MTLE Mathematics (Grades 5-12) key terms

A number expressible as a fraction a/b where a and b are integers and b is not zero. Its decimal form either terminates or repeats a fixed block of digits forever.

A real number that cannot be written as a ratio of two integers. Its decimal expansion is infinite and non-repeating, as with the square root of 2 or pi.

The ordered field of all rational and irrational numbers, closed under the four basic operations and satisfying the completeness property that every nonempty bounded set has a least upper bound.

The property that between any two distinct real numbers there lies another rational number. It explains why fractions can approximate any real value to arbitrary precision.

Rules for adding, subtracting, multiplying, and dividing positive and negative whole numbers, including how subtraction is rewritten as adding the opposite and how sign pairs determine products.

The number that, when added to a given number, yields zero. For any real number a the additive inverse is negative a, which supports reasoning about integer subtraction.

A visual representation placing numbers at positions on a line, used to show order, magnitude, and operations such as adding a positive value by moving right and a negative value by moving left.

The distance of a number from zero on the number line, always nonnegative. Formally it is a when a is nonnegative and negative a when a is negative.

A compact way to write very large or very small numbers as a product of a number between 1 and 10 and a power of ten, useful for comparing magnitudes and performing estimation.

Laws governing powers, including that multiplying like bases adds exponents, dividing subtracts them, and a negative exponent indicates a reciprocal. These rules extend to rational exponents and radicals.

An expression containing a root symbol, such as a square or cube root. Simplifying requires factoring out perfect powers and rationalizing denominators when appropriate.

Reasoning about relationships in which two quantities vary at a constant ratio, supporting setup of proportions, unit-rate comparisons, and scaling in measurement and percent contexts.

A rate expressed per single unit of one quantity, found by dividing one measure by the other. It enables direct comparison of prices, speeds, or densities across contexts.

The ratio of an amount of increase or decrease to the original amount, expressed as a percent. It supports problems on discounts, markups, growth, and error analysis.

Familiar reference values such as one half, one quarter, and three quarters used to estimate the size of unfamiliar fractions and to judge reasonableness of computed results.

A rule assigning each input in a domain exactly one output in a codomain. Multiple representations include tables, graphs, equations, and verbal descriptions of the relationship.

The domain is the set of all allowable inputs of a function; the range is the set of all outputs actually produced. Restrictions arise from division by zero and even roots.

A function whose graph is a straight line and whose rule can be written as f of x equals m x plus b, where m is the constant rate of change and b the initial value.

The amount one quantity changes relative to another, computed as a difference in outputs divided by the corresponding difference in inputs. For nonlinear functions it varies across intervals.

The equation y equals m x plus b, where m gives the slope and b the y-intercept. It makes graphing and interpreting linear models in context straightforward.

A function of degree two, graphed as a parabola. Key features include the vertex, axis of symmetry, intercepts, and whether the leading coefficient opens the graph upward or downward.

A quadratic written as a times the square of x minus h, plus k, where the point h comma k is the vertex. It reveals maximum or minimum values directly.

If a product of factors equals zero, then at least one factor equals zero. It underlies solving factored polynomial equations and finding x-intercepts of graphs.

A technique rewriting a quadratic expression as a perfect square plus a constant, used to derive the quadratic formula, convert to vertex form, and solve equations.

A function of the form a times b to the x power, where the base b is positive and not one. It models constant percent growth or decay rather than constant additive change.

A function that reverses another, so composing the two returns the original input. It exists only when the original function is one-to-one over the relevant domain.

Two or more equations considered together, solved by graphing, substitution, or elimination. Solutions correspond to points shared by the graphs and may be none, one, or infinitely many.

Shifts, reflections, and stretches applied to a parent graph. Translations add or subtract inside or outside the rule; reflections and scalings multiply inputs or outputs.

A relationship in which two figures have equal corresponding angles and proportional corresponding side lengths. It justifies indirect measurement and scale-drawing reasoning.

The constant ratio comparing corresponding lengths of similar figures. Areas scale by its square and volumes by its cube, a frequent source of student error.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. Its converse tests for right angles, and it underlies the distance formula.

The distance between two points in the coordinate plane equals the square root of the sum of the squared differences of their coordinates, derived from the Pythagorean theorem.

The study of geometric figures using coordinates and algebra, connecting slopes, midpoints, distances, and equations of lines and circles to geometric properties.

Surface area measures the total area of a solid's faces; volume measures the space it occupies. Formulas differ by solid, and scaling affects each differently.

Connections among radius, diameter, circumference, area, central angles, inscribed angles, and arc length, including that an inscribed angle is half its intercepted arc.

Summary values describing a typical data point, chiefly mean, median, and mode. The mean responds to every value, while the median resists extreme observations.

Statistics describing variability, including range, interquartile range, variance, and standard deviation. They complement center measures when comparing distributions.

An observation lying far from the bulk of the data, often flagged by the one-and-a-half interquartile range rule. Outliers can strongly shift the mean but not the median.

A display showing minimum, first quartile, median, third quartile, and maximum. It highlights spread and skew but conceals sample size and gaps within the data.

A bar graph of quantitative data grouped into intervals, where bar heights show frequency or relative frequency. Shape reveals modality, skew, and possible outliers.

A graph of paired quantitative variables, with correlation measuring the strength and direction of a linear association. Correlation does not establish causation.

A straight line summarizing a linear trend in bivariate data, often found by least squares. It supports prediction while inviting caution about extrapolation.

An assignment of probabilities to outcomes in a sample space, satisfying that each probability lies between zero and one and the total equals one.

The set of all possible outcomes of a random experiment. Listing it systematically supports computing probabilities by counting favorable outcomes.

The probability of an event given that another event has occurred, written as the probability of A given B. It equals the joint probability divided by the probability of B.

A relationship in which the occurrence of one event does not change the probability of another. Formally, the joint probability equals the product of the separate probabilities.

The long-run average outcome of a random variable, computed by multiplying each value by its probability and summing. It anchors fair-game and decision analyses.

A chain of justified claims leading to a conclusion, using definitions, established results, and valid inference. It may be informal in classrooms but must avoid unsupported leaps.

Disproving a universal claim by producing a single case in which it fails. It is a legitimate form of refutation and a powerful tool for testing student conjectures.

Inductive reasoning generalizes from observed cases and yields conjectures; deductive reasoning derives conclusions from premises and yields certainty. Both are essential in mathematics classrooms.

The cycle of representing a real situation mathematically, analyzing the model, interpreting results in context, and revising. Assumptions and limitations must be examined explicitly.

Producing an approximate answer through rounding, compatible numbers, or front-end strategies, then judging reasonableness. It supports number sense and error detection.

Expressing the same mathematical idea numerically, graphically, symbolically, or verbally, then translating among forms. Fluency across representations deepens conceptual understanding.

Purposeful effort on challenging tasks that students can solve without direct instruction, building persistence and conceptual connections. Teachers withhold answers while offering scaffolded questions.

Ongoing gathering of evidence about student understanding to adjust instruction in real time, through exit tickets, questioning, or observation rather than graded summative tests.

Temporary instructional supports, such as models, prompts, or partial worked examples, that enable students to succeed on tasks slightly beyond independent reach and are gradually removed.

A coherent but incorrect model of a mathematical idea, such as treating multiplication as always enlarging. Effective instruction surfaces and confronts it rather than merely correcting answers.

Structured classroom talk in which students explain reasoning, compare strategies, and critique arguments. Teacher moves include revoicing, pressing for justification, and selecting sequences of student work.