NYSTCE Math CST key terms

The iterative process of translating a real situation into mathematics, solving, interpreting results in context, and revising the model when it fails to fit. Candidates should describe each stage and justify decisions made at each step.

Explaining why a statement is true using definitions, properties, and logical reasoning rather than examples alone. On constructed-response items, answers must show a valid chain of reasoning, not just a correct final value.

A single case that makes a general statement false. One counterexample disproves a claim, while confirming examples can never prove a universal statement; this distinction is frequently tested in reasoning items.

A general statement believed true based on patterns or partial evidence but not yet proven. Instructionally, students test conjectures with additional cases and attempt to justify or refute them mathematically.

Drawing a general conclusion from observed patterns or specific cases. It generates conjectures but does not establish certainty, so results must later be verified by deductive argument.

Deriving conclusions from accepted premises, definitions, and previously proven statements using valid logical steps. Unlike inductive reasoning, a valid deductive argument guarantees its conclusion when premises are true.

Attending to accurate terminology, symbols, labels, and unit consistency when stating results. Answers must specify units, rounding, and referents so a solution is unambiguous in its real-world context.

A number expressible as a ratio of two integers with nonzero denominator. Its decimal expansion either terminates or eventually repeats, which distinguishes it from irrational numbers on the real number line.

A real number that cannot be written as a fraction of integers, such as the square root of 2 or pi. Its decimal form is infinite and nonrepeating, and it is not closed under the four operations.

The ordered field of all rational and irrational numbers, closed under addition and multiplication and satisfying order properties. Instruction emphasizes subsets, density, and the number line representation.

The property that between any two distinct real numbers there exists a rational number, and likewise an irrational one. This supports arguments that no smallest positive rational exists.

A number of the form a + bi where a and b are real and i squared equals negative one. Arithmetic follows familiar field rules, and geometric interpretation uses the complex plane.

For a + bi, the conjugate is a - bi. Multiplying a complex number by its conjugate yields the real number a squared plus b squared, a technique used to divide complex numbers and rationalize denominators.

The distance from the origin to the point representing a + bi in the complex plane, equal to the square root of a squared plus b squared. It generalizes absolute value to complex numbers.

A quantity with both magnitude and direction, such as displacement or force, represented by directed segments or components. Operations include addition, scalar multiplication, and dot product.

For vectors with components, the sum of the products of corresponding components; geometrically it equals the product of magnitudes times the cosine of the included angle. It detects perpendicularity when zero.

A quantity described by magnitude alone, such as mass, temperature, or speed, without direction. Distinguishing scalars from vectors is essential when setting up physical and geometric applications.

A correspondence assigning each input in the domain exactly one output. The definition emphasizes single-valuedness, which underlies vertical line tests, inverse relations, and modeling relationships between quantities.

The domain is the set of permitted inputs; the range is the set of resulting outputs. Restriction decisions must reflect context, such as nonnegative time or integer counts of objects.

A function that reverses another, existing only when the original is one-to-one on its domain. Graphs of inverses reflect across the line y equals x, and composition returns the input.

The ratio describing how one quantity changes relative to another, computed as change in output over change in input. For linear functions it is constant; for others it may be averaged over an interval.

The difference in function values divided by the difference in inputs across an interval, equivalent to the slope of the secant line. It summarizes overall behavior without claiming constancy.

The constant rate of change of a linear relationship, measured as rise over run. It indicates steepness and direction, and appears in slope-intercept form as the coefficient of the input variable.

A function with constant rate of change, written as f of x equals m x plus b. Its graph is a line, and it models proportional or additive relationships with constant growth or decline.

A function of the form a times b to the x, where the base is positive and not one. It models repeated proportional change, with constant percent growth or decay per unit interval.

A function of degree two, whose graph is a parabola with a vertex and axis of symmetry. Standard, vertex, and factored forms reveal different features such as intercepts and maximum or minimum values.

The turning point of a quadratic graph, representing the minimum when the parabola opens upward and the maximum when it opens downward. Its coordinates come from completing the square or the vertex formula.

The expression b squared minus four a c from the quadratic formula. Its sign determines whether a quadratic has two, one, or no real solutions, linking algebra to the number of x-intercepts.

An ordered list in which consecutive terms differ by a constant amount called the common difference. It corresponds to a linear function defined on the natural numbers and has an explicit and recursive rule.

An ordered list in which each term after the first is a constant multiple, the common ratio, of the previous term. It corresponds to exponential behavior and can model repeated percent change.

Changes to a parent graph produced by translations, reflections, stretches, and compressions. Parameters inside the function affect horizontal behavior, while those outside affect vertical behavior, often in inverse-seeming ways.

A relationship in which figures have equal corresponding angles and proportional corresponding side lengths. It justifies proportional reasoning with scale factors and underlies trigonometric ratios in right triangles.

The ratio comparing corresponding lengths of similar figures. Areas scale by its square and volumes by its cube, a relationship frequently tested in measurement and modeling problems.

When a solid is enlarged by scale factor k, surface area multiplies by k squared and volume by k cubed. This explains why doubling dimensions multiplies volume eightfold while surface area only quadruples.

Ratios of side lengths in a right triangle: sine, cosine, and tangent, defined relative to an acute angle. They connect angle measures to side lengths and extend to all angles via the unit circle.

The circle of radius one centered at the origin, used to define sine and cosine for all angles as coordinates of points on the circle. It explains periodicity, signs by quadrant, and radian measure.

An angle measure defined as the ratio of intercepted arc length to radius, making a full turn two pi radians. Radians simplify arc length, sector area, and calculus-related formulas.

Relationships extending trigonometry to non-right triangles. The law of sines relates sides to opposite angles; the law of cosines generalizes the Pythagorean theorem to include the included angle.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. Its converse identifies right triangles, and it underlies the distance formula and trigonometric identities.

The distance between two points in the coordinate plane, derived from the Pythagorean theorem. In three dimensions it extends with a third squared difference under the radical.

The coordinates of the point halfway between two endpoints, found by averaging corresponding coordinates. It supports coordinate proofs about bisectors, medians, and symmetry.

Portions of a circle's circumference and interior determined by a central angle. With angle in radians, arc length equals radius times angle and sector area equals one-half radius squared times angle.

Circumference equals two pi times radius, and area equals pi times radius squared. Distinguishing linear from square units is a common source of error in measurement items.

Summary values including mean, median, and mode that describe a typical observation. The mean responds to skew and outliers, while the median is resistant, so choice depends on distribution shape.

Statistics describing variability, including range, interquartile range, variance, and standard deviation. Standard deviation measures typical distance from the mean and is expressed in the data's original units.

The square root of variance, quantifying how far values typically lie from the mean. Larger values indicate greater dispersion, and it supports comparisons between distributions and z-score calculations.

Skewness describes asymmetry in a distribution; outliers are unusually distant observations. In right-skewed data the mean typically exceeds the median, and outliers strongly influence the mean but not the median.

A display showing minimum, first quartile, median, third quartile, and maximum, with the box spanning the interquartile range. It highlights spread and outliers but hides sample size and fine distribution shape.

A symmetric, bell-shaped continuous distribution defined by mean and standard deviation. Roughly 68 percent of values fall within one standard deviation of the mean and about 95 percent within two.

The number of standard deviations a value lies from the mean, computed by subtracting the mean and dividing by the standard deviation. It permits comparison across different scales and distributions.

A number between negative one and one summarizing the strength and direction of a linear association. It measures linear relationship only and does not by itself establish causation.

A regression line minimizing squared vertical distances from data points, used to predict one variable from another. Slope and intercept are interpreted in context, with caution about extrapolation.

A strong association between variables does not prove that one causes the other. Confounding variables and study design must be examined before drawing causal conclusions from observational data.

Fundamental principles including complement, addition, and multiplication rules. Mutually exclusive events cannot co-occur, while independent events have joint probability equal to the product of their probabilities.

The probability of an event given that another event has occurred, computed as the joint probability divided by the probability of the condition. It underlies tree diagrams and Bayes-style reasoning.

Events whose occurrence does not change each other's probabilities, so the joint probability factors into a product. This differs from mutually exclusive events, which cannot happen simultaneously.

The long-run average outcome of a random variable, computed by summing each value times its probability. It supports fair-game and decision-making analyses and need not equal any possible outcome.

Selecting a subset to represent a population, where random selection supports generalization. Bias arises from nonrandom selection, voluntary response, or poorly worded questions and distorts inference.

A bound describing plausible sampling variability around an estimate, shrinking as sample size grows. It quantifies uncertainty from random sampling but not from bias or flawed design.

Tracking units through computations so that conversion factors cancel and results carry correct units. It catches setup errors and is essential in rate, density, and measurement conversion problems.