PSAT/NMSQT Math: Study Guide & Free Lessons
Short lessons, worked examples, and practice that connects.
Translate a linear relationship
A linear relationship changes by the same amount for each equal increase in its input. Identify what changes and what is already present at the start.
Connect tables and equations
A table, an equation and a graph can represent the same relationship. Translating between them is often more useful than doing a long calculation in just one representation.
Use two conditions together
A system asks for values that make every stated condition true. Two independent observations can determine two unknown quantities.
Work inside a feasible range
An inequality describes a set of allowed values. A budget, capacity or minimum requirement creates a boundary rather than always an exact equality.
Read a quadratic feature
A quadratic can reveal different information in different forms. Factored form highlights zeros; a completed square highlights the turning point.
Exponents and nonlinear rules
Repeated multiplication produces exponential change. Powers and radicals are linked operations, but their domains and signs still matter.
Percentages with a clear base
Every percentage describes a part relative to a particular base. Naming that base prevents a common family of errors in discounts and comparisons.
Ratios with visible units
A ratio is meaningful only after its quantities and units are identified. Unit conversion is multiplication by a representation of one.
Summarize and interpret data
A data summary answers a particular question about a distribution. The mean uses the total, while the median uses position after sorting.
Use counts to describe chance
A probability compares favorable cases with the relevant sample space. Conditions restrict which outcomes belong in that denominator.
Measure areas and volumes
Area, perimeter and volume measure different attributes of an object. Decide which attribute is requested before selecting a formula.
Angles, similarity and right triangles
Triangle relationships connect angles, lengths and ratios. Similarity preserves angles and ratios; a right-angle condition enables the Pythagorean theorem.
Read parameters and equivalent linear forms
A parameter changes an equation’s structure. Solving for a coefficient, preserving every solution and finding one unknown value are different tasks.
Reason inside simultaneous constraints
A feasible choice must satisfy every restriction at once. An inequality boundary is only one part of the region, and integer requirements can change the final answer.
Distinguish intersections and solution counts
Systems can have one, none or infinitely many solutions. Proportional coefficients describe parallel or identical lines, while a nonlinear intersection can create several candidates.
Rewrite powers and polynomial expressions
Equivalent expressions agree on their shared domain. Exponent laws depend on the operation, and rational simplification must retain the original excluded values.
Keep valid solutions after nonlinear steps
Squaring and absolute values change how equations encode signs. Candidate roots must satisfy the original equation and any contextual restrictions.
Recover a model from changing inputs
A model links the input interval to the way outputs change. Constant ratios suggest exponential structure; a horizontal shift changes which input reaches a known output.
Let units control multistep proportions
A compound rate combines quantities with different dimensions. A conversion is trustworthy when unwanted units cancel and the requested unit remains.
Track the base of every percentage
A percentage is incomplete without its base. Percentage-point differences compare two percentages directly, while relative change divides by the original rate.
Compare spread without losing frequencies
Center, spread and frequency describe different features. Repeated observations must retain their multiplicity, and translating every value does not change distances between observations.
Use tables to test models and sample estimates
A numerical model should be checked against observations and the collection process. A table can encode predictions, subgroup shares or conditional sample spaces.
Connect angle facts to similarity
Parallel-line and triangle relationships turn geometry into equations. Similarity preserves angles and scales lengths, while areas depend on the square of the length factor.
Recover lengths from volume and right triangles
Solid geometry connects a two-dimensional base to a third length. A space diagonal may require two right triangles, and complementary acute angles exchange opposite and adjacent roles.
400 parameterized questions across 96 families, not 400 independent designs
A clear route through the material
Follow the topics in order or use the prerequisites to choose a starting point. Read the worked example before focused practice, then use review cards to revisit the idea.
Continue your preparation
Scope: Digital PSAT/NMSQT Math foundations