What Is on the PSAT Math Section? Every Skill, and Why It Is Familiar

By Dr. Brink, Ph.D., 32 years teaching mathematics

Here is the part most students do not realize: almost nothing on the PSAT is new. If you are in 9th or 10th grade, the skills the test is built on are the exact ones you are working through in class right now, and many of them go back to Grade 5, 6 and 7. This page lays out both halves, what the test looks like and how it is weighted, then the 280 foundational skills every PSAT math question is quietly built on. Get these solid in 9th and 10th grade and the PSAT, and the SAT a year later, mostly takes care of itself.

How the section is built

Two modules, 22 questions each. The first module is the same for everyone; the second module adapts, getting harder if you did well on the first and easier if you did not. The clock is 35 minutes per module and it does not pause between questions. The on-screen Desmos calculator is available for every question, and a formula reference sheet is one click away, so you never have to memorize the area of a circle or the Pythagorean theorem.

The four content areas, and how they are weighted

Most of the test is algebra in one form or another. Here is roughly how the 44 questions split.

• Algebra about 13 to 15 questions

  Linear equations, inequalities, and systems. The single biggest area, and the heart of 9th grade algebra.

• Advanced Math about 13 to 15 questions

  Quadratics, polynomials, exponents, and functions. Nearly as large as Algebra, and where 10th grade math is heading.

• Problem-Solving and Data Analysis about 5 to 7 questions

  Ratios, percentages, rates, and reading data from tables and graphs.

• Geometry and Trigonometry about 5 to 7 questions

  Area and volume, triangles, circles, and the first taste of trigonometry.

The skills underneath: the DNA of the PSAT Math section

Those four areas are the labels. This is what is really being tested. Below are the 280 foundational skills the whole PSAT math section is built from, the ones a question assumes you already have before it even begins. They are grouped the way math actually builds, from arithmetic up through algebra, functions, geometry and statistics, and the grade tag on each one shows how early it first shows up. For a 9th or 10th grader, this is not a list of things to learn someday. It is a checklist of what you are doing in class now, and what the PSAT will quietly test you on.

Arithmetic 2

The number sense everything else stands on.

• Order of operations (PEMDAS/BODMAS)Grade 5
• Simplifying fractionsGrade 5

Pre-algebra 31

Signed numbers, fractions and the first move into symbols.

• Concept of a negative number (number line)Grade 5
• Concept of absolute value as distance from zero on the number lineGrade 6
• Dividing integers (sign rules)Grade 6
• Integer arithmetic (adding, subtracting integers of mixed sign)Grade 6
• Multiplying integers (sign rules)Grade 6
• Reciprocal fractions (two fractions whose product is 1)Grade 6
• Subtracting integers (rewrite as adding the opposite)Grade 6
• Concept of a proportion: two ratios set equal ($a : b = c : d$)Grade 7
• Cross-multiplication to solve a proportionGrade 7
• Domain restriction in contextGrade 7
• Solving ratio problems involving fractions and percentGrade 7
• Negative reciprocal of a number (sign flip + invert)Grade 8
• Choosing direction of a conversion factor (which unit goes on top)
• Concept of an exponent as repeated multiplication
• Converting a percent to a decimal (and back) for arithmetic
• Converting compound units (e.g., mph to ft/s)
• Dimensional analysis: multiplying by conversion factors equal to 1
• Distinguishing counts from percentages in two-way tables
• Distinguishing discrete from continuous data
• Identifying perfect-square factors in a radicand
• Part-to-part vs part-to-whole ratios
• Percent change is always relative to the original, not the new value
• Percent increase: $\text{new} = \text{original} \times (1 + r)$; percent decrease: $\text{new} = \text{original} \times (1 - r)$
• Percent of a quantity: $(\text{percent as a decimal}) \times \text{quantity}$
• Reading axis scale and units before reading a graph value
• Reading the correct row and column of a two-way table
• Reverse percent: dividing by $(1 + r)$ or $(1 - r)$ to find the original
• Scaling a ratio to find missing values
• Sign of a power of a negative base: even exponent gives positive, odd gives negative
• Successive percent changes do not add - they compound
• Unit rate as a quantity per single unit (e.g., miles per 1 hour)

Algebra 148

The largest strand by far, and the spine of the whole test.

• Concept of a variable as a placeholder for a numberGrade 6
• Identifying ordered pairs $(x, y)$ as points on the coordinate planeGrade 6
• Reading algebraic notation ($3x$ means $3 \cdot x$, implicit multiplication)Grade 6
• Associative property of addition and multiplicationGrade 7
• Checking a solution by substitutionGrade 7
• Coefficient of a variableGrade 7
• Combining like termsGrade 7
• Commutative property of addition and multiplicationGrade 7
• Concept of a solution to an equationGrade 7
• Concept of an equation (left side $=$ right side)Grade 7
• Concept of an inequality ($>$, $<$, $\geq$, $\leq$)Grade 7
• Distributing a negative across parentheses (or factoring out a $-1$ from a binomial)Grade 7
• Distributive property: $a(b + c) = ab + ac$Grade 7
• Equation balance: same operation on both sides preserves equalityGrade 7
• Evaluating an algebraic expression by substitutionGrade 7
• Expanding products like $2(3x + 5)$Grade 7
• Inverse operations (addition and subtraction, multiplication and division)Grade 7
• Like and unlike algebraic termsGrade 7
• Solving $ax = b$ (one-step, divide)Grade 7
• Solving $x + a = b$ (one-step, add or subtract)Grade 7
• Solving $x \div a = b$ (one-step, multiply)Grade 7
• Concept of a quadratic expression ($ax^2 + bx + c$)Grade 8
• Concept of a square root and what it representsGrade 8
• Concept of a system of equations (two equations, two unknowns, solved simultaneously)Grade 8
• Elimination method (add or subtract equations to cancel a variable)Grade 8
• Factoring a common factor out of a binomialGrade 8
• Flipping the inequality sign when multiplying or dividing by a negativeGrade 8
• Multiplying two binomials (FOIL)Grade 8
• Point-slope form: $y - y_1 = m(x - x_1)$Grade 8
• Recognizing equivalent equation formsGrade 8
• Sign tracking across multiple stepsGrade 8
• Slope as rate of change (units per unit, e.g. dollars per hour)Grade 8
• Slope formula: rise over runGrade 8
• Slope-intercept form: $y = mx + b$Grade 8
• Solution set of an inequality (range of values, not single value)Grade 8
• Solving equations requiring distribution firstGrade 8
• Solving equations with combining like terms on one sideGrade 8
• Solving equations with the variable on both sidesGrade 8
• Solving systems graphically (intersection of two lines)Grade 8
• Solving two-step equations like $ax + b = c$Grade 8
• Standard form of a linear equation: $Ax + By = C$Grade 8
• Substitution method for solving systemsGrade 8
• Time-distance-speed problemsGrade 8
• Unit consistency in word problems (matching units across the equation)Grade 8
• $y$-intercept as initial value in context (starting amount when $x = 0$)Grade 8
• $|\text{expression}| = \text{negative}$ has no solutionGrade 8
• $|x| = a$ has two solutions: $x = a$ and $x = -a$ (for $a > 0$)Grade 8
• Axis of symmetry as a vertical lineGrade 9
• Cancelling a common factor in a rational expressionGrade 9
• Cancelling factors, not termsGrade 9
• Clearing fractions by multiplying both sides by the LCDGrade 9
• Compound inequalities (and / or, written as $a < x < b$ or two separate inequalities)Grade 9
• Converting between linear equation forms (slope-intercept, point-slope, standard)Grade 9
• Distinguishing $f(0)$ from $f(x) = 0$Grade 9
• Equivalent terms for the solutions of a quadratic: roots, solutions, $x$-intercepts, and zerosGrade 9
• Factor before simplifyingGrade 9
• Factoring a difference of squares ($a^2 - b^2 = (a - b)(a + b)$)Grade 9
• Factoring a perfect-square trinomial ($a^2 \pm 2ab + b^2 = (a \pm b)^2$)Grade 9
• Factoring a trinomial of the form $x^2 + bx + c$ (find two numbers that sum to $b$ and multiply to $c$)Grade 9
• Factoring a trinomial with leading coefficient $\neq 1$ ($ax^2 + bx + c$)Grade 9
• Horizontal-shift sign in vertex formGrade 9
• Identifying constraints in a word problem (positive only, integer only, within a range, etc.)Grade 9
• Identifying when a system has infinitely many solutions (same line)Grade 9
• Identifying when a system has no solution (parallel lines, contradictory equations)Grade 9
• Maximum vs minimum from the sign of the leading coefficientGrade 9
• Multiplying an entire equation by a constant to align coefficients for eliminationGrade 9
• Parallel lines have equal slopesGrade 9
• Perpendicular lines have negative reciprocal slopes ($m_1 \cdot m_2 = -1$)Grade 9
• Quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$Grade 9
• Rational expression as a quotient of polynomialsGrade 9
• Reading a question for what's being solved (variable vs parameter)Grade 9
• Reading zeros directly from factored formGrade 9
• Real-world meaning of a zeroGrade 9
• Recognizing common factoring patternsGrade 9
• Recognizing infinite-solution equations (e.g. $0 = 0$)Grade 9
• Recognizing no-solution equations (e.g. $3 = 5$)Grade 9
• Representing the solution of an inequality algebraically, graphically (on a number line), or in interval notationGrade 9
• Setting a quadratic equation equal to zero before factoringGrade 9
• Setting up the two cases for $|ax + b| = c$ (isolate the absolute value first, then split)Grade 9
• Simplifying expressions involving square rootsGrade 9
• Solving equations with decimal coefficientsGrade 9
• Solving equations with fractional coefficientsGrade 9
• Squaring a binomial: $(a + b)^2 = a^2 + 2ab + b^2$, not $a^2 + b^2$Grade 9
• Translating word problems into linear equationsGrade 9
• Translations of function graphsGrade 9
• Vertex form of a quadraticGrade 9
• Zero of a function as an $x$-interceptGrade 9
• Degree and leading coefficient determine end behaviour of polynomialsGrade 10
• Discriminant ($b^2 - 4ac$) and what it tells you about the rootsGrade 10
• Domain of a rational expressionGrade 10
• Factor theoremGrade 10
• Finding the vertex from standard form using $x = -\dfrac{b}{2a}$Grade 10
• Leading term dominance for large $|x|$Grade 10
• Limit-arrow notation in plain languageGrade 10
• Local extremum vs absolute extremumGrade 10
• No real roots when a squared expression equals a negativeGrade 10
• Number of real zeros is at most the degreeGrade 10
• Restriction from a cancelled factorGrade 10
• Roots, multiplicity, and $x$-intercepts of polynomialsGrade 10
• Sign-flip cancellation: $\dfrac{a - b}{b - a} = -1$Grade 10
• Solving equations with parameters (solving for $k$ given $x$)Grade 10
• Symmetry of function values about the axis of symmetryGrade 10
• Solving systems with matricesGrade 11
• Boundary value of an inequality (the value where strict inequality becomes equality)
• Completing the square to rewrite $ax^2 + bx + c$ as $a(x - h)^2 + k$
• Concavity from the sign of a (parabola opens up or down)
• Converting between standard form and scientific notation
• Distinguishing growth ($b > 1$) from decay ($0 < b < 1$)
• Distinguishing rate of change from initial value in a linear context (slope vs $y$-intercept)
• Equivalent forms of a quadratic: standard, vertex, factored
• Exponent rules require same bases
• Extraneous solution check for absolute value equations (RHS must be non-negative)
• Extraneous solutions from clearing denominators
• Extraneous solutions from squaring both sides
• Fractional exponent $\dfrac{1}{n}$ means the $n$th root
• Fractional exponent $\dfrac{m}{n}$: take the root, then the power
• Growth factor vs growth rate: $b = 1 + r$ for growth, $b = 1 - r$ for decay
• Inequality balance: same operation on both sides preserves the inequality (with sign-flip caveat)
• Initial value of an exponential model: $a = y$ when $x = 0$
• Isolating the radical before squaring
• LCD of rational expressions (factor denominators first)
• Like and unlike radicals
• Linear functions add a constant amount per unit; exponential functions multiply by a constant factor per unit
• Maximum height / maximum profit interpretation from a parabola vertex
• Multiplying and dividing in scientific notation
• Negative exponents mean reciprocals
• Number of intersection points of a line and a parabola from the discriminant
• Pattern recognition: identifying a hidden quadratic form (e.g., $x^4 - 5x^2 + 4$ is quadratic in $x^2$)
• Power of a power: multiply the exponents
• Power of a product: $(xy)^a = x^a \cdot y^a$
• Product rule for exponents: multiplying powers adds the exponents
• Product rule for square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
• Quotient rule for exponents: dividing powers subtracts the exponents
• Rationalizing a denominator with a single radical
• Repeated percent change as compound exponential
• Rewriting bases to match (e.g., $8 = 2^3$, $27 = 3^3$, $16 = 2^4$)
• Scientific notation: $a \times 10^n$ form ($1 \leq |a| < 10$)
• Sign of an algebraic expression containing a parameter (e.g., $-\dfrac{16}{p}$ when $p < 0$)
• Solution to a system of two linear equations corresponds to the intersection point of their graphs
• Squaring both sides to eliminate a radical
• Standard form of exponential model: $y = a \cdot b^x$
• Substituting a known point into an equation to solve for an unknown coefficient
• Substitution to simplify a complicated expression (let $u = \ldots$)
• Two-case structure of absolute value equations ($|A| = B$ means $A = B$ or $A = -B$)
• Vertical or horizontal segment length as a difference of coordinates
• $x$-intercept: set $y = 0$ and solve for $x$; $y$-intercept: set $x = 0$ and solve for $y$
• Zero exponent: $x^0 = 1$ (for $x \neq 0$)
• Zero product property: if $AB = 0$, then $A = 0$ or $B = 0$

Functions 26

Reading equations, graphs and tables as one connected idea.

• Basics of $f(x)$ notation ($f(a) = b$ means the point $(a, b)$; $f(0)$ is the $y$-intercept; $f(x) = 0$ gives the $x$-intercept)Grade 8
• Generating an ordered pair from a complete equation by choosing an $x$ and computing $y$Grade 8
• Inter-relating algebraic equations, function notation, the coordinate plane, the graphical representation of functions (equations) and tables of $x$- and $y$-valuesGrade 8
• Relating the input ($x$-value, independent variable) of an equation to the output ($y$-value, dependent variable)Grade 8
• Sideways parabola: $x = a(y - k)^2 + h$ has horizontal axis of symmetry, vertex $(h, k)$, opens left/rightGrade 11
• Combining domain restrictions from multiple sources
• Distinguishing a function from a non-function (vertical line test)
• Domain restriction from a denominator (set denominator not equal to 0)
• Domain restriction from an even-index radical (set radicand $\geq 0$)
• Evaluating a function from a table or graph (not just an equation)
• Function as a rule assigning exactly one output to each input
• Function composition is not multiplication: $f(g(x))$ is not $f(x) \times g(x)$
• Function composition: $f(g(x))$ means evaluate $g$ first, then $f$
• Horizontal stretch and compression: $f(bx)$
• Independent vs dependent variables
• Interpreting $f(a) = b$ in a word problem context
• Interpreting $f(a) = k$ as a condition on $x$
• Interpreting $f(x + h)$ in context
• Order of transformations matters for combined transformations
• Piecewise function: different rules apply on different intervals of the domain
• Range as the set of possible output values
• Reading a piecewise function in context (tiered pricing, tax brackets, etc.)
• Reflection across the $x$-axis: $-f(x)$ vs reflection across the $y$-axis: $f(-x)$
• Solving for the base of a power or exponential from a known value by taking an nth root, then evaluating
• Units of input vs units of output in function notation
• Vertical stretch and compression: $a \cdot f(x)$

Geometry and Trigonometry 42

Shape, space, angle and the trig the PSAT actually asks for.

• Exterior Angle Theorem: an exterior angle of a triangle equals the sum of the two remote interior anglesGrade 8
• Relationship between two circles from distance between centers vs sum and difference of radiiGrade 10
• 30-60-90 triangle side ratios: $1 : \sqrt{3} : 2$
• 45-45-90 triangle side ratios: $1 : 1 : \sqrt{2}$
• Arc length as a fraction of circumference: $\dfrac{\theta}{360} \times 2\pi r$
• Area formula for a circle: $A = \pi r^2$
• Area formula for a rectangle: $A = \text{length} \times \text{width}$
• Area formula for a triangle: $A = \dfrac{1}{2} \times \text{base} \times \text{height}$
• Breaking a composite figure into simpler shapes
• Circumference of a circle: $C = 2\pi r = \pi d$
• Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25) and their multiples
• Complementary angle identity: $\sin\theta = \cos(90^\circ - \theta)$
• Completing the square to convert general form of a circle to standard form
• Converting degrees to radians: multiply by $\dfrac{\pi}{180}$
• Converting radians to degrees: multiply by $\dfrac{180}{\pi}$
• Distance formula: the Pythagorean theorem on the coordinate plane
• Distinguishing arc length (1D) from sector area (2D)
• Distinguishing radius from diameter
• Distinguishing volume from surface area
• Horizontal vs vertical distance on the coordinate plane
• Hypotenuse is the side opposite the right angle (longest side)
• Identifying opposite and adjacent sides relative to a specific angle
• Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) to find an angle from a ratio
• Linear pair: two angles forming a straight line sum to 180 degrees
• Midpoint formula: average the two $x$-values and the two $y$-values
• Parallel lines cut by a transversal: corresponding, alternate interior, alternate exterior, co-interior
• Pythagorean theorem: $a^2 + b^2 = c^2$ (right triangles only)
• Radian definition: arc length divided by radius
• Reading center and radius from standard form of a circle
• Scaling factor $k$ effects: perimeter scales by $k$, area by $k^2$, volume by $k^3$
• Sector area as a fraction of full circle area: $\dfrac{\theta}{360} \times \pi r^2$
• Setting up a similarity proportion with corresponding sides
• Similar triangles have proportional corresponding sides and equal corresponding angles
• Similarity ratio vs area ratio: areas scale as the square of the linear ratio
• SOH-CAH-TOA: remembering the sine, cosine, and tangent ratios
• Standard form of a circle: $(x - h)^2 + (y - k)^2 = r^2$
• Supplementary angles sum to 180 degrees; complementary angles sum to 90 degrees
• Triangle angle sum is 180 degrees
• Vertical angles are equal
• Volume of a cylinder: $V = \pi r^2 h$
• Volume of a prism: $V = (\text{area of base}) \times \text{height}$
• Volume of a rectangular prism: $V = \text{length} \times \text{width} \times \text{height}$

Statistics and Data 27

Reading data, rates, spread and what a graph is really saying.

• Standard deviation under linear transformation: adding a constant leaves SD unchanged; multiplying every value by $k$ scales SD by $|k|$Grade 10
• Addition for mutually exclusive OR events
• Adjusting denominators for dependent (without replacement) events
• Choosing a model type from a scatterplot's shape (linear / quadratic / exponential)
• Choosing mean vs median based on skew or outliers
• Comparing standard deviations qualitatively from a graph
• Comparing two distributions by shape, center, and spread
• Complementary probability: $P(\text{not } A) = 1 - P(A)$
• Computing mean from a frequency table (weighted average)
• Correlation does not imply causation
• Correlation: positive, negative, or no association
• Counting outcomes with the multiplication principle
• Extrapolation vs interpolation
• Line of best fit minimizes the overall vertical distance from the data points
• Mean: sum of values divided by count
• Median: middle value when data is ordered (or average of two middles)
• Multiplication for independent AND events
• Outlier in a scatterplot
• Predicting $y$ from $x$ using the line of best fit equation
• Probability as a ratio: favorable outcomes out of total outcomes
• Probability from a frequency table or two-way table
• Probability range: $0 \leq P(A) \leq 1$
• Range: maximum minus minimum
• Reading boxplots (Q1, median, Q3, whiskers)
• Reading histograms (bar height = frequency in a range)
• Shape of a distribution: symmetric, skewed left, skewed right, uniform
• Standard deviation as a measure of spread, not center

Problem-Solving Habits 4

The moves a strong test-taker makes on any question.

• Determining whether a quantity must be integer, positive, or non-zero
• Eliminating impossible answer choices first
• Estimating an answer to check reasonableness
• Plugging answer choices back into the problem when stuck

Already have a PSAT score? Read what counts as a good PSAT math score, with the official benchmarks and roughly where each score ranks.