IB Mathematics Applications and Interpretation
132 of 132 topics written so far. Greyed-out topics are coming soon.
Units marked HL are for Higher Level only. Everything else is for both SL and HL.
Unit 1: Number and Algebra
- 1Scientific Notation
Patterns in powers of 10, how the sign and size of an exponent change a power, and writing, converting and calculating with very large and very small numbers.
Draft - 2Arithmetic Sequences
Sequences that add the same amount each time — the common difference, the general term t_n = a + (n − 1)d, and problems.
Draft - 3Arithmetic Series
Adding the terms of an arithmetic sequence quickly with S_n = n/2 (2a + (n − 1)d), and why the formula works.
Draft - 4Sigma Notation
Reading and writing sums with Σ, counting terms, evaluating arithmetic and geometric sums written in sigma form, changing the index, and checking with a GDC.
Draft - 5Geometric Sequences
Sequences that multiply by the same amount each time — the common ratio, the general term t_n = ar^(n − 1), and problems.
Draft - 6Geometric Series
Adding the terms of a geometric sequence with S_n = a(r^n − 1)/(r − 1), and why the formula works.
Draft - 7Compound Interest
Interest on interest — A = P(1 + i)^n, compounding periods, present value, and finding the rate or time.
Draft - 8Appreciation and Depreciation
How things gain or lose value over time — straight-line and percent depreciation, percent appreciation, year-by-year tables, reading value graphs, and buying new vs. used.
Draft - 9Exponent Laws
The product, quotient and power laws, what zero and negative exponents mean, and how the graph of y = 2^x compares with y = x².
Draft - 10Introduction to Logarithms
A logarithm is an exponent — evaluating logs, switching between exponential and logarithmic form, and estimating logs between whole numbers.
Draft - 11Rounding, Bounds and Error
Decimal places and significant figures, upper and lower bounds of rounded numbers and of calculations, percentage error, and checking that answers are reasonable.
Draft - 12Annuities: Future Value
Regular deposits that earn compound interest — the future value formula, why it's a geometric series, and finding the payment.
Draft - 13Annuities: Present Value
How much a series of future payments is worth today — loans, regular withdrawals, and the present value formula.
Draft - 14Loan Amortization
Using a GDC finance solver (N, I%, PV, PMT, FV, P/Y, C/Y) for loans and annuities, building amortization tables, finding balances and total interest, and comparing loan options.
Draft - 15Systems and Polynomials with Technology
Using a GDC to solve systems of linear equations in up to three variables and polynomial equations, setting them up from contexts, and interpreting the results.
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Unit 2: Number and Algebra HL
- 1Laws of Logarithms
The product, quotient, and power laws, the change of base formula, and using them to simplify and evaluate expressions.
Draft - 2Rational Exponents
What fractional exponents like 8^(2/3) mean, how to evaluate and simplify them, and how to rewrite powers in a different base.
Draft - 3Infinite Geometric Series
When an infinite geometric series converges, its sum a/(1 − r), and how to use it for repeating decimals, shifted indices, and bouncing-ball problems.
Draft - 4Introduction to Complex Numbers
The number i, Cartesian form a + bi, real and imaginary parts, conjugate, modulus and argument, the Argand diagram, complex arithmetic, and quadratics with a negative discriminant.
Draft - 5Polar and Euler Form of Complex Numbers
Writing complex numbers as r cis θ and re^(iθ), converting between forms, multiplying and dividing as rotations and stretches, and adding sinusoidal waves with complex numbers.
Draft - 6Introduction to Matrices
What a matrix is (elements, rows, columns, order), matrix equality, addition, subtraction and scalar multiplication, matrix multiplication and its properties, identity and zero matrices, and practical uses.
Draft - 7Determinants and Inverse Matrices
Determinants and inverses of 2 × 2 matrices by hand and larger matrices with technology, singular matrices, solving systems AX = B with an inverse matrix, and coding and decoding messages.
Draft - 8Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors of 2 × 2 matrices, the characteristic polynomial, diagonalization, and using A to the power n = PDⁿP⁻¹ to predict long-term behaviour.
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Unit 3: Functions
- 1Equations of Lines
The slope formula, zero and undefined slopes, finding a line's equation from a graph, a table, a point and slope, or two points, and switching between y = mx + b, Ax + By + C = 0 and Ax + By = D.
Draft - 2Parallel and Perpendicular Lines
Parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes. Testing pairs of lines and finding equations of parallel and perpendicular lines through a point.
Draft - 3Function Notation
Writing f(x), evaluating functions, and solving f(x) = a value — from equations, tables, and graphs.
Draft - 4Domain and Range
Finding the set of possible inputs and outputs of a function from graphs, equations, and real-world situations.
Draft - 5Inverse Functions
Undoing a function — finding, graphing, and checking inverses.
Draft - 6Key Features of Graphs
Sketching graphs from information or a GDC screen, finding intercepts, maximums and minimums, symmetry and asymptotes with technology, intersecting curves, and graphing sums and differences of functions.
Draft - 7Vertex Form and Transformations
Graphing y = a(x − h)² + k as a transformation of y = x², sketching by hand with the step pattern, and finding the equation of a parabola from its graph.
Draft - 8Exponential Growth and Decay
Modelling populations, depreciation, half-life, and doubling with exponential functions, and solving real-world problems.
Draft - 9Direct and Inverse Variation
Models of the form f(x) = axⁿ — direct variation, inverse variation and inverse-square laws — plus cubic models, finding parameters, and the modelling cycle (IB AI SL 2.5, 2.6).
Draft - 10Sinusoidal Modelling
Building and using sinusoidal models for Ferris wheels, tides, temperatures, and daylight, in degrees.
Draft - 11Function Modelling
Building linear, quadratic, exponential, sinusoidal, and polynomial models from data, checking how well they fit, and using them to predict and interpret real situations.
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Unit 4: Functions HL
- 1Composition of Functions
Putting one function inside another — finding f(g(x)) from tables, graphs, and equations, its domain and range, decomposing functions, and real-world chains.
Draft - 2Combining Transformations
Graphing y = af(k(x − d)) + c by applying stretches, reflections, and translations in the right order.
Draft - 3Exponential and Logarithmic Applications
Using logarithms to find the time in growth and decay problems — doubling time, half-life, compound interest, populations, and cooling — and reading answers from graphs.
Draft - 4Logistic Models
The logistic differential equation dP/dt = kP(1 − P/L) — carrying capacity, long-run behaviour, where growth is fastest, and the solution formula.
Draft - 5Piecewise Models
Functions defined by different rules on different intervals — tax brackets, shipping costs, phone plans and speed profiles — including evaluating, graphing, making the pieces join, and solving for inputs (IB AI AHL 2.9).
Draft - 6Logarithmic Scales
pH, earthquake magnitude, and decibels — why each step on a logarithmic scale means multiplying by 10, and how to compare and solve with them.
Draft - 7Linearizing Data with Logarithms
Using logarithms to scale data, and semi-log and log-log graphs to decide whether data is exponential or a power law and to find the parameters of the model (IB AI AHL 2.10).
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Unit 5: Geometry and Trigonometry
- 1Vectors in Three Dimensions
The right-handed 3-D coordinate system, plotting points, vectors [x, y, z] and the unit vectors i, j, k, magnitude, distance between points, operations in component form, and collinearity.
Draft - 2Surface Area of 3-D Objects
Using nets to find the surface area of prisms, cylinders, square-based pyramids and cones, including slant height and composite objects.
Draft - 3Volume of 3-D Objects
Volume of prisms, cylinders, pyramids and cones (and spheres), the one-third relationship, capacity in millilitres and litres, and composite objects.
Draft - 4Spheres, Cones, Pyramids and Composite Solids
Volume and surface area of spheres, hemispheres, right pyramids and right cones, and of solids built from them, with answers to 3 significant figures.
Draft - 5Angles in 3-D Solids
Finding the angle between two lines and between a line and a plane in cuboids, pyramids and cones, by spotting the right triangle, with 3-D distance and midpoint along the way.
Draft - 6The Primary Trigonometric Ratios
Sine, cosine and tangent in right triangles — naming the sides, SOH CAH TOA, and finding missing sides and angles.
Draft - 7The Sine Law
Solving any triangle when you know two angles and a side, using a/sin A = b/sin B = c/sin C.
Draft - 8The Cosine Law
Solving triangles from two sides and the angle between them, or from three sides, using c² = a² + b² − 2ab cos C.
Draft - 9Area of a Triangle Using Sine
The formula Area = ½ab sin C, where it comes from, using it with the sine and cosine rules, and finding an angle from a known area.
Draft - 10Right Triangle Problems
Angles of elevation and depression, clinometers, ladders, ramps, navigation, and problems with two right triangles.
Draft - 11Trig Problems in Three Dimensions
Solving 3-D problems — towers, cliffs, boxes, and pyramids — by splitting them into 2-D right and oblique triangles.
Draft - 12Arc Length and Sector Area
Arc length, sector area and sector perimeter with the angle in degrees or radians, and the area of a segment as a sector minus a triangle.
Draft - 13Right Bisectors and Distance to a Line
Finding the equation of the right bisector of a segment, the shortest distance from a point to a line, and the circumcentre of a triangle.
Draft - 14Voronoi Diagrams
Sites, cells, edges and vertices; edges as perpendicular bisectors; adding a site; nearest-neighbour interpolation; and the toxic waste dump problem.
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Unit 6: Geometry and Trigonometry HL
- 1Radian Measure
What a radian is, converting between degrees and radians, arc length a = rθ, and angular velocity.
Draft - 2Trig Ratios in Radians
Primary and reciprocal trig ratios of angles in radians, with a calculator and exactly for the special angles, using the CAST rule and related acute angles.
Draft - 3The Ambiguous Case
When two sides and a non-included angle (SSA) give no triangle, one triangle, or two — and how to find them all.
Draft - 4Solving Linear Trig Equations
Solving equations like 2 sin x + 1 = 0 for 0 ≤ x ≤ 2π in radians, with exact answers from special angles, decimal answers from related angles, and equations with kx.
Draft - 5Matrix Transformations
Using 2 × 2 matrices and vectors to reflect, stretch, enlarge, rotate and translate points in the plane, combining transformations in the right order, and the determinant as the area scale factor.
Draft - 6Introduction to Vectors
Scalars vs vectors, magnitude and direction, vector notation, equal and opposite vectors, and describing directions with bearings and angles.
Draft - 7Cartesian Vectors
Writing 2-D vectors in component form [x, y], position vectors, converting between components and magnitude-direction, the vector between two points, unit vectors i and j, and operations in component form.
Draft - 8Scalar Multiplication of Vectors
Multiplying a vector by a number, collinear vectors, unit vectors, the distributive properties, linear combinations, and simple geometric proofs with vectors.
Draft - 9Equations of Lines in 3-Space
Vector, parametric, and symmetric equations of a line in 3-space, why a line has no single scalar equation, and writing a line as the intersection of two planes.
Draft - 10Vector Kinematics
Position, velocity, and speed for objects moving in a straight line with constant velocity, when and where two objects meet, closest approach, and (AI HL) motion with variable velocity in 2-D.
Draft - 11The Dot Product
Multiplying two vectors to get a number — the geometric and component formulas, the angle between vectors, perpendicular vectors, properties, and work.
Draft - 12The Cross Product
Multiplying two vectors in 3-space to get a perpendicular vector — the component formula, the determinant shortcut, the right-hand rule, the magnitude formula, and properties.
Draft - 13Vector Projections
The scalar and vector projection of one vector onto another, splitting a vector into parallel and perpendicular parts, direction cosines, and applications to forces and work.
Draft - 14Introduction to Graph Theory
Vertices, edges, degree and the handshake lemma, simple, complete, weighted and directed graphs, subgraphs and trees.
Draft - 15Adjacency Matrices
Representing graphs with adjacency matrices and weighted tables, counting walks with powers of the matrix, and building transition matrices for random walks on a graph.
Draft - 16Eulerian and Hamiltonian Paths
Walks, trails, paths, circuits and cycles; when a graph has an Eulerian trail or circuit (the odd-vertex test); and Hamiltonian paths and cycles.
Draft - 17Minimum Spanning Trees
Spanning trees, and finding a minimum spanning tree with Kruskal's algorithm and Prim's algorithm, including Prim's matrix (table) method.
Draft - 18The Chinese Postman Problem
Finding the shortest closed route that travels along every edge of a weighted graph, by pairing the odd vertices and repeating the shortest paths between them.
Draft - 19The Travelling Salesman Problem
Finding a Hamiltonian cycle of least weight; tables of least distances, an upper bound from the nearest-neighbour algorithm, and a lower bound from the deleted-vertex algorithm.
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Unit 7: Statistics and Probability
- 1Types of Data
Why statistical studies collect data, why data varies, and how to classify variables and data sets.
Draft - 2Sampling Methods
Populations and samples, random and non-random sampling methods, and organizing data in a spreadsheet.
Draft - 3Bias in Sampling
How sampling bias, non-response bias, response bias, and measurement bias distort results, and how to reduce them.
Draft - 4Displaying One-Variable Data
Choosing the right graph for the data — bar and circle graphs, histograms, stem-and-leaf plots, and boxplots — and spotting graphs that mislead.
Draft - 5Cumulative Frequency
Cumulative frequency tables and graphs for grouped data, reading off the median, quartiles, percentiles and IQR, and turning the results into a box-and-whisker diagram.
Draft - 6Quartiles and Percentiles
Quartiles, the interquartile range, the five-number summary, the 1.5 × IQR rule for outliers, boxplots, and percentiles.
Draft - 7Measures of Central Tendency
Mean, median, and mode; weighted means; estimating the mean from grouped data; the effect of outliers; and choosing the best measure.
Draft - 8Standard Deviation
Measuring spread with the range, variance, and standard deviation — population vs sample formulas, by hand and with technology.
Draft - 9Scatter Plots and Correlation
Independent and dependent variables, scatter plots, describing a relationship, the correlation coefficient r, and side-by-side boxplots.
Draft - 10Linear Regression
The line of best fit by least squares, interpreting slope and intercept, interpolation and extrapolation, residuals, and the effect of outliers.
Draft - 11Probability and Sample Spaces
Outcomes, sample spaces, and events; theoretical probability for equally likely outcomes; and probability distributions.
Draft - 12Complements and Mutually Exclusive Events
The complement rule, mutually exclusive events, the additive principle for P(A or B), and Venn diagrams.
Draft - 13Independent and Dependent Events
When one event affects another, multiplying probabilities for "and", drawing with and without replacement, and tree diagrams.
Draft - 14Conditional Probability
The probability of B given A, from formulas, two-way tables, and tree diagrams — and why the order matters.
Draft - 15Discrete Random Variables
Random variables, probability distributions in tables, probability histograms, and the uniform distribution.
Draft - 16Expected Value
The long-run average of a random variable — computing E(X), fair games, raffles, and the link to the weighted mean.
Draft - 17Binomial Distribution
Counting successes in independent trials — the binomial conditions and formula, tables and histograms, E(X) = np, and how the shape changes with n.
Draft - 18The Normal Distribution
The bell-shaped normal model, its properties, the notation X ~ N(μ, σ²), and estimating probabilities with the 68–95–99.7 rule.
Draft - 19Spearman's Rank Correlation
Ranking data (including tied ranks), finding Spearman's rank correlation coefficient with technology, and choosing between Pearson's r and Spearman's rs.
Draft - 20Introduction to Hypothesis Testing
Null and alternative hypotheses, significance levels, p-values, critical values, one-tailed and two-tailed tests, and how to write a conclusion in context.
Draft - 21Chi-Squared Tests
Observed and expected frequencies, the χ² goodness-of-fit test, and the χ² test for independence in contingency tables — degrees of freedom, critical values, p-values and conclusions in context.
Draft - 22The t-Test
Using the pooled two-sample t-test to compare the means of two populations — hypotheses, one- and two-tailed tests, p-values from technology, assumptions and conclusions in context.
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Unit 8: Statistics and Probability HL
- 1Survey and Experiment Design
Randomization, replication, and control in experiments; and writing fair, ethical surveys with clear questions.
Draft - 2Non-linear Regression
Fitting quadratic, cubic, exponential, power and sine regression curves with technology, measuring fit with the sum of square residuals and the coefficient of determination R², and choosing between models (IB AI AHL 4.13).
Draft - 3Linear Combinations of Random Variables
How the mean and variance change when you scale, shift, add, or subtract random variables — and why the sample mean and s²ₙ₋₁ are unbiased estimates.
Draft - 4The Central Limit Theorem
Sums and differences of independent normal variables, the distribution of the sample mean, and why sample means are approximately normal for large samples from any population.
Draft - 5Confidence Intervals for the Mean
Estimating a population mean with an interval — the z-interval when σ is known, the t-interval when it isn't, what the confidence level really means, and how sample size and level affect the width.
Draft - 6The Poisson Distribution
Modelling the number of random events in a fixed interval — the Poisson conditions and formula, mean and variance both equal to m, changing the interval, sums of Poisson variables, and choosing between binomial, Poisson, and normal models.
Draft - 7Hypothesis Tests for Parameters
Critical values and critical regions; testing a mean (z-test and one-sample or paired t-test), a binomial proportion, and a Poisson mean; testing whether ρ = 0 with technology; and calculating the probabilities of Type I and Type II errors.
Draft - 8Markov Chains
Transition diagrams and matrices, initial state matrices, powers of transition matrices, regular Markov chains, and steady-state (long-term) probabilities.
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Unit 9: Calculus
- 1Introduction to Limits
The idea of a limit — what a function approaches near a point, limit notation, one-sided limits, and when a limit does not exist.
Draft - 2The Definition of the Derivative
The derivative as a limit, derivative notation, tangent lines, and estimating derivatives from tables and graphs.
Draft - 3First Derivative Test
Finding where a function increases and decreases from the sign of f', building sign charts, and using the first derivative test to classify relative extrema.
Draft - 4The Power Rule and Basic Derivative Rules
Differentiating powers of x (including negative and fractional exponents), constants, sums, differences, and constant multiples.
Draft - 5Tangents and Normals
Equations of the tangent and normal to a curve at a point, finding where the tangent has a given gradient, tangents through an outside point, and using technology for gradients.
Draft - 6Antiderivatives and Indefinite Integrals
Finding antiderivatives with the basic rules (power, 1/x, exponential, trig, inverse trig), the + C, and solving initial-value problems.
Draft - 7Evaluating Definite Integrals with the FTC
The Fundamental Theorem of Calculus, part 2 — evaluating a definite integral as F(b) − F(a) using an antiderivative, and the net change theorem.
Draft - 8Critical Points and Extrema
Absolute and relative maximums and minimums, the Extreme Value Theorem, and finding critical points where f'(x) = 0 or f'(x) does not exist.
Draft - 9Optimization
Setting up and solving optimization problems with derivatives — variables, constraint, objective function, domain — and justifying the answer for fencing, boxes, cans, cost, and distance.
Draft - 10Riemann Sums
Approximating the area under a curve with left, right, midpoint, and trapezoidal sums, from graphs, tables, and formulas.
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Unit 10: Calculus HL
- 1Derivatives of Sine, Cosine, eˣ, and ln x
The derivatives of sin x, cos x, e^x, and ln x, why radians matter, and using them with the basic derivative rules.
Draft - 2The Chain Rule
Differentiating composite functions with f'(g(x)) · g'(x) and dy/du · du/dx — powers, trig, exponential, and log compositions, tables of values, and choosing which rules to combine.
Draft - 3The Product Rule
Differentiating a product of two (or three) functions, including from tables of values and in context.
Draft - 4The Quotient Rule and Trig Derivatives
Differentiating quotients, and using the quotient rule to find the derivatives of tan x, cot x, sec x, and csc x.
Draft - 5Related Rates Problems
A step-by-step strategy for multi-step related rates — sliding ladders, filling cones, moving shadows, and approaching cars.
Draft - 6Higher-Order Derivatives
Second, third, and higher derivatives — notation like f''(x), d²y/dx², and y'', what the second derivative means, and second derivatives of implicit relations.
Draft - 7Concavity and the Second Derivative Test
Concave up and concave down from the sign of f'', points of inflection, and the second derivative test for relative extrema — including when it's inconclusive.
Draft - 8Integration by Substitution
Undoing the chain rule with u-substitution, for indefinite and definite integrals (including changing the limits), and choosing an integration technique.
Draft - 9Area Between Curves
Finding the area between two curves with vertical slices — top minus bottom, finding intersection points, and splitting the integral when the curves cross more than twice.
Draft - 10Area Between Curves Using Horizontal Slices
Integrating with respect to y — right minus left, rewriting curves as x in terms of y, and choosing the slice direction that makes the problem easier.
Draft - 11Volumes of Revolution — the Disc Method
Spinning a region around a line to make a solid — discs with radius R, revolving about the x-axis, the y-axis, and other horizontal or vertical lines.
Draft - 12Straight-Line Motion with Derivatives
Position, velocity, and acceleration of a particle on a line — speed, direction, changing direction, and speeding up versus slowing down.
Draft - 13Motion Along a Line with Integrals
Going backwards from velocity to position — initial conditions, displacement versus total distance, and velocity from acceleration.
Draft - 14Separation of Variables
Solving separable differential equations, finding particular solutions from initial conditions, and stating where a solution is defined.
Draft - 15Slope Fields
Sketching slope fields from a differential equation, matching fields to equations, and drawing and reading solution curves.
Draft - 16Euler's Method
Approximating a solution of a differential equation step by step with tangent lines, organizing the work in a table, and deciding whether the estimate is too high or too low.
Draft - 17Euler's Method for Coupled Systems
Using Euler's method on coupled differential equations such as predator–prey models, setting up step-by-step tables and spreadsheets, and solving second-order equations by rewriting them as a coupled system.
Draft - 18Phase Portraits
Coupled linear systems dx/dt = ax + by, dy/dt = cx + dy — solving with eigenvalues and eigenvectors, and sketching phase portraits for saddles, nodes, spirals and centres to describe long-term behaviour.
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