Math AA Syllabus

IB Math AA Syllabus

Enhanced representation of the official content syllabus from the IB Subject Guide.

[2019 (first assessment 2021) spec]

SL Content Recommended teaching hours: 120

HL Content Recommended teaching hours: +90

“…” means I was too lazy to copy the full thing from the official IB subject guide.

I bolded key words/phrases in the descriptions to make it easier to glance over. I also grouped and named subunits by topic wherever appropriate (Topic column).

Last update: 23^rd Apr 2024

#	Topic	Content	Notes
UNIT 1: Number & Algebra
1.1	Scientific form	Operations with numbers in the form where and is an integer.	Calculator or computer notation is not acceptable. For example, 5.2E30 is not acceptable and should be written as .
1.2	Arithmetic sequences and series	Use of the formulae for the ^th term and the sum of the first terms of the sequence. Use of sigma notation for sums of sequences.	Spreadsheets, GDCs and graphing software may be used to generate and display sequences in several ways. If technology is used in examinations, students will be expected to identify the first term and the common difference.
		Applications.	Examples: simple interest over a number of years.
		Analysis, interpretation, and prediction where a model is not perfectly arithmetic in real life.	Students will need to approximate common Differences.
1.3	Geometric sequences and series	Use of the formulae for the ^th term and the sum of the first terms of the sequence. Use of sigma notation for sums of sequences.	If technology is used in examinations, students will be expected to identify the first term and the ratio. Link to: models/functions in topic 2 and regression in topic 4.
1.3		Applications.	Examples: spread of disease, salary increase and decrease, and population growth.
1.4		Financial applications of geometric sequences and series: · compound interest · annual depreciation.	Examination questions may require the use of technology, including built-in financial packages. Calculate the real value of an investment with an interest rate and an inflation rate. In examinations, questions that ask students to derive the formula will not be set. Compound interest can be calculated yearly, half-yearly, quarterly, or monthly. Link to: exponential models/functions in topic 2.
1.5	Integer indices	Laws of exponents with integer exponents.
1.5	Integer indices	Introduction to logarithms with base 10 and . Numerical evaluation of logarithms using technology.	Awareness that is equivalent to , that , and .
1.6	Simple deductive proofs	Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity.	Example: Show that . Show that the algebraic generalisation of this is LHS to RHS proofs require students to begin with the left-hand side expression and transform this using known algebraic steps into the expression on the right-hand side (or vice versa). Example: Show that . Students will be expected to show how they can check a result including a check of their own results.
1.7	Logarithms and fractional indices	Laws of exponents with rational exponents.	, if is even this refers to the positive root. Example:
		Laws of logarithms. …	…
		Change of base of a logarithm:	…
		Solving exponential equations, including using logarithms.	…
1.8	Sum of convergent geometric sequences		Use of and modulus notation. Link to: geometric sequences and series (1.3).
1.9	Binomial theorem	Expansion of	Counting principles may be used in the development of the theorem.
1.9	Binomial theorem	Use of Pascal’s Triangle Combinations;	should be found using both the formula and technology. Example: Find when , using a table of values generated with technology.
1.10 (HL)	Counting principles	Permutations and Combinations	Not required: Permutations where some objects are identical. Circular arrangements.
1.10 (HL)	Counting principles	Combinations () with rational ; Extension of the binomial theorem to fractional and negative indices:	Link to: power series expansions (5.19 HL) Not required: Proof of binomial theorem
1.11 (HL)	Partial fractions		Maximum of two distinct linear terms in the denominator, with degree of numerator less than the degree of the denominator. …
1.12 (HL)	Complex numbers (basics & forms)	where . Cartesian form ; the terms real part, imaginary part, conjugate, modulus, and argument.
1.12 (HL)		The complex plane.	The complex plane is also known as the Argand diagram. Link to: vectors (3.12 HL)
1.13 (HL)		Modulus–argument (polar) form: Euler form: Sums, products, and quotients in Cartesian, polar or Euler forms and their geometric interpretation.	The ability to convert between Cartesian, modulus-argument (polar) and Euler form is expected.
1.14 (HL)		Complex conjugate roots of quadratic and polynomial equations with real coefficients.	Complex roots occur in conjugate pairs.
1.14 (HL)		De Moivre’s theorem and its extension to rational exponents. Powers and roots of complex numbers.	Includes proof by induction for the case where ; awareness that it is true for Link to: sum and product of roots of polynomial equations (2.12 HL), compound angle identities (3.10 HL).
1.15 (HL)	Proofs	Proof by Mathematical Induction.	Proof should be incorporated throughout the course where appropriate. Mathematical induction links specifically to a wide variety of topics, for example complex numbers, differentiation, sums of sequences and divisibility.
		Proof by contradiction.	…
		Use of a counterexample to show that a statement is not always true.	…
1.16 (HL)	Simultaneous equations	Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions or no solution.	…

UNIT 2: Functions
2.1	Linear functions	Different forms of the equation of a straight line. Gradient; intercepts. Lines with gradients and Parallel when: . Perpendicular when:	(gradient-intercept form). (general form). ) (point-gradient form). Calculate gradients of inclines such as mountain roads, bridges, etc
2.2	Basic functional concepts	Concept of a function, domain, range, and graph. Function notation, for example , , . The concept of a function as a mathematical model.	…
2.2	Basic functional concepts	Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line , and the notation	…
2.3	Graphing	The graph of a function; its equation .	Students should be aware of the difference between the command terms “draw” and “sketch”.
2.3		Creating a sketch from information given or a context, including transferring a graph from screen to paper. Using technology to graph functions including their sums and differences.	All axes and key features should be labelled. This may include functions not specifically mentioned in topic 2.
2.4		Determine key features of graphs.	Maximum and minimum values; intercepts; symmetry; vertex; zeros of functions or roots of equations; vertical and horizontal asymptotes using graphing technology.
2.4		Finding the point of intersection of two curves or lines using technology.
2.5	Composite functions	Composite functions.
2.5	Composite functions	Identity function. Finding the inverse function	The existence of an inverse for one-to-one functions. Link to: concept of inverse function as a reflection in the line (2.2).
2.6	Quadratic functions	The quadratic function Its graph; y-intercept ; axis of symmetry. Intercept form ; x-intercepts and . Vertex form ; vertex .	A quadratic graph is also called a parabola. Link to: transformations (2.11). Candidates are expected to be able to change from one form to another.
2.7		Solution of quadratic equations and inequalities. The quadratic formula.	Using factorization, completing the square (vertex form), and the quadratic formula. Solutions may be referred to as roots or zeros
2.7		The discriminant and the nature of the roots: two distinct real roots, two equal real roots, and no real roots	…
2.8	Fractional linear functions	The reciprocal function } Its graph and self-inverse nature.
2.8	Fractional linear functions	Rational functions of the form and their graphs. Equations of vertical and horizontal asymptotes.	Sketches should include all horizontal and vertical asymptotes and any intercepts with the axes. Link to: transformations (SL2.11). Vertical asymptote: Horizontal asymptote:
2.9	Exponential functions	Exponential functions and their graphs … Logarithmic functions and their graphs …	… Exponential and logarithmic functions are inverses of each other
2.10	Solving equations	Solving equations, both graphically and analytically.
		Use of technology to solve equations with no valid analytic approach.
		Application of graphing and solving skills to real-life situations.	Link to: exponential growth (2.9)
2.11	Trans-formations	Transformations of graphs. Translations, reflections (on axes), horizontal stretch, and vertical stretch.	Students should be aware of the relevance of the order in which transformations are performed. Dynamic graphing packages could be used to investigate these transformations.
2.11	Trans-formations	Composite transformations.	Example: Using y = x 2 to sketch y = 3x 2 + 2 Link to: composite functions (SL2.5). Not required at SL: transformations of the form
2.12 (HL)	Polynomial functions	Polynomial functions, their graphs, and equations; zeros, roots, and factors. The factor and remainder theorems.
2.12 (HL)	Polynomial functions	Sum and product of the roots of polynomial equations.	The sum is The product is Link to: complex roots of quadratic and polynomial equations (1.14 HL).
2.13 (HL)	Fractional polynomial functions	Rational functions in the form of and Their graphs and asymptotes	The reciprocal function is a particular case. Graphs should include all asymptotes (horizontal, vertical and oblique) and any intercepts with axes. Dynamic graphing packages could be used to investigate these functions. Link to: rational functions (SL 2.8).
2.14 (HL)	Properties of functions	Odd and even functions	Even: Odd: Includes periodic functions.
		Finding the inverse, including domain restriction
		Self-inverse functions
2.15 (HL)	Functional inequalities	Solutions of , Both graphically and analytically.	Graphical or algebraic methods for simple polynomials up to degree 3. Use of technology for these and other functions.
2.16 (HL)	Modulus functions	The graphs of the functions , , , , and	Dynamic graphing packages could be used to investigate these transformations.
2.16 (HL)	Modulus functions	Solution of modulus equations and inequalities.	Example:

UNIT 3: Geometry & Trigonometry
3.1	Basic geometry	The distance between two points in three-dimensional space, and their midpoint. Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane.	In SL examinations, only right-angled trigonometry questions will be set in reference to three-dimensional shapes. In problems related to these topics, students should be able to identify relevant right-angled triangles in three-dimensional objects and use them to find unknown lengths and angles.
3.2	Trigonometry I (basics)	Use of sine, cosine, and tangent ratios to find the sides and angles of right-angled triangles.	In all areas of this topic, students should be encouraged to sketch well-labelled diagrams to support their solutions. Link to: inverse functions (2.2) when finding angles
3.2		Sine rule: Cosine rule: Area:	This section does not include the ambiguous case of the sine rule.
3.3		Application of right- and non-right- angled trigonometry, including Pythagoras’s theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements.	Contexts may include use of bearings.
3.4	Angle measure	The circle: radian measure of angles; length of an arc; area of a sector.	Radian measure may be expressed as exact multiples of , or decimals.
3.5	Standard angles	Definition of and in terms of the unit circle.	Includes the relationship between angles in different quadrants. Examples:
		Definition of as .	…
		Standard angles; Exact values of trigonometric ratios of: 0, , , , , and their multiples.	…
		Extension of the sine rule to the ambiguous case.
3.6	Trigonometry II (identities and equations)	The Pythagorean Identity Double angle identities for sine and cosine.	Simple geometrical diagrams and dynamic graphing packages may be used to illustrate the double angle identities (and other trigonometric identities).
3.6		The relationship between trigonometric ratios.	…
3.7		The circular functions , , and ; amplitude, their periodic nature, and their graphs. Composite trig functions of the form .	Trigonometric functions may have domains given in degrees or radians. …
		Transformations.	…
		Real-life contexts.	Examples: height of tide, motion of a Ferris wheel.
3.8		Solving trigonometric equations in a finite interval, both graphically and analytically.	…
3.8		Equations leading to quadratics containing trig functions.	… Not required: The general solution of trigonometric equations
3.9 (HL)	Trigonometry III (more functions & identities)	Definition of the reciprocal trigonometric ratios , and . Pythagorean Identities The inverse functions , etc.; their domains, ranges, and graphs.
3.10 (HL)		Compound angle identities. Double angle identity for tangent.	Derivation of double angle identities from compound angle identities. Link to: De Moivre’s theorem (1.14 HL).
3.11 (HL)		Relationships between trigonometric functions and the symmetry properties of their graphs.	Link to: the unit circle (SL3.5), odd and even functions (HL2.14), compound angles (HL3.10).
3.12 (HL)	Vectors I (basics)	Concept of a vector; position vectors; displacement vectors. Representation of vectors using directed line segments. Base/unit vectors , , . Notation:
3.12 (HL)		Algebraic and geometric approaches to the following: - sum and difference of vectors - the zero vector, the vector - multiplication by a scalar, ; parallel vectors - magnitude of a vector, - unit vectors, - position vectors: , - displacement vectors; Proofs of geometrical properties using vectors.
3.13 (HL)		The scalar/dot product of two vectors. The angle between two vectors. Perpendicular vectors; parallel vectors.	Applications of the properties of the dot product ; ; ; . , where is the angle between and . For non-zero vectors, is equivalent to the vectors being perpendicular. For parallel vectors, .
3.14 (HL)	Vector linear equations	Vector equation of a line in two and three dimensions: Parametric form: Cartesian form:	Relevance of (initial position) and (direction).
		The angle between two lines.	Using the scalar product of the two direction vectors.
		Simple applications to kinematics.	Interpretation of as time and as velocity, with representing speed.
3.15 (HL)		Coincident, parallel, intersecting, and skew lines, distinguishing between these cases. Points of intersection.	Skew lines are non-parallel lines that do not intersect in three-dimensional space.
3.16 (HL)	Vectors II (cross product)	Vector/cross product.	“Vector product” = “Cross product”. , where is the angle between and , and is the unit normal vector whose direction is given by the right hand screw rule.
		Properties of the cross product.	; ; ; . For non-zero vectors, is equivalent to the vectors being parallel.
		Geometric interpretation of .	Use of to find the area of a parallelogram (and hence a triangle).
3.17 (HL)	Planes (not birds)	Vector equations of a plane: where and are non-parallel vectors within the plane. where is a normal to the plane and is the position vector of a point on the plane. Cartesian form:
3.18 (HL)	Planes (not birds)	Intersections of: a line with a plane; two planes; three planes. Angle between: a line and a plane; two planes.	Finding intersections by solving equations; geometrical interpretation of solutions. Link to: solutions of systems of linear equations (HL 1.16).

UNIT 4: Statistics & Probability
4.1	Sampling	Concepts of population, sample, random sample, discrete and continuous data.	This is designed to cover the key questions that students should ask when they see a data set/ analysis.
		Reliability of data sources and bias in sampling.	Dealing with missing data, errors in the recording of data.
		Interpretation of outliers.	Outlier is defined as a data item which is more than 1.5 × interquartile range (IQR) from the nearest quartile. Awareness that, in context, some outliers are a valid part of the sample, but some outlying data items may be an error in the sample. Link to: box and whisker diagrams (4.2) and measures of dispersion (4.3).
		Sampling techniques and their effectiveness.	Simple random, convenience, systematic, quota and stratified sampling methods.
4.2	Statistical graphs	Presentation of data (discrete and continuous): frequency distributions (tables).	Class intervals will be given as inequalities, without gaps.
		Histograms. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range, and interquartile range (IQR).	Frequency histograms with equal class intervals. Not required: Frequency density histograms.
		Production and understanding of box and whisker diagrams.	Use of box and whisker diagrams to compare two distributions, using symmetry, median, interquartile range, or range. Outliers should be indicated with a cross. Determining whether the data may be normally distributed by consideration of the symmetry of the box and whiskers.
4.3	Statistics	Measures of central tendency (mean, median and mode). Estimation of mean from grouped data.	Calculation of mean using formula and technology. Students should use mid-interval values to estimate the mean of grouped data.
		Modal class.	For equal class intervals only.
		Measures of dispersion (interquartile range, standard deviation, and variance).	Calculation of standard deviation and variance of the sample using only technology; however, hand calculations may enhance understanding. Variance is the square of the standard deviation.
		Effect of constant changes on the original data.	Examples: If three is subtracted from the data items, then the mean is decreased by three, but the standard deviation is unchanged. If all the data items are doubled, the mean is doubled, and the standard deviation is also doubled.
		Quartiles of discrete data.	Using technology. Awareness that different methods for finding quartiles exist and therefore the values obtained using technology and by hand may differ.
4.4	Bivariate analysis	Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient, .	Technology should be used to calculate . However, hand calculations of r may enhance understanding. Critical values of will be given where appropriate. Students should be aware that Pearson’s product moment correlation coefficient () is only meaningful for linear relationships.
		Scatter diagrams; lines of best fit, by eye, passing through the mean point.	Positive, zero, negative; strong, weak, no correlation. Students should be able to make the distinction between correlation and causation and know that correlation does not imply causation.
		Equation of the regression line of on . Use of the equation of the regression line for prediction purposes. Interpret the meaning of the parameters, and , in a linear regression	Technology should be used to find the equation. Students should be aware: · of the dangers of extrapolation · that they cannot always reliably make a prediction of from a value of , when using a on line.
4.5	Probability basics	Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space () and event. The probability of an event is . The complementary events and (not-).	Sample spaces can be represented in many ways, for example as a table or a list. Experiments using coins, dice, cards and so on, can enhance understanding of the distinction between experimental (relative frequency) and theoretical probability. Simulations may be used to enhance this topic.
4.5	Probability basics	Expected number of occurrences.	Example: If there are 128 students in a class and the probability of being absent is 0.1, the expected number of absent students is 12.8.
4.6	Probability calculations	Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.
		Combined events: . Mutually exclusive events: .	The non-exclusivity of “or”.
		Conditional probability: .	An alternate form of this is: . Problems can be solved with the aid of a Venn diagram, tree diagram, sample space diagram or table of outcomes without explicit use of formulae. Probabilities with and without replacement.
		Independent events: .
4.7	Discrete random variables I (basics)	Concept of discrete random variables and their probability distributions. Expected value (mean), for discrete data. Applications.	…
4.8	Binomial distribution	Binomial distribution. Mean and variance of the binomial distribution.	Situations where the binomial distribution is an appropriate model. In examinations, binomial probabilities should be found using available technology. Not required: Formal proof of mean and variance. Link to: expected number of occurrences (4.5).
4.9	Normal distribution	The normal distribution and curve. Properties of the normal distribution. Diagrammatic representation.	Awareness of the natural occurrence of the normal distribution. Students should be aware that approximately 68% of the data lies between μ ± σ, 95% lies between μ ± 2σ and 99.7% of the data lies between μ ± 3σ.
		Normal probability calculations.	Probabilities and values of the variable must be found using technology
		Inverse normal calculations	For inverse normal calculations mean and standard deviation will be given. This does not involve transformation to the standardized normal variable .
4.10	Linear regression	Equation of the regression line of on .
4.10	Linear regression	Use of the equation for prediction purposes.	Students should be aware that they cannot always reliably make a prediction of from a value of , when using an on line.
4.11	Conditional probabilities	Formal definition and use of the formulae for conditional probabilities, and for independent events.	An alternate form of this is: . Testing for independence.
4.12	Standardized normal variables	Standardization of normal variables (-values).	Probabilities and values of the variable must be found using technology. The standardized value () gives the number of standard deviations from the mean.
4.12	Standardized normal variables	Inverse normal calculations where mean and standard deviation are unknown.	Use of -values to calculate unknown means and standard deviations.
4.13 (HL)	Bayes’ theorem	Use of Bayes’ theorem for a maximum of three events.	Link to: independent events (4.6)
4.14 (HL)	Discrete random variables II (statistics)	Variance of a discrete random variable.	Link to: discrete random variables (4.7)
		Continuous random variables and their probability density functions.	…
		Mode and median of continuous random variables.	…
		Mean, variance and standard deviation of both discrete and continuous random variables.	…
		The effect of linear transformations of .	…

UNIT 5: Calculus
5.1	Calculus fundamentals	Introduction to the concept of a limit.	Estimation of the value of a limit from a table or graph.
5.1		Derivative interpreted as gradient function and as rate of change.	Forms of notation …
5.2		Increasing and decreasing functions. Graphical interpretation of , ,	Identifying intervals on which functions are increasing or decreasing.
5.3	Basic differentiation	Power rule (integer powers): Derivative of is . Derivatives of (Laurent) polynomials.
5.4	Basic differentiation	Tangents and normals at a given point, and their equations.	Use of both analytic approaches and technology.
5.5	Basic integration	Introduction to integration as anti-differentiation… Of (Laurent) polynomials ().
		Anti-differentiation with a boundary condition to determine the constant term.	Example: if and when , then .
		Definite integrals using technology. Area of a region enclosed by a curve and the x-axis, where .	Students are expected to first write a correct expression before calculating the area, … The use of dynamic geometry or graphing software is encouraged in the development of this concept.
5.6	Analytical differentiation methods	Power rule for Derivatives of , , and . Differentiation of sums and multiples of these functions.
		Chain rule.	Example: …
		Product and quotient rules.	Link to: composite functions (SL2.5).
5.7	Applications of derivatives	The second derivative. Graphical behaviour of functions, including the relationship between the graphs of , and .	…
5.8		Local maximum and minimum points. Testing for maximum and minimum.	…
		Optimization.	Examples of optimization may include profit, area, and volume.
		Points of inflexion with zero and non-zero gradients.	At a point of inflexion, and changes sign (concavity change). …
5.9	Kinematics	Kinematic problems involving displacement , velocity , acceleration , and total distance travelled.	; Displacement from to is given by . Distance between to is given by . Speed is the magnitude of velocity.
5.10	Analytical integration methods	Reverse-power rule for . Indefinite integrals of , , , and .
		The composites of any of these with the linear function .	Example: …
		Integration by inspection (reverse chain rule) or by substitution for expressions of the form:	Examples: …
5.11	Definite integrals	Definite integrals, including analytical approach.	…
5.11	Definite integrals	Areas of a region enclosed by a curve and the x-axis, where can be positive or negative, without the use of technology. Areas between curves.
5.12 (HL)	Limits and differentiation	Informal understanding of continuity and differentiability of a function at a point.	In examinations, students will not be asked to test for continuity and differentiability.
		Understanding of limits (convergence and divergence). Definition of derivative from first principles …	Link to: infinite geometric sequences (SL1.8). Use of this definition for polynomials only.
		Higher derivatives.	Familiarity with the notations and . Link to: proof by mathematical induction (1.15HL).
5.13 (HL)	L’Hôpital’s rule	Evaluation of limits of the form and using l’Hôpital’s rule or the Maclaurin series.	The indeterminate forms and . Example: . Link to: horizontal asymptotes (SL2.8).
5.13 (HL)	L’Hôpital’s rule	Repeated use of l’Hôpital’s rule.
5.14 (HL)	Related rates	Implicit differentiation. Related rates of change. Optimisation problems.	Appropriate use of the chain rule or implicit differentiation, including cases where the optimum solution is at the end point.
5.15 (HL)	Advanced analytical calculus	Derivatives of all trig and exponential functions and their inverses.	, , , , , , , , and .
		Indefinite integrals of the derivatives of any of the above functions. The composites of any of these with a linear function.	Indefinite integral interpreted as a family of curves. Examples:
		Use of partial fractions to rearrange the integrand.	Link to: partial fractions (HL1.11)
5.16 (HL)		Integration by substitution.	On examination papers, substitutions will be provided if the integral is not of the form . Link to: integration by substitution (SL5.10).
		Integration by parts.	…
		Repeated integration by parts.	…
5.17 (HL)	Area and volume	Area of the region enclosed by a curve and the y axis in a given interval. Volumes of revolution about the x-axis or y-axis.
5.18 (HL)	Differential equations	First order differential equations (1^st order linear ODE). Numerical solution of using Euler’s method.	, where is a constant.
		Separable differential equations. (Solving them.)	Example: the logistic equation Link to: partial fractions (HL1.11) and use of partial fractions to rearrange the integrand (HL5.15).
		Homogeneous differential equations; Solving 1^st order linear ODEs of the form using the substitution .
		Integrating factor method; Solving 1^st order linear ODEs of the form …	…using an “integrating factor” . For certain , .
5.19 (HL)	Maclaurin series	Maclaurin series to obtain expansions for , , , and .
		Use of simple substitution, products, integration, and differentiation to obtain other series.	Example: for substitution: replace with to define the Maclaurin series for . Example: the expansion of .
		Maclaurin series developed from differential equations.
END OF SYLLABUS :)
#	Topic	Content	Notes