Math Terms Explained - TheMaths.Online

Kindergarten

Counting: Saying numbers in order to find how many there are.
Number: A word or symbol that represents a specific quantity.
Zero: The number that represents a count of no objects.
Compare: To see if a group of objects has more, fewer, or the same number as another group.
More than (>): A relationship where a quantity is larger than another.
Less than (<): A relationship where a quantity is smaller than another.
Equal To (=): Having the same amount or value.
Shape: The form of an object or its external boundary.
Circle: A round shape with no corners or edges.
Square: A shape with four equal straight sides and four right angles.
Triangle: A shape with three straight sides and three angles.
Rectangle: A shape with four straight sides and four right angles, with opposite sides of equal length.
Hexagon: A shape with six straight sides and six angles.
Above/Below: Terms describing the relative position of objects.
In front of/Behind: Terms describing the relative position of objects.
Next to/Beside: Terms describing the relative position of objects.
Sort: To group objects based on a shared characteristic like color, size, or shape.
Pattern: A repeating arrangement of numbers, shapes, or colors.

1st Grade

Addition: The process of combining two or more numbers (addends) to find their total (sum).
Subtraction: The process of taking one number away from another to find the difference.
Addend: Any of the numbers that are added together.
Sum: The result of an addition problem.
Difference: The result of a subtraction problem.
Fact Family: A group of related math facts involving the same numbers (e.g., 3+4=7, 4+3=7, 7-3=4, 7-4=3).
Tens and Ones: The two digits that make up a two-digit number, representing its place value.
Number Line: A line with numbers placed in their correct position, useful for counting, adding, and subtracting.
Half: One of two equal parts of a whole.
Quarter/Fourth: One of four equal parts of a whole.
Clock (Analog/Digital): A device used to measure and show time, distinguishing between hour and minute hands.
Data: A collection of information gathered by observation, questioning or measurement.
Bar Graph: A chart that uses bars to show comparisons between categories of data.
Tally Mark: A form of counting using vertical marks, with the fifth mark crossing the previous four.
Length: The measurement of something from end to end.
Greater Than (>): Symbol used to compare two numbers, with the greater number given first.
Less Than (<): Symbol used to compare two numbers, with the lesser number given first.

2nd Grade

Place Value: The value of a digit based on its position in a number (e.g., ones, tens, hundreds, thousands).
Regrouping: Making groups of tens when adding or subtracting two-digit numbers (or more), also known as carrying or borrowing.
Even Number: A whole number that can be divided by 2 into two equal whole numbers.
Odd Number: A whole number that cannot be divided exactly by 2.
Array: An arrangement of objects in rows and columns, used to understand multiplication.
Measurement: Finding a number that shows the size or amount of something.
Inch/Centimeter: Standard units for measuring length.
Foot/Meter: Larger standard units for measuring length. 1 foot = 12 inches.
Money (Coins/Bills): Understanding the value of different denominations of currency and making change.
Polygon: A closed, flat shape with three or more straight sides.
Cube: A three-dimensional shape with six square faces, eight vertices, and twelve edges.
Sphere: A three-dimensional object shaped like a ball.
Cylinder: A three-dimensional shape with two identical flat circular ends and one curved side.
Cone: A three-dimensional shape with a circular base and a vertex.
Expanded Form: A way of writing numbers to see the math value of individual digits (e.g., 347 = 300 + 40 + 7).
Skip Counting: Counting forward by numbers other than 1, such as by 2s, 5s, or 10s.

3rd Grade

Multiplication: A mathematical operation of scaling one number by another, essentially repeated addition.
Division: The process of splitting a number into equal parts and finding how many equal parts can be made.
Factor: Numbers that are multiplied together to get a product.
Product: The result of a multiplication problem.
Dividend: The number that is being divided.
Divisor: The number by which another number is divided.
Quotient: The result of a division problem.
Remainder: The amount 'left over' after a division that is not exact.
Fraction: A number that represents a part of a whole.
Numerator: The top number in a fraction that shows how many parts you have.
Denominator: The bottom number in a fraction that shows how many equal parts the whole is divided into.
Equivalent Fractions: Fractions that have different numerators and denominators but represent the same value.
Area: The amount of space inside a two-dimensional shape, measured in square units ($Area = length \times width$).
Perimeter: The total distance around the outside of a two-dimensional shape.
Rounding: Approximating a number to its nearest place value (e.g., nearest ten or hundred).
Right Angle: An angle that measures exactly 90 degrees.
Parallel Lines: Two lines on a plane that never meet; they are always the same distance apart.
Perpendicular Lines: Two lines that intersect to form right angles (90 degrees).

4th Grade

Prime Number: A whole number greater than 1 that has only two factors: one and itself.
Composite Number: A whole number greater than 1 that has more than two factors.
Decimal: A number containing a decimal point, which separates the whole number part from the fractional part.
Decimal Point: A dot separating the whole number from the fraction in a decimal number.
Mixed Number: A number consisting of a whole number and a proper fraction.
Improper Fraction: A fraction where the numerator is greater than or equal to the denominator.
Angle: The figure formed by two rays sharing a common endpoint, called the vertex.
Degree: A unit used to measure angles.
Protractor: A tool used for measuring or drawing angles.
Acute Angle: An angle that measures less than 90 degrees.
Obtuse Angle: An angle that measures more than 90 degrees but less than 180 degrees.
Straight Angle: An angle that measures exactly 180 degrees, forming a straight line.
Symmetry: When one shape becomes exactly like another if you flip, slide, or turn it.
Line of Symmetry: A line that divides a figure into two mirror images.
Vertex: A point where two or more lines or edges meet; a corner.
Long Division: A standard method for dividing multi-digit numbers.

5th Grade

Order of Operations (PEMDAS/BODMAS): The rules (Parentheses/Brackets, Exponents/Orders, Multiplication/Division, Addition/Subtraction) that state the sequence to solve an expression.
Exponent: A number that shows how many times a base number is to be multiplied by itself.
Volume: The amount of space a three-dimensional object occupies, measured in cubic units ($Volume = length \times width \times height$).
Coordinate Plane: A two-dimensional plane formed by the intersection of a horizontal x-axis and a vertical y-axis.
Axes (x-axis and y-axis): The two perpendicular number lines on a coordinate plane.
Origin: The point (0,0) on the coordinate plane where the x-axis and y-axis intersect.
Ordered Pair (x, y): A pair of numbers used to locate a point on a coordinate plane.
Expression: A mathematical phrase that can contain numbers, variables, and operators, but no equal sign.
Equation: A statement that two mathematical expressions are equal.
Multiplying & Dividing Fractions: The processes for finding the product and quotient of two fractions.
Scaling: Resizing a number or object in relation to another number or object.
Line Plot: A graph that shows the frequency of data along a number line.
Classification of Triangles: Understanding triangles by their sides (scalene, isosceles, equilateral) and angles (acute, obtuse, right).
Classification of Quadrilaterals: Understanding the properties of and relationships between squares, rectangles, parallelograms, trapezoids, and rhombuses.
Numerical Pattern: A sequence of numbers that has been created based on a rule.

6th Grade

Ratio: A comparison of two quantities by division.
Rate: A ratio that compares two quantities with different units of measure.
Unit Rate: A rate with a denominator of 1.
Percent: A special ratio that compares a number to 100.
Integer: The set of whole numbers and their opposites (...-3, -2, -1, 0, 1, 2, 3...).
Rational Number: A number that can be written as a fraction, including integers, terminating decimals, and repeating decimals.
Absolute Value: The distance a number is from zero on a number line, always a non-negative value. Notation: $|x|$.
Variable: A symbol, usually a letter, used to represent a number that can change or is unknown.
Coefficient: A number used to multiply a variable.
Term: A single number or variable, or numbers and variables multiplied together.
Like Terms: Terms whose variables (and their exponents) are the same.
Distributive Property: A property stating that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. $a(b+c) = ab + ac$.
Statistical Question: A question that can be answered by collecting data that vary.
Median: The middle value in a sorted list of numbers.
Mean: The average of a set of numbers, found by summing the values and dividing by the count of values.
Mode: The value that appears most frequently in a data set.
Range: The difference between the highest and lowest values in a data set.
Box Plot: A standardized way of displaying the distribution of data based on a five-number summary (minimum, first quartile, median, third quartile, and maximum).
Histogram: A graphical display of data using bars of different heights, where each bar groups numbers into ranges.

7th Grade

Proportion: An equation stating that two ratios are equivalent.
Constant of Proportionality: The constant value of the ratio of two proportional quantities y and x; usually written y = kx, where k is the constant.
Scale Drawing: A drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (the scale).
Inequality: A mathematical sentence that uses symbols like >, <, ≥, or ≤ to compare two unequal expressions.
Probability: The measure of the likelihood that an event will occur, expressed as a number between 0 and 1.
Simple Interest: A quick method of calculating the interest charge on a loan or principal. Formula: $I = prt$.
Compound Interest: Interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods.
Markup/Markdown: The amount added to the cost price of goods to cover overhead and profit / A reduction of price.
Circumference: The distance around a circle. Formula: $C = 2\pi r$ or $C = \pi d$.
Area of a Circle: The space enclosed by a circle. Formula: $A = \pi r^2$.
Complementary Angles: Two angles whose measures have a sum of 90 degrees.
Supplementary Angles: Two angles whose measures have a sum of 180 degrees.
Vertical Angles: A pair of opposite angles made by intersecting lines. They are always equal.
Adjacent Angles: Two angles that have a common vertex and a common side but do not overlap.
Random Sample: A subset of a statistical population in which each member of the subset has an equal probability of being chosen.
Inference: The process of using data analysis to deduce properties of an underlying probability distribution.
Surface Area: The total area of the surface of a three-dimensional object.

8th Grade

Linear Equation: An equation between two variables that gives a straight line when plotted on a graph, typically in the form $y = mx + b$.
Slope: A measure of the steepness of a line, calculated as 'rise over run' ($m = \frac{y_2 - y_1}{x_2 - x_1}$).
Y-intercept: The point where a line or curve crosses the y-axis of a graph.
Function: A relation where each input value is paired with exactly one output value.
Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides ($a^2 + b^2 = c^2$).
Irrational Number: A number that cannot be expressed as a simple fraction, like $\pi$ or $\sqrt{2}$.
Real Number: The set of all rational and irrational numbers.
Scientific Notation: A way of writing very large or very small numbers as a product of a number between 1 and 10 and a power of 10.
Laws of Exponents: Rules that govern how to perform operations on expressions with exponents.
Square Root: A value that, when multiplied by itself, gives the original number.
Cube Root: A value that, when multiplied by itself three times, gives the original number.
Transformation: A process that changes the position or orientation of a figure. Includes translations, rotations, reflections, and dilations.
Dilation: A transformation that changes the size of a figure but not its shape.
Congruence: When two figures have the same size and shape. Transformations like translation, rotation, and reflection preserve congruence.
Similarity: When two figures have the same shape but possibly different sizes. Dilations produce similar figures.
System of Linear Equations: A set of two or more linear equations containing the same variables, which can be solved to find a common solution (the point of intersection).
Scatter Plot: A graph in which the values of two variables for a set of data are plotted as points on a coordinate plane.
Line of Best Fit: A straight line drawn through the center of a group of data points on a scatter plot that best represents the data's trend.

Algebra 1

Polynomial: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Monomial/Binomial/Trinomial: Polynomials with one, two, or three terms, respectively.
Degree of a Polynomial: The highest exponent of its variable.
Quadratic Equation: An equation where the highest exponent of the variable is a square (degree 2), typically in the form $ax^2 + bx + c = 0$.
Factoring: The process of breaking down a polynomial into a product of simpler polynomials (its factors).
System of Equations: A set of two or more equations with the same set of variables; the solution is the set of values that satisfy all equations simultaneously.
System of Inequalities: A set of two or more inequalities with the same variables.
Radical: An expression that uses a root, such as a square root ($\sqrt{x}$), cube root, etc.
Domain: The set of all possible input values (x-values) for which a function is defined.
Range: The set of all possible output values (y-values) of a function.
Function Notation: A way of writing functions where the function is named, e.g., $f(x)$, which is read as 'f of x'.
Quadratic Formula: A formula that provides the solution(s) to a quadratic equation: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
Discriminant: The part of the quadratic formula under the square root, $b^2-4ac$, which determines the number of real solutions.
Vertex of a Parabola: The highest or lowest point on a parabola.
Axis of Symmetry: A line that divides a parabola into two mirror images.
Exponential Function: A function of the form $y = ab^x$, where b is a positive real number not equal to 1, representing growth or decay.
Measures of Central Tendency: Values that describe the center of a data set: mean, median, and mode.
Measures of Spread: Values that describe how spread out a data set is: range, interquartile range (IQR), and standard deviation.

Geometry

Postulate/Axiom: A statement that is accepted as true without proof.
Theorem: A statement that has been proven to be true based on postulates, definitions, and other theorems.
Corollary: A theorem that follows on from another theorem.
Congruent: Geometric figures having the exact same size and shape.
Similar: Geometric figures having the same shape but not necessarily the same size.
Rigid Motion: A transformation of the plane that preserves distance. Includes translations, rotations, and reflections.
Sine (sin): In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.
Cosine (cos): In a right triangle, the ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent (tan): In a right triangle, the ratio of the length of the opposite side to the length of the adjacent side.
Cosecant (csc): The reciprocal of sine: $\frac{hypotenuse}{opposite}$.
Secant (sec): The reciprocal of cosine: $\frac{hypotenuse}{adjacent}$.
Cotangent (cot): The reciprocal of tangent: $\frac{adjacent}{opposite}$.
Chord: A line segment whose endpoints both lie on a circle.
Arc: A portion of the circumference of a circle.
Sector: The region enclosed by two radii of a circle and their intercepted arc.
Tangent (to a circle): A line that intersects a circle at exactly one point.
Conditional Probability: The probability of an event occurring, given that another event has already occurred.
Volume of a Sphere/Cylinder/Cone/Prism: Formulas to calculate the three-dimensional space occupied by various geometric solids.

Algebra 2

Matrix: A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Determinant: A scalar value that can be computed from the elements of a square matrix.
Logarithm: The exponent to which a specified base must be raised to obtain a given number. It is the inverse operation of exponentiation.
Natural Logarithm (ln): A logarithm to the base $e$ (approximately 2.718).
Complex Number: A number that can be expressed in the form $a + bi$, where a and b are real numbers and $i$ is the imaginary unit, satisfying $i^2 = -1$.
Conic Section: A curve obtained as the intersection of the surface of a cone with a plane (e.g., circle, ellipse, parabola, hyperbola).
Rational Expression: An algebraic fraction whose numerator and denominator are polynomials.
Asymptote (Vertical/Horizontal/Slant): A line that a graph approaches but never crosses as it heads towards infinity.
Inverse Function: A function that reverses another function. If $f(a) = b$, then the inverse function $f^{-1}(b) = a$.
Composition of Functions: The process of combining two functions by applying one function to the results of another, denoted $(f \circ g)(x) = f(g(x))$.
Sequence (Arithmetic/Geometric): An ordered list of numbers. An arithmetic sequence has a common difference; a geometric sequence has a common ratio.
Series: The sum of the terms of a sequence.
Summation Notation (Sigma): A notation (using the Greek letter $\Sigma$) to represent the sum of the terms in a sequence.
Radical Equation: An equation in which a variable is under a radical.
Extraneous Solution: A solution derived from an original equation that is not a true solution to the original equation.
Polynomial Division (Long/Synthetic): Methods for dividing a polynomial by another polynomial of the same or lower degree.

Pre-Calculus

Trigonometric Identity: An equation involving trigonometric functions that is true for all values of the involved variables for which both sides of the equality are defined.
Pythagorean Identity: A fundamental trigonometric identity: $\sin^2(\theta) + \cos^2(\theta) = 1$.
Unit Circle: A circle with a radius of one, centered at the origin, used to define sine and cosine for all real numbers.
Radian: A unit of angle measure, based on the radius of a circle. $2\pi$ radians equals 360 degrees.
Periodic Function: A function that repeats its values at regular intervals or periods.
Amplitude: The height from the center line to the peak or trough of a periodic function.
Phase Shift: A horizontal shift of a periodic function.
Vector: A quantity that has both magnitude (size) and direction.
Polar Coordinates: A two-dimensional coordinate system where each point is determined by a distance from a reference point (pole) and an angle from a reference direction (polar axis).
Limit: The value that a function or sequence 'approaches' as the input or index 'approaches' some value.
Continuity: A property of a function where there are no interruptions, jumps, or holes in its graph.
Parametric Equations: A set of equations that express a set of quantities as explicit functions of a number of independent variables, known as 'parameters'.
Law of Sines: A relationship between the sides and angles of non-right triangles: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.
Law of Cosines: A relationship between the sides and angles of non-right triangles: $c^2 = a^2 + b^2 - 2ab\cos(C)$.
Dot Product: An algebraic operation that takes two equal-length sequences of numbers and returns a single number.

Calculus

Derivative: The instantaneous rate of change of a function with respect to one of its variables, or the slope of the tangent line to the function's graph.
Integral: A mathematical object that can be interpreted as an area or a generalization of area. The reverse process of differentiation is integration.
Definite Integral: An integral with upper and lower limits, representing the accumulated value or area over an interval.
Indefinite Integral: An integral without upper and lower limits, representing a family of functions whose derivative is the integrand.
Chain Rule: A formula to compute the derivative of a composite function. If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$.
Product Rule: A formula used to find the derivatives of products of two or more functions.
Quotient Rule: A formula used to find the derivatives of quotients of two functions.
Differential Equation: An equation that relates one or more functions and their derivatives.
Series (Convergence/Divergence): The sum of the terms of a sequence. Determining if the sum approaches a finite limit (convergence) or not (divergence).
Power Series: A series of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$.
Taylor Series: A representation of a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point.
L'Hôpital's Rule: A rule that uses derivatives to help evaluate limits involving indeterminate forms.
Fundamental Theorem of Calculus: A theorem that links the concepts of differentiating a function and integrating a function.
Optimization: The process of finding the maximum or minimum value of a function, often under certain constraints.
Related Rates: Problems that involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known.
Implicit Differentiation: A procedure used to find the derivative of a function that is not defined explicitly.

Statistics

Standard Deviation: A measure of the amount of variation or dispersion of a set of values.
Variance: The square of the standard deviation, another measure of data spread.
Correlation: A statistical measure that expresses the extent to which two variables are linearly related.
Regression (Linear/Non-linear): A statistical method used to model the relationship between a dependent variable and one or more independent variables.
Hypothesis Testing: A method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Null Hypothesis ($H_0$): A general statement that there is no relationship between two measured phenomena, or no association among groups.
Alternative Hypothesis ($H_a$ or $H_1$): The hypothesis that is contrary to the null hypothesis.
P-value: The probability of obtaining results at least as extreme as the observed results of a statistical hypothesis test, assuming that the null hypothesis is correct.
Confidence Interval: A range of estimates for an unknown parameter, calculated from the sample data.
Margin of Error: A statistic expressing the amount of random sampling error in the results of a survey.
Sample: A subset of individuals from a larger statistical population.
Population: The entire group of individuals that you want to draw conclusions about.
Normal Distribution: A probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean (bell curve).
Binomial Distribution: A discrete probability distribution of the number of successes in a sequence of n independent experiments.
Chi-Square Test: A statistical test used to compare observed results with expected results.
Type I & Type II Errors: Errors in hypothesis testing: Type I is the rejection of a true null hypothesis (false positive), while Type II is the non-rejection of a false null hypothesis (false negative).