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Mathematics
From counting to calculus — numbers, algebra, geometry, trigonometry, calculus and statistics.
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Mathematics on LearnViz HighSchoolEdu has 200 interactive topics. Each one has a plain-English explanation, a live animation you can control, key ideas to remember and a quick quiz. Number and algebra skills are used in science, engineering, budgeting and computing.
Number Sense
1Natural Numbers and IntegersNatural numbers count things; integers add zero and the negatives.2Place Value and RoundingThe position of a digit decides its value, and rounding trims numbers to a chosen place.3Addition and SubtractionAddition combines amounts while subtraction finds the difference or removes some.4Multiplication as Repeated AdditionMultiplying means adding the same number several times.5Division and RemaindersDivision splits a quantity into equal groups and may leave a remainder.6Order of OperationsBrackets, exponents, multiplication and division, then addition and subtraction.7Factors and MultiplesFactors divide a number exactly while multiples are its times table.8Prime NumbersA prime has exactly two factors: 1 and itself.9Prime FactorizationEvery composite number is a unique product of primes.10Greatest Common DivisorThe GCD is the largest number that divides two numbers exactly.11Least Common MultipleThe LCM is the smallest number that both numbers divide into.12Divisibility RulesQuick tests tell whether a number divides evenly without dividing.13Negative NumbersNegative numbers lie left of zero and represent debts, drops and depth.14Absolute ValueAbsolute value measures distance from zero, so it is never negative.15FractionsA fraction shows parts of a whole as numerator over denominator.16Equivalent FractionsDifferent fractions can name the same amount.17Adding and Subtracting FractionsFractions need a common denominator before combining.18Multiplying and Dividing FractionsMultiply across; to divide, multiply by the reciprocal.19Mixed Numbers and Improper FractionsA mixed number combines a whole number and a fraction.20DecimalsDecimals extend place value to the right of the point using tenths, hundredths and more.21PercentagesA percent is a fraction out of 100.22Percent ChangePercent change compares the change to the original value.23RatiosA ratio compares two quantities by division.24ProportionsA proportion says two ratios are equal.25Rates and Unit RatesA rate compares different units; a unit rate has a denominator of 1.26ExponentsAn exponent says how many times to multiply a base by itself.27Square RootsA square root undoes squaring.28Scientific NotationScientific notation writes very large or small numbers as a × 10ⁿ.29Rational and Irrational NumbersRational numbers are fractions; irrational numbers cannot be written as fractions.30Number Line and Real NumbersEvery real number has a place on the continuous number line.Algebra Basics
31Variables and ExpressionsA variable is a letter that stands for an unknown or changing number.32Evaluating ExpressionsSubstitute numbers for variables then follow order of operations.33Like Terms and SimplifyingOnly terms with identical variable parts can be combined.34Distributive PropertyMultiplication distributes over addition: a(b + c) = ab + ac.35One-Step EquationsUse an inverse operation to isolate the variable.36Two-Step EquationsUndo addition or subtraction first, then multiplication or division.37Multi-Step EquationsSimplify each side, collect variables, then isolate.38Variables on Both SidesGather variable terms on one side and constants on the other.39Literal Equations and FormulasRearrange a formula to solve for a chosen variable.40Linear InequalitiesInequalities describe ranges of solutions using <, >, ≤ or ≥.41Compound InequalitiesTwo inequalities joined by and or or.42Absolute Value Equations∣x∣ = a splits into two cases: x = a or x = −a.Linear Functions
43Coordinate PlaneThe plane uses an x-axis and y-axis to locate points as ordered pairs.44SlopeSlope measures steepness as rise over run.45Slope-Intercept Formy = mx + b gives the slope m and y-intercept b.46Point-Slope Formy − y₁ = m(x − x₁) builds a line from a point and slope.47Standard Form of a LineAx + By = C keeps x and y on one side.48Parallel and Perpendicular LinesParallel lines share slope; perpendicular slopes are negative reciprocals.49Graphing Linear EquationsPlot the intercept then use the slope to find more points.50Systems by GraphingThe solution of a system is where the lines cross.51Substitution MethodSolve one equation for a variable and plug it into the other.52Elimination MethodAdd or subtract equations to cancel a variable.53Linear Inequalities on a GraphShade the half-plane that satisfies the inequality.54Functions and RelationsA function gives exactly one output for each input.55Function Notationf(x) names a function and shows its input.56Domain and RangeDomain is all valid inputs; range is all outputs.Sequences
57Arithmetic SequencesEach term differs from the last by a constant.58Geometric SequencesEach term is the previous times a constant ratio.59Series and Sigma NotationA series adds up the terms of a sequence.Exponents and Polynomials
60Exponent RulesLaws of exponents let you simplify products, quotients and powers.61Zero and Negative Exponentsa⁰ = 1 and a⁻ⁿ is the reciprocal of aⁿ.62Polynomials and DegreeA polynomial is a sum of terms with whole-number exponents.63Adding and Subtracting PolynomialsCombine like terms; distribute the negative when subtracting.64Multiplying PolynomialsMultiply each term by each term, then combine like terms.65Special ProductsSquares of binomials and the difference of squares follow patterns.66Factoring Out the GCFPull the greatest common factor out of every term.67Factoring TrinomialsFind two numbers that multiply to c and add to b.68Difference of Squaresa² − b² factors as (a + b)(a − b).69Factoring by GroupingGroup four terms in pairs and factor each pair.70Polynomial Long DivisionDivide polynomials the way you divide numbers.71Remainder and Factor TheoremsEvaluating p(a) gives the remainder when dividing by (x − a).72Rational Root TheoremPossible rational roots are factors of the constant over factors of the leading coefficient.73Polynomial End BehaviorLeading term decides what the graph does at far left and right.Quadratics
74Quadratic FunctionsQuadratics make parabolas with the form y = ax² + bx + c.75Graphing ParabolasFind the vertex, axis of symmetry and intercepts, then sketch.76Vertex Formy = a(x − h)² + k shows the vertex (h, k) directly.77Solving by FactoringSet the quadratic to 0, factor, then use the zero product property.78Square Root MethodIf x² = k then x = ±√k.79Completing the SquareAdd (b/2)² to turn a quadratic into a perfect square.80Quadratic Formulax = (−b ± √(b² − 4ac)) / 2a solves every quadratic.81Discriminantb² − 4ac tells how many real roots exist.82Quadratic Word ProblemsQuadratics model areas, projectile heights and profit.83Quadratic InequalitiesSolve the equation first, then test intervals.84Complex NumbersComplex numbers add an imaginary unit i where i² = −1.85Complex PlaneComplex numbers are points on a plane with real and imaginary axes.Exponentials and Logarithms
86Exponential FunctionsExponentials multiply by a constant factor each step: y = a·bˣ.87Exponential Growth and DecayQuantities that change by a fixed percent follow exponential models.88Compound InterestInterest on interest makes money grow exponentially.89The Number ee ≈ 2.718 is the natural base of continuous growth.90LogarithmsA logarithm answers: what power gives this number?91Properties of LogarithmsLogs turn multiplication into addition.92Solving Exponential EquationsTake logs of both sides to bring the exponent down.93Natural Logarithmln x is the logarithm with base e.94Radical ExpressionsRadicals are roots written with the √ symbol.95Rational Exponentsaᵐ/ⁿ means the nth root of a raised to m.Functions in Depth
96Piecewise FunctionsA piecewise function uses different rules on different intervals.97Transformations of FunctionsShifts, stretches and reflections change a parent graph.98Function CompositionComposing functions feeds the output of one into another.99Inverse FunctionsAn inverse undoes a function and swaps inputs with outputs.100Even and Odd FunctionsEven functions mirror across the y-axis; odd ones have origin symmetry.101Rational FunctionsA rational function is a ratio of polynomials.102Direct and Inverse VariationDirect: y = kx. Inverse: y = k/x.103Average Rate of ChangeThe slope of the secant line between two points.104MatricesMatrices are rectangular arrays used to organise and transform data.105Systems of Three EquationsSolve three unknowns by eliminating variables step by step.106Binomial TheoremExpand (a + b)ⁿ using Pascal's triangle coefficients.107Mathematical InductionProve a statement for all n by a base case and an inductive step.Geometry Foundations
108Points, Lines and PlanesGeometry starts with undefined ideas: points, lines and planes.109Angles and MeasurementAn angle is formed by two rays sharing an endpoint.110Angle RelationshipsComplementary angles add to 90° and supplementary angles to 180°.111Parallel Lines and TransversalsA transversal crossing parallel lines creates equal and supplementary angles.112Triangle ClassificationTriangles are named by their sides and by their angles.113Triangle Angle SumThe interior angles of every triangle add to 180°.114Exterior Angle TheoremAn exterior angle equals the sum of the two remote interior angles.115Triangle InequalityAny two sides of a triangle must sum to more than the third.116Congruent TrianglesCongruent triangles have the same shape and size.117Triangle Congruence ShortcutsSSS, SAS, ASA and AAS prove triangles congruent.118Similar TrianglesSimilar triangles have equal angles and proportional sides.119Pythagorean TheoremIn a right triangle, a² + b² = c².120Pythagorean TriplesWhole-number sets that satisfy a² + b² = c².121Special Right Triangles45-45-90 and 30-60-90 triangles have fixed side ratios.122Midsegments of TrianglesA midsegment joins midpoints of two sides and is parallel to the third.123Medians, Altitudes and CentroidsSpecial segments inside a triangle meet at famous points.124Polygons and Angle SumsInterior angles of an n-sided polygon sum to (n − 2) × 180°.125QuadrilateralsFour-sided shapes are classified by sides and angles.126ParallelogramsOpposite sides are parallel and equal.127Rectangles, Rhombi and SquaresSpecial parallelograms with extra properties.128Perimeter and AreaPerimeter is the distance around; area is the space inside.129Area of Triangles and TrapezoidsTriangles take half of a base × height box.130Circles: Parts and TermsA circle is every point at a fixed distance from the center.131Circumference and Area of a CircleC = 2πr and A = πr².132Arc Length and SectorsAn arc and a sector are fractions of the whole circle.133Central and Inscribed AnglesAn inscribed angle is half the central angle on the same arc.134Chords, Tangents and SecantsLines meeting circles have power relationships.135Equation of a Circle(x − h)² + (y − k)² = r² has center (h, k) and radius r.136Distance and Midpoint FormulasCoordinate geometry measures segments on a grid.Solids and Transformations
137Surface Area of PrismsSurface area is the total area of all faces.138Volume of Prisms and CylindersVolume = base area × height.139Pyramids and ConesPyramids and cones hold one third of the matching prism or cylinder.140SpheresA sphere is every point in space at a fixed distance from the center.141Cross-Sections and Solids of RevolutionSlicing a solid reveals 2D shapes; spinning a shape builds a solid.142TranslationsA translation slides every point the same distance and direction.143ReflectionsA reflection flips a figure over a line of reflection.144RotationsA rotation turns a figure around a fixed point by an angle.145DilationsA dilation resizes a figure by a scale factor from a center.146SymmetryA shape has symmetry if a transformation maps it onto itself.147TessellationsShapes that tile the plane with no gaps or overlaps.148Geometric ProofsA proof is a chain of logical reasons from givens to a conclusion.Trigonometry
149Right Triangle TrigonometrySine, cosine and tangent relate angles to side ratios.150Solving Right TrianglesUse trig ratios and inverses to find sides and angles.151Angles of Elevation and DepressionAngles measured up or down from the horizontal.152RadiansRadians measure angles by arc length on a unit circle.153The Unit CircleA circle of radius 1 where cos θ and sin θ are the coordinates.154Special Angle Values30°, 45° and 60° have exact trig values.155Reference AnglesThe acute angle between the terminal side and the x-axis.156Graph of SineThe sine curve oscillates between −1 and 1 with period 2π.157Graph of CosineThe cosine curve is sine shifted left by π/2.158Graph of TangentTangent repeats every π and has vertical asymptotes.159Amplitude, Period and Phase Shifty = A sin(B(x − C)) + D transforms the wave.160Pythagorean Identitiessin²θ + cos²θ = 1 for every angle.161Sum and Difference IdentitiesFormulas for sin or cos of A ± B.162Double-Angle IdentitiesFormulas for sin 2θ and cos 2θ.163Inverse Trig Functionsarcsin, arccos and arctan return angles from ratios.164Solving Trigonometric EquationsIsolate the trig function, find the reference angle, list all solutions.165Law of Sinesa/sin A = b/sin B = c/sin C.166Law of Cosinesc² = a² + b² − 2ab cos C generalises Pythagoras.167Polar CoordinatesLocate a point by its distance r and angle θ.168Simple Harmonic MotionOscillations like springs and pendulums follow sine waves.Calculus
169Introduction to LimitsA limit describes what a function approaches near a point.170Limit LawsLimits of sums, products and quotients follow simple rules.171Limits at InfinityLimits at infinity describe end behavior and horizontal asymptotes.172ContinuityA function is continuous if you can draw it without lifting the pencil.173The DerivativeThe derivative measures instantaneous rate of change: the slope of the tangent.174Power Ruled/dx xⁿ = n·xⁿ⁻¹.175Product and Quotient RulesRules for differentiating products and ratios.176Chain RuleDifferentiate the outside function then multiply by the derivative of the inside.177Derivatives of Trig and Exponential Functionssin and cos trade derivatives; eˣ is its own derivative.178Implicit DifferentiationDifferentiate both sides when y is tangled with x.179Related RatesLink the rates of changing quantities with a shared equation.180Critical Points and Curve SketchingWhere f′ = 0 or is undefined, the graph may peak or dip.181OptimizationUse derivatives to find the best value of a quantity.182Mean Value TheoremSomewhere between two points, the instantaneous slope equals the average slope.183Riemann Sums and the Definite IntegralAdd the areas of thin rectangles to find area under a curve.184Fundamental Theorem of CalculusDifferentiation and integration are inverse operations.185Antiderivatives and Integration RulesAn antiderivative reverses differentiation.186Area Between CurvesIntegrate top minus bottom to find the area between graphs.Statistics and Probability
187Mean, Median and ModeThree ways to find the center of a data set.188Range, Quartiles and Box PlotsSpread describes how far data stretches.189Standard DeviationStandard deviation measures typical distance from the mean.190Histograms and Bar ChartsBars show how often values occur.191Scatterplots and CorrelationScatterplots show how two variables move together.192Line of Best FitA line that summarises a trend in scattered data.193Basic ProbabilityProbability measures how likely an event is on a scale from 0 to 1.194Counting PrincipleMultiply the number of choices at each step.195PermutationsPermutations count ordered arrangements.196CombinationsCombinations count selections where order does not matter.197Conditional ProbabilityP(A given B) is the chance of A given that B happened.198Independent and Dependent EventsIndependent events don't affect each other.199Normal DistributionThe bell curve describes many natural measurements.200Sampling and BiasGood samples represent the population.No topics match.
Frequently asked questions
What topics are in Mathematics?
There are 200 topics, including Natural Numbers and Integers, Place Value and Rounding, Addition and Subtraction, Multiplication as Repeated Addition, Division and Remainders, and more.
Who is Mathematics on LearnViz for?
High school students who want to understand mathematics visually, teachers who want a quick classroom demonstration, and parents helping with homework.
Do I have to study the topics in order?
No. The list follows the order of the course, but every topic has its own page, so you can start wherever you need help.
How do I track my progress?
Open a topic and press “Mark as learned”. This subject page then shows how many topics you have completed. Progress is saved in your own browser.
How is Mathematics used in real life?
Number and algebra skills are used in science, engineering, budgeting and computing.