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International Baccalaureate IB Diploma Programme Mathematics: Applications and Interpretation

Syllabus DP Mathematics: Applications and Interpretation · First assessments for SL and HL—2021 · sourced from International Baccalaureate Organization, Diploma Programme, Mathematics: applications and interpretation guide, first assessment 2021 (published February 2019, updated August 2019/May 2020/August 2020/November 2020), verified 2026-09-08.

  1. 1. Number and algebra

    • 1.1 Operations with numbers in the form a × 10^k
    • 1.2 Arithmetic sequences and series
    • 1.3 Geometric sequences and series
    • 1.4 Financial applications of geometric sequences and series (compound interest, depreciation)
    • 1.5 Laws of exponents with integer exponents; introduction to logarithms
    • 1.6 Approximation: decimal places, significant figures, upper/lower bounds, percentage error, estimation
    • 1.7 Amortization and annuities using technology
    • 1.8 Use technology to solve systems of linear equations and polynomial equations
    • 1.9 Laws of logarithms (AHL only)
    • 1.10 Simplifying expressions involving rational exponents (AHL only)
    • 1.11 The sum of infinite geometric sequences (AHL only)
    • 1.12 Complex numbers: Cartesian form, the complex plane, quadratic equations with complex roots (AHL only)
    • 1.13 Complex numbers: modulus-argument (polar) and exponential (Euler) forms (AHL only)
    • 1.14 Matrices: definition, algebra, multiplication, determinants, inverses, solving linear systems (AHL only)
    • 1.15 Eigenvalues, eigenvectors, characteristic polynomial and diagonalization of 2×2 matrices (AHL only)

  2. 2. Functions

    • 2.1 Different forms of the equation of a straight line; gradient, parallel and perpendicular lines
    • 2.2 Concept of a function, domain, range and graph; inverse function
    • 2.3 The graph of a function; sketching from information or a context
    • 2.4 Determining key features of graphs; points of intersection of two curves using technology
    • 2.5 Modelling with linear, quadratic, exponential, direct/inverse variation, cubic and sinusoidal functions
    • 2.6 Modelling skills: developing, fitting, testing and using a model
    • 2.7 Composite functions in context; inverse function including domain restriction (AHL only)
    • 2.8 Transformations of graphs: translations, reflections, stretches and composite transformations (AHL only)
    • 2.9 Further modelling: exponential (half-life), natural logarithmic, sinusoidal, logistic and piecewise models (AHL only)
    • 2.10 Scaling large or small numbers with logarithms; linearizing data; log-log and semi-log graphs (AHL only)

  3. 3. Geometry and trigonometry

    • 3.1 3D distance and midpoint; volume and surface area of pyramids, cones, spheres and combinations; angles between lines/planes
    • 3.2 Right-angled trigonometry; the sine rule, cosine rule and area of a triangle
    • 3.3 Applications of right- and non-right-angled trigonometry, including Pythagoras’ theorem and bearings
    • 3.4 The circle: length of an arc and area of a sector
    • 3.5 Equations of perpendicular bisectors
    • 3.6 Voronoi diagrams: sites, vertices, edges, cells; nearest-neighbour interpolation; the "toxic waste dump" problem
    • 3.7 Radian measure; area of sector and arc length using radians (AHL only)
    • 3.8 Definitions of sinθ/cosθ via the unit circle; Pythagorean identity; ambiguous case of the sine rule (AHL only)
    • 3.9 Matrix transformations: reflections, stretches, enlargements, translations, rotations; determinant as area scale factor (AHL only)
    • 3.10 Vectors: concept, components, base vectors, magnitude, position vectors, normalization (AHL only)
    • 3.11 Vector equation of a line in two and three dimensions (AHL only)
    • 3.12 Vector applications to kinematics: linear motion with constant and variable velocity (AHL only)
    • 3.13 Scalar and vector product of two vectors; angle between vectors; components of vectors (AHL only)
    • 3.14 Graph theory: graphs, vertices, edges, degree, simple/complete/weighted/directed graphs, subgraphs and trees (AHL only)
    • 3.15 Adjacency matrices, walks, weighted adjacency tables, transition matrices for strongly-connected graphs (AHL only)
    • 3.16 Tree and cycle algorithms: Kruskal’s and Prim’s MST, Chinese postman and travelling salesman problems (AHL only)

  4. 4. Statistics and probability

    • 4.1 Population, sample, sampling techniques, reliability and bias, interpretation of outliers
    • 4.2 Presentation of data: frequency distributions, histograms, cumulative frequency graphs, box-and-whisker diagrams
    • 4.3 Measures of central tendency and dispersion; effect of constant changes on data; quartiles
    • 4.4 Linear correlation of bivariate data; Pearson’s product-moment correlation coefficient; regression line of y on x
    • 4.5 Concepts of trial, outcome, sample space, event, probability and expected number of occurrences
    • 4.6 Venn diagrams, tree diagrams; combined, mutually exclusive, conditional and independent events
    • 4.7 Discrete random variables and their probability distributions; expected value
    • 4.8 The binomial distribution; its mean and variance
    • 4.9 The normal distribution and curve; normal probability and inverse normal calculations
    • 4.10 Spearman’s rank correlation coefficient; limitations of Pearson’s and Spearman’s coefficients
    • 4.11 Hypothesis testing: null/alternative hypotheses, significance levels, p-values, χ² tests, the t-test
    • 4.12 Design of valid data collection methods; reliability and validity tests (AHL only)
    • 4.13 Non-linear regression; sum of square residuals; coefficient of determination R² (AHL only)
    • 4.14 Linear transformation and combination of random variables; unbiased estimators of μ and σ² (AHL only)
    • 4.15 Linear combinations of normal random variables; the central limit theorem (AHL only)
    • 4.16 Confidence intervals for the mean of a normal population (AHL only)
    • 4.17 The Poisson distribution, its mean and variance; sum of independent Poisson distributions (AHL only)
    • 4.18 Critical values/regions; hypothesis tests for mean, proportion and correlation; Type I and II errors (AHL only)
    • 4.19 Transition matrices, regular Markov chains, steady-state and long-term probabilities (AHL only)

  5. 5. Calculus

    • 5.1 Introduction to the concept of a limit; derivative as gradient function and rate of change
    • 5.2 Increasing and decreasing functions; graphical interpretation of f′(x)
    • 5.3 Derivative of f(x) = ax^n and sums of such terms
    • 5.4 Tangents and normals at a given point
    • 5.5 Introduction to integration as anti-differentiation; definite integrals and area using technology
    • 5.6 Values where the gradient is zero; local maximum and minimum points
    • 5.7 Optimisation problems in context
    • 5.8 Approximating areas using the trapezoidal rule
    • 5.9 Derivatives of sin x, cos x, tan x, e^x, ln x, x^n (n rational); chain, product and quotient rules; related rates of change (AHL only)
    • 5.10 The second derivative; concavity and points of inflexion (AHL only)
    • 5.11 Definite and indefinite integration of further functions; integration by inspection or substitution (AHL only)
    • 5.12 Area enclosed by a curve and an axis; volumes of revolution about the x- or y-axis (AHL only)
    • 5.13 Kinematic problems involving displacement, velocity and acceleration (AHL only)
    • 5.14 Setting up and solving first order differential equations by separation of variables (AHL only)
    • 5.15 Slope fields and their diagrams (AHL only)
    • 5.16 Euler’s method for numerical solutions of first order and coupled differential equations (AHL only)
    • 5.17 Phase portraits for coupled linear differential equations; qualitative analysis by eigenvalue type (AHL only)
    • 5.18 Second order differential equations by Euler’s method and via phase portraits (AHL only)