SAT Math Study Guide

The overall SAT guide covers modules, adaptivity and scoring. This guide is about the math itself. College Board's Assessment Framework splits the 40 scored questions by domain and by format, which shows where typed answers (student-produced responses, or SPRs) appear.

Domain	Share	Scored questions	Multiple choice	SPR
Algebra	About 35%	13 to 15	10 to 11	3 to 4
Advanced Math	About 35%	13 to 15	10 to 11	3 to 4
Problem-Solving and Data Analysis	About 15%	5 to 7	4 to 5	1 to 2
Geometry and Trigonometry	About 15%	5 to 7	4 to 5	1 to 2

Every module mixes all four domains, easiest to hardest. Roughly 30% are word problems, most of them 50 words or fewer, so the reading load is light.

Read the Directions Once, Before Test Day

The Bluebook Math directions state four assumptions that apply unless a question says otherwise:

• All variables and expressions represent real numbers.
• Figures provided are drawn to scale.
• All figures lie in a plane.
• The domain of a function f is the set of all real numbers x for which f(x) is a real number.

Because figures are drawn to scale by default, you can estimate an angle or length from the picture to eliminate impossible choices.

Algebra on the SAT means linear relationships, tested in five skills. The framework's verbs repeat: create, solve, interpret, and connect a table, an equation and a graph.

• Linear equations in one variable: solve fluently, write one from a context, interpret a constant or solution, and decide when an equation has no solution, exactly one, or infinitely many.
• Linear equations in two variables: model a constraint such as a budget, interpret the graph of Ax + By = C in context, and write the equation of a line from two points, a point and a slope, or a point and a parallel or perpendicular line.
• Linear functions: write the rule from two input/output pairs or from one pair and the rate of change, interpret slope and intercept, and move between table, rule and graph.
• Systems of two linear equations: solve, model, and find the conditions for no solution, one solution or infinitely many.
• Linear inequalities in one or two variables: model constraints and decide whether a point in the xy-plane is in the solution set.

Interpreting Slope and Intercept

In a linear model, the coefficient of the variable is a rate (units of output per one unit of input) and the constant is the starting value (the output when the input is 0). Say the units out loud: "dollars per hour" is a rate; "dollars" alone is a fixed amount.

Worked Example: Infinitely Many Solutions

In the system below, a and b are constants. For which values does the system have infinitely many solutions?

Infinitely many solutions means the two equations describe the same line, so one must be a multiple of the other. Multiplying the first equation by 2 gives 6x - 4y = 16, so a = -4 and b = 16. If a were -4 but b were anything other than 16, the lines would be parallel and distinct: no solution. Any other value of a gives different slopes: exactly one solution.

Worked Example: A Perpendicular Line

Line k passes through (2, 5) and is perpendicular to the line y = (2/3)x - 4. Perpendicular slopes are negative reciprocals, so k has slope -3/2.

Worked Example: Writing an Inequality

A club has at most 60 dollars for supplies. Notebooks cost 4 dollars and pens cost 1.50 dollars. If n is the number of notebooks and p the number of pens, the constraint is 4n + 1.5p ≤ 60. "At most" means ≤; "at least" means ≥. To check a point such as (9, 16), substitute: 36 + 24 = 60, which satisfies ≤ 60, so buying 9 notebooks and 16 pens is possible.

College Board lists three Advanced Math skills:

• Equivalent expressions: factor out common factors, difference of two squares, trinomials into two binomials, and harder polynomials; rewrite simple rational expressions; convert rational exponents to radical form; add, subtract and multiply polynomials.
• Nonlinear equations in one variable and systems in two variables: quadratics in many forms, linear absolute value equations, simple rational and radical equations, polynomials in factored form, systems of a linear and a nonlinear equation, and the conditions for a quadratic to have no, one or two real solutions. It also includes solving a formula for one variable in terms of the others.
• Nonlinear functions: build and interpret quadratic and exponential models, choose the form of a function that displays a key feature, connect tables, graphs and equations (including transformations), and use function notation. The x-intercepts of y = f(x) are the solutions of f(x) = 0, and f(0) is the y-intercept.

The Three Forms of a Quadratic

The framework specifically tests choosing "the most suitable form" of a function to display a key feature. For quadratics, each form shows something different:

To get vertex form from standard form, complete the square: half of -8 is -4, and (x - 4)² = x² - 8x + 16, so subtract 9 to get back to +7. The vertex's x-coordinate is also the midpoint of the zeros: (1 + 7)/2 = 4. In general the vertex of y = ax² + bx + c is at x = -b/(2a).

Exponential Models

An exponential function has the form f(t) = a(b)^t, where a is the initial value and b is the growth factor. College Board's framework explicitly tests the link between percent change and growth factor: a 5% increase is a factor of 1.05, and a 15% decrease is a factor of 0.85. Watch the time unit in the exponent:

Worked Example: A Linear and a Quadratic System

How many solutions does this system have? Set the expressions for y equal:

Exactly one solution, (3, -2): the line is tangent to the parabola. The discriminant gives the same answer without solving: (-6)² - 4(1)(9) = 0. A positive discriminant would mean two intersection points; a negative one, none.

Worked Example: Radical Equations and Extraneous Solutions

Solve √(x + 7) = x - 5. Squaring both sides gives x + 7 = x² - 10x + 25, so x² - 11x + 18 = 0 and (x - 2)(x - 9) = 0. Now check both in the original equation: x = 9 gives √16 = 4 and 9 - 5 = 4, which works. x = 2 gives √9 = 3 but 2 - 5 = -3, which fails, because a square root is never negative. The only solution is 9. Squaring and clearing variable denominators can both create extraneous solutions.

Equivalent Expressions Worth Recognising on Sight

Seven skills share only 5 to 7 scored questions, so each appears once or twice at most.

Ratios, Rates, Proportions and Units

Expect scale drawings, derived units (kilowatt-hours, people per square kilometre) and one-step or multistep unit conversions. Write units on every number and multiply by conversion factors equal to 1, so the unwanted units cancel:

Percentages

The framework includes discounts, interest, taxes and tips, percent increases and decreases, and percentages of 100% or more. The reliable method is the multiplier: new = original × (1 ± rate). That makes reverse problems easy. If a price after a 25% increase is 90 dollars, the original is 90 / 1.25 = 72 dollars, not 90 × 0.75 = 67.50.

One-Variable Data: Centre and Spread

You will read frequency tables, histograms, dot plots and box plots, and compare mean, median and range. Two ideas come up repeatedly:

• Outliers pull the mean toward them but barely move the median. Replace the largest value with a much larger one and the mean rises while the median stays put.
• Standard deviation is never computed by hand on the SAT; you compare it. The data set whose values cluster more tightly around the mean has the smaller standard deviation.

Two-Variable Data: Models and Scatterplots

Fit and interpret linear, quadratic and exponential models to scatterplots, use a line of best fit to predict, and compare linear growth (adds the same amount each step) with exponential growth (multiplies by the same factor each step).

Probability and Conditional Probability

Most use a two-way table, and the key is the denominator. "Given that the student rides the bus" means the denominator is the bus total, not the grand total.

Inference and Margin of Error

Use a sample mean or proportion to estimate the population value, and interpret a margin of error as a range of plausible values. If a random sample of 400 voters shows 62% support with a margin of error of 4 percentage points, plausible population values run from 58% to 66%. College Board's framework states the relationship directly: a larger sample size generally leads to a smaller margin of error.

Evaluating Statistical Claims

Two rules decide almost every study-design question:

• Results can be generalised only to the population that was randomly sampled. A random sample of students at one school says nothing reliable about all students nationally.
• Only random assignment to treatment groups supports a cause-and-effect conclusion. An observational study can show an association, not causation.

Area and Volume

Most formulas are on the reference sheet, so these questions test choosing and applying the right one. The framework names scale factor explicitly: if every length is multiplied by k, area is multiplied by k² and volume by k³. Triple the radius and height of a cylinder and its volume becomes 27 times larger.

Lines, Angles and Triangles

Know the triangle angle sum (180 degrees), vertical angles (equal), and the angles formed when a transversal cuts parallel lines (corresponding and alternate interior angles are equal; same-side interior angles add to 180). Similar triangles have equal angles and proportional sides; scaling never changes angle measures.

Right Triangles and Trigonometry

Use the Pythagorean theorem, the special right triangles (on the reference sheet) and SOH CAH TOA. Because similar right triangles have the same ratios, sine, cosine and tangent depend only on the angle. The complementary-angle relationship appears often:

So if sin(x°) = cos(32°) and both angles are acute, x = 58.

Circles

This skill covers radii, diameters, tangents, angles, arc lengths, sector areas, radian measure, the unit circle, and circle equations in the xy-plane. The reference sheet gives you 360 degrees and 2π radians in a circle; from that you get every proportion you need:

A tangent is perpendicular to the radius at the point of tangency.

Worked Example: Completing the Square on a Circle

Find the centre and radius of x² + y² + 4x - 10y + 20 = 0.

The Reference Sheet

The sheet in Bluebook contains:

• Circle: A = πr² and C = 2πr
• Rectangle: A = lw; triangle: A = ½bh
• Pythagorean theorem: c² = a² + b²
• Special right triangles: 30-60-90 with sides x, x√3, 2x and 45-45-90 with sides s, s, s√2
• Volumes: box V = lwh, cylinder V = πr²h, sphere V = (4/3)πr³, cone V = (1/3)πr²h, pyramid V = (1/3)lwh
• A circle has 360 degrees and 2π radians of arc; a triangle's angles sum to 180 degrees

Everything else in this guide, including slope, the quadratic formula, vertex form, the circle equation, arc length and the trig ratios, you must know from memory.

The Built-In Desmos Calculator

Bluebook embeds a Desmos calculator with graphing and scientific modes, and you can switch between them at any time in the Math section. You may bring your own approved calculator instead, but calculators with computer algebra system (CAS) functionality are not allowed. Practise with the calculator you will use on test day. These standard Desmos moves are worth drilling:

• Solve any equation by graphing both sides. Type the left side as one expression and the right side as another; click the intersection to read x. This works for radical, absolute value and exponential equations that are slow by hand.
• Read key features by clicking. Graph a quadratic and Desmos marks the vertex, intercepts and intersections as grey points you can click.
• Count solutions of a system. Graph both equations and count intersections. Zoom out first so you do not miss one.
• Use sliders for unknown constants. If a question says "for what value of k does the system have no solution", type the equation with k, add the slider, and drag until the lines become parallel. Then confirm algebraically.
• Run a regression. Enter a table of data, then type y₁ ~ mx₁ + b (or an exponential model) to fit it.

Student-Produced Response Rules

SPR questions are 8 to 12 of the 40 scored questions. The Bluebook directions set these rules:

• If you find more than one correct answer, enter only one.
• Up to 5 characters for a positive answer and up to 6 for a negative answer, including the negative sign.
• A fraction that does not fit: enter its decimal equivalent.
• A decimal that does not fit: truncate or round at the fourth digit.
• A mixed number such as 3½: enter 7/2 or 3.5. Typing "3 1/2" or "31/2" will not receive credit.
• Do not enter symbols such as a percent sign, comma or dollar sign.

College Board's examples: for -1/3, the entries -1/3, -.3333 and -0.333 are accepted, but -.33 and -0.33 are not. A short rounded decimal is too imprecise.

College Board's framework says each multiple-choice distractor "represents a common error that students might reasonably make". That means wrong answers are predictable. These are the ones that catch the most students:

• Wrong quantity: the question asks for 2x, x + y or f(3), and the choice for x alone is waiting.
• Reverse percent done forward: undo an increase by dividing by the multiplier.
• Unchecked roots: in radical and rational equations, "both solutions" is often the trap.
• Scaling: doubling lengths multiplies area by 4 and volume by 8.