GTLE Mathematics Specialism key terms

Any number that can be written as a fraction of two integers with a nonzero denominator, including terminating and repeating decimals. It is closed under addition, subtraction, multiplication, and division by nonzero values.

A real number that cannot be expressed as a ratio of two integers, so its decimal expansion never terminates or repeats. Familiar examples include the square root of two and pi.

The ordered field of all rational and irrational numbers, complete in the sense that every nonempty bounded set has a least upper bound. It underpins limits, functions, and coordinate geometry.

The property stating that every nonempty subset of real numbers bounded above has a least upper bound in the reals. It distinguishes the reals from the rationals and guarantees convergence of bounded monotone sequences.

The fact that between any two distinct real numbers there exists a rational number, and likewise an irrational one. It shows rationals cannot be listed in order of size without gaps.

A number of the form a + bi where a and b are real and i squared equals negative one. Complex numbers form a field closed under all four operations and include all real solutions to polynomial equations.

Two complex numbers a + bi and a - bi that share a real part and opposite imaginary parts. Their product is the real number a squared plus b squared, which is useful for rationalizing complex denominators.

The distance of a complex number a + bi from the origin, computed as the square root of a squared plus b squared. It generalizes absolute value and multiplies under complex multiplication.

A rule assigning exactly one output value to each input in a specified domain. Mastery requires distinguishing functions from relations and reasoning about domain, range, and representation.

The domain is the set of all permitted inputs to a function; the range is the set of outputs actually produced. Both depend on the rule and any stated restrictions.

A function that reverses another, existing only when the original is one-to-one on its domain. Graphs of inverses are reflections across the line y equals x.

Applying one function to the output of another, written f of g of x. Composition is generally not commutative, and its domain excludes inputs where the inner function is undefined.

A function of the form f of x equals a x squared plus b x plus c with a nonzero. Its graph is a parabola whose vertex, axis of symmetry, and intercepts follow from the coefficients.

Writing a quadratic as a times the square of x minus h, plus k, which displays the vertex at (h, k) directly. Converting from standard form uses completing the square.

The quantity b squared minus four a c in a quadratic equation. Its sign determines whether the equation has two real roots, one repeated real root, or a conjugate pair of complex roots.

Rewriting a quadratic expression as a perfect square plus a constant by adding and subtracting the square of half the linear coefficient. It derives the quadratic formula and reveals vertices.

Change modeled by a base raised to a variable exponent, where the base exceeds one for growth and lies between zero and one for decay. Constant percentage change per unit time characterizes such models.

The exponent to which a fixed base must be raised to produce a given number. Logarithms invert exponentials and convert multiplication into addition, enabling solution of exponential equations.

An ordered list of numbers with a constant difference between consecutive terms. Its nth term is linear in n, and partial sums form a quadratic pattern.

An ordered list of numbers with a constant ratio between consecutive terms. Its nth term grows or shrinks exponentially, and finite sums follow a closed formula when the ratio is not one.

The sum of the terms of a sequence, often written with sigma notation. Finite series have definite sums; infinite series converge only when partial sums approach a limit.

Figures with equal corresponding angles and proportional corresponding side lengths. Their areas compare as the square of the scale factor and volumes as its cube.

The relationship between figures identical in shape and size, related by rigid motions. Triangle congruence can be established by side-angle-side, angle-side-angle, or side-side-side criteria.

A translation, rotation, or reflection that preserves distances and angles. Sequences of rigid motions formalize congruence, while dilations added to them define similarity.

The constant ratio relating corresponding lengths in similar figures or a dilation. It multiplies lengths directly but affects area and volume by higher powers.

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 coordinate points equals the square root of the sum of squared differences of their coordinates. It follows directly from the Pythagorean theorem.

The midpoint of a segment has coordinates equal to the averages of the endpoints' x and y coordinates. It generalizes to finding points that divide segments in given ratios.

A ratio of side lengths in a right triangle, such as sine, cosine, or tangent, defined relative to an acute angle. These ratios extend to all angles through 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 it. It explains periodicity and radian measure.

An angle measure defined as the ratio of intercepted arc length to radius. A full turn equals two pi radians, linking arc length and angle without arbitrary units.

The distance along a circle's curve between two points, computed as radius times the central angle in radians. It scales linearly with both radius and angle.

The portion of a circle's area enclosed by two radii and their intercepted arc. It equals one half times radius squared times the central angle in radians.

Summary values such as mean, median, and mode that describe a typical observation. The mean responds to every value, while the median resists extreme outliers.

Statistics such as range, interquartile range, variance, and standard deviation that describe how dispersed data are. Spread complements center when comparing distributions.

The square root of the average squared deviation from the mean, expressed in the original units. It quantifies typical distance of observations from the mean.

An observation lying far from the bulk of the data, often flagged by the one-and-a-half interquartile range rule. Outliers can strongly distort the mean and standard deviation.

A graph displaying the minimum, first quartile, median, third quartile, and maximum of a data set. It highlights spread, skew, and outliers compactly.

A symmetric, bell-shaped probability distribution determined by its mean and standard deviation. Roughly 68 percent of values fall within one standard deviation of the mean.

The number of standard deviations a value lies above or below the mean. Standardizing with z-scores allows comparison across distributions with different centers and spreads.

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

Two events are independent when the occurrence of one does not change the probability of the other, so their joint probability equals the product of their probabilities.

A grid classifying cases by two categorical variables, with counts in cells and totals in margins. It supports computing joint, marginal, and conditional relative frequencies.

A single case that satisfies a statement's hypotheses but violates its conclusion, thereby disproving a universal claim. One counterexample suffices; confirming examples never prove a general statement.

A proof technique that assumes the negation of the desired conclusion and derives an impossibility. The contradiction establishes the original statement, as in proving irrationality of the square root of two.

A mathematical statement believed true based on evidence but not yet proven. Reasoning about conjectures develops argumentation, and a counterexample refutes one decisively.

A proof method for statements about all natural numbers: establish a base case, assume the statement for n, and prove it for n plus one. It formalizes recursive patterns.

General strategies such as drawing a diagram, working backward, solving a simpler case, or looking for patterns. They guide exploration when no routine procedure is apparent.

Expressing a mathematical idea through symbols, graphs, tables, diagrams, or words. Flexible movement among representations deepens understanding and supports problem solving.

Structured classroom talk in which learners explain reasoning, justify claims, and critique others' arguments. It builds communication skills and surfaces misconceptions for instruction.

Ongoing gathering of evidence about student understanding during instruction to adjust teaching. It contrasts with summative assessment, which evaluates learning at the end of a unit.

Purposeful effort by students on challenging tasks before direct instruction, fostering persistence and conceptual sense-making. Teachers support it with scaffolds rather than immediate answers.