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30-60-90 Triangle

A 30-60-90 triangle is a right triangle whose angles measure 30°, 60° and 90°. Its sides are always in the ratio 1 : square root of 3 : 2: if the short leg (opposite 30°) is x, the long leg is xsquare root of 3 and the hypotenuse is 2x. So knowing one side gives the other two.

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New in the dictionary 300

Terms added in the November 2026 expansion, each with a full illustrated entry.

30-60-90 Triangle45-45-90 TriangleAcute AngleAddendAdditionAlternate Interior AnglesArc LengthArea ModelArea of a CircleArithmetic SequenceArrayAssociative PropertyAsymptoteAt Least and At MostAverageAverage Rate of ChangeAverage SpeedAxis of SymmetryCalculusCategorical DataCausationCentral AngleChain RuleChordCircleCombinationCombining Like TermsCommon DenominatorCommon DifferenceCommon RatioCommutative PropertyComplementary AnglesCompleting the SquareComplex NumberComposition of FunctionsCompound EventCompound InequalityCompound InterestConditional ProbabilityConeCongruentConsecutive IntegersConstant TermConversion FactorCorrelationCorrelation CoefficientCorresponding AnglesCosineCounterexampleCounting NumbersCritical PointCross MultiplicationCube RootCylinderDataDecimal PointDecomposeDegreeDegree of a PolynomialDensityDependent EventsDerivativeDifferenceDifference of SquaresDigitDilationDimensional AnalysisDirectly ProportionalDiscriminantDistance FormulaDistance-Rate-Time FormulaDividendDivisibility RulesDivisionDivisorDomainDot PlotElapsed TimeEmpirical RuleEnd BehaviorEqual SignEqual ToEquation of a CircleEquilateral TriangleEquivalent ExpressionsEuler's NumberEvaluateEven NumberEventExpanded FormExponential DecayExponential FunctionExponential GrowthFact FamilyFactor CompletelyFactor PairFactorialFOIL MethodFootFormula SheetFractionFunction NotationGeometric SequenceGramGreater ThanGreater Than or Equal ToHeightHundredthsIdentity PropertyImaginary NumberIn Terms OfInchIndependent EventsInfinityIntegralInterior AngleInterquartile RangeInterval NotationInverse FunctionInverse OperationsInversely ProportionalIsosceles TriangleKilogramLaw of CosinesLaw of SinesLaws of ExponentsLeading CoefficientLeast Common DenominatorLengthLess ThanLess Than or Equal ToLimitLineLine of Best FitLine of SymmetryLine SegmentLinear FunctionLogarithmLong DivisionMatrixMean Absolute DeviationMedian of a TriangleMetric SystemMidpointMidpoint FormulaMileMinuendMixture ProblemsMonomialMultiplicationMultiplication TableMutually Exclusive EventsNatural LogarithmNatural NumberNegative ExponentNegative NumberNo More ThanNonnegativeNormal DistributionNumberNumber LineObtuse AngleOdd NumberOddsOrdinal NumberOriginOutcomeParabolaParallelogramParent FunctionPEMDASPercent DecreasePercent IncreasePercentilePerfect SquarePeriodPermutationPiPiecewise FunctionPlanePointPoint-Slope FormPolygonPolygon Angle SumPopulationPositive IntegerPowerPower RulePrime FactorizationPrincipalProductProofProper FractionProperties of LogarithmsProportional RelationshipPyramidQuadrantQuadratic FormulaQuadratic FunctionQuadrilateralQuartileQuotientRadianRadicalRange of a FunctionRateRate of ChangeRational ExponentRational ExpressionRational FunctionRayReal NumberReal SolutionReciprocalRectangular PrismRecursive FormulaReflectionRegroupingRegular PolygonRelationRelative FrequencyRemainderRemainder TheoremRepeated AdditionRhombusRight AngleRight TriangleRootsRotationRounded to the NearestSampleSample SpaceSampling BiasScale DrawingScalene TriangleSecantSequenceSeriesSetSideSigma NotationSimple InterestSimplest FormSimplifySineSkip CountingSOH CAH TOASpecial Right TrianglesSphereSquareSquare NumberStandard AlgorithmStandard DeviationStandard FormStandard Form of a Linear EquationStandard Form of a QuadraticStatisticsSubsetSubstituteSubtractionSubtrahendSumSupplementary AnglesSymmetrySynthetic DivisionTally MarkTangentTenthsTermTheoremTransformation of FunctionsTranslating Words into MathTranslationTransversalTrapezoidTriangle Inequality TheoremTrigonometryTrinomialTwo-Column ProofUndefinedUnit CircleUnit ConversionUnit FractionVarianceVectorVenn DiagramVertex FormVertex of a ParabolaVertical AnglesVertical Line TestWeighted AverageWord FormWork Rate ProblemsX-InterceptY-AxisYardZ-ScoreZero ExponentZero of a FunctionZero Property of Multiplication

Guided study sets

Short lists to learn together, picked for a grade band or a test.

Grade 3-5 foundations 13 words

Place ValueRoundingWhole NumberFactorMultiplePrime NumberComposite NumberNumeratorDenominatorEquivalent FractionsDecimalCentimeterMeter
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Middle school readiness 12 words

RatioUnit RateProportionPercentIntegerAbsolute ValueOppositeExpressionEquationInequalityCoordinate PlaneOrdered Pair
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Grade 8 algebra bridge 12 words

SlopeSlope-Intercept FormLinear EquationSystem of EquationsFunctionInputOutputPythagorean TheoremScientific NotationScatter PlotTwo-Way TableJoint Frequency
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GED test vocabulary 13 words

Order of OperationsPercentRatioUnit PriceAreaPerimeterVolumeMeanMedianModeProbabilityWord ProblemReasonableness
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Algebra 1 toolkit 11 words

VariableCoefficientConstantLike TermsDistributive PropertyFactoringPolynomialQuadratic EquationSubstitution MethodElimination MethodSolution
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Geometry and measurement 12 words

AngleParallel LinesPerpendicular LinesAreaSurface AreaVolumeRadiusDiameterCircumferenceScale FactorSimilar FiguresCongruent Figures
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Topic trails

Every term, grouped by the part of math it belongs to.

Calculus 7

Limits, derivatives and integrals: the math of change and accumulation.

CalculusChain RuleCritical PointDerivativeIntegralLimitPower Rule

Symbols and Notation 8

How to read comparison signs and the phrases tests use.

Equal SignEqual ToGreater ThanGreater Than or Equal ToInfinityInterval NotationLess ThanLess Than or Equal To

Grade paths

The words students meet in each grade or course, from Kindergarten to Calculus.

Kindergarten: 14 words

See all 14 words in the A to Z list

Precalculus: 50 words

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Test paths

Vocabulary that shows up on the math sections of SAT, ACT, GED, ASVAB, ACCUPLACER, TABE, HiSET, PSAT, AP and TEAS.

ACCUPLACER: 133 words

See all 133 words in the A to Z list

Math words A to Z

3

  • 30-60-90 TriangleNewA 30-60-90 triangle is a right triangle whose angles measure 30°, 60° and 90°. Its sides are always in the ratio 1 : square root of 3 : 2: if the short leg (opposite 30°) is x, the long leg is xsquare root of 3 and the hypotenuse is 2x. So knowing one side gives the other two.Geometry · Geometry, Precalculus

4

  • 45-45-90 TriangleNewA 45-45-90 triangle is a right triangle with two 45° angles and two equal legs. Its sides are always in the ratio 1 : 1 : square root of 2: the hypotenuse is a leg times square root of 2. Legs of 5 give a hypotenuse of 5square root of 2. It is half of a square cut along a diagonal.Geometry · Grade 8, Geometry +

A

  • Absolute ValueUpdatedThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7 +
  • Acute AngleNewAn acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.Geometry · Grade 4, Grade 5 +
  • AddendNewAn addend is any number that is added to another number. In 8 + 5 = 13, the addends are 8 and 5, and 13 is the sum. An addition can have two or more addends, and you can add them in any order. When one addend is missing, subtract the known addend from the sum to find it.Operations · Grade 1, Grade 2 +
  • AdditionNewAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1 +
  • Additive InverseUpdatedThe additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.Number System · Grade 6, Grade 7 +
  • Alternate Interior AnglesNewAlternate interior angles are two angles that lie between two lines and on opposite sides of a transversal that crosses them. When the two lines are parallel, alternate interior angles are equal. They often trace a Z shape: in the diagram, angles 3 and 6 form one pair and angles 4 and 5 the other.Geometry · Grade 8, Geometry
  • AngleUpdatedAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5 +
  • Arc LengthNewArc length is the distance along a curved part of a circle, measured in units of length. An arc with central angle θ in a circle of radius r is θ/360 of the circumference: s = θ/360 × 2π r when θ is in degrees, or simply s = rθ when θ is in radians.Geometry · Geometry, Precalculus
  • AreaUpdatedArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle's area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4 +
  • Area ModelNewAn area model is a rectangle used to multiply or divide, with each side broken into place-value parts. To find 23 × 14, split the sides into 20 + 3 and 10 + 4, find the area of each smaller box (200, 30, 80, 12) and add these partial products: 322.Operations · Grade 3, Grade 4 +
  • Area of a CircleNewThe area of a circle is the amount of flat space inside the circle, measured in square units. It equals pi times the radius squared: A = π r^2. A circle with radius 5 cm has area 25π, about 78.5 square centimeters. If you know the diameter, halve it first to get the radius.Geometry · Grade 7, Grade 8 +
  • Arithmetic SequenceNewAn arithmetic sequence is a list of numbers in which you add the same amount each time to get the next term. That fixed amount is the common difference, d. In 4, 7, 10, 13, … the common difference is 3. The nth term is an = a1 + (n − 1)d.Functions · Grade 8, Algebra 1 +
  • ArrayNewAn array is an arrangement of objects in equal rows and equal columns, forming a rectangle. A 4 by 6 array has 4 rows of 6, so it holds 4 × 6 = 24 objects. Arrays show multiplication as equal groups, and turning one sideways shows that 4 × 6 = 6 × 4.Operations · Grade 2, Grade 3 +
  • Associative PropertyNewThe associative property says that when you add or multiply three or more numbers, the way you group them does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). It lets you pick the easiest pair first. It does not work for subtraction or division.Operations · Grade 1, Grade 3 +
  • AsymptoteNewAn asymptote is a line that a graph gets closer and closer to as it heads off toward infinity. A vertical asymptote x = a appears where the function blows up, often at a zero denominator. A horizontal asymptote y = b shows the value it approaches far left or right. y = 1/x has both: x = 0 and y = 0.Functions · Algebra 2, Precalculus +
  • At Least and At MostNewAt least and at most are word-problem phrases for limits that include the boundary. "At least 3" means 3 or more, written x ≥ 3. "At most 3" means 3 or less, written x ≤ 3. In both cases the number 3 itself is allowed, so it gets a filled dot on a number line.Problem Solving · Grade 6, Grade 7 +
  • AverageNewAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6 +
  • Average Rate of ChangeNewThe average rate of change of a function over an interval is the change in output divided by the change in input: (f(b) − f(a))/(b − a). It tells how fast the function changes, on average, from x = a to x = b. On a graph it is the slope of the line through the two endpoints.Functions · Algebra 1, Algebra 2 +
  • Average SpeedNewAverage speed is the total distance traveled divided by the total time taken. If a car goes 150 miles in 3 hours, its average speed is 150 ÷ 3 = 50 miles per hour, even if it sped up and slowed down along the way. It is not the average of the separate speeds unless each speed lasted equally long.Problem Solving · Grade 6, Grade 7 +
  • Axis of SymmetryNewThe axis of symmetry of a parabola is the vertical line through its vertex that splits it into two mirror-image halves. For y = ax^2 + bx + c, it is the line x = −b/2a. Points at equal distances on either side of this line have the same y-value.Functions · Algebra 1, Algebra 2

B

  • Bar GraphUpdatedA bar graph uses bars to compare amounts in different categories. The taller or longer the bar, the larger the value it represents. Each bar stands for one category, such as a favorite sport, and its length is read against a number scale. Bars can be vertical or horizontal, with gaps between them.Statistics · Grade 3, Grade 4 +
  • BaseUpdatedA base is the number or expression that is repeatedly multiplied when using an exponent. In 4^5 the base is 4 and the exponent is 5, so 4^5 = 4 × 4 × 4 × 4 × 4 = 1024. The base is the big number written on the line; the exponent is the small raised number.Algebra · Grade 6, Grade 7 +
  • BinomialUpdatedA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a^2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2
  • Box PlotUpdatedA box plot displays a data set through five numbers: minimum, first quartile, median, third quartile, and maximum. The box holds the middle half of the data, a line inside it marks the median, and whiskers reach out to the smallest and largest values. It makes spread easy to compare between sets.Statistics · Grade 6, Grade 7 +

C

  • CalculusNewCalculus is the branch of mathematics that studies how quantities change. Its two main tools are the derivative, which measures the rate of change at an instant (the slope of a curve at a point), and the integral, which adds up infinitely many tiny pieces (the area under a curve). Both are built on limits.Calculus · Calculus
  • Categorical DataNewCategorical data are data that sort people or things into groups, or categories, such as eye color, favorite sport or a yes/no answer. The values are labels, not measurements, so you count how many fall in each category and show the counts with a bar graph, a pie chart or a two way table.Statistics · Grade 6, Grade 8 +
  • CausationNewCausation means a change in one variable directly produces a change in another: the first is the cause and the second is the effect. Two variables can be correlated without causation, because a lurking variable or chance may link them. A well-designed randomized experiment is the standard way to show causation.Statistics · Grade 8, Algebra 1 +
  • CentimeterUpdatedA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4 +
  • Central AngleNewA central angle is an angle whose vertex is the center of a circle and whose sides are two radii. It cuts off an arc, and the arc's degree measure equals the central angle. A 90° central angle marks off a quarter of the circle, and the central angles around the center add to 360 degrees.Geometry · Grade 7, Geometry
  • Chain RuleNewThe chain rule is the rule for the derivative of a composite function: if y = f(g(x)), then y′ = f′(g(x)) · g′(x). Differentiate the outer function, leave the inner function inside, then multiply by the derivative of the inner function. For (x^2 + 1)^3, the derivative is 3(x^2 + 1)^2 · 2x.Calculus · Calculus
  • ChordNewA chord is a line segment whose two endpoints both lie on a circle. The diameter is the longest chord, because it passes through the center. A chord at distance d from the center of a circle with radius r has length 2square root of (r^2 − d^2), so chords closer to the center are longer.Geometry · Grade 7, Geometry
  • CircleNewA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r^2.Geometry · Kindergarten, Grade 1 +
  • CircumferenceUpdatedCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle's perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Geometry · Grade 6, Grade 7 +
  • CoefficientUpdatedA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7 +
  • CombinationNewA combination is a selection of items from a group in which the order does not matter. Picking Ana, Ben and Cy for a team is the same combination as Cy, Ana and Ben. The number of ways to choose r items from n is C(n, r) = (n!)/(r!(n − r)!); for example, C(5, 2) = 10.Probability · Algebra 2, Precalculus +
  • Combining Like TermsNewCombining like terms is simplifying an expression by adding or subtracting the terms that have exactly the same variable part. You add their coefficients and keep the variable part: 3x + 2x = 5x. Constants combine with constants. So 3x + 5 + 2x − 1 simplifies to 5x + 4.Algebra · Grade 6, Grade 7 +
  • Common DenominatorNewA common denominator is a number that can be used as the denominator of two or more fractions at once. It is any common multiple of the denominators. For 1/3 and 1/4, 12 works: 4/12 and 3/12. You need one to add, subtract or compare fractions with different denominators.Number System · Grade 4, Grade 5 +
  • Common DifferenceNewThe common difference is the fixed amount added to each term of an arithmetic sequence to get the next term, written d. Find it by subtracting any term from the term after it: in 7, 11, 15, 19, d = 11 − 7 = 4. A positive d makes the terms increase, a negative d makes them decrease.Functions · Grade 8, Algebra 1 +
  • Common RatioNewThe common ratio is the fixed number you multiply each term of a geometric sequence by to get the next term. Find it by dividing any term by the term before it: r = an/(an − 1). In 5, 20, 80, 320, the common ratio is 4. It can be a fraction or negative, but never 0.Functions · Algebra 1, Algebra 2
  • Commutative PropertyNewThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3 +
  • Complementary AnglesNewComplementary angles are two angles whose measures add up to 90 degrees. Each angle is the complement of the other, so the complement of an angle x is 90° − x. They can share a side and form a right angle, or sit apart. For example, 35° and 55° are complementary.Geometry · Grade 4, Grade 7 +
  • Completing the SquareNewCompleting the square is a method that rewrites x^2 + bx as a perfect square by adding (b/2)^2: x^2 + bx + (b/2)^2 = (x + b/2)^2. It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.Algebra · Algebra 1, Algebra 2 +
  • Complex NumberNewA complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = −1. The real part is a and the imaginary part is b. Examples: 3 + 2i and −4i. Complex numbers let you take square roots of negatives, such as square root of −9 = 3i.Number System · Algebra 2, Precalculus
  • Composite NumberUpdatedA composite number is a whole number greater than 1 that has more than two factors. It can be divided evenly by a number other than 1 and itself: 12 is composite because 1, 2, 3, 4, 6 and 12 all divide it. Every whole number above 1 is either prime or composite; 1 is neither.Number System · Grade 4, Grade 5 +
  • Composition of FunctionsNewA composition of functions applies one function to the output of another. The composite f(g(x)), also written (f ∘ g)(x), means: find g(x) first, then put that result into f. If g(x) = x + 1 and f(x) = x^2, then f(g(2)) = f(3) = 9. Order matters.Functions · Algebra 2, Precalculus
  • Compound EventNewA compound event is an event that combines two or more simple events, such as flipping heads and rolling a 6, or drawing a red card or a king. Find its probability by counting outcomes in a list, table or tree diagram, or with rules such as P(A and B) = P(A) × P(B) for independent events.Probability · Grade 7, Grade 8 +
  • Compound InequalityNewA compound inequality is two inequalities joined by "and" or "or". With "and", a number must make both true, as in −2 < x ≤ 3, which is one segment on a number line. With "or", it only has to make at least one true, as in x < −1 or x ≥ 4, which is often two separate pieces.Algebra · Algebra 1, Algebra 2
  • Compound InterestNewCompound interest is interest computed on the principal plus all the interest already earned, so the balance grows exponentially. The formula is A = P( 1 + r/n )^(nt), where P is the principal, r the annual rate, n the compoundings per year and t the years. It beats simple interest after the first period.Functions · Grade 7, Algebra 1 +
  • Conditional ProbabilityNewConditional probability is the probability that an event happens given that another event has already happened. It is written P(A ∣ B), read "the probability of A given B", and found by P(A ∣ B) = (P(A and B))/(P(B)). Knowing B shrinks the possible outcomes to only those in B.Probability · Algebra 2, Statistics
  • ConeNewA cone is a three-dimensional solid with one flat circular base and a curved surface that narrows to a single point, the apex, like a party hat. Its volume is one third of a cylinder with the same base and height: V = 1/3π r^2 h, where r is the base radius and h the height.Geometry · Kindergarten, Grade 1 +
  • CongruentNewCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB bar ≅ CD bar reads "segment AB is congruent to segment CD."Geometry · Grade 8, Geometry
  • Congruent FiguresUpdatedCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8
  • Consecutive IntegersNewConsecutive integers are integers that follow each other in order, each 1 more than the one before, such as 7, 8, 9 or −2, −1, 0. In algebra, write them as n, n + 1, n + 2. Consecutive even or consecutive odd integers are 2 apart: n, n + 2, n + 4.Algebra · Grade 7, Grade 8 +
  • ConstantUpdatedA constant is a fixed value that does not change. In an expression, a constant is a number without a variable, such as 8 in x + 8 or −3 in 4a − 3. Special numbers such as π are constants too. Changing the variable changes the other terms, but never the constant.Algebra · Grade 6, Grade 7 +
  • Constant of ProportionalityUpdatedThe constant of proportionality is the constant ratio between two proportional quantities, k = y/x. It is the k in y = kx and stays the same for every pair of values, which makes the graph a straight line through the origin. If 3 notebooks cost 9 dollars, k = 3 dollars per notebook.Ratios and Proportions · Grade 7, Grade 8 +
  • Constant TermNewA constant term is a term in an expression or equation that has no variable, so its value never changes. In 3x^2 − 5x + 7, the constant term is 7. Its sign belongs to it: in x^2 + 4x − 9 it is −9. In a word problem it is usually a fixed amount, such as a one-time fee.Algebra · Grade 6, Grade 7 +
  • Conversion FactorNewA conversion factor is a fraction whose top and bottom are the same amount written in different units, such as (12 in)/(1 ft), so it equals 1. Multiplying a measurement by it changes the units without changing the amount: 5 ft × (12 in)/(1 ft) = 60 in. Pick the version that cancels the unit you start with.Measurement · Grade 6, Grade 7 +
  • Coordinate PlaneUpdatedThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6 +
  • CorrelationNewCorrelation is a measure of how strongly two variables move together in a straight-line pattern. In a positive correlation both rise together; in a negative correlation one rises as the other falls. The correlation coefficient r runs from −1 to 1: values near ± 1 are strong and values near 0 are weak. Correlation does not prove causation.Statistics · Grade 8, Algebra 1 +
  • Correlation CoefficientNewThe correlation coefficient, r, is a number from −1 to 1 that measures how closely the points of a scatter plot follow a straight line. The sign gives the direction (positive slopes up, negative slopes down), and the size gives the strength: near ± 1 is strong, near 0 is weak or none.Statistics · Algebra 1, Statistics
  • Corresponding AnglesNewCorresponding angles are two angles in the same relative position at each place where a transversal crosses two lines, such as the top-left corner at both crossings. When the two lines are parallel, corresponding angles are equal. A transversal across two lines makes four pairs of corresponding angles.Geometry · Grade 8, Geometry
  • CosineNewCosine is the trigonometric ratio that compares the side adjacent to an angle with the hypotenuse of a right triangle: cos θ = adjacent/hypotenuse. On the unit circle, cos θ is the x-coordinate of the point at angle θ. The cosine of an angle equals the sine of its complement.Trigonometry · Geometry, Algebra 2 +
  • CounterexampleNewA counterexample is a single example that shows a general statement is false. It fits all the conditions of the statement but does not have the claimed result. To disprove 'every prime number is odd,' one counterexample is enough: 2 is prime and even. Examples that agree with a claim never prove it, but one counterexample disproves it.Geometry · Grade 8, Algebra 1 +
  • Counting NumbersNewCounting numbers are the numbers you say when you count objects one by one: 1, 2, 3, 4, 5 and so on, without end. They start at 1, not 0, and include no fractions, decimals or negative numbers. Adding 0 to the counting numbers gives the whole numbers.Number System · Kindergarten, Grade 1 +
  • Critical PointNewA critical point of a function is an input x = c in its domain where the derivative is 0 or does not exist. These are the only places a smooth graph can turn around, so every local maximum or minimum happens at a critical point. For f(x) = x^2 − 4x, f′(x) = 2x − 4 = 0 at x = 2.Calculus · Calculus
  • Cross MultiplicationNewCross multiplication is a shortcut for solving a proportion: if a/b = c/d, then ad = bc. Multiply each numerator by the denominator on the other side and set the two products equal. To solve x/6 = 10/15, write 15x = 60, so x = 4. It works only when one fraction equals one fraction.Ratios and Proportions · Grade 7, Grade 8 +
  • Cube RootNewThe cube root of a number is the value that, used as a factor three times, gives that number. It is written cube root of x. Because 4 × 4 × 4 = 64, cube root of 64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: cube root of −27 = −3.Number System · Grade 8, Algebra 1 +
  • CylinderNewA cylinder is a three-dimensional solid with two parallel, equal circular bases joined by one curved surface, like a soup can. Its size is set by the radius r of a base and the height h between the bases. Its volume is π r^2 h, the base area times the height.Geometry · Grade 1, Grade 8 +

D

  • DataNewData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3 +
  • DecimalUpdatedA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75/100, and 0.5 is one half.Number System · Grade 4, Grade 5 +
  • Decimal PointNewA decimal point is the dot that separates the whole-number part of a number from its fractional part. Digits to its left are ones, tens and hundreds; digits to its right are tenths, hundredths and thousandths. In 46.25 the point sits between 6 ones and 2 tenths, and you read it aloud as "and".Number System · Grade 4, Grade 5 +
  • DecomposeNewDecompose means to break a number or shape into smaller parts that together make the whole. You can decompose 8 into 5 + 3, decompose 347 into 300 + 40 + 7, or decompose 3/4 into 1/4 + 2/4. Most numbers can be decomposed in many ways, and the parts always add back to the original.Number System · Kindergarten, Grade 1 +
  • DegreeNewA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1/360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5 +
  • Degree of a PolynomialNewThe degree of a polynomial is the highest degree of any of its terms. For one variable, that is the largest exponent: 4x^3 − 2x^5 + 7x has degree 5. For a term with several variables, add their exponents, so 3x^2y^4 has degree 6. A nonzero constant such as 9 has degree 0.Algebra · Algebra 1, Algebra 2
  • DenominatorUpdatedThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 3/5, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4 +
  • DensityNewDensity is the amount of mass packed into each unit of volume, found by dividing mass by volume: D = m/V. Its units combine the two, such as grams per cubic centimeter. A 40 g block with a volume of 16 cm³ has D = 40 ÷ 16 = 2.5 g/cm³. Population density works the same way: people per unit of area.Problem Solving · Grade 8, Algebra 1 +
  • Dependent EventsNewDependent events are two events where the outcome of the first changes the probability of the second. Drawing two cards without putting the first one back is the classic example. For dependent events, P(A and B) = P(A) × P(B ∣ A), where P(B ∣ A) is the probability of B once A has happened.Probability · Grade 7, Algebra 2 +
  • Dependent VariableUpdatedA dependent variable is the output value that changes based on the input. It responds to changes in the independent variable: in y = 2x + 1, y is the dependent variable because its value depends on x. On a graph it goes on the vertical axis, because its value depends on what the input was.Functions · Grade 6, Grade 7 +
  • DerivativeNewA derivative measures how fast a function's output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x^2 is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.Calculus · Calculus
  • DiameterUpdatedThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Geometry · Grade 6, Grade 7 +
  • DifferenceNewA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. "The difference of a and b" means a − b, in that order.Operations · Grade 1, Grade 2 +
  • Difference of SquaresNewA difference of squares is an expression of the form a^2 − b^2: one perfect square subtracted from another. It always factors as (a + b)(a − b). For example, x^2 − 49 = (x + 7)(x − 7). A sum of squares, a^2 + b^2, does not factor over the real numbers.Algebra · Algebra 1, Algebra 2
  • DigitNewA digit is one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 that we use to write every number. The number 507 has three digits: 5, 0 and 7. What a digit is worth depends on its place: in 507, the 5 stands for 5 hundreds, or 500.Number System · Grade 1, Grade 2 +
  • DilationNewA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Geometry · Grade 8, Geometry
  • Dimensional AnalysisNewDimensional analysis is a way to solve problems by tracking the units: you multiply by conversion factors, cancel any unit that appears on both the top and the bottom, and check that the units left over are the ones the answer needs. It is also called the factor-label method or unit analysis.Measurement · Grade 7, Grade 8 +
  • Directly ProportionalNewDirectly proportional describes two quantities whose ratio never changes: y = kx for a fixed nonzero number k, the constant of proportionality. If x doubles, y doubles; if x is cut in half, so is y. The graph is a straight line through the origin, and y/x = k for every pair.Ratios and Proportions · Grade 7, Grade 8 +
  • DiscriminantNewThe discriminant of a quadratic equation ax^2 + bx + c = 0 is the number b^2 − 4ac. Its sign tells how many real solutions the equation has without solving it: positive means two, zero means one (a repeated root), and negative means none, because the parabola never touches the x-axis.Algebra · Algebra 1, Algebra 2
  • Distance FormulaNewThe distance formula finds the straight-line length between two points on the coordinate plane: d = square root of ((x2 − x1)^2 + (y2 − y1)^2). It is the pythagorean theorem in disguise: the horizontal and vertical differences are the legs of a right triangle, and the distance is the hypotenuse.Geometry · Grade 8, Algebra 1 +
  • Distance-Rate-Time FormulaNewThe distance-rate-time formula is d = rt: distance equals rate (speed) times time, for anything moving at a constant rate. Solve it for the other two letters to get r = d/t and t = d/r. Driving at 45 miles per hour for 3 hours covers 45 × 3 = 135 miles.Problem Solving · Grade 6, Grade 7 +
  • Distributive PropertyUpdatedThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7 +
  • DividendNewThe dividend is the number being divided in a division problem. In 56 ÷ 8 = 7, the dividend is 56, the divisor is 8 and the quotient is 7. It is the total you start with before splitting it into equal groups. In a fraction such as 56/8, the dividend is the numerator.Operations · Grade 3, Grade 4 +
  • Divisibility RulesNewDivisibility rules are shortcuts that tell whether one whole number divides another with no remainder, without doing the division. A number is divisible by 2 if its last digit is even, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5, and by 10 if it ends in 0.Number System · Grade 4, Grade 5 +
  • DivisionNewDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4 +
  • DivisorNewA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4 +
  • DomainNewThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1 +
  • Domain and RangeUpdatedDomain and range describe a function: the domain is the set of all inputs it accepts, and the range is the set of all outputs it produces. For the points (1, 4) and (2, 5), the domain is {1, 2} and the range is {4, 5}. Restrictions usually come from division by zero or square roots of negatives.Functions · Grade 8, Algebra 1 +
  • Dot PlotNewA dot plot is a graph that shows each data value as a dot above its place on a number line. Dots for repeated values stack up in a column, so the tallest stack shows the mode, and you can see the center, spread, gaps and outliers of the data at a glance.Statistics · Grade 3, Grade 4 +

E

  • Elapsed TimeNewElapsed time is the amount of time that passes between a start time and an end time. To find it, count forward from the start in easy jumps (to the next hour, then whole hours, then the leftover minutes) and add the jumps. From 9:40 to 11:15, the elapsed time is 1 hour 35 minutes.Measurement · Grade 3, Grade 4 +
  • Elimination MethodUpdatedThe elimination method solves a system of equations by adding or subtracting them so one variable cancels. Scaling one equation first is often needed to make the coefficients match. After one variable is eliminated, solve for the other variable, then substitute back to find the first.Algebra · Grade 8, Algebra 1
  • Empirical RuleNewThe empirical rule says that in a bell-shaped (normal) distribution about 68% of the data fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. It is also called the 68-95-99.7 rule, and it only works for data that are roughly symmetric and bell-shaped.Statistics · Algebra 2, Statistics
  • End BehaviorNewEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x^3 + x rises on the left and falls on the right.Functions · Algebra 2, Precalculus
  • Equal SignNewThe equal sign (=) is the symbol that shows two expressions have the same value. In 3 + 5 = 2 + 6, both sides equal 8, so the statement is true. The equal sign does not mean "the answer comes next"; it works like a balanced scale, and a number sentence with = is called an equation.Symbols and Notation · Kindergarten, Grade 1 +
  • Equal ToNewEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 1/2 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1 +
  • EquationUpdatedAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7 +
  • Equation of a CircleNewThe equation of a circle with center (h, k) and radius r is (x − h)^2 + (y − k)^2 = r^2. It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x^2 + y^2 = r^2.Geometry · Geometry, Algebra 2 +
  • Equilateral TriangleNewAn equilateral triangle is a triangle whose three sides all have the same length. Its three angles are also equal, so each one measures 60 degrees. With side length a, its perimeter is 3a and its area is (square root of 3)/4a^2. Every equilateral triangle is also isosceles and acute.Geometry · Grade 4, Grade 5 +
  • Equivalent ExpressionsNewEquivalent expressions are expressions that have the same value for every value of the variable, even though they look different. For example, 3(x + 2) and 3x + 6 are equivalent: both equal 15 when x = 3 and 0 when x = −2. You build them with the properties of operations, such as the distributive property.Algebra · Grade 6, Grade 7 +
  • Equivalent FractionsUpdatedEquivalent fractions are fractions that name the same value, like 1/2, 2/4 and 3/6. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4 +
  • EstimateUpdatedAn estimate is an answer that is close to the exact one, and to estimate is to find an answer that is close enough, usually by rounding or using friendly numbers. For example, 49 × 21 is about 50 × 20 = 1000. Estimation helps you check whether an exact answer is reasonable.Problem Solving · Grade 3, Grade 4 +
  • Euler's NumberNewEuler's number, written e, is an irrational constant approximately equal to 2.71828. It is the number that ( 1 + 1/n )^n gets closer and closer to as n grows. It is the base of natural exponential growth, e^x, and of the natural logarithm, natural log x.Functions · Algebra 2, Precalculus +
  • EvaluateNewEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7 +
  • Even NumberNewAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3 +
  • EventNewAn event is a set of one or more outcomes of a chance experiment, such as "rolling an even number" on a number cube. It is part of the sample space. When outcomes are equally likely, the probability of an event is the number of outcomes in it divided by the number of outcomes in the sample space.Probability · Grade 7, Grade 8 +
  • Expanded FormNewExpanded form is a way of writing a number as the sum of the values of its digits. Each digit is multiplied by its place value, so 4072 = 4000 + 70 + 2, or 4 × 1000 + 7 × 10 + 2 × 1. It shows what every digit is worth. Zeros add nothing, so they are usually left out.Number System · Grade 2, Grade 3 +
  • ExponentUpdatedAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 5^3, the base is 5 and the exponent is 3, so 5^3 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7 +
  • Exponential DecayNewExponential decay is a decrease by the same percent in each equal time period, so the amount drops quickly at first and then more and more slowly. It is modeled by y = a(1 − r)^t, where a is the starting amount and r is the decay rate. The amount approaches 0 but never reaches it.Functions · Algebra 1, Algebra 2
  • Exponential FunctionNewAn exponential function is a function in which the input is the exponent: f(x) = a · b^x, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2 +
  • Exponential GrowthNewExponential growth is an increase by the same percent, or the same factor, in each equal time period, so the amount grows faster and faster. It is modeled by y = a(1 + r)^t, where a is the starting amount and r is the growth rate. Growing 10% a year multiplies the amount by 1.1 every year.Functions · Algebra 1, Algebra 2
  • ExpressionUpdatedAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7 +

F

  • Fact FamilyNewA fact family is a group of related math facts that use the same three numbers. With 6, 9 and 15 you get 6 + 9 = 15, 9 + 6 = 15, 15 − 6 = 9 and 15 − 9 = 6. Multiplication and division make fact families too, such as 4, 7 and 28. They show that operations undo each other.Operations · Grade 1, Grade 2 +
  • FactorUpdatedA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5 +
  • Factor CompletelyNewFactor completely means to write a polynomial as a product of factors that cannot be factored any further. First take out the greatest common factor, then factor what is left: 2x^2 − 18 = 2(x^2 − 9) = 2(x + 3)(x − 3). Stopping at 2(x^2 − 9) is not complete, because x^2 − 9 still factors.Algebra · Algebra 1, Algebra 2
  • Factor PairNewA factor pair is two whole numbers that multiply to give a certain number. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. Listing factor pairs is a sure way to find every factor of a number: start at 1 and stop once the pairs begin to repeat.Number System · Grade 4, Grade 5 +
  • FactorialNewThe factorial of a whole number n, written n!, is the product of all the whole numbers from n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. It counts the ways to arrange n different objects in a row. By agreement, 0! = 1.Probability · Algebra 2, Precalculus +
  • FactoringUpdatedFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x^2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7 +
  • FOIL MethodNewThe FOIL method is a memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms and the Last terms, then combine like terms. For (x + 2)(x + 5): x^2 + 5x + 2x + 10 = x^2 + 7x + 10. It is the distributive property applied twice.Algebra · Algebra 1
  • FootNewA foot is a US customary unit of length equal to 12 inches, about the length of a standard school ruler. Its abbreviation is ft, and its plural is feet. There are 3 feet in a yard and 5,280 feet in a mile, and 1 foot is exactly 30.48 centimeters.Measurement · Grade 2, Grade 3 +
  • FormulaUpdatedA formula is a rule written with symbols that shows how quantities are related. It is an equation with two or more variables, such as A = lw for the area of a rectangle. Formulas help solve problems the same way each time: substitute the values you know, then compute the one you need.Problem Solving · Grade 5, Grade 6 +
  • Formula SheetNewA formula sheet is a page of math formulas that a test or class gives you, so you do not have to memorize them: area and volume formulas, slope and the quadratic formula. The GED math test provides one with lines such as A = π r^2 and d = rt. You still have to choose the right formula and use it correctly.Problem Solving · Grade 7, Grade 8 +
  • FractionNewA fraction is a number that names equal parts of a whole, written a/b. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 3/4 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4 +
  • FunctionUpdatedA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1 +
  • Function NotationNewFunction notation is the way of writing a function as f(x), read "f of x": f names the function, x is the input, and f(x) is the output. Writing f(x) = 2x + 5 means the same as y = 2x + 5, and f(3) = 11 says that the input 3 gives the output 11.Functions · Grade 8, Algebra 1 +

G

  • Geometric SequenceNewA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is an = a1 · r^(n − 1). The terms grow when |r| > 1 and shrink when |r| < 1.Functions · Algebra 1, Algebra 2 +
  • GramNewA gram is a metric unit of mass, about the mass of a paper clip. Its symbol is g. There are 1,000 grams in a kilogram and 1,000 milligrams in a gram. Grams measure light things, such as food portions, coins and letters; heavier things are measured in kilograms.Measurement · Grade 3, Grade 4 +
  • GraphUpdatedA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6 +
  • Greater ThanNewGreater than means that one number is bigger than another, and its symbol is >. The statement 7 > 3 reads "seven is greater than three". The open side of the symbol faces the bigger number and the point aims at the smaller one. On a number line, the greater number is always farther to the right.Symbols and Notation · Grade 1, Grade 2 +
  • Greater Than or Equal ToNewGreater than or equal to means a value is either bigger than a number or exactly equal to it, and its symbol is ≥. The inequality x ≥ 3 is true for 3 and every number above it. It means "at least 3". On a number line it is graphed with a filled dot, because the endpoint is included.Symbols and Notation · Grade 6, Grade 7 +
  • Greatest Common FactorUpdatedThe greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.Number System · Grade 5, Grade 6 +

H

  • HeightNewHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5 +
  • HistogramUpdatedA histogram uses bars to show how numerical data are grouped into intervals. The data are split into equal-width intervals, called bins, and each bar's height is the count of values in that interval. The bars touch, because the intervals are continuous, which is what distinguishes a histogram from a bar graph.Statistics · Grade 6, Grade 7 +
  • HundredthsNewHundredths are the parts you get when one whole is split into 100 equal parts; one hundredth is 1/100 or 0.01. The hundredths place is the second digit to the right of the decimal point, so in 3.47 the 7 means 7 hundredths. One cent is one hundredth of a dollar.Number System · Grade 4, Grade 5 +
  • HypotenuseUpdatedThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a^2 + b^2 = c^2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1

I

  • Identity PropertyNewThe identity property says a number keeps its value when you add 0 or multiply by 1: a + 0 = a and a × 1 = a. So 0 is the additive identity and 1 is the multiplicative identity. For example, 15 + 0 = 15 and 15 × 1 = 15. It is how equivalent fractions work.Operations · Grade 1, Grade 3 +
  • Imaginary NumberNewAn imaginary number is a real number multiplied by i, the imaginary unit, where i = square root of −1 and i^2 = −1. Examples are 3i, −1/2i and square root of −25 = 5i. Imaginary numbers give square roots of negative numbers a value, so equations like x^2 = −9 have solutions: x = ± 3i.Number System · Algebra 2, Precalculus
  • Improper FractionUpdatedAn improper fraction has a numerator that is greater than or equal to its denominator, so its value is 1 or more. Examples are 7/4, 9/2 and 5/5. Every improper fraction can be rewritten as a whole number or a mixed number: 7/4 = 1 3/4. Improper does not mean wrong.Number System · Grade 4, Grade 5 +
  • In Terms OfNewIn terms of means the answer should be written using a given variable or symbol. "Write the perimeter in terms of w" wants an expression with w, such as 4w + 10. "Solve for y in terms of x" means get y alone, with only x on the other side. "In terms of π" keeps π as a symbol: 9π, not 28.26.Problem Solving · Grade 7, Grade 8 +
  • InchNewAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3 +
  • Independent EventsNewIndependent events are two events where the outcome of one does not change the probability of the other. Flipping heads on a coin does not affect rolling a 5 on a die. For independent events, multiply: P(A and B) = P(A) × P(B), so P(heads and 5) = 1/2 × 1/6 = 1/12.Probability · Grade 7, Algebra 1 +
  • Independent VariableUpdatedAn independent variable is an input value that can be chosen or changed. The dependent variable changes based on it. In y = 3x, x is the independent variable and y is the dependent one. On a graph the independent variable goes on the horizontal axis; in an experiment it is the factor you change on purpose.Functions · Grade 6, Grade 7 +
  • InequalityUpdatedAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7 +
  • InfinityNewInfinity is the idea of something without any end or bound, written with the symbol ∞. It is not a real number, so you cannot add, subtract or divide with it like an ordinary number. Math uses it for endless sets, quantities that grow without limit, and intervals such as (3, ∞).Symbols and Notation · Grade 6, Algebra 1 +
  • InputUpdatedAn input is the value you put into a function or a rule. The rule acts on the input to produce the output: for f(x) = x + 2, the input 5 gives the output 7. For many functions, the input is the x-value. In an input-output table, the inputs fill the first column.Functions · Grade 6, Grade 7 +
  • IntegerUpdatedAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 1/2 are not integers.Number System · Grade 6, Grade 7 +
  • IntegralNewAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral integral from a to b of f(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: integral of 2x dx = x^2 + C.Calculus · Calculus
  • Interior AngleNewAn interior angle is an angle inside a polygon, formed at a vertex by two sides that meet there. A polygon with n sides has n interior angles, and they add up to (n − 2) × 180 degrees. A triangle's interior angles total 180 degrees, and a rectangle's four right angles total 360 degrees.Geometry · Grade 7, Grade 8 +
  • Interquartile RangeNewThe interquartile range (IQR) is the spread of the middle half of a data set: the third quartile minus the first quartile, IQR = Q3 − Q1. Because it ignores the lowest and highest quarters of the data, outliers barely change it. In a box plot it is the length of the box.Statistics · Grade 6, Grade 7 +
  • Interval NotationNewInterval notation is a short way to write a set of numbers between two endpoints, using brackets and parentheses. A bracket, [ or ], means the endpoint is included; a parenthesis, ( or ), means it is not. So −2 ≤ x < 3 is written [−2, 3), and x > 4 is written (4, ∞).Symbols and Notation · Algebra 1, Algebra 2 +
  • Inverse FunctionNewAn inverse function undoes another function: if f sends a to b, its inverse f^(−1) sends b back to a. For f(x) = 2x + 3, the inverse is f^(−1)(x) = (x − 3)/2. Only one-to-one functions have inverses, and the graph of f^(−1) is the mirror image of f across y = x.Functions · Algebra 2, Precalculus
  • Inverse OperationsNewInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3 +
  • Inversely ProportionalNewInversely proportional describes two quantities whose product never changes: y = k/x, or xy = k, for a fixed nonzero number k. When one quantity doubles, the other is cut in half. On a 120-mile trip, speed and time are inversely proportional: 60 mph takes 2 hours and 30 mph takes 4 hours.Ratios and Proportions · Grade 8, Algebra 1 +
  • Irrational NumberUpdatedAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are square root of 2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1
  • Isosceles TriangleNewAn isosceles triangle is a triangle with at least two sides of equal length. The equal sides are called legs, and the third side is the base. The two angles touching the base, the base angles, are always equal. If the angle between the legs is 40 degrees, each base angle is 70 degrees.Geometry · Grade 4, Grade 5 +

J

  • Joint FrequencyUpdatedJoint frequency is the number of data values that fit two categories at the same time, such as students who are in Grade 7 and play an instrument. In a two-way table, a joint frequency is one of the inside cells, not the totals. Dividing it by the grand total gives the joint relative frequency.Statistics · Grade 8, Algebra 1

K

  • KilogramNewA kilogram is the base unit of mass in the metric system (SI), equal to 1,000 grams. Its symbol is kg. A liter of water has a mass of about 1 kilogram, and 1 kilogram is about 2.2 pounds. Kilograms measure heavier things, such as people, bags of flour or luggage.Measurement · Grade 3, Grade 4 +
  • KilometerUpdatedA kilometer (km) is a metric unit of length equal to 1,000 meters, roughly six tenths of a mile. It measures long distances, such as the length of a road race or the distance between towns. To change kilometers to meters, multiply by 1,000: 3 km is 3,000 m.Measurement · Grade 4, Grade 5 +

L

  • Law of CosinesNewThe law of cosines relates the three sides of any triangle to one of its angles: c^2 = a^2 + b^2 − 2ab cos C, where C is the angle across from side c. Use it to find the third side when you know two sides and the angle between them, or to find an angle when you know all three sides.Trigonometry · Geometry, Algebra 2 +
  • Law of SinesNewThe law of sines says that in any triangle, each side divided by the sine of its opposite angle gives the same number: a/(sin A) = b/(sin B) = c/(sin C). Use it to solve triangles that have no right angle when you know two angles and a side, or two sides and an angle opposite one of them.Trigonometry · Geometry, Algebra 2 +
  • Laws of ExponentsNewThe laws of exponents are the rules for simplifying powers that share a base or an exponent. Multiply powers of the same base by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them: x^2 · x^5 = x^7, (x^6)/(x^2) = x^4, (x^3)^2 = x^6.Algebra · Grade 8, Algebra 1 +
  • Leading CoefficientNewThe leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x^2 − 5x^4 + 2x, the highest power is x^4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph's ends point.Algebra · Algebra 1, Algebra 2
  • Least Common DenominatorNewThe least common denominator (LCD) of two or more fractions is the smallest number that every denominator divides into evenly. It is the least common multiple of the denominators. For 1/4 and 5/6, the LCD is 12, so you rewrite them as 3/12 and 10/12 before you add, subtract or compare.Number System · Grade 5, Grade 6 +
  • Least Common MultipleUpdatedThe least common multiple (LCM) is the smallest positive multiple shared by two or more numbers. The LCM of 6 and 9 is 18: it is the first number on both lists of multiples, 6, 12, 18, … and 9, 18, …. The LCM of the denominators is the least common denominator for adding fractions.Number System · Grade 5, Grade 6 +
  • LengthNewLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2 +
  • Less ThanNewLess than means smaller in value, and its symbol is <. The statement 4 < 9 reads "4 is less than 9." The pointed end of the symbol always faces the smaller number. On a number line, the smaller number is farther left, so −3 < 2 even though 3 is bigger than 2. Less than does not include equal.Symbols and Notation · Grade 1, Grade 2 +
  • Less Than or Equal ToNewLess than or equal to is a comparison meaning one value is smaller than another or exactly equal to it, written with the symbol ≤. The statement x ≤ 5 is true for 5 itself and for every number below 5, such as 4.9, 0 and −3. In words, ≤ means at most or no more than.Symbols and Notation · Grade 6, Grade 7 +
  • Like TermsUpdatedLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a^2 and −2a^2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7 +
  • LimitNewA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limit as x → a of f(x) = L. The function need not even be defined at a. For f(x) = (x^2 − 1)/(x − 1), f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, Calculus
  • LineNewA line is a perfectly straight path of points that goes on forever in both directions. It has no thickness and no endpoints, so it has no length you can measure. Through any two points there is exactly one line, which is named by those two points, as in line AB, or by one lowercase letter.Geometry · Grade 4, Grade 8 +
  • Line of Best FitNewA line of best fit is a straight line drawn through a scatter plot that shows the overall trend of the data as closely as possible. It passes through the middle of the points, with about as many above it as below. Its equation, y hat = mx + b, lets you predict one variable from the other.Statistics · Grade 8, Algebra 1 +
  • Line of SymmetryNewA line of symmetry is a line that divides a figure into two halves that are mirror images of each other. If you fold the figure along that line, the two halves match exactly. A square has 4 lines of symmetry, a rectangle has 2, and a parallelogram that is not a rectangle has none.Geometry · Grade 4, Grade 8 +
  • Line PlotUpdatedA line plot shows data values along a number line, often with marks above each value. Each X or dot stands for one data value, and repeated values stack in a column, so the tallest stack is the mode. Line plots make it easy to see clusters, gaps, outliers and the spread of a small data set.Statistics · Grade 3, Grade 4 +
  • Line SegmentNewA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB bar. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Geometry · Grade 4, Grade 5 +
  • Linear EquationUpdatedA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8 +
  • Linear FunctionNewA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1
  • LogarithmNewA logarithm is the exponent you need to raise a base to in order to get a given number. log base b of x = y means b^y = x, so log base 2 of 32 = 5 because 2^5 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, Precalculus
  • Long DivisionNewLong division is a step-by-step written method for dividing large numbers. You work through the dividend one digit at a time from the left, repeating four steps: divide, multiply, subtract, bring down. For 872 ÷ 4, the steps give the quotient 218 with a remainder of 0.Operations · Grade 4, Grade 5 +

M

  • Mathematical ModelUpdatedA mathematical model represents a real situation using numbers, equations, diagrams, or graphs, well enough to make predictions. The equation C = 15 + 9m can model the cost of a subscription after m months. Every model simplifies, and knowing what a model leaves out is as important as knowing what it includes.Problem Solving · Grade 6, Grade 7 +
  • MatrixNewA matrix is a rectangular array of numbers arranged in rows and columns, written inside square brackets. A matrix with m rows and n columns has dimensions m × n, read "m by n". Each number is an entry, named by its row and then its column: a23 is in row 2, column 3. The plural is matrices.Algebra · Algebra 2, Precalculus
  • MeanUpdatedThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.Statistics · Grade 5, Grade 6 +
  • Mean Absolute DeviationNewThe mean absolute deviation (MAD) is the average distance between each data value and the mean. Find the mean, find how far each value is from it, then average those distances. A small MAD means the data cluster close to the mean; a large MAD means they are spread out.Statistics · Grade 6, Grade 7
  • MedianUpdatedThe median is the middle value when data are listed in order. If there is an even number of values, the median is the average of the two middle values. Half the data lie at or below the median and half at or above it, and outliers barely move it.Statistics · Grade 5, Grade 6 +
  • Median of a TriangleNewA median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side. Every triangle has three medians, and they always meet at one point, the centroid. The centroid divides each median in a 2 : 1 ratio, with the longer part next to the vertex.Geometry · Geometry
  • MeterUpdatedA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4 +
  • Metric SystemNewThe metric system is a system of measurement in which units are related by powers of 10. Its base units are the meter for length, the gram for mass and the liter for capacity. Prefixes name larger and smaller units: kilo- means 1,000, centi- means one hundredth and milli- means one thousandth, so 1 km = 1,000 m.Measurement · Grade 4, Grade 5 +
  • MidpointNewA midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2).Geometry · Grade 8, Algebra 1 +
  • Midpoint FormulaNewThe midpoint formula gives the point halfway between (x1, y1) and (x2, y2): M = ((x1 + x2)/2, (y1 + y2)/2). You average the two x-coordinates and, separately, the two y-coordinates. For (2, 1) and (10, 7), the midpoint is (6, 4).Geometry · Grade 8, Algebra 1 +
  • MileNewA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5 +
  • MinuendNewThe minuend is the number you subtract from: the first number in a subtraction. In 17 − 6 = 11, the minuend is 17, the subtrahend is 6 and the difference is 11. The minuend is the whole you start with, so it equals the difference plus the subtrahend.Operations · Grade 1, Grade 2 +
  • Mixed NumberUpdatedA mixed number has a whole number and a fraction together, such as 2 3/4, which means 2 + 3/4. It represents a value greater than one whole when the whole-number part is positive. Every mixed number can be rewritten as an improper fraction, and back: 2 3/4 = 11/4.Number System · Grade 4, Grade 5 +
  • Mixture ProblemsNewMixture problems are word problems in which two or more ingredients with different values, prices or strengths are combined. Each ingredient adds amount × value to the total, and the parts add up: 3 lb of 4-dollar nuts plus 2 lb of 9-dollar nuts make 5 lb worth 12 + 18 = 30 dollars, which is 6 dollars per pound.Problem Solving · Algebra 1
  • ModeUpdatedThe mode is the value that appears most often in a data set. To find it, count how many times each value occurs and pick the most frequent one. A data set can have one mode, more than one mode, or no mode at all when every value appears only once. The mode works for categories, too.Statistics · Grade 5, Grade 6 +
  • MonomialNewA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x^2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Algebra · Grade 8, Algebra 1
  • MultipleUpdatedA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5 +
  • MultiplicationNewMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3 +
  • Multiplication TableNewA multiplication table is a grid that lists the products of pairs of numbers, usually 1 to 10 or 1 to 12. To read it, find one factor along the top row and the other down the left column; the box where they meet is the product. Row 6 meets column 7 at 42, since 6 × 7 = 42.Operations · Grade 2, Grade 3 +
  • Mutually Exclusive EventsNewMutually exclusive events are events that cannot happen at the same time, so P(A and B) = 0. Rolling a 2 and rolling a 5 on one roll of a number cube are mutually exclusive. For such events the addition rule is simple: P(A or B) = P(A) + P(B).Probability · Grade 7, Algebra 2 +

N

  • Natural LogarithmNewThe natural logarithm of a positive number x, written natural log x, is the power you must raise e ≈ 2.718 to in order to get x. So natural log x = y means e^y = x. For example, natural log 1 = 0, natural log e = 1, and natural log 20 ≈ 3.0 because e^3 ≈ 20.1.Functions · Algebra 2, Precalculus +
  • Natural NumberNewA natural number is one of the numbers 1, 2, 3, 4 and so on, used for counting and ordering. Natural numbers are positive and whole, so they include no fractions, decimals, negatives or zero in most US school math. Some books also include 0, so check the convention. The set is written ℕ.Number System · Grade 5, Grade 6 +
  • Negative ExponentNewA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a^(−n) = 1/(a^n) for a ≠ 0. So 2^(−3) = 1/8 and 10^(−2) = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1
  • Negative NumberNewA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −2/3. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7
  • No More ThanNewNo more than means less than or equal to, written ≤. The limit itself is allowed: you may spend no more than 20 dollars means 20 dollars or any smaller amount, so x ≤ 20. It means the same as at most. The reverse phrase, no less than, means greater than or equal to, ≥.Problem Solving · Grade 6, Grade 7 +
  • NonnegativeNewNonnegative means greater than or equal to zero: such a number is either 0 or positive, so x ≥ 0. The nonnegative integers are 0, 1, 2, 3 and so on. It is not the same as positive, because 0 is nonnegative but not positive. Distances, absolute values and principal square roots are always nonnegative.Problem Solving · Grade 6, Grade 7 +
  • Normal DistributionNewA normal distribution is a symmetric, bell-shaped distribution of data centered at the mean, where the mean, median and mode are equal. Its spread is set by the standard deviation σ. About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ. Heights and test scores are often close to normal.Statistics · Algebra 2, Statistics
  • NumberNewA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1 +
  • Number LineNewA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3 +
  • NumeratorUpdatedThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 3/5, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4 +

O

  • Obtuse AngleNewAn obtuse angle is an angle that measures more than 90 degrees and less than 180 degrees. It opens wider than a square corner but does not flatten into a straight line. Angles of 100, 135 and 179 degrees are obtuse. A triangle can have at most one obtuse angle.Geometry · Grade 4, Grade 5 +
  • Odd NumberNewAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3 +
  • OddsNewOdds compare the number of ways an event can happen with the number of ways it cannot, written as a ratio such as 3 to 2 or 3:2. Odds in favor are favorable : unfavorable, and odds against are the reverse. Probability instead divides favorable outcomes by all outcomes, so 3:2 odds mean a probability of 3/5.Probability · Grade 7, Statistics
  • OppositeUpdatedThe opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.Number System · Grade 6, Grade 7 +
  • Order of OperationsUpdatedThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6 +
  • Ordered PairUpdatedAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6 +
  • Ordinal NumberNewAn ordinal number tells the position of something in an ordered list: first, second, third, fourth and so on. It answers "Which place?" rather than "How many?". Ordinals are written as words or as a numeral with an ending: 1st, 2nd, 3rd, 4th, 11th, 21st, 22nd.Number System · Kindergarten, Grade 1 +
  • OriginNewThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6 +
  • OutcomeNewAn outcome is one possible result of a chance experiment, such as rolling a 4 on a number cube or getting tails on a coin flip. The set of all outcomes is the sample space, and a group of outcomes is an event. When outcomes are equally likely, each one has probability 1 divided by the number of outcomes.Probability · Grade 7, Grade 8 +
  • OutlierUpdatedAn outlier is a data value that is far away from most of the other values in a set. In 4, 5, 6, 7, 50, the value 50 is an outlier. A common test flags any value more than 1.5 times the interquartile range below the first quartile or above the third. Outliers pull the mean strongly.Statistics · Grade 6, Grade 7 +
  • OutputUpdatedAn output is the value a function or rule produces from a given input. You put a number in, apply the rule, and the result that comes out is the output. For y = 2x, the input 3 gives the output 6. In an input-output table the output sits in the second column.Functions · Grade 6, Grade 7 +

P

  • ParabolaNewA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x^2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2 +
  • Parallel LinesUpdatedParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5 +
  • ParallelogramNewA parallelogram is a four-sided shape whose opposite sides are parallel. Its opposite sides are also equal in length, its opposite angles are equal, and its diagonals cut each other in half. Rectangles, rhombuses and squares are special parallelograms. Its area is base times height: A = bh.Geometry · Grade 3, Grade 5 +
  • Parent FunctionNewA parent function is the simplest function of a family, before any shifts, stretches or reflections. For example, y = x^2 is the parent of every quadratic, and y = |x| is the parent of every absolute value function. Every other member of the family is a transformation of its parent's graph.Functions · Algebra 1, Algebra 2 +
  • PEMDASNewPEMDAS is an acronym for remembering the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Multiplication and division share one step and are done from left to right, and so are addition and subtraction. Many students learn it as "Please Excuse My Dear Aunt Sally."Algebra · Grade 5, Grade 6 +
  • PercentUpdatedA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25/100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Number System · Grade 5, Grade 6 +
  • Percent ChangeUpdatedPercent change measures how much a value grew or shrank relative to where it started: the difference divided by the original amount, written as a percent. From 50 to 60 is a 20% increase; from 80 to 60 is a 25% decrease. The original value is always the base, which is the step people get wrong.Ratios and Proportions · Grade 7, Grade 8 +
  • Percent DecreaseNewPercent decrease is how much a quantity went down, written as a percent of its original value. Subtract the new value from the original, divide by the original, and multiply by 100. A price that drops from 80 dollars to 60 dollars has a decrease of 20 dollars, and 20/80 = 25%.Ratios and Proportions · Grade 7, Grade 8 +
  • Percent IncreaseNewPercent increase is how much a quantity grew, written as a percent of its original value. Subtract the original from the new value, divide by the original, and multiply by 100: from 80 to 100 is 20/80 = 25%. To increase a number by r%, multiply it by 1 + r/100, so 80 × 1.25 = 100.Ratios and Proportions · Grade 7, Grade 8 +
  • PercentileNewA percentile tells what percent of the values in a data set are at or below a given value. If your test score is at the 80th percentile, about 80% of the scores are at or below yours. The 25th, 50th and 75th percentiles are the first quartile, the median and the third quartile.Statistics · Grade 7, Algebra 1 +
  • Perfect SquareNewA perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and square root of 49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.Number System · Grade 6, Grade 7 +
  • PerimeterUpdatedPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4 +
  • PeriodNewThe period of a repeating function is the horizontal length of one complete cycle: the smallest distance after which the graph repeats exactly. y = sin x and y = cos x have period 2π, or 360°, and y = tan x has period π. For y = sin (Bx) or y = cos (Bx), the period is 2π/(|B|).Trigonometry · Algebra 2, Precalculus
  • PermutationNewA permutation is an arrangement of objects in a specific order, so ABC and CBA count as different. The number of ways to arrange r objects chosen from n different objects is P(n, r) = (n!)/((n − r)!). For example, picking a president and vice president from 5 people gives 5 × 4 = 20 permutations.Probability · Algebra 2, Precalculus +
  • Perpendicular LinesUpdatedPerpendicular lines are lines that intersect to form right angles: their intersection makes a 90-degree angle. The symbol ⊥ means is perpendicular to. On a coordinate plane, two perpendicular lines that are not vertical have slopes that multiply to −1, such as 2 and −1/2.Geometry · Grade 4, Grade 5 +
  • PiNewPi (π) is the number you get when you divide any circle's circumference by its diameter. It is the same for every circle, about 3.14159. Pi is an irrational number: its decimal never ends or repeats, so we round it to 3.14 or use 22/7.Geometry · Grade 7, Grade 8 +
  • Piecewise FunctionNewA piecewise function is a function defined by two or more formulas, each used on its own interval of the domain. To find an output, first check which interval the input belongs to, then use that piece's formula. Absolute value, step functions and tiered prices are common examples.Functions · Algebra 1, Algebra 2 +
  • Place ValueUpdatedPlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4 +
  • PlaneNewA plane is a perfectly flat surface that has no thickness and extends forever in every direction, like a tabletop that never ends. It is one of the undefined terms of geometry, along with point and line. Any three points that are not on one line lie in exactly one plane.Geometry · Grade 7, Grade 8 +
  • PointNewA point is an exact location in space. It has no length, no width and no thickness, so it has no size at all. You draw a point as a dot and name it with a capital letter, such as point A. On a coordinate plane, an ordered pair such as (3, 2) names a point.Geometry · Grade 4, Grade 5 +
  • Point-Slope FormNewPoint-slope form is a way to write the equation of a line from its slope m and one point (x1, y1) on it: y − y1 = m(x − x1). It is the fastest form to use when you know a point and the slope, or two points. Multiplying it out gives slope-intercept form.Algebra · Grade 8, Algebra 1
  • PolygonNewA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3 +
  • Polygon Angle SumNewThe polygon angle sum is the total of the interior angles of a polygon. For a polygon with n sides it equals (n − 2) × 180°, because diagonals from one vertex cut the polygon into n − 2 triangles. A pentagon's angles add to 540°, and a hexagon's add to 720°.Geometry · Grade 8, Geometry
  • PolynomialUpdatedA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x^2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2
  • PopulationNewA population in statistics is the whole group of people, animals or objects you want to learn about, such as every student in a school or every bulb a factory makes. Measuring all of them is often too costly, so you study a sample, a part of the population, and use it to draw conclusions about the whole group.Statistics · Grade 7, Statistics +
  • Positive IntegerNewA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ^+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7 +
  • PowerNewA power is a number written as a base with an exponent, such as 5^3, or the value that results, 5^3 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or "5 to the third power". The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6 +
  • Power RuleNewThe power rule is the shortcut for the derivative of a power of x: if f(x) = x^n, then f′(x) = nx^(n − 1). Bring the exponent down as a coefficient and subtract 1 from it, so x^5 becomes 5x^4. It works for any real exponent, including negative and fractional ones.Calculus · Calculus
  • Prime FactorizationNewPrime factorization is writing a whole number as a product of prime numbers only, such as 72 = 2 × 2 × 2 × 3 × 3 = 2^3 × 3^2. Find it with a factor tree or repeated division. Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors.Number System · Grade 4, Grade 5 +
  • Prime NumberUpdatedA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5 +
  • PrincipalNewThe principal is the original amount of money that is borrowed, deposited or invested, before any interest is added. In the simple interest formula I = Prt, the letter P stands for the principal. Deposit 2,000 dollars at 3% simple interest for 4 years: the principal is 2,000 dollars, the interest is 240 dollars, and the total is 2,240 dollars.Problem Solving · Grade 7, Algebra 1
  • ProbabilityUpdatedProbability measures how likely an event is, from 0 for impossible to 1 for certain. For equally likely outcomes it is the number of favorable outcomes divided by the total number of outcomes: for a fair coin, the probability of heads is 1/2. Probabilities can be written as fractions, decimals or percents.Probability · Grade 6, Grade 7 +
  • ProductNewThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words "product of" always mean multiply, so "the product of 3 and 7" is 21.Operations · Grade 3, Grade 4 +
  • ProofNewA proof is a step-by-step logical argument that shows a math statement is true in every case it covers. It starts from given facts and uses definitions, postulates and statements already proved, with a reason for each step. A statement that has been proved is called a theorem.Geometry · Geometry, Algebra 1
  • Proper FractionNewA proper fraction is a fraction whose numerator is smaller than its denominator, such as 3/4 or 2/9. Because fewer parts are taken than make one whole, a positive proper fraction is always less than 1. If the numerator is equal to or greater than the denominator, the fraction is improper instead.Number System · Grade 3, Grade 4 +
  • Properties of LogarithmsNewThe properties of logarithms are rules for rewriting logs: the log of a product is a sum, log base b of (MN) = log base b of M + log base b of N; the log of a quotient is a difference; and the log of a power is a multiple, log base b of (M^p) = p log base b of M. They come from the rules of exponents.Functions · Algebra 2, Precalculus
  • ProportionUpdatedA proportion is an equation that says two ratios are equal, such as 1/2 = 3/6. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7 +
  • Proportional RelationshipNewA proportional relationship is a relationship between two quantities in which one is always the same multiple of the other, so y = kx for a fixed number k. The ratio y/x is the same for every pair, and the graph is a straight line through the origin. Example: at 3 dollars per pound, cost = 3 × pounds.Ratios and Proportions · Grade 7, Grade 8 +
  • PyramidNewA pyramid is a solid figure with one polygon as its base and triangular faces that rise from the base edges and meet at a single top point, the apex. A pyramid is named by its base, such as a square pyramid. Its volume is one third of the base area times the height: V = 1/3Bh.Geometry · Grade 6, Grade 7 +
  • Pythagorean TheoremUpdatedThe Pythagorean theorem says that in a right triangle, a^2 + b^2 = c^2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1

Q

  • QuadrantNewA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point's coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6 +
  • Quadratic EquationUpdatedA quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2
  • Quadratic FormulaNewThe quadratic formula gives the solutions of any quadratic equation ax^2 + bx + c = 0 with a ≠ 0: x = (−b ± square root of (b^2 − 4ac))/2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra · Algebra 1, Algebra 2
  • Quadratic FunctionNewA quadratic function is a function that can be written f(x) = ax^2 + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola, which opens up when a > 0 and down when a < 0. Its highest or lowest point is the vertex.Functions · Algebra 1, Algebra 2 +
  • QuadrilateralNewA quadrilateral is a polygon with four straight sides, four vertices and four angles. Its four interior angles always add to 360 degrees. Squares, rectangles, rhombuses, parallelograms, trapezoids and kites are special quadrilaterals, each named for extra properties of its sides or angles.Geometry · Grade 2, Grade 3 +
  • QuartileNewA quartile is one of three values that split an ordered data set into four parts with about the same number of values. The first quartile, Q1, is the median of the lower half; Q2 is the median of all the data; Q3 is the median of the upper half. The interquartile range is Q3 − Q1.Statistics · Grade 6, Grade 7 +
  • QuotientNewA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4 +

R

  • RadianNewA radian is a unit for measuring angles: one radian is the central angle that cuts off an arc exactly as long as the circle's radius. A full turn is 2π radians, so π radians = 180° and 1 radian is about 57.3°. To change degrees to radians, multiply by π/180.Trigonometry · Geometry, Algebra 2 +
  • RadicalNewA radical is an expression that uses the root symbol square root of (), such as square root of 25 or cube root of x. The number under the symbol is the radicand, and the small number in the notch is the index, which tells which root to take. With no index, it is a square root: square root of 25 = 5.Number System · Grade 8, Algebra 1 +
  • RadiusUpdatedThe radius is a line segment from the center of a circle to any point on the circle, or the length of that segment. The radius is half the diameter: if the diameter is 10, the radius is 5. All radii in the same circle have equal length, and r is the radius in C = 2π r.Geometry · Grade 6, Grade 7 +
  • RangeUpdatedThe range of a data set is the difference between the greatest and least values: maximum minus minimum. For 2, 5, 9, the range is 9 − 2 = 7. Range is a simple measure of spread: it tells how far apart the extreme values are, though one outlier can make it misleading about the rest of the data.Statistics · Grade 5, Grade 6 +
  • Range of a FunctionNewThe range of a function is the set of all output values (usually y-values) the function actually produces from its domain. For f(x) = x^2, every output is 0 or more, so the range is y ≥ 0. On a graph, the range is how far the graph reaches down and up.Functions · Grade 8, Algebra 1 +
  • RateNewA rate is a ratio that compares two quantities measured in different units, such as 150 miles in 3 hours or 12 dollars for 4 pounds. The word "per" signals a rate. Dividing gives the unit rate, the amount for 1 unit: 150 miles in 3 hours is 50 miles per hour.Ratios and Proportions · Grade 6, Grade 7
  • Rate of ChangeNewA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Functions · Grade 8, Algebra 1 +
  • RatioUpdatedA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 3/5. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7 +
  • Rational ExponentNewA rational exponent is an exponent written as a fraction, and it stands for a root. The denominator is the root and the numerator is the power: a^(1/n) = nth root of a and a^(m/n) = ( nth root of a )^m. For example, 9^(1/2) = 3 and 27^(2/3) = 3^2 = 9.Algebra · Algebra 1, Algebra 2
  • Rational ExpressionNewA rational expression is a fraction whose numerator and denominator are polynomials, such as (x + 3)/(x − 2) or 5/(x^2). The denominator cannot equal 0, so values that make it 0 are excluded. You simplify, add, subtract, multiply and divide rational expressions the same way you work with numerical fractions.Functions · Algebra 1, Algebra 2
  • Rational FunctionNewA rational function is a function that can be written as a fraction of two polynomials, f(x) = (p(x))/(q(x)), where q(x) is not the zero polynomial. It is undefined wherever q(x) = 0, so its graph often has vertical asymptotes or holes. Examples: f(x) = 1/x and g(x) = (x + 2)/(x − 5).Functions · Algebra 2, Precalculus
  • Rational NumberUpdatedA rational number is any number that can be written as a fraction of two integers, a/b, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 3/4, −5 = −5/1, 1.6 and 0.333… all qualify. Numbers such as square root of 2 and π are irrational.Number System · Grade 7, Grade 8 +
  • RayNewA ray is part of a line that starts at one endpoint and goes on forever in one direction, like a beam of light from a flashlight. It is named by its endpoint first and then any other point on it, so ray AB starts at A and passes through B. Two rays sharing an endpoint form an angle.Geometry · Grade 4, Geometry
  • Real NumberNewA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as square root of 2 and π). Square roots of negative numbers, like square root of −4, are not real.Number System · Grade 8, Algebra 1
  • Real SolutionNewA real solution of an equation is a solution that is a real number, so it lies on the number line, such as x = 3 or x = −square root of 2. Some equations have none: x^2 + 4 = 0 has only imaginary solutions. For a quadratic, the discriminant b^2 − 4ac tells how many there are: 2, 1 or 0.Algebra · Algebra 1, Algebra 2
  • ReasonablenessUpdatedReasonableness is checking whether an answer makes sense for the problem before you accept it. Estimate first to see about how big the answer should be, then check the units and the context: a negative length, 3.5 people or 800 dollars for one pencil is not reasonable. A correctly calculated number can still fail this check.Problem Solving · Grade 3, Grade 4 +
  • ReciprocalNewThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 3/4 is 4/3, and the reciprocal of 5 is 1/5. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6 +
  • Rectangular PrismNewA rectangular prism is a solid shape with six rectangular faces, where opposite faces match and every corner is a right angle. A shoebox and a brick are rectangular prisms. It has 12 edges and 8 vertices, and its volume is length times width times height: V = lwh.Geometry · Grade 5, Grade 6 +
  • Recursive FormulaNewA recursive formula defines a sequence by giving its first term and a rule that builds each next term from the term or terms before it. For example, a1 = 5 and an = an − 1 + 4 make 5, 9, 13, 17. To find a later term, you compute every term before it, one step at a time.Functions · Algebra 1, Algebra 2
  • ReflectionNewA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, Geometry
  • RegroupingNewRegrouping is trading 10 of one place value for 1 of the next larger place, or 1 for 10 of the next smaller place, so that column addition or subtraction works. In 47 + 38, the 15 ones become 1 ten and 5 ones; in 52 − 27, 1 ten becomes 10 ones. It is also called carrying and borrowing.Operations · Grade 1, Grade 2 +
  • Regular PolygonNewA regular polygon is a polygon whose sides are all the same length and whose angles are all the same size. The equilateral triangle, the square and the stop-sign octagon are regular. Each interior angle of a regular n-gon is ((n − 2) × 180°)/n, and each exterior angle is (360°)/n.Geometry · Grade 5, Grade 8 +
  • RelationNewA relation is any set of ordered pairs that links inputs to outputs, such as {(1, 4), (2, 5), (2, 7)}. It can be shown as a list, table, mapping diagram or graph. A relation in which every input has exactly one output is a function; this one is not, because 2 has two outputs.Functions · Grade 8, Algebra 1
  • Relative FrequencyNewRelative frequency is how often a value or event occurs compared with the total number of observations: frequency ÷ total. It can be a fraction, a decimal or a percent. If 12 of 40 students choose apples, the relative frequency of apples is 12/40 = 0.3, or 30%. All the relative frequencies in a table add to 1.Statistics · Grade 7, Grade 8 +
  • RemainderNewA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5 +
  • Remainder TheoremNewThe Remainder Theorem says that when a polynomial p(x) is divided by x − a, the remainder equals p(a). So substituting x = a replaces the division. For p(x) = x^2 + 1 divided by x − 3, the remainder is p(3) = 10. A remainder of 0 means x − a is a factor.Algebra · Algebra 2, Precalculus
  • Repeated AdditionNewRepeated addition is adding the same number again and again. It is one way to understand multiplication: 3 + 3 + 3 + 3 = 4 × 3 = 12. The number of times you add is one factor, and the number being added is the other. It only works when all the addends are equal.Operations · Grade 2, Grade 3
  • RhombusNewA rhombus is a quadrilateral with four sides of equal length. It is a special parallelogram, so its opposite sides are parallel and its opposite angles are equal. Its diagonals cross at right angles and cut each other in half, and its area is half the product of the diagonals.Geometry · Grade 3, Grade 4 +
  • Right AngleNewA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3 +
  • Right TriangleNewA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the pythagorean theorem: a^2 + b^2 = c^2.Geometry · Grade 4, Grade 6 +
  • RootsNewThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x^2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2
  • RotationNewA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, Geometry
  • Rounded to the NearestNewRounded to the nearest tells you which place value to keep when you round an answer. Nearest tenth keeps one decimal place; nearest whole number keeps none. Look at the next digit to the right: 5 or more rounds up, 4 or less rounds down. So 3.14159 becomes 3.14 to the nearest hundredth.Problem Solving · Grade 3, Grade 4 +
  • RoundingUpdatedRounding replaces a number with a nearby number that is easier to use, to a chosen place value. Look at the digit immediately to the right of that place: 5 or more rounds up, 4 or less rounds down. So 47 rounds to 50 to the nearest ten, and 3.68 rounds to 3.7 to the nearest tenth.Number System · Grade 3, Grade 4 +

S

  • SampleNewA sample in statistics is a part of a population that is actually observed or measured, such as 60 students surveyed out of a school of 900. You use the sample to estimate facts about the whole population. A random sample, where every member has the same chance to be chosen, tends to represent the population fairly.Statistics · Grade 7, Statistics +
  • Sample SpaceNewThe sample space of a chance experiment is the set of all its possible outcomes, usually written S. For one coin flip, S = {H, T}; for one number cube, S = {1, 2, 3, 4, 5, 6}. Organized lists, tables and tree diagrams help you write it out, and the probabilities of all its outcomes add to 1.Probability · Grade 7, Grade 8 +
  • Sampling BiasNewSampling bias happens when the way a sample is chosen makes some members of the population more likely to be picked than others, so the sample does not represent the population. Its results lean the same wrong way every time. Surveying only gym-goers about exercise, or counting only volunteers who reply, are classic examples.Statistics · Grade 7, Algebra 1 +
  • Scale DrawingNewA scale drawing is a drawing of a real object in which every length is multiplied by the same number, so it has the same shape but a different size. Its scale, such as 1 cm = 2 m, links a length on the drawing to the actual length. Maps, floor plans and blueprints are scale drawings.Ratios and Proportions · Grade 7
  • Scale FactorUpdatedA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 1/2 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · Grade 7, Grade 8 +
  • Scalene TriangleNewA scalene triangle is a triangle whose three sides all have different lengths. Because no two sides match, no two of its angles are equal either. A scalene triangle can be acute, right or obtuse: the 3-4-5 right triangle is scalene, and so is a triangle with sides 4, 6 and 7.Geometry · Grade 4, Grade 5 +
  • Scatter PlotUpdatedA scatter plot displays paired numerical data as points on a coordinate plane, one point for each pair. It helps show whether two quantities have a pattern or relationship: points rising from left to right show a positive association, points falling show a negative association, and a shapeless cloud shows no association.Statistics · Grade 8, Algebra 1
  • Scientific NotationUpdatedScientific notation writes a number as a value from 1 to less than 10 times a power of 10, such as 5.4 × 10^3 for 5,400. The exponent records how far the decimal point moved: positive for large numbers and negative for numbers between 0 and 1, as in 0.0062 = 6.2 × 10^(−3).Number System · Grade 8, Algebra 1
  • SecantNewSecant (sec) is the trigonometric function equal to 1 divided by cosine: sec θ = 1/(cos θ). In a right triangle it is the hypotenuse divided by the side adjacent to the angle. Secant is undefined wherever cosine is 0, such as at 90°, and its values are always at least 1 or at most −1.Trigonometry · Algebra 2, Precalculus +
  • SequenceNewA sequence is an ordered list of numbers that follows a rule, such as 3, 7, 11, 15, …. Each number is a term, and its position is 1st, 2nd, 3rd and so on. You can describe a sequence with a formula for the nth term, an, or by how each term comes from the one before it.Functions · Grade 8, Algebra 1 +
  • SeriesNewA series is the sum of the terms of a sequence. The sequence 3, 7, 11, 15 is a list; the series 3 + 7 + 11 + 15 = 36 is what you get by adding it. A finite series stops after n terms, written Sn. An infinite series has a sum only if its partial sums approach a fixed number.Functions · Algebra 2, Precalculus +
  • SetNewA set is a well-defined collection of distinct objects, called elements or members. Sets are written with curly braces and named with capital letters, as in A = {2, 4, 6}. Order does not matter and repeated items count once, so {1, 2, 3} and {3, 1, 2} are the same set.Probability · Grade 6, Algebra 1 +
  • SideNewA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1 +
  • Sigma NotationNewSigma notation is a compact way to write a long sum using the Greek capital letter Σ (sigma). In sum from k=1 to 4 of k^2, the index k starts at the number below sigma, stops at the number above it, and each value is put into the rule k^2: 1 + 4 + 9 + 16 = 30.Functions · Algebra 2, Precalculus +
  • Similar FiguresUpdatedSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8 +
  • Simple InterestNewSimple interest is interest paid only on the original amount of money, called the principal, at a fixed yearly rate. Find it with I = Prt: multiply the principal by the rate written as a decimal and by the time in years. The interest is the same every year, so the balance grows in a straight line.Ratios and Proportions · Grade 7, Algebra 1
  • Simplest FormNewSimplest form is the way of writing a fraction so its numerator and denominator have no common factor other than 1. To get there, divide the top and bottom by their greatest common factor: 6/8 = (6 ÷ 2)/(8 ÷ 2) = 3/4. The value stays the same; only the numbers get smaller. It is also called lowest terms.Number System · Grade 4, Grade 5 +
  • SimplifyNewSimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 18/24 = 3/4. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5 +
  • SineNewSine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = opposite/hypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.Trigonometry · Geometry, Algebra 2 +
  • Skip CountingNewSkip counting is counting forward or backward by the same number each time instead of by 1, such as 2, 4, 6, 8 or 5, 10, 15, 20. Each jump adds the same amount, so skip counting by 5 six times reaches 6 × 5 = 30. It is a quick way to count groups and learn multiplication facts.Number System · Kindergarten, Grade 1 +
  • SlopeUpdatedSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1
  • Slope-Intercept FormUpdatedSlope-intercept form is the linear equation form y = mx + b, where m is the slope and b is the y intercept. You can read both values directly from the equation without rearranging: in y = 2x + 5, the slope is 2 and the line crosses the y-axis at (0, 5).Algebra · Grade 8, Algebra 1
  • SOH CAH TOANewSOH CAH TOA is a memory aid for the three basic trigonometric ratios in a right triangle: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent. Name the sides from the angle you are using, choose the ratio that links the sides you know and need, then solve.Trigonometry · Geometry, Algebra 2 +
  • SolutionUpdatedA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7 +
  • Special Right TrianglesNewSpecial right triangles are the two right triangles whose angles are 45°-45°-90° or 30°-60°-90°. Their sides always come in fixed ratios: x : x : xsquare root of 2 for the 45-45-90 triangle and x : xsquare root of 3 : 2x for the 30-60-90 triangle. From one side you can find the other two without the pythagorean theorem.Geometry · Grade 8, Geometry +
  • SphereNewA sphere is a perfectly round solid shape, like a ball, in which every point on the surface is the same distance from the center. That distance is the radius. A sphere has no faces, edges or vertices. Its volume is V = 4/3π r^3 and its surface area is 4π r^2.Geometry · Kindergarten, Grade 8 +
  • SquareNewA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s^2, and each diagonal is ssquare root of 2.Geometry · Kindergarten, Grade 2 +
  • Square NumberNewA square number is the result of multiplying a whole number by itself. The first square numbers are 1, 4, 9, 16, 25 and 36, because 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on. That many dots fill a square exactly, like 16 dots in a 4 by 4 square.Operations · Grade 3, Grade 4 +
  • Square RootUpdatedA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1
  • Standard AlgorithmNewThe standard algorithm is the usual written column method for adding, subtracting, multiplying or dividing multi-digit numbers. You line up digits by place value, work one place at a time, and regroup (carry or borrow) when a place overflows or runs short. For 478 + 365 it gives 843 in three short steps.Operations · Grade 4, Grade 5 +
  • Standard DeviationNewStandard deviation is a number that measures how spread out data values are from their mean: roughly, the typical distance of a value from the mean. A small standard deviation means the values cluster near the mean; a large one means they are spread out. It is the square root of the variance.Statistics · Algebra 1, Statistics
  • Standard FormNewStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3 +
  • Standard Form of a Linear EquationNewThe standard form of a linear equation is Ax + By = C, where A, B and C are integers and A and B are not both 0. Both variables sit on the left side. It makes the intercepts easy to find: the x-intercept is C/A, the y-intercept is C/B, and the slope is −A/B.Algebra · Grade 8, Algebra 1
  • Standard Form of a QuadraticNewThe standard form of a quadratic is ax^2 + bx + c, with the terms in descending powers of x and a ≠ 0. As an equation it is ax^2 + bx + c = 0; as a function, y = ax^2 + bx + c. It shows a, b and c directly, which the quadratic formula needs, and c is the y-intercept.Algebra · Algebra 1, Algebra 2
  • StatisticsNewStatistics is the branch of math that collects, organizes, analyzes and interprets data to answer questions. Descriptive statistics summarize data with graphs and numbers such as the mean and median. Inferential statistics use a sample to draw conclusions about a whole population, with a stated amount of uncertainty.Statistics · Grade 6, Grade 7 +
  • SubsetNewA subset is a set whose elements all belong to another set. If every element of A is also in B, then A is a subset of B, written A ⊆ B. Every set is a subset of itself, the empty set is a subset of every set, and a set with n elements has 2^n subsets.Probability · Algebra 1, Statistics
  • SubstituteNewSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7 +
  • Substitution MethodUpdatedThe substitution method solves a system of equations by replacing one variable with an equivalent expression. Solve one equation for a variable, put that expression into the other equation, and solve the new one-variable equation. Then substitute back to find the other variable. It is easiest when one equation is already solved for a variable.Algebra · Grade 8, Algebra 1
  • SubtractionNewSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2 +
  • SubtrahendNewThe subtrahend is the number being subtracted: the number written after the minus sign. In 12 − 5 = 7, the subtrahend is 5. It is taken away from the minuend (12) to leave the difference (7). To find a missing subtrahend, subtract the difference from the minuend: 12 − 7 = 5.Operations · Grade 1, Grade 2 +
  • SumNewA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2 +
  • Supplementary AnglesNewSupplementary angles are two angles whose measures add up to 180 degrees. Each one is the supplement of the other, so the supplement of an angle x is 180° − x. When they share a side, they form a straight line, called a linear pair. For example, 118° and 62° are supplementary.Geometry · Grade 7, Grade 8 +
  • Surface AreaUpdatedSurface area is the total area of all outside faces of a three-dimensional figure, measured in square units. To find it, unfold the solid into a flat net, find the area of each face and add them. A box with length l, width w and height h has surface area 2lw + 2lh + 2wh.Geometry · Grade 6, Grade 7 +
  • SymmetryNewSymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8 +
  • Synthetic DivisionNewSynthetic division is a shortcut for dividing a polynomial by a linear factor of the form x − k. You write only the coefficients, bring down the first one, then repeat two moves: multiply by k and add. The bottom row gives the quotient's coefficients, and its last number is the remainder, which equals the polynomial's value at k.Algebra · Algebra 2, Precalculus
  • System of EquationsUpdatedA system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.Algebra · Grade 8, Algebra 1

T

  • Tally MarkNewA tally mark is a short line drawn to record one item as you count. Marks are grouped in fives: four vertical lines with a fifth line drawn across them, so a group is easy to see and count by 5s. Tally marks are used to collect data quickly, as in a tally chart of votes.Statistics · Kindergarten, Grade 1 +
  • TangentNewTangent (tan) is the trigonometric ratio of the side opposite an acute angle to the side adjacent to it in a right triangle: tan θ = opposite/adjacent. For legs 3 and 4, tan θ = 3/4 = 0.75. It is the TOA in SOH CAH TOA, and it equals (sin θ)/(cos θ).Trigonometry · Geometry, Algebra 2 +
  • TenthsNewTenths are the parts you get when one whole is split into 10 equal parts; each part is one tenth, written 1/10 or 0.1. The tenths place is the first digit to the right of the decimal point, so in 4.7 the 7 means 7 tenths, or 0.7.Number System · Grade 3, Grade 4 +
  • TermNewA term is a single number, a variable, or numbers and variables multiplied together, such as 7, x or −4xy^2. In an expression, terms are the parts separated by plus and minus signs, so 5x^2 − 3x + 8 has three terms. The sign in front of a term belongs to it.Algebra · Grade 6, Grade 7 +
  • TheoremNewA theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.Geometry · Grade 8, Geometry
  • TransformationUpdatedA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8 +
  • Transformation of FunctionsNewTransformation of functions is changing a function's rule so that its graph moves, flips or resizes. Adding k outside, f(x) + k, shifts the graph up; replacing x with x − h shifts it right; −f(x) reflects it over the x-axis; and a · f(x) stretches it vertically. The basic graph being changed is the parent function.Functions · Algebra 1, Algebra 2 +
  • Translating Words into MathNewTranslating words into math means turning a sentence into an expression, equation or inequality by matching words to symbols: sum means +, difference −, product ×, quotient ÷, and is means =. For example, 5 less than twice a number becomes 2n − 5. Watch the order, because less than and subtracted from reverse it.Problem Solving · Grade 6, Grade 7 +
  • TranslationNewA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, Geometry
  • TransversalNewA transversal is a line that crosses two or more other lines at different points. Crossing two lines makes eight angles, four at each intersection. When the two lines are parallel, those angles come in equal groups: corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to 180 degrees.Geometry · Grade 8, Geometry
  • TrapezoidNewA trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases, and the other two sides are the legs. Its area is the average of the bases times the height: A = (b1 + b2)/2 × h. In British English it is called a trapezium.Geometry · Grade 3, Grade 5 +
  • TriangleUpdatedA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4 +
  • Triangle Inequality TheoremNewThe triangle inequality theorem says that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side. So 3, 4 and 5 can form a triangle, but 2, 3 and 6 cannot, because 2 + 3 < 6. Shortcut: the two shorter sides must add to more than the longest.Geometry · Grade 7, Geometry
  • TrigonometryNewTrigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.Trigonometry · Geometry, Algebra 2 +
  • TrinomialNewA trinomial is a polynomial with exactly three terms, such as x^2 + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax^2 + bx + c, which students learn to factor into two binomial factors.Algebra · Algebra 1, Algebra 2
  • Two-Column ProofNewA two-column proof is a way of writing a proof as a table: numbered statements in the left column and the reason for each one in the right column. It begins with the given facts and ends with the statement to be proved. Every reason must be a given, a definition, a postulate, a property or a proved theorem.Geometry · Geometry
  • Two-Way TableUpdatedA two-way table shows counts for two categorical variables collected from the same group, with one variable in the rows and the other in the columns. Each inside cell is a joint frequency, and the totals in the margins are marginal frequencies. Dividing a cell by a row or column total gives relative frequencies that reveal associations.Statistics · Grade 8, Algebra 1

U

  • UndefinedNewUndefined describes an expression that has no value, most often because it divides by zero. 7/0 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3/(x − 2) is undefined at x = 2. Undefined does not mean zero: 0/7 = 0, but 7/0 has no value.Problem Solving · Grade 6, Grade 8 +
  • Unit CircleNewThe unit circle is the circle with radius 1 centered at the origin, x^2 + y^2 = 1. An angle θ measured from the positive x-axis meets the circle at the point (cos θ, sin θ). This lets sine and cosine work for every angle, including obtuse and negative angles, not just the acute angles of right triangles.Trigonometry · Algebra 2, Precalculus
  • Unit ConversionNewUnit conversion is rewriting a measurement in a different unit without changing the amount, such as writing 3 feet as 36 inches. You multiply by a conversion factor, a fraction equal to 1 such as (12 in)/(1 ft). Converting to a smaller unit gives a larger number; converting to a larger unit gives a smaller number.Measurement · Grade 4, Grade 5 +
  • Unit FractionNewA unit fraction is a fraction with a numerator of 1, such as 1/2, 1/5 or 1/12. It names one equal part of a whole split into that many parts. The bigger the denominator, the smaller the unit fraction, and every fraction a/b is a copies of 1/b.Number System · Grade 3, Grade 4 +
  • Unit PriceUpdatedA unit price is the cost of one unit of something, found by dividing the total price by the number of units. For example, 18 dollars for 6 pounds is 18 ÷ 6 = 3 dollars per pound. It is the fairest way to compare products sold in different package sizes: the lower unit price costs less per unit.Ratios and Proportions · Grade 6, Grade 7 +
  • Unit RateUpdatedA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7 +

V

  • VariableUpdatedA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7 +
  • VarianceNewVariance is a measure of spread: the average of the squared distances of the data values from their mean. Find each value's distance from the mean, square it, add the squares, and divide by n for a population (or n − 1 for a sample). Larger variance means more spread. Its square root is the standard deviation.Statistics · Algebra 1, Statistics
  • VectorNewA vector is a quantity that has both a size, called its magnitude, and a direction. It is drawn as an arrow from an initial point to a terminal point and written in components, such as v = ⟨ 4, 3 ⟩: 4 units right and 3 units up. Velocity and force are vectors; speed and mass are not.Trigonometry · Precalculus
  • Venn DiagramNewA Venn diagram is a picture that uses overlapping circles inside a rectangle to show how groups, or sets, are related. Each circle is a set, the overlap holds items in both sets, and the rectangle holds everything being considered. Venn diagrams make it easy to count "and", "or" and "not" cases and to find probabilities.Probability · Grade 7, Algebra 2 +
  • VertexUpdatedA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5 +
  • Vertex FormNewVertex form is a way to write a quadratic function as y = a(x − h)^2 + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)^2 + 1 has its vertex at (3, 1).Algebra · Algebra 1, Algebra 2
  • Vertex of a ParabolaNewThe vertex of a parabola is its turning point: the lowest point when the parabola opens up, or the highest point when it opens down. For y = ax^2 + bx + c, the vertex has x-coordinate x = −b/2a; substitute that value to get the y-coordinate. The vertex sits on the axis of symmetry.Algebra · Algebra 1, Algebra 2
  • Vertical AnglesNewVertical angles are the pair of opposite angles formed when two lines intersect. They share a vertex but no sides, and they are always equal in measure. Two crossing lines make two pairs of vertical angles: if one angle is 120°, the angle across from it is also 120°.Geometry · Grade 7, Grade 8 +
  • Vertical Line TestNewThe vertical line test is a way to tell from a graph whether it shows a function: if any vertical line crosses the graph more than once, it is not a function. If every vertical line crosses it at most once, each input x has only one output y, so the graph is a function.Functions · Grade 8, Algebra 1 +
  • VolumeUpdatedVolume measures the amount of space inside a three-dimensional figure, in cubic units such as cubic centimeters or cubic feet. You can find it by counting the unit cubes that fill the solid. For a rectangular prism, V = l × w × h; for any prism, it equals the area of the base multiplied by the height.Geometry · Grade 5, Grade 6 +

W

  • Weighted AverageNewA weighted average is an average in which some values count more than others. Multiply each value by its weight, add those products, and divide by the sum of the weights. Course grades, GPAs and combined groups use it, because a plain mean would treat every value as equally important.Statistics · Grade 7, Grade 8 +
  • Whole NumberUpdatedA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 3/4 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4 +
  • WidthUpdatedWidth is the distance across a shape or object from side to side. In a rectangle, width is one of the two side measurements used with length, usually the shorter one, and multiplying length by width gives the area of a rectangle. Width is measured in length units such as inches, feet, centimeters or meters.Measurement · Grade 3, Grade 4 +
  • Word FormNewWord form is a way of writing a number using words instead of digits, as you would say it aloud: 4,306 is written "four thousand, three hundred six". Read each three-digit period, then say its name (million, thousand). For decimals, say "and" at the decimal point: 6.25 is "six and twenty-five hundredths".Number System · Grade 2, Grade 3 +
  • Word ProblemUpdatedA word problem describes a math situation in everyday language instead of only symbols. To solve one, identify what the problem is asking you to find, translate the words into an expression or equation, solve it, and check that the answer makes sense in the story. It is also called a story problem or a real-world problem.Problem Solving · Grade 3, Grade 4 +
  • Work Rate ProblemsNewWork rate problems ask how long a job takes when people or machines work together. Each worker's rate is the part of the job done per hour: 1 divided by the time alone. Rates add: 1/a + 1/b = 1/t. If one pump fills a tank in 6 hours and another in 3 hours, together they need 2 hours.Problem Solving · Algebra 1, Algebra 2

X

  • X-AxisUpdatedThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6 +
  • X-InterceptNewAn x-intercept is a point where a graph crosses or touches the x axis. Its y-coordinate is always 0, so it is written (a, 0). To find it, set y = 0 and solve for x. A non-horizontal line has one x-intercept, while a parabola can have two, one or none.Algebra · Grade 8, Algebra 1 +

Y

  • Y-AxisNewThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x axis at the origin, (0, 0). A point's y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6 +
  • Y-InterceptUpdatedThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1
  • YardNewA yard (yd) is a US customary unit of length equal to 3 feet or 36 inches. It is a little shorter than a meter: 1 yard is exactly 0.9144 meter. Yards measure fabric, rooms, gardens and the length of a football field, which is 100 yards between the goal lines.Measurement · Grade 2, Grade 3 +

Z

  • Z-ScoreNewA z-score tells how many standard deviations a value is above or below the mean. Find it with z = (x − μ)/σ: subtract the mean, then divide by the standard deviation. A positive z-score is above the mean, a negative one is below, and z = 0 is exactly at the mean.Statistics · Algebra 2, Statistics
  • ZeroUpdatedZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Number System · Grade 3, Grade 4 +
  • Zero ExponentNewA zero exponent is an exponent of 0, and any nonzero number raised to the power 0 equals 1. So 7^0 = 1, (−3)^0 = 1 and x^0 = 1 for x ≠ 0. The rule keeps the quotient rule working: (5^3)/(5^3) = 5^0, and a number divided by itself is 1.Algebra · Grade 8, Algebra 1
  • Zero of a FunctionNewA zero of a function is an input that makes the output 0: a number a with f(a) = 0. For f(x) = x − 5, the zero is 5. On a graph, the real zeros are the x-values where the curve meets the x-axis. You find zeros by solving f(x) = 0.Functions · Algebra 1, Algebra 2 +
  • Zero Property of MultiplicationNewThe zero property of multiplication says that any number multiplied by 0 equals 0: a × 0 = 0 and 0 × a = 0. For example, 58 × 0 = 0. If even one factor in a product is 0, the whole product is 0. It explains why 5 ÷ 0 has no answer.Operations · Grade 3, Grade 4 +

How each entry works

Every word follows the same plan, so you always know where to look. Rebuilt entries are marked Updated.

  1. Definition cardA one-paragraph answer in plain words, with the Spanish term, other names, grade level and Common Core standards.
  2. PictureA labeled diagram drawn for that word, with a text description for screen readers.
  3. Formal definition and formulaThe precise version, the symbols and how to read them aloud, and a formula box when there is one.
  4. Try itAn interactive you can drag, type into or tap, with a live explanation of what changed.
  5. Worked examplesTwo or three problems, from warm-up to test-style, revealed one step at a time.
  6. Common mistakesThe errors students really make, why they happen and what to do instead.
  7. Quick checkThree questions with instant feedback and an explanation for every answer.
  8. Related terms and flashcardWhat to learn before and after, with previews on hover or tap, plus a printable flashcard.

Updated November 2026·Reviewed by the Effortless Math team·Definitions checked against the Common Core State Standards and OpenStax textbooks

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