IB Maths AA Syllabus Checklist

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This Interactive syllabus checklist for IB Maths AA is broken down by topics, then subtopics. Your progress is saved site-wide and you can update it as you move throughout the IB Maths AA syllabus.

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1: Number & Algebra 2: Functions 3: Geometry & Trigonometry 4: Stats & Probability 5: Calculus

1 - Number & Algebra

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Master IB Maths AA Topic 1: Number & Algebra. Build a solid mathematical foundation with comprehensive notes on sequences, series, exponents, logarithms, and the binomial theorem. Unlock advanced HL concepts including complex numbers, partial fractions, systems of equations, and rigorous deductive proofs.

Subtopic

Understanding

1.1: Scientific notation

Operations with numbers in the form

a \times10^{k}

where

1 \leq a < 10

and

k

is an integer

1.2: Arithmetic sequences

Arithmetic sequences and series

Use of the formulae for the

n

th term and the sum of the first

n

terms of the sequence

Use of sigma notation for sums of arithmetic sequences

1.3: Geometric sequences

Geometric sequences and series

Use of the formulae for the

n

th term and the sum of the first

n

terms of the sequence

Use of sigma notation for the sums of geometric sequences

Applications of the skills above

Sum of infinite convergent geometric sequences

1.4: Interest

Simple interest and applications of arithmetic sequences and series

Financial applications of geometric sequences and series (compound interest and annual depreciation)

Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life

1.5: Exponents & logs

Laws of exponents with integer exponents

Introduction to logarithms with base

10

and

e

Numerical evaluation of logarithms using technology

Laws of exponents with rational exponents

Laws of logarithms

Change of base of a logarithm

Solving exponential equations, including using logarithms.

1.6: 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 notations for equality and identity

1.7: Binomial theorem

Use of Pascal's triangle and

nCr

The binomial theorem

1.8: Counting principles (HL)

Counting principles, including permutations and combinations

Extension of the binomial theorem to fractional and negative indices

1.9: Partial fractions (HL)

Partial fractions

1.10: Complex numbers (HL)

Complex numbers: the number

i

, where

i^{2} = -1

Cartesian form

z = a + bi

; the terms real part, imaginary part, conjugate, modulus and argument

The complex plane

Modulus-argument (polar) form

Euler form

Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation

Complex conjugate roots of quadratic and polynomial equations with real coefficients

De Moivre's theorem and its extension to rational exponents

Powers and roots of complex numbers

1.11: Additional proofs (HL)

Proof by mathematical induction

Proof by contradiction

Use of a counterexample to show that a statement is Example: Consider the set

P

of numbers of the form not always true

1.12: Systems of equations (HL)

Solutions of systems of linear equations (a maximum of

3

equations in

3

unknowns) including cases where there is a unique solution, an infinite number of solutions or no solution

2 - Functions

Mark Complete

Excel in IB Maths AA Topic 2: Functions. Dive deep into graphing, solving, and transforming mathematical functions. Explore linear, quadratic, exponential, and logarithmic models, and master HL topics like polynomial functions, rational functions, inequalities, and advanced graph transformations.

Subtopic

Understanding

2.1: Forms of lines

Different forms of the equation of a straight line

Gradient; intercepts

Lines with gradients

m_{1}

and

m_{2}

Parallel lines

m_{1} = m_{2}

Perpendicular lines

m_{1} \times m_{2} = -1

2.2: Functions

Concept of a function, domain, range and graph

Function notation

The concept of a function as a mathematical model

Informal concept that an inverse function reverses or undoes the effect of a function

Inverse function as a reflection in the line

y = x

, and the notation

f^{-1}(x)

The graph of a function

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

Determine key features of a graph

10:

Finding the point of intersection of two curves or lines using technology

11:

Composite functions

12:

Identity function. Finding the inverse function

13:

Odd and even functions

14:

Finding the inverse function, including domain restriction

15:

Self-inverse functions

16:

The quadratic function, its graph, and

y

-intercept

17:

The quadratic function

f(x) = ax^{2} + bx + c

: its graph,

y

-intercept

(0, c)

. Axis of symmetry

18:

The form

f(x) = a(x - p)(x - q)

x

-intercepts

(p, 0)

and

(q, 0)

19:

The form

f(x) = a(x - h)^{2} + k

, vertex

(h, k)

20:

Solution of quadratic equations and inequalities. The quadratic formula

21:

The discriminant

\Delta = b^{2} - 4ac

and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots

22:

The reciprocal function

f(x) = \frac{1}{x}

x \neq 0

: its graph and self-inverse nature

23:

Rational functions of the form

f(x) = \frac{ax+b}{cx+d}

and their graphs

24:

Equations of vertical and horizontal asymptotes

25:

Exponential functions and their graphs:

f(x) = a^{x}

a > 0

f(x) = e^{x}

26:

Logarithmic functions and their graphs:

f(x) = \log_{a}{x}

x > 0

f(x) = \ln{x}

x > 0

2.3: Solving functions

Solving equations, both graphically and analytically

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach

Applications of graphing skills and solving equations that relate to real-life situations

2.4: Graph transformations

Transformations of graphs

Translations:

y = f(x) + b

;

y = f(x - a)

Reflections (in both axes):

y = -f(x)

;

y = f(-x)

Vertical stretch with scale factor

p

y = pf(x)

Horizontal stretch with scale factor

\frac{1}{q}

y = f(qx)

Composite transformations.

2.5: Advanced functions (HL)

Polynomial functions, their graphs and equations; zeros, roots and factors

The factor and remainder theorems

Sum and product of the roots of polynomial equations

Rational functions

2.6: Inequalities (HL)

Solutions of

g(x) \geq f(x)

, both graphically and analytically

Solution of modulus equations and inequalities

2.7: Additional transformations (HL)

The graphs of the functions,

y = |f(x)|

and

y = f(|x|)

y = \frac{1}{f(x)}

y = f(ax + b)

y = [f(x)]^{2}

3 - Geometry & Trigonometry

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Navigate IB Maths AA Topic 3: Geometry & Trigonometry. Master spatial reasoning with essential guides on 3D geometry, circular functions, trigonometric identities, and the unit circle. Advance your HL studies with comprehensive resources on compound angles, vector algebra, lines, and planes.

Subtopic

Understanding

3.1: Fundamentals of 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

3.2: Introduction to trigonometry

Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles

The sine and cosine rule

Area of a triangle

3.3: Applied trigonometry

Applications of right and non-right angled trigonometry, including Pythagoras's theorem

Angles of elevation and depression

Construction of labelled diagrams from written statements

3.4: Circles

The circle: radian measure of angles; length of an arc; area of a sector

3.5: Unit circle

Definition of

\cos{\theta}

\sin{\theta}

in terms of the unit circle

Definition of

\tan{\theta}

\frac{\sin{\theta}}{\cos{\theta}}

Exact values of trigonometric ratios of

0

\frac{\pi}{6}

\frac{\pi}{4}

\frac{\pi}{3}

\frac{\pi}{2}

and their multiples

Extension of the sine rule to the ambiguous case

Relationships between trigonometric functions and the symmetry properties of their graphs

3.6: Trigonometric identities

The Pythagorean identity

\cos^{2}{\theta} + \sin^{2}{\theta} = 1

Double angle identities for sine and cosine

The relationship between trigonometric ratios

Solving trigonometric equations in a finite interval, both graphically and analytically

Equations leading to quadratic equations in

\sin{x}

\cos{x}

\tan{x}

3.7: Trigonometric graphs

The circular functions

\sin{x}

\cos{x}

, and

\tan{x}

; amplitude, their periodic nature, and their graphs

Composite functions of the form

f(x) = a\sin(b(x + c)) + d

Transformations

3.8: Advanced trigonometry (HL)

Definition of the reciprocal trigonometric ratios

\sec{\theta}

\csc{\theta}

and

\cot{\theta}

Pythagorean identities

The inverse functions

f(x) = \arcsin{x}

f(x) = \arccos{x}

f(x) = \arctan{x}

; their domains and ranges; their graphs

3.9: Compound angles (HL)

Compound angle identities

Double angle identity for

\tan

3.10: Vectors (HL)

Concept of a vector; position vectors; displacement vectors

Representation of vectors using directed line segments

Base vectors

i

j

k

Components of a vector

Algebraic and geometric approaches to the sum and difference of two vectors, the zero vector

0

, the vector

-v

, multiplication by a scalar, parallel vectors, magnitude of a vector,

|v|

; unit vectors,

\frac{v}{|v|}

, position vectors, and displacement vectors

Proofs of geometrical properties using vectors.

The definition of the scalar product of two vectors

The angle between two vectors

Perpendicular vectors; parallel vectors

10:

The definition of the vector product of two vectors

11:

Properties of the vector product

12:

Geometric interpretation of

|v \times w|

3.11: Lines (HL)

Vector equation of a line in two and three dimensions:

r = a + \lambda b

The angle between two lines

Simple applications to kinematics

Coincident, parallel, intersecting and skew lines, distinguishing between these cases

Points of intersection

3.12: Planes (HL)

Vector equations of a plane:

r = a + \lambda b + \mu c

, where

b

and

c

are non-parallel vectors within the plane;

r \cdot n = a \cdot n

, where

n

is a normal to the plane and

a

is the position vector of a point on the plane

Cartesian equation of a plane

ax + by + cz = d

Intersections of: a line with a plane; two planes; three planes

Angle between: a line and a plane; two planes

4 - Stats & Probability

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Ace IB Maths AA Topic 4: Statistics & Probability. Learn to analyze data effectively with notes on data visualization, bivariate statistics, and probability rules. Master binomial and normal distributions, and tackle complex HL topics including Bayes' theorem, probability density functions, and advanced random variables.

Subtopic

Understanding

4.1: Fundamentals of statistics

Concepts of population, sample, random sample, discrete and continuous data

Reliability of data sources and bias in sampling

Sampling techniques and their effectiveness

Presentation of data (discrete and continuous): frequency distributions (tables)

Interpretation of outliers

Measures of central tendency (mean, median and mode)

Estimation of mean from grouped data

Modal class

Quartiles of discrete data

4.2: Visualising statistics

Histograms

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR)

Production and understanding of box and whisker diagrams.

4.3: Distribution properties

Measures of dispersion (interquartile range, standard deviation and variance)

Effect of constant changes on the original data.

4.4: Linear statistics

Linear correlation of bivariate data

Pearson's product-moment correlation coefficient,

r

Scatter diagrams; lines of best fit, by eye, passing through the mean point

Equation of the regression line of

y

x

Use of the equation of the regression line for prediction purposes

Interpret the meaning of the parameters,

a

and

b

, in a linear regression

y = ax + b

Equation of the regression line of

x

y

Use of the

x

y

equation for prediction purposes

4.5: Probability

Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (

U

) and event

The probability of an event

The complementary events

A

and

A'

(not

A

)

Expected number of occurrences

Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities

Combined events

Mutually exclusive events

Conditional probability

Independent events:

P(A \cap B) = P(A)P(B)

10:

Formal definition and use of the formulae:

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

for conditional probabilities, and

P(A \mid B) = P(A) = P(A \mid B')

for independent events.

4.6: Probability distributions

Concept of discrete random variables and their probability distributions

Expected value (mean), for discrete data

Applications of above concepts

Binomial distribution

Mean and variance of the binomial distribution

The normal distribution and curve

Properties of the normal distribution

Diagrammatic representation

Normal probability calculations

10:

Inverse normal calculations

11:

Standardization of normal variables (

z

-values)

12:

Inverse normal calculations where mean and standard deviation are unknown

4.7: Bayes theorem (HL)

Use of Bayes' theorem for a maximum of three events

4.8: Advanced distribution properties (HL)

Variance of a discrete random variable

Mean, variance and standard deviation of both discrete and continuous random variables

Continuous random variables and their probability density functions

Mode and median of continuous random variables

The effect of linear transformations of

X

5 - Calculus

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Conquer IB Maths AA Topic 5: Calculus. Unlock the mathematics of change with detailed study materials on limits, differentiation, integration, and kinematics. Enhance your HL calculus skills with advanced integration techniques, differential equations, implicit differentiation, l'Hôpital's rule, and Maclaurin series.

Subtopic

Understanding

5.1: Introduction to derivatives

Introduction to the concept of a limit

Derivative interpreted as gradient function and as rate of change

Increasing and decreasing functions

Graphical interpretation of

f'(x) > 0

f'(x) = 0

f'(x) < 0

Derivative of

f(x) = ax^{n}

f'(x) = anx^{n-1}

n \in \mathbb{Z}

The derivative of functions of the form

f(x) = ax^{n} + bx^{n-1} + \dots

where all exponents are integers

5.2: Applying derivatives

Tangents and normals at a given point, and their equations

The second derivative

Graphical behaviour of functions, including the relationship between the graphs of

f

f'

and

f''

Local maximum and minimum points

Testing for maximum and minimum

Points of inflexion with zero and non-zero gradients

Optimization

5.3: Advanced derivatives

Derivative of

x^{n}

(

n \in \mathbb{Q}

\sin{x}

\cos{x}

e^{x}

and

\ln{x}

Differentiation of a sum and a multiple of these functions

The chain rule for composite functions

The product and quotient rules

5.4: Introduction to integration

Introduction to integration as anti-differentiation of functions of the form

f(x) = ax^{n} + bx^{n-1} + \dots

, where

n \in \mathbb{Z}

n \neq -1

Anti-differentiation with a boundary condition to determine the constant term

Definite integrals using technology

Area of a region enclosed by a curve

y = f(x)

and the

x

-axis, where

f(x) > 0

Definite integrals, including analytical approach

Areas of a region enclosed by a curve

y = f(x)

and the

x

-axis, where

f(x)

can be positive or negative, without the use of technology

Areas between curves

5.5: Advanced integration

Indefinite integrals

Composites

Reverse chain rule

5.6: Kinematics

Kinematic problems involving displacement

s

, velocity

v

, acceleration

a

and total distance travelled

5.7: Advanced limits (HL)

Informal understanding of continuity and differentiability of a function at a point

Understanding of limits (convergence and divergence)

Higher derivatives

Evaluation of limits

Repeated use of l'Hôpital's rule

5.8: Implicit differentiation (HL)

Implicit differentiation

Related rates of change

Optimisation problems

5.9: Challenging derivatives (HL)

Derivatives of

\tan{x}

\sec{x}

\csc{x}

\cot{x}

a^{x}

\log_{a}{x}

\arcsin{x}

\arccos{x}

\arctan{x}

5.10: Challenging integration (HL)

Indefinite integrals of the derivatives of any of the above functions

The composites of any of the above with a linear function

Use of partial fractions to rearrange the integrand

Integration by substitution

Integration by parts

Repeated integration by parts

5.11: Applying integration (HL)

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

5.12: Differential equations (HL)

First order differential equations

Numerical solution of

\frac{dy}{dx} = f(x,y)

using Euler's rule

Variables separable

Homogeneous differential equation

\frac{dy}{dx} = f\left(\frac{y}{x}\right)

using the substitution

y = vx

Solution of

y' + P(x)y = Q(x)

, using the integrating factor

5.13: Maclaurin series (HL)

Maclaurin series to obtain expansions for

e^{x}

\sin{x}

\cos{x}

\ln(1 + x)

(1 + x)^{p}

p \in \mathbb{Q}

Use of simple substitution, products, integration, differentiation to obtain other series

Maclaurin series developed from differential equation

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