IB Maths AA HL

This topic explores the representation of patterns, equivalencies, and generalizations to model real-world situations and solve mathematical problems using numerical concepts and algebraic abstraction.

• Scientific notationStudents learn to perform operations with numbers expressed in scientific notation (form a × 10^k), understanding the constraints on 'a' and 'k'.
• Arithmetic sequences and seriesThis subtheme covers the properties of arithmetic sequences and series, including the use of formulae for the nth term, the sum of n terms, and sigma notation.
• Geometric sequences and seriesStudents will study geometric sequences and series, applying formulae for the nth term, the sum of n terms, and using sigma notation.
• Financial applicationsThis subtheme focuses on the real-world application of geometric sequences to financial situations, including compound interest and annual depreciation.
• Exponents and logarithmsStudents are introduced to the laws of exponents with integer and rational exponents, and the fundamental laws and properties of logarithms.
• Deductive proofThis subtheme introduces simple deductive proof, including numerical and algebraic proofs, and the proper notation for equality and identity.
• Infinite geometric seriesStudents learn to calculate the sum of infinite convergent geometric sequences, utilizing the condition for convergence.
• Binomial theoremThis subtheme covers the expansion of binomials of the form (a + b)^n for positive integer values of n, using Pascal's triangle and combinations.
• Counting principles and permutationsStudents learn advanced counting principles, including permutations and combinations, and extend the binomial theorem to fractional and negative indices.
• Partial fractionsThis subtheme introduces the decomposition of rational expressions into partial fractions, limited to cases with two distinct linear terms in the denominator.
• Complex numbersStudents are introduced to complex numbers, including the imaginary unit 'i', Cartesian form (z = a + bi), and representation on the complex (Argand) plane.
• Polar and Euler forms of complex numbersThis subtheme covers the modulus-argument (polar) and Euler forms of complex numbers, and operations like sums, products, and quotients in these forms.
• De Moivre's theoremStudents will learn and apply De Moivre's theorem to find powers and roots of complex numbers, and understand its extension to rational exponents.
• Advanced proof techniquesThis subtheme covers formal proof methods, including proof by mathematical induction, proof by contradiction, and the use of counterexamples.
• Systems of linear equationsStudents learn to solve systems of linear equations with up to three variables, considering cases with unique, infinite, or no solutions.

This topic focuses on representing and modeling real-life events using functions, equations, and graphs, and exploring the relationships between variables.

• Linear functionsStudents study different forms of the equation of a straight line, including gradient-intercept, general, and point-gradient forms, and the properties of parallel and perpendicular lines.
• Function conceptsThis subtheme introduces the core concepts of a function, including domain, range, graph, function notation, and the informal concept of an inverse function.
• Graphing and analyzing functionsStudents learn to create and interpret graphs of functions, identify key features like intercepts and asymptotes, and find points of intersection using technology.
• Composite and inverse functionsThis subtheme covers the creation of composite functions and the process of finding the inverse function f⁻¹(x) algebraically and graphically.
• Quadratic functionsStudents analyze quadratic functions in various forms (standard, vertex, intercept), identifying features like the axis of symmetry, vertex, and intercepts.
• Quadratic equations and inequalitiesThis subtheme focuses on solving quadratic equations and inequalities using methods such as factorization, completing the square, and the quadratic formula, and analyzing the nature of roots using the discriminant.
• Rational and reciprocal functionsStudents study the graphs and properties of the reciprocal function and simple rational functions, including identifying vertical and horizontal asymptotes.
• Exponential and logarithmic functionsThis subtheme explores the graphs and properties of exponential and logarithmic functions, including their inverse relationship.
• Solving equations with technologyStudents learn to solve various equations both graphically and analytically, with an emphasis on using technology for complex cases.
• Transformations of graphsThis subtheme covers the transformation of function graphs, including translations, reflections, and vertical/horizontal stretches.
• Polynomial functionsStudents will analyze polynomial functions, their graphs, roots, and factors, and apply the factor and remainder theorems.
• Advanced rational functionsThis subtheme extends the study of rational functions to more complex forms, including those with oblique asymptotes.
• Properties of functionsStudents learn to classify functions as odd or even, find inverse functions with domain restrictions, and identify self-inverse functions.
• Solving function inequalitiesThis subtheme focuses on solving inequalities of the form g(x) ≥ f(x) using both graphical and analytical methods for polynomials up to degree 3.
• Modulus functionsStudents explore the graphs of functions involving the absolute value, such as y = |f(x)| and y = f(|x|), and solve modulus equations and inequalities.

This topic provides tools for analysis, measurement, and transformation of quantities in two and three dimensions, enhancing spatial awareness.

• 3D coordinate geometryStudents learn to calculate the distance and midpoint between two points in three-dimensional space and solve problems involving 3D solids.
• Trigonometry of trianglesThis subtheme covers the use of sine, cosine, and tangent ratios in right-angled triangles, and the application of the sine and cosine rules for non-right-angled triangles.
• Radian measure and circular functionsStudents are introduced to radian measure, calculating arc length and sector area, and defining trigonometric ratios in terms of the unit circle.
• Trigonometric identities and equationsThis subtheme focuses on the Pythagorean identity (sin²θ + cos²θ = 1), double angle identities, and their use in solving trigonometric equations.
• Graphs of trigonometric functionsStudents will analyze the graphs of sine, cosine, and tangent functions, including their amplitude and periodic nature, and apply transformations to them.
• Reciprocal and inverse trigonometric functionsThis subtheme introduces the reciprocal trigonometric ratios (sec, csc, cot), related Pythagorean identities, and the inverse trigonometric functions (arcsin, arccos, arctan).
• Compound angle identitiesStudents learn and apply compound angle identities for sine, cosine, and tangent to solve problems and derive other identities.
• Vector concepts and operationsThis subtheme introduces vector concepts including position and displacement vectors, representation using base vectors (i, j, k), and algebraic operations such as addition, subtraction, and scalar multiplication.
• Scalar (dot) productStudents learn the definition and properties of the scalar product of two vectors and use it to find the angle between vectors and determine if they are perpendicular.
• Vector equation of a lineThis subtheme covers the vector, parametric, and Cartesian forms of the equation of a line in two and three dimensions.
• Vector (cross) productStudents learn the definition and properties of the vector product of two vectors and use it to find a vector perpendicular to two given vectors and calculate the area of a parallelogram.
• Vector equations of a planeThis subtheme covers the vector and Cartesian equations of a plane in three-dimensional space.
• Intersections of lines and planesStudents learn to find the intersections of lines and planes by solving systems of equations and to determine the geometric relationship between them (coincident, parallel, intersecting, skew).

This topic concerns the collection, analysis, and interpretation of data, and uses probability theory to quantify likelihood, evaluate risk, and make predictions.

• Statistical concepts and samplingStudents are introduced to fundamental concepts such as population, sample, and data types, as well as sampling techniques and sources of bias.
• Data presentationThis subtheme covers methods of presenting data, including frequency distributions, histograms, cumulative frequency graphs, and box and whisker diagrams.
• Measures of central tendency and dispersionStudents learn to calculate and interpret measures of central tendency (mean, median, mode) and dispersion (range, interquartile range, standard deviation, variance).
• Linear correlation and regressionThis subtheme focuses on analyzing bivariate data using scatter diagrams, calculating Pearson's product-moment correlation coefficient, and finding the equation of the regression line of y on x.
• Probability fundamentalsStudents learn basic probability concepts, including sample space, events, complementary events, and calculating theoretical and experimental probabilities.
• Combined and conditional probabilityThis subtheme covers calculating probabilities of combined events using Venn and tree diagrams, and introduces conditional and independent events.
• Discrete random variablesStudents are introduced to discrete random variables, their probability distributions, and the calculation of expected value (mean).
• Binomial distributionThis subtheme explores the binomial distribution as a model for a fixed number of independent trials, and its mean and variance.
• Normal distributionStudents study the properties of the normal distribution and curve, and use technology to perform probability and inverse calculations.
• Standardized normal variablesThis subtheme covers the standardization of normal variables (z-values) to compare distributions and solve for unknown means or standard deviations.
• Bayes' theoremStudents learn to use Bayes' theorem to solve problems involving conditional probability for a maximum of three events.
• Continuous random variablesThis subtheme introduces continuous random variables and their probability density functions, including the calculation of mode, median, mean, and variance.

This topic explores rates of change between variables and the accumulation of limiting areas, enabling the modeling and analysis of real-world problems and the behavior of functions.

• Limits and first principlesStudents are introduced to the concept of a limit and the definition of a derivative from first principles, establishing the foundation of differential calculus.
• Differentiation rulesThis subtheme covers the rules for differentiating functions, including polynomials, sums, products, quotients, and composite functions (chain rule).
• Derivatives of standard functionsStudents learn to find the derivatives of key functions, including x^n, sin(x), cos(x), e^x, and ln(x).
• Applications of differentiationThis subtheme focuses on using derivatives to find tangents and normals, analyze function behavior (increasing/decreasing), and solve optimization problems.
• The second derivativeStudents learn to find and interpret the second derivative to determine concavity and identify points of inflexion.
• Introduction to integrationThis subtheme introduces integration as anti-differentiation, indefinite integrals of standard functions, and finding the constant of integration from a boundary condition.
• Definite integrals and areaStudents learn to calculate definite integrals and apply them to find the area of regions enclosed by curves and the x-axis, or between two curves.
• KinematicsThis subtheme applies calculus to solve kinematic problems involving displacement, velocity, and acceleration.
• Evaluation of limits with L'Hôpital's ruleStudents learn to evaluate limits of indeterminate forms (0/0 and ∞/∞) using l'Hôpital's rule.
• Advanced differentiationThis subtheme covers advanced techniques such as implicit differentiation and related rates of change, and finds derivatives of a wider range of trigonometric, logarithmic, and inverse trigonometric functions.
• Advanced integration techniquesStudents learn advanced methods of integration, including integration by substitution and integration by parts.
• Applications of integrationThis subtheme extends the application of integration to finding the area enclosed by a curve and the y-axis, and calculating volumes of revolution.
• Differential equationsStudents are introduced to first-order differential equations, including solving them by separating variables, using an integrating factor, and applying Euler's method for numerical solutions.
• Maclaurin seriesThis subtheme covers the derivation and use of Maclaurin series to obtain expansions for standard functions like e^x, sin(x), and cos(x).