GACE Middle Grades Mathematics (4-8) (704) key terms

A number that can be written as a fraction of two integers with a nonzero denominator. Its decimal form either terminates or repeats a block of digits forever.

A real number that cannot be expressed as a ratio of two integers. Its decimal expansion never terminates and never settles into a repeating pattern, such as the square root of 2 or pi.

Any whole number from the set of positive counting numbers, their negatives, and zero. Integers contain no fractional or decimal part and are closed under addition, subtraction, and multiplication.

Any point on the continuous number line, including both rational and irrational values. Every real number has a position, an order relative to others, and a distance from zero.

The distance of a number from zero on the number line, always nonnegative. Written with vertical bars, it removes sign, so the absolute value of negative seven is seven.

A compact way to write very large or very small numbers as a factor between one and ten multiplied by a power of ten. Useful for comparing magnitudes and for estimation.

A superscript number showing how many times a base is used as a factor. Rules for combining exponents include adding powers when multiplying like bases and subtracting when dividing.

A value that, when multiplied by itself, produces a given nonnegative number. The principal square root is the nonnegative one; perfect squares have whole-number roots.

The largest whole number that divides two or more given numbers with no remainder. Often used to simplify fractions or factor algebraic expressions.

The smallest positive number that is a multiple of two or more given numbers. It gives a common denominator when adding or comparing fractions.

A whole number greater than one whose only positive divisors are one and itself. Primes are the building blocks of all other whole numbers through multiplication.

Solving problems by recognizing that two quantities change at a constant ratio. Setting up and solving a proportion, or using a unit rate, are common strategies.

A rate expressed per single unit of one quantity, found by dividing one quantity by the other. It appears as the constant of proportionality in proportional relationships.

A symbol, usually a letter, that stands for an unknown or changing number. Variables allow general statements and equations to describe many cases at once.

A mathematical phrase combining numbers, variables, and operations without an equals sign. It can be simplified or evaluated but not solved.

A statement that two expressions have equal value, joined by an equals sign. Solving means finding values of the variable that make the statement true.

A statement comparing two expressions using less than, greater than, or their or-equal-to forms. Its solution is a range of values, often graphed on a number line.

A function whose graph is a straight line and whose rule can be written in slope-intercept form. It has a constant rate of change over equal intervals.

The measure of steepness of a line, computed as vertical change divided by horizontal change between two points. It equals the constant rate of change of a linear function.

The point where a graph crosses the vertical axis, occurring where the input value is zero. In a real context it often represents an initial or starting amount.

A function whose rule includes a squared variable term as its highest power. Its graph is a parabola, and its key features include a vertex, axis of symmetry, and intercepts.

The U-shaped curve that is the graph of a quadratic function. It opens upward when the squared term has a positive coefficient and downward when it is negative.

The turning point of a parabola, where it reaches its maximum or minimum value. It lies on the axis of symmetry and can be found from the quadratic rule or graph.

A rule assigning each input exactly one output. Graphs of functions pass the vertical line test, and function notation names the rule and its input together.

The complete set of input values for which a function is defined. Restrictions arise from division by zero, even roots of negatives, and real-world constraints.

The complete set of output values a function actually produces. It is read from the vertical spread of the graph, not from the inputs.

Two or more equations considered together, whose solution satisfies every equation at once. Graphically the solution is a point of intersection of the graphs.

Expressions that produce the same value for every substitution of the variables. Properties of operations and combining like terms transform one into the other without changing meaning.

The rule that multiplying a sum by a factor gives the same result as multiplying each addend by that factor and adding. It underlies expanding and factoring expressions.

Figures with exactly the same size and shape, related by a sequence of rigid motions: translations, rotations, and reflections. Corresponding sides and angles are equal.

Figures with the same shape but possibly different size, related by a sequence including a dilation. Corresponding angles are equal and corresponding sides share a constant ratio.

The multiplier that enlarges or reduces a figure during a dilation. Lengths scale by this factor, while areas scale by its square and volumes by its cube.

A change in a figure's position, size, or orientation, such as a translation, rotation, reflection, or dilation. Rigid motions preserve size and shape.

The relationship in a right triangle stating that the squares of the two legs add to the square of the hypotenuse. Used to find missing side lengths and distances.

A rule giving the straight-line distance between two coordinate points, derived from the Pythagorean theorem. It squares the differences in coordinates, adds them, and takes the square root.

The two-dimensional grid formed by perpendicular number lines intersecting at an origin. Points are located by ordered pairs and figures can be transformed on it.

The amount of space a three-dimensional figure occupies, measured in cubic units. Formulas differ by solid, and scaling a solid by a factor changes volume by its cube.

The total area of all faces or curved surfaces of a three-dimensional figure, measured in square units. Nets help reveal every face when computing it.

The arithmetic average of a data set, found by adding all values and dividing by how many there are. It is pulled toward extreme values more than the median is.

The middle value of an ordered data set, or the average of the two middle values when the count is even. It resists distortion by unusually large or small values.

A data value far from the rest of the distribution. Outliers can strongly affect the mean and range while leaving the median largely unchanged.

The difference between the third and first quartiles of a data set, covering the middle half of the values. It measures spread while ignoring extreme values.

A number from zero to one expressing how likely an event is. It can be estimated from data, computed from equally likely outcomes, or built from a model.

The set of all possible outcomes of a random experiment. Listing it systematically helps compute probabilities of compound events.

Events whose outcomes do not affect each other's likelihood. For such events, the probability that both occur is the product of their individual probabilities.

A subset drawn from a population so that every member has a known chance of selection. It supports drawing valid inferences about the whole population.

The entire group about which information is wanted in a statistical study. Conclusions rest on samples drawn from it, and its size matters for inference.

Habits of mind describing how students engage with mathematics, such as making sense of problems, reasoning, modeling, and using tools strategically. They apply across all content domains.

Representing a real situation with mathematical structures, then interpreting results back in context. Models are simplified, so results must be checked for reasonableness.

Ongoing gathering of evidence about student understanding during instruction to adjust teaching. It differs from summative testing, which judges learning at the end.

Showing the same mathematical idea through tables, graphs, equations, or verbal descriptions. Moving among representations builds conceptual understanding and supports problem solving.