even root requires the inside to be nonnegative (in real numbers).
A context might require whole numbers or positive values.
Nonlinear models you’ll see
Quadratic: curved parabola.
Exponential (less frequent but possible in modeling contexts): repeated multiplication.
Rational: ratios that can create asymptotes.
You’re rarely asked for deep theory; instead you interpret key features: intercepts, growth/decay, maximum/minimum, or constraints.
Worked example: evaluate a function
Given:
f(x) = 2x^2 - 3x + 4
Find f(-2).
Substitute carefully with parentheses:
f(-2) = 2(-2)^2 - 3(-2) + 4
= 2(4) + 6 + 4
= 18
The main danger here is writing -2^2 (which equals -(2^2)) instead of (-2)^2.
Worked example: domain restriction from a rational expression
For:
g(x) = \frac{1}{x - 7}
The domain excludes x = 7 because the denominator would be zero.
Exam Focus
Typical question patterns:
Evaluate a function at a given input or solve f(x) = k.
Determine domain restrictions from an expression or context.
Interpret a nonlinear graph’s intercepts/turning point.
Common mistakes:
Treating f(x) like multiplication.
Substitution errors with negative inputs and exponents.
Ignoring domain restrictions (especially on student-produced responses).
Geometry: Lines, Angles, Triangles, and Coordinate Geometry
SAT geometry focuses on a core set of relationships—many of which are provided on the formula reference, but you still must know when and how to use them.
Angle relationships
Key facts:
A straight line is 180^\circ.
A full circle is 360^\circ.
Vertical angles are equal.
Adjacent angles on a line add to 180^\circ.
For parallel lines cut by a transversal, corresponding angles are equal and alternate interior angles are equal.
Why it matters: These are the building blocks behind many diagram-based SAT questions.
Triangles and similarity
A triangle’s interior angles sum to:
180^\circ
Similar triangles have equal corresponding angles and proportional corresponding side lengths.
Similarity is a powerful SAT tool because it turns geometry diagrams into proportion problems.
The Pythagorean theorem and distance
For a right triangle with legs a and b and hypotenuse c:
a^2 + b^2 = c^2
In the coordinate plane, the distance between \left(x1, y1\right) and \left(x2, y2\right) is:
d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
That formula is the Pythagorean theorem applied to horizontal and vertical differences.
Circle basics
A circle with radius r has:
\text{circumference} = 2\pi r
\text{area} = \pi r^2
Coordinate form (center-radius form):
(x - h)^2 + (y - k)^2 = r^2
where \left(h, k\right) is the center.
Worked example: similar triangles
A smaller triangle is similar to a larger triangle. A side of length 6 in the small corresponds to 15 in the large. Another side in the small is 8. Find the corresponding side in the large.
Scale factor from small to large:
\frac{15}{6} = \frac{5}{2}
Multiply the corresponding side:
8 \cdot \frac{5}{2} = 20
So the corresponding side is 20.
Worked example: distance in the coordinate plane
Find the distance between \left(-1, 4\right) and \left(5, -2\right).
Compute differences:
x2 - x1 = 5 - (-1) = 6
y2 - y1 = -2 - 4 = -6
Distance:
d = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
Exam Focus
Typical question patterns:
Use angle rules to find unknown measures from a diagram.
Use similarity to set proportions for missing side lengths.
Apply Pythagorean theorem or distance formula in coordinate geometry.
Common mistakes:
Assuming triangles are similar without matching corresponding angles/sides.
Mixing up radius and diameter in circle formulas.
Arithmetic slips in squaring and square-root simplification.
Geometry: Area, Volume, and Composite Figures
Area and volume problems test both formula knowledge and modeling—deciding what shapes are present and how they combine.
Area concepts
Area measures two-dimensional “coverage.” Key formulas:
Rectangle:
A = lw
Triangle:
A = \frac{1}{2}bh
Circle:
A = \pi r^2
A frequent SAT challenge is identifying the correct base and height—especially in slanted triangles where the height is perpendicular to the base.
Volume concepts
Volume measures three-dimensional “space.” Common formulas:
Rectangular prism:
V = lwh
Cylinder:
V = \pi r^2h
You’ll sometimes use volume plus density or unit conversion (like cubic inches to cubic feet). The SAT typically keeps conversions simple but expects careful unit tracking.
Composite figures
A composite figure is made of simpler shapes. The reliable method is:
Break the figure into familiar pieces.
Compute each area/volume.
Add (or subtract cut-outs).
Worked example: composite area
A shape consists of a rectangle of width 10 and height 6, with a semicircle of radius 3 attached along the height (so the semicircle’s diameter is 6). Find the total area.
Rectangle area:
A_{\text{rect}} = 10 \cdot 6 = 60
Semicircle area is half a circle:
A_{\text{semi}} = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi(3^2) = \frac{9\pi}{2}
Total:
60 + \frac{9\pi}{2}
Worked example: cylinder volume
A cylinder has radius 4 and height 9. Volume:
V = \pi r^2 h = \pi(4^2)(9) = 144\pi
Exam Focus
Typical question patterns:
Compute area/volume directly from given dimensions.
Composite shapes: add/subtract standard areas.
Use geometry formulas inside a word problem (paint needed, capacity, material cost).
Common mistakes:
Using diameter where radius is required.
Using slanted side length as height in triangle area.
Forgetting “half” in semicircle or triangular prism-like decompositions.
Trigonometry (Right Triangles): Sine, Cosine, Tangent
SAT trigonometry is mostly right-triangle trigonometry—a set of ratios that connect angles to side relationships.
The meaning of trig ratios
In a right triangle, for an acute angle \theta:
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
These ratios matter because they let you compute unknown distances indirectly—exactly how surveying, navigation, and physics problems work.
A useful memory aid is the phrase “opposite over hypotenuse” etc., but what really prevents errors is drawing the triangle and labeling relative to the specific angle \theta.
Special right triangles
Two special triangles appear frequently:
45^\circ-45^\circ-90^\circ triangle: legs equal; hypotenuse is leg times \sqrt{2}.
30^\circ-60^\circ-90^\circ triangle: sides in the ratio 1 : \sqrt{3} : 2.
These help you avoid calculator dependence and recognize exact values.
Worked example: tangent in a right triangle
In a right triangle, angle \theta has opposite side 12 and adjacent side 5. Find \tan(\theta).
By definition:
\tan(\theta) = \frac{12}{5}
Worked example: using sine to find a missing side
A right triangle has hypotenuse 10 and an acute angle \theta where \sin(\theta) = 0.6. Find the length of the opposite side.
Since:
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
we have:
0.6 = \frac{o}{10}
So:
o = 6
Exam Focus
Typical question patterns:
Identify or compute \sin(\theta), \cos(\theta), \tan(\theta) from a labeled triangle.
Use a given trig value to find a missing side.
Recognize special right triangles and their side ratios.
Common mistakes:
Mixing up opposite and adjacent relative to the chosen angle.
Using hypotenuse in the wrong ratio (only sine and cosine use hypotenuse).
Forgetting special triangle ratios and overcomplicating with extra steps.
Word Problems and Modeling: Turning Situations into Equations
Nearly every SAT Math topic becomes harder (or easier) depending on how well you translate words into math. Modeling is the skill of choosing variables, writing relationships, and interpreting results.
A reliable modeling process
Define variables clearly. For example, let t be time in hours.
Translate one sentence at a time into an equation piece.
Check units—they often reveal the correct operations.
Solve using algebra.
Interpret the solution: does it answer the question, and does it make sense?
Common SAT modeling structures
“Starting amount plus rate times time” becomes a linear function.
“Part of a whole” becomes a percent or proportion.
“Two constraints” becomes a system.
“Area depends on side length” often becomes quadratic.
Worked example: linear model from a context
A taxi charges a base fee of 3 dollars plus 2.50 dollars per mile. Write a function for cost C in terms of miles m, and find the cost for 12 miles.
The base fee is the intercept, and the per-mile charge is the slope:
C = 3 + 2.5m
Evaluate at m = 12:
C = 3 + 2.5(12) = 3 + 30 = 33
So the ride costs 33 dollars.
Worked example: system from ticket sales
A school sold 120 tickets for a play. Student tickets cost 5 and adult tickets cost 8. Total revenue was 840. How many adult tickets were sold?
Let s be student tickets and a be adult tickets.
Count equation:
s + a = 120
Revenue equation:
5s + 8a = 840
Use substitution from the first: s = 120 - a.
Substitute:
5(120 - a) + 8a = 840
600 - 5a + 8a = 840
600 + 3a = 840
3a = 240
a = 80
So 80 adult tickets were sold.
Exam Focus
Typical question patterns:
Build a linear equation from an initial value and a constant rate.
Translate a two-variable scenario into a system.
Use units to decide whether to multiply/divide and what the answer represents.
Common mistakes:
Solving correctly but answering the wrong quantity (for example, finding s when asked for a).
Mixing units (miles vs kilometers, minutes vs hours).
Skipping interpretation—accepting a negative or non-integer result when the context forbids it.
Tool Use on the Digital SAT: Algebraic Reasoning with a Graphing Calculator
The Digital SAT provides an on-screen graphing calculator, which is best treated as a tool for verification and visualization, not a replacement for reasoning.
When graphing helps most
Checking solutions by graphing both sides of an equation and finding intersections.
Visualizing a quadratic’s vertex and intercepts.
Regressions and scatterplot reasoning may appear via interpretation tasks; the key is understanding what the model means.
When algebra is faster
Simple linear equations and systems with clean coefficients.
Simplifying expressions and factoring.
Exact-value tasks (especially with fractions and radicals).
A practical habit: “predict, then compute”
Before you compute with a calculator, predict what the answer should look like—positive/negative, bigger/smaller than a reference, reasonable units. This reduces careless calculator-entry errors.
Worked example: verify an intersection conceptually
Suppose you solve a system and get \left(3, 5\right). A fast check is substitution: plug into both equations and verify both are true. This is often faster and more reliable than graphing if the arithmetic is manageable.
Exam Focus
Typical question patterns:
Use graphs to interpret solutions, intercepts, and turning points.
Choose between algebraic manipulation and calculator-based solving.
Spot which expression form best matches a graph.
Common mistakes:
Relying on approximate graph intersections when an exact value is required.
Calculator entry errors (missing parentheses for negatives or fractions).
Using the tool without interpreting what the result means in context.
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Heart of Algebra: Linear Equations, Inequalities, and Systems
The SAT Math section, as designed by the College Board for the Digital SAT, is divided into two modules and covers four major content domains: Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry. Algebra is the single most heavily tested domain, accounting for roughly 13–15 of the 44 math questions you will see. Mastering it is non-negotiable.
Linear Equations in One Variable
A linear equation is any equation that can be written in the form ax + b = c where a, b, and c are constants and x is a variable raised to the first power. The word "linear" literally means "line" — if you graphed every possible solution, you would get a straight line.
To solve a linear equation, your goal is to isolate the variable on one side. You do this by performing the same operation on both sides of the equation — adding, subtracting, multiplying, or dividing. The golden rule is: whatever you do to one side, you must do to the other.
Consider the equation 3x - 7 = 14. Add 7 to both sides to get 3x = 21, then divide both sides by 3 to get x = 7.
The SAT loves to dress up simple linear equations in complicated clothing. You might see something like:
\frac{2(x + 3)}{5} = 4
Multiply both sides by 5: 2(x + 3) = 20. Distribute: 2x + 6 = 20. Subtract 6: 2x = 14. Divide by 2: x = 7.
A common trap is the no-solution or infinitely many solutions scenario. If simplifying both sides of an equation leads to a false statement like 0 = 5, there is no solution. If it leads to a true identity like 0 = 0, there are infinitely many solutions. The SAT tests this concept regularly — for instance, asking you for which value of a constant k the equation has no solution.
Linear Inequalities
Linear inequalities work exactly like linear equations with one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign. This is the single most common mistake students make with inequalities.
For example, to solve -2x + 5 > 11, subtract 5 from both sides: -2x > 6. Divide both sides by -2 and flip the sign: x < -3.
The solution to a linear inequality is a range of values, not a single value. On a number line, x < -3 means every number to the left of -3, not including -3 itself.
Systems of Linear Equations
A system of linear equations consists of two or more equations with the same variables. On the SAT, you will almost always deal with a system of two equations in two unknowns. There are two primary methods for solving them:
Substitution: Solve one equation for one variable, then plug that expression into the other equation. This works best when one variable already has a coefficient of 1 or -1.
Elimination (also called combination): Add or subtract the equations — or multiply one equation by a constant first — so that one variable cancels out.
For example, consider:
2x + y = 10
3x - y = 5
Adding these equations eliminates y: 5x = 15, so x = 3. Substituting back: 2(3) + y = 10, so y = 4.
A system can have one solution (the lines intersect at exactly one point), no solution (the lines are parallel — same slope, different intercepts), or infinitely many solutions (the lines are the same line). The SAT frequently asks you to determine how many solutions a system has, or to find a constant that makes a system have no solution. Two lines are parallel when their slopes are equal: if a1 / b1 = a2 / b2 but c1 \neq c2 (in the standard form ax + by = c), there is no solution.
Graphing Linear Equations
The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). The slope measures steepness — it is the ratio of the vertical change to the horizontal change between any two points on the line:
m = \frac{y2 - y1}{x2 - x1}
A positive slope means the line goes upward from left to right; a negative slope means it goes downward. A slope of zero is a horizontal line, and an undefined slope is a vertical line.
The standard form of a linear equation is Ax + By = C. You can convert between forms freely. The point-slope form is y - y1 = m(x - x1) and is useful when you know the slope and one point.
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other — if one line has slope m, a line perpendicular to it has slope -1/m.
Exam Focus
Typical question patterns: You will be asked to solve for a variable in a linear equation, find the number of solutions to a system, interpret the slope or y-intercept in a real-world context (e.g., "What does the slope represent in terms of the situation described?"), or write a linear equation from a word problem.
Common mistakes: Forgetting to flip the inequality sign when dividing by a negative; distributing a negative sign incorrectly across parentheses; confusing "no solution" with "x = 0" — these are very different things.
Advanced Math: Quadratics, Polynomials, and Nonlinear Equations
Advanced Math accounts for roughly 13–15 questions on the SAT. This domain focuses on your ability to work with expressions that are not linear — primarily quadratics, but also polynomials of higher degree, rational expressions, radical expressions, and exponential functions.
Quadratic Equations
A quadratic equation has the general form ax^2 + bx + c = 0 where a \neq 0. The graph of a quadratic function is a parabola — a U-shaped curve that opens upward if a > 0 and downward if a < 0.
There are three main methods for solving quadratic equations:
Factoring: Express the quadratic as a product of two binomials. For example, x^2 + 5x + 6 = 0 factors as (x + 2)(x + 3) = 0. By the zero product property, if the product of two factors is zero, at least one factor must be zero, giving x = -2 or x = -3.
Factoring requires you to find two numbers that multiply to c and add to b (when a = 1). When a \neq 1, you can use the AC method: find two numbers that multiply to ac and add to b, then rewrite the middle term and factor by grouping.
The Quadratic Formula: When factoring is difficult or the roots are not integers, use:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
This formula always works. The expression under the square root, b^2 - 4ac, is called the discriminant. It tells you the nature of the solutions:
If b^2 - 4ac > 0: two distinct real solutions
If b^2 - 4ac = 0: exactly one real solution (a repeated root)
If b^2 - 4ac < 0: no real solutions (the parabola does not cross the x-axis)
The SAT will ask questions about the discriminant — for example, asking for which values of a constant k the equation has no real solutions.
Completing the Square: This method converts ax^2 + bx + c = 0 into vertex form a(x - h)^2 + k = 0, which reveals the vertex of the parabola at the point (h, k). To complete the square for x^2 + 6x + 2 = 0: take half of the coefficient of x (which is 3), square it (which is 9), and add and subtract it: (x^2 + 6x + 9) - 9 + 2 = 0, giving (x + 3)^2 - 7 = 0.
Key Properties of Parabolas
The vertex is the minimum point (when a > 0) or maximum point (when a < 0) of the parabola. In standard form, the x-coordinate of the vertex is:
x = \frac{-b}{2a}
The axis of symmetry is the vertical line x = -b/(2a). The parabola is symmetric about this line, meaning the two roots (if they exist) are equidistant from it.
The y-intercept is the point where x = 0, which is simply the value c in ax^2 + bx + c.
Polynomials and Factoring
A polynomial is an expression with one or more terms where each term is a constant multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is the highest power of the variable.
Key facts about polynomials that the SAT tests:
A polynomial of degree n has at most n real roots (x-intercepts).
If (x - r) is a factor of a polynomial, then r is a root (and vice versa — this is the Factor Theorem).
The Remainder Theorem states that when a polynomial p(x) is divided by (x - r), the remainder is p(r).
You should be comfortable multiplying polynomials (using distribution/FOIL), recognizing special products like (a + b)^2 = a^2 + 2ab + b^2 and (a - b)(a + b) = a^2 - b^2, and performing polynomial long division or synthetic division.
Exponential Functions and Growth/Decay
An exponential function has the form f(x) = a \cdot b^x where a is the initial value and b is the base (the growth or decay factor). If b > 1, the function represents exponential growth; if 0 < b < 1, it represents exponential decay.
For percent increase or decrease problems, the base is b = 1 + r for growth or b = 1 - r for decay, where r is the rate expressed as a decimal. For example, a population that grows by 5% per year from an initial size of 1000 is modeled by P(t) = 1000(1.05)^t.
The SAT frequently asks you to interpret exponential functions in context — for instance, identifying what the base or coefficient represents in a real-world scenario.
Rational and Radical Expressions
A rational expression is a fraction where the numerator and/or denominator are polynomials. To simplify, factor both and cancel common factors. Always note what values make the denominator zero — these are excluded values (the expression is undefined there).
To add or subtract rational expressions, you need a common denominator, just like with numerical fractions.
A radical expression involves square roots (or other roots). The key rules are:
\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
When solving radical equations, isolate the radical and square both sides. Always check your solutions — squaring both sides can introduce extraneous solutions that don't actually satisfy the original equation.
Exam Focus
Typical question patterns: Solving a quadratic by factoring or using the quadratic formula; determining the vertex of a parabola; interpreting parts of an exponential function in context; simplifying rational expressions; finding for which value of a constant an equation has a specific number of solutions.
Common mistakes: Sign errors when using the quadratic formula (especially with the -b term); forgetting to check for extraneous solutions with radical or rational equations; confusing the vertex form a(x - h)^2 + k by getting the sign of h wrong (the vertex is at x = h, but it appears as x - h in the formula, so (x + 3)^2 means h = -3).
Problem-Solving and Data Analysis
This domain accounts for roughly 5–7 questions on the SAT and focuses on your ability to work with real-world data — ratios, proportions, percentages, statistics, and probability. These questions appear only in the calculator-allowed context of the digital SAT.
Ratios, Rates, and Proportional Relationships
A ratio compares two quantities. It can be written as a:b, a/b, or "a to b." A proportion is an equation stating that two ratios are equal: a/b = c/d. You solve proportions by cross-multiplying: ad = bc.
A rate is a ratio that compares two quantities with different units — for example, miles per hour, dollars per item, or people per square mile. A unit rate is a rate with a denominator of 1.
The SAT commonly sets up word problems where you need to use proportional reasoning. For example: "If 3 workers can build 5 walls in 8 hours, how many walls can 6 workers build in 12 hours?" These problems require you to set up the correct proportional relationships (often using the idea that doubling workers doubles output, and multiplying time by 1.5 multiplies output by 1.5).
Percentages
Percent means "per hundred." So 35% means 35/100 = 0.35. There are three fundamental percentage calculations:
Finding a percent of a number: x\% of n is (x/100) \cdot n
Percent change: \text{Percent change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100
Finding the whole from a part and percent: If 30% of a number is 60, then 0.30 \cdot x = 60, so x = 200.
A very common SAT pattern is successive percent changes. If a price increases by 20% and then decreases by 20%, the final price is NOT the original price. An increase of 20% multiplies by 1.20, and a decrease of 20% multiplies by 0.80. So the net effect is 1.20 \times 0.80 = 0.96, which is a 4% decrease overall. Many students mistakenly think these cancel out.
Statistics: Center and Spread
The SAT tests your understanding of statistical measures, not your ability to calculate them from huge data sets. Know what these measures mean:
Mean (average): The sum of all values divided by the number of values. The mean is sensitive to outliers — one extremely high or low value can pull it significantly.
\text{Mean} = \frac{\sum x_i}{n}
Median: The middle value when the data is ordered from least to greatest. If there is an even number of values, the median is the average of the two middle values. The median is resistant to outliers — it is not affected much by extreme values.
Mode: The most frequently occurring value.
Range: The difference between the maximum and minimum values.
Standard deviation: A measure of how spread out the data is from the mean. You will not need to calculate standard deviation on the SAT, but you need to understand it conceptually. A larger standard deviation means more spread; a smaller one means data is clustered closer to the mean. If every value in a data set is the same, the standard deviation is zero.
When the SAT shows you two data sets and asks which has a greater standard deviation, look at which data set is more spread out from its center.
Interpreting Data: Tables, Graphs, and Scatterplots
You will encounter scatterplots (showing the relationship between two variables), bar graphs, histograms, line graphs, and two-way frequency tables.
For scatterplots, understand line of best fit (also called the regression line or trend line). The SAT may give you the equation of a line of best fit and ask you to interpret the slope (the predicted change in y for each unit increase in x) or the y-intercept (the predicted value of y when x = 0). Be careful — always say "predicted" or "estimated" because the line of best fit does not give exact values for individual data points.
A two-way frequency table organizes data by two categorical variables. You might be asked to find conditional probabilities, marginal totals, or relative frequencies from such a table.
Probability
The probability of an event is:
P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}
Probability ranges from 0 (impossible) to 1 (certain). The SAT typically tests basic probability and conditional probability — "given that a student is in 10th grade, what is the probability they prefer math?" For this, you restrict your sample to only 10th graders and find the fraction who prefer math.
Study Design and Inference
The SAT asks conceptual questions about how conclusions can be drawn from data. Key ideas:
A random sample allows you to generalize results to the population.
A randomized controlled experiment (with random assignment to groups) allows you to draw cause-and-effect conclusions.
An observational study (no random assignment) can only establish association, not causation.
If a question describes a survey of volunteers, you should recognize that the results cannot be generalized to a broader population (because the sample is not random).
Exam Focus
Typical question patterns: Interpreting the slope of a line of best fit in context; calculating probability from a two-way table; determining whether a study supports a causal conclusion; percent increase/decrease problems.
Common mistakes: Confusing "association" with "causation"; using the wrong denominator when calculating conditional probability (the denominator should be the size of the given group, not the entire table total); applying percent change with the wrong base value.
Geometry and Trigonometry
This domain covers roughly 5–7 questions on the SAT. It includes a mix of coordinate geometry, plane geometry (lines, angles, triangles, circles), area/volume, and basic trigonometry.
Lines and Angles
Some fundamental angle facts that appear constantly:
Supplementary angles add to 180°.
Complementary angles add to 90°.
Vertical angles (formed by two intersecting lines) are equal.
When a transversal crosses two parallel lines, it creates several angle relationships: corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles are supplementary.
The sum of the interior angles of a triangle is always 180°. The sum of the interior angles of any polygon with n sides is:
180(n - 2)
Triangles
Triangles are the most tested geometric shape on the SAT. Key facts:
The Pythagorean Theorem applies to right triangles: a^2 + b^2 = c^2 where c is the hypotenuse (the side opposite the right angle) and a and b are the legs.
Common Pythagorean triples worth memorizing: 3, 4, 5 (and multiples like 6, 8, 10); 5, 12, 13; 8, 15, 17; 7, 24, 25.
Special right triangles appear frequently:
The 45-45-90 triangle has sides in the ratio 1 : 1 : \sqrt{2}. If each leg has length s, the hypotenuse is s\sqrt{2}.
The 30-60-90 triangle has sides in the ratio 1 : \sqrt{3} : 2. The side opposite 30° is the shortest; the side opposite 60° is \sqrt{3} times the shortest side; the hypotenuse is twice the shortest side.
The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the third side.
Similar triangles have the same shape but not necessarily the same size — their corresponding angles are equal, and their corresponding sides are in proportion. If triangle ABC is similar to triangle DEF with a scale factor of k, then every side of DEF is k times the corresponding side of ABC.
The area of a triangle is:
A = \frac{1}{2}bh
where b is the base and h is the height measured perpendicular to that base.
Circles
The equation of a circle in the coordinate plane with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Be careful: the equation has r^2 on the right side, not r. If you see (x - 3)^2 + (y + 1)^2 = 25, the center is (3, -1) and the radius is 5 (because \sqrt{25} = 5).
Sometimes the SAT gives you the equation in expanded (general) form, such as x^2 + y^2 + 6x - 4y + 4 = 0. You need to complete the square for both x and y to convert it to standard form.
Key formulas for circles:
Circumference: C = 2\pi r
Area: A = \pi r^2
Arc length (for a central angle of \theta degrees): \text{Arc length} = \frac{\theta}{360} \times 2\pi r
Sector area: \text{Sector area} = \frac{\theta}{360} \times \pi r^2
If you work in radians (which the SAT sometimes uses), the arc length is simply s = r\theta and the sector area is A = \frac{1}{2}r^2\theta.
A key property: a central angle is equal in measure to its intercepted arc. An inscribed angle (vertex on the circle) is half the measure of its intercepted arc.
Also, a tangent line to a circle is perpendicular to the radius drawn to the point of tangency.
Area and Volume
The SAT provides a reference sheet of formulas, but knowing them from memory saves time:
Rectangle area: A = lw
Parallelogram area: A = bh
Trapezoid area: A = \frac{1}{2}(b1 + b2)h
Rectangular prism volume: V = lwh
Cylinder volume: V = \pi r^2 h
Sphere volume: V = \frac{4}{3}\pi r^3
Cone volume: V = \frac{1}{3}\pi r^2 h
Pyramid volume: V = \frac{1}{3}Bh where B is the area of the base
Right Triangle Trigonometry
Trigonometry on the SAT is limited primarily to right triangle trig and the unit circle basics. The three main trigonometric ratios are:
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
The mnemonic SOH-CAH-TOA is the classic way to remember these: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.
A crucial relationship that the SAT tests: the sine of an angle equals the cosine of its complement (and vice versa). If two angles are complementary (they add to 90°), then:
\sin(x) = \cos(90° - x)
So if \sin(30°) = 0.5, then \cos(60°) = 0.5 as well.
You should also know the Pythagorean identity:
\sin^2(\theta) + \cos^2(\theta) = 1
This is simply the Pythagorean theorem written in trigonometric terms, and the SAT may ask you to use it to find one trig value given another.
Radians
Radians are an alternative way to measure angles. A full circle is 2\pi radians (which equals 360°). Therefore:
\pi \text{ radians} = 180°
To convert from degrees to radians, multiply by \pi / 180. To convert from radians to degrees, multiply by 180 / \pi.
Some key equivalences: 90° = \pi/2, 60° = \pi/3, 45° = \pi/4, 30° = \pi/6.
Exam Focus
Typical question patterns: Finding arc length or sector area given a central angle; using the Pythagorean theorem in a multi-step problem; converting a circle equation from general form to standard form; applying \sin(x) = \cos(90° - x); calculating the volume of a composite solid.
Common mistakes: Forgetting that the equation of a circle uses r^2 (not r) — so if r = 5, the equation has 25 on the right side, and if the equation has 25, r is 5, not 25; mixing up the legs and hypotenuse in trigonometric ratios; using degree formulas when the problem gives angles in radians (or vice versa).
Essential SAT Math Strategies and Crosscutting Skills
Beyond the four content domains, several skills cut across all SAT math questions. These are not separate topics but habits and techniques that make you faster and more accurate.
Translating Word Problems into Math
Many SAT math questions are word problems. The key to translating them is knowing the mathematical meaning of common English phrases:
English Phrase | Mathematical Operation |
|---|---|
"is," "equals," "is the same as" | = |
"more than," "sum of," "increased by" | + |
"less than," "decreased by," "fewer than" | - |
"of," "times," "product of" | \times |
"per," "for each," "divided by" | \div |
"what," "a number" | variable (e.g., x) |
Be especially careful with "less than" — "5 less than x" is x - 5, not 5 - x. The order is reversed from how you read it in English.
Plugging In and Back-Solving
When a problem has variables in the answer choices, you can often plug in a specific number for the variable, calculate the result, and see which answer choice matches. Choose a number that is easy to work with but not 0 or 1 (since these have special properties that can make multiple answers appear correct).
When a problem asks for a specific numerical answer and gives you numerical answer choices, you can back-solve: plug each answer choice into the problem and see which one works. Start with choice B or C (the middle values) so you can determine which direction to go if it is not correct.
These strategies are not shortcuts for avoiding math — they are legitimate problem-solving techniques that can save you time, especially on complex algebra problems.
Reading Graphs and Interpreting Context
The SAT consistently asks you to interpret mathematical results in context. If you calculate that the slope of a line is 3.5, and the problem describes a relationship between hours studied and test score, the question might ask: "What does the slope represent?" The answer would be something like: "For each additional hour studied, the predicted test score increases by 3.5 points."
Always pay attention to:
Units — what are the units on each axis or in the problem?
The word 'predicted' or 'estimated' — models give approximations, not exact values.
What the variables represent — don't just solve; understand what your answer means in the scenario.
Working with Equivalent Expressions
A major skill on the SAT is recognizing that the same expression can be written in different forms, and each form reveals different information. For a quadratic function:
Standard form ax^2 + bx + c: reveals the y-intercept (c)
Factored form a(x - r1)(x - r2): reveals the x-intercepts/roots (r1 and r2)
Vertex form a(x - h)^2 + k: reveals the vertex ((h, k)) and the axis of symmetry (x = h)
The SAT may ask you to rewrite an expression in a specific form to extract certain information. Being fluent in moving between forms is essential.
Function Notation
If f(x) = 2x + 3, then f(5) means you replace every x with 5: f(5) = 2(5) + 3 = 13. The SAT tests composite functions — for example, if g(x) = x^2, then f(g(3)) = f(9) = 2(9) + 3 = 21. Work from the inside out.
The domain of a function is the set of all valid inputs (x-values). For most polynomials, the domain is all real numbers. For rational functions, exclude values where the denominator is zero. For square root functions, the expression under the radical must be non-negative.
Absolute Value
The absolute value of a number is its distance from zero, always non-negative. |x| = x if x \geq 0, and |x| = -x if x < 0.
To solve |x - 3| = 7, set up two cases:
x - 3 = 7 \quad \Rightarrow \quad x = 10
x - 3 = -7 \quad \Rightarrow \quad x = -4
For absolute value inequalities: |x| < a means -a < x < a (the solution is between -a and a). |x| > a means x < -a or x > a (the solution is outside that interval).
Complex Numbers
The SAT occasionally tests basic operations with complex numbers. The imaginary unit i is defined as:
i = \sqrt{-1}
Therefore i^2 = -1. A complex number has the form a + bi where a is the real part and b is the imaginary part.
To add or subtract complex numbers, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i^2 with -1. To divide, multiply the numerator and denominator by the conjugate of the denominator (if the denominator is a + bi, the conjugate is a - bi).
The powers of i cycle every four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then the pattern repeats. To find i^n for a large n, divide n by 4 and use the remainder to determine the value.
Exam Focus
Typical question patterns: Interpreting the meaning of a constant or coefficient in a given equation within a real-world context; evaluating a function at a given input or solving for when a function equals a given output; rewriting an expression in an equivalent form; operations with complex numbers.
Common mistakes: With word problems, writing an equation that reverses the relationship (e.g., writing 5 - x instead of x - 5); with functions, confusing f(x) = 0 (finding the x-values where the output is zero) with f(0) (finding the output when x is zero); with absolute value, forgetting to consider both the positive and negative cases.
Putting It All Together: The Structure of SAT Math
The Digital SAT Math section consists of two modules, each with 22 questions and a 35-minute time limit — so 44 questions in 70 minutes total. The second module adapts in difficulty based on your performance in the first module. About 75% of questions are multiple choice (four answer options), and about 25% are student-produced response (SPR) questions where you type in your answer.
For SPR questions, be aware of these rules: you can enter negative numbers, fractions, and decimals. If you get a repeating decimal, enter as many digits as the space allows or convert to a fraction. If a question says "one possible value," there may be a range of correct answers — you only need to give one.
Time management is crucial. You have roughly 1 minute and 35 seconds per question on average. If a question is taking too long, mark it and move on. There is no penalty for guessing on the SAT, so never leave a question blank — always enter an answer.
Finally, the digital SAT provides a built-in graphing calculator (Desmos) for both modules. You can also bring your own approved calculator. Learning to use Desmos effectively — graphing equations to find intersection points, checking your algebra visually, or evaluating complex expressions — can be a significant advantage. For example, to solve a system of equations, you can graph both equations and find the intersection point directly. To find the solutions of a quadratic, graph it and identify the x-intercepts. This is not a replacement for understanding the math, but it is a powerful verification tool.
The path to a strong SAT Math score is built on deep understanding of the four domains covered in these notes. Algebra and Advanced Math together account for roughly 60–70% of the questions, so prioritize these while ensuring you are comfortable with the data analysis and geometry concepts that round out the test. Practice applying these concepts to a variety of question types, and always focus on understanding why the math works — not just memorizing procedures.