Lessons | IB Math AAHL

Concept Lessons

Access comprehensive explorations for all concepts in your objective, including problems, helpful visuals, and AI features.

What you'll find in every lesson:

High-quality explanations: Each lesson is carefully crafted to help you truly understand the concept, not just memorize it.

Checkpoint exercises: Practice as you learn with interactive problems designed to reinforce your understanding.

AI-powered discussions: Engage with thought-provoking questions that help reveal your misconceptions.

What you'll find in every lesson:

High-quality explanations: Each lesson is carefully crafted to help you truly understand the concept, not just memorize it.

Checkpoint exercises: Practice as you learn with interactive problems designed to reinforce your understanding.

AI-powered discussions: Engage with thought-provoking questions that help reveal your misconceptions.

Concept Lessons

Access comprehensive explorations for all concepts in your objective, including problems, helpful visuals, and AI features.

100 Lessons Available

What you'll find in every lesson:

High-quality explanations: Each lesson is carefully crafted to help you truly understand the concept, not just memorize it.

Checkpoint exercises: Practice as you learn with interactive problems designed to reinforce your understanding.

AI-powered discussions: Engage with thought-provoking questions that help reveal your misconceptions.

What you'll find in every lesson:

High-quality explanations: Each lesson is carefully crafted to help you truly understand the concept, not just memorize it.

Checkpoint exercises: Practice as you learn with interactive problems designed to reinforce your understanding.

AI-powered discussions: Engage with thought-provoking questions that help reveal your misconceptions.

Cartesian plane & lines

Distance between two points, midpoint of two points, and finding gradient using two points

Distance, Midpoint, & Gradient

Gradient-intercept form, point-gradient form, vertical lines, horizontal lines, standard form of a line

Equations of a Line

Parallel and perpendicular lines, intersection of 2 straight lines, solving a two-variable system of equations

Line Intersections & Systems of Equations

Quadratics

Quadratic expressions and functions in general form ax2+bx+c, factored form a(x−α)(x−β), and vertex form a(x−h)2+k, including factoring, completing the square, the quadratic formula, roots/x-intercepts, vertex, axis of symmetry x=−2ab, and concavity from the sign of a.

Foundations of Quadratics

Using the discriminant Δ=b2−4ac to determine the number of real roots, Vieta's formulas α+β=−ab and αβ=ac, solving quadratic inequalities, and optimization problems modeled by quadratic functions.

Applications of Quadratics

Exponents & Logarithms

Exponential notation and algebra, including laws for zero and negative exponents, multiplying and dividing powers with the same base, products and quotients with the same exponent, powers of powers, equating exponents in exponential equations, and rationalizing denominators when needed.

Exponential Algebra

Square roots and nth roots, including n√a=an1, rational exponents anm, simplifying radicals, roots of negative numbers, and rationalizing denominators using roots or conjugates.

Radicals and Roots

Logarithm algebra including the definition logab=x⇔ax=b, base 10 and natural logs, evaluating logs exactly or with technology, and using the rules logax+logay=loga(xy), logax−logay=loga(x/y), loga(xm)=mlogax, change of base, and logarithms to solve exponential equations.

Logarithm algebra

Exponential functions f(x)=ax and logarithmic functions f(x)=logax, their domains and ranges, inverse relationship and symmetry in y=x, and graphing/making sense of exponential growth and decay models such as cax+k or Abt+c.

Exp & Log functions (Plus Only)

Function Theory

The concept of a function as a mathematical machine, the notation f(x)=…, and function tables / diagrams.

Functions and their properties

Graphing functions with a calculator, characteristics of a function, even and odd functions, x and y intercepts, horizontal and vertical asymptotes, maxima and minima

Function Graphs

The domain of a function is the set of possible inputs, and the range is the set of possible outputs.

Domain and range

Composing functions like f∘g

Function Composition (Plus Only)

Even functions satisfy f(−x)=f(x) and have symmetry in the y-axis, while odd functions satisfy f(−x)=−f(x) and have origin symmetry.

Even and odd functions

The concept of an inverse function, its graph as a reflection in the line y=x, finding the inverse of a specific value, and domain and range of inverse functions. When inverse functions exist, and how to find them.

Inverse Functions

Rounding & Error

Rounding, significant figures, decimal places

Rounding Numbers

Converting numbers to and from scientific notation (a×10k)

Scientific Notation

Sequences & Series

Introduction to the definition, properties, and general term of an arithmetic sequence

Arithmetic Sequences

Introduction to definition, identification, and general term of geometric sequences

Geometric Sequences

Definition of series, formula for arithmetic series

Arithmetic Series (Plus Only)

Summation notation, properties of sums

Σ summation notation (Plus Only)

Finite geometric series with sum Sn=1−ru1(1−rn), and infinite geometric series convergence when ∣r∣<1 so that S∞=1−ru1.

Geometric Series (Plus Only)

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Working with sequences on a calculator, computations and applications of a calculator

Sequence Mode (Calculator) (Plus Only)

Thumbnail for Compounding (Appreciation & Depreciation)

Compound growth, depreciation, interest, inflation, real value

Compounding (Appreciation & Depreciation) (Plus Only)

Counting & Binomials

Pascal's triangle and binomial coefficients, using factorials and nCr=r!(n−r)!n! to find combinations and the coefficients in binomial expansions.

Pascal's Triangle and nCr

Expansion of (a+b)n using binomial coefficients, including the general term (nr)an−rbr and finding the coefficient of a specified term.

Binomial Theorem (Plus Only)

Expansion of (a+b)n using binomial coefficients, including the general term (nr)an−rbr and finding the coefficient of a specified term.

Counting (Plus Only)

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Counting arrangements by splitting into subcases and using complementary counts, including treating items that must stay together as one block and placing items in separate slots to keep them apart.

Counting with Restrictions (Plus Only)

Proof and Reasoning

Strict logical proofs by deduction, including parity arguments with even and odd integers written as n=2k and n=2k+1, and direct proofs that two expressions are equivalent by showing LHS≡RHS for all variables.

Proof by deduction

Proof by induction for statements about integers, divisibility, inequalities, summations and derivatives: establish a base case, assume the result for n=k, then use the inductive hypothesis to prove n=k+1, often by comparing successive cases, splitting a sum, or using the product rule.

Proof by induction (Plus Only)

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Proof by contradiction, where assuming the negation of a statement leads to an inconsistency, and counterexamples used to disprove claims, often with examples involving prime numbers and rational numbers.

Contradiction and Counterexamples (Plus Only)

Transformations & asymptotes

Linear transformations of graphs and points, including translations by vectors, vertical and horizontal stretches, vertical and horizontal translations, reflections in the axes, and combined transformations such as y=f(ax+b).

Linear Transformations

Absolute value (modulus) ∣x∣, and solving modulus equations and inequalities such as ∣f(x)∣=x−3 and g(x)≥f(x) algebraically or graphically, including changing the inequality direction when multiplying by a negative number.

Modulus & Inequalties (Plus Only)

Graphs and asymptotes of rational functions, including the reciprocal function f(x)=x1, linear rational functions cx+dax+b, forms with oblique asymptotes from polynomial division, and rationals with quadratic denominators.

Rational functions (Plus Only)

Graph transformations of functions including combined horizontal translations and stretches f(ax+b), and the effects of modulus, squaring and reciprocals: f(∣x∣), ∣f(x)∣, [f(x)]2, and f(x)1.

Further Transformations (Plus Only)

Polynomials

Polynomial definition in x with coefficients and degree, identifying terms like akxk, finding degrees of polynomial products and specific coefficients in products, and matching coefficients in polynomial identities that hold for all x∈R.

Polynomial Basics

Polynomial division with quotient and remainder, including long division by a linear divisor, the degree condition that degR<degD, and the remainder and factor theorems (P(a) is the remainder on division by (x−a), and (x−a) is a factor iff P(a)=0).

Polynomial division, factors and remainders

Polynomial graphs and how the sign of the leading coefficient affects the broad shape of cubic and quartic graphs, and how real roots appear as x-intercepts where the curve touches or crosses the axis depending on whether the root is repeated or unique.

Polynomial graphs

A degree n polynomial has n roots counting multiplicity, and its roots may be real or complex; for a polynomial with real coefficients, complex roots occur in conjugate pairs, and the sum and product of roots are given by the coefficient relationships an−an−1 and an(−1)na0.

Polynomial roots

2D & 3D Geometry

Pythagoras's theorem, SOHCAHTOA, finding side length from angles

Right angled triangles

Area of non-right-angled triangles using A=21absinC, the sine rule sinAa=sinBb=sinCc, the cosine rule c2=a2+b2−2abcosC, and the ambiguous case when solving triangles.

Non-right-angled triangles (Plus Only)

Thumbnail for Circles: Radians, arcs and sectors

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=rθ, A=21θr2, C=2πr, and A=πr2.

Circles: Radians, arcs and sectors (Plus Only)

Spheres, cylinders, prisms, right cones, right pyramids, and combinations of solids

3D space & solids (Plus Only)

Angles of elevation and depression, true bearings, projections and more

Applied triangle geometry

Trig equations & identities

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=rθ, A=21θr2, C=2πr, and A=πr2.

Circles: Radians, arcs and sectors (Plus Only)

The unit circle: using cosθ and sinθ as the x- and y-coordinates of a point on the circle, key exact values and quadrants, symmetry rules, periodicity, tangent as tanθ=cosθsinθ, and sin2θ+cos2θ=1.

The Unit Circle

Radian measure, the sine and cosine functions with domain x∈R and range (−1,1), the tangent function tanx=cosxsinx with its asymptotes and roots, and sinusoidal functions of the form asin(b(x+c))+d or acos(b(x+c))+d.

Trigonometric Functions

Reciprocal trig functions secθ=cosθ1, cosecθ=sinθ1 and cotθ=tanθ1, including the graph, domain and range of secx, cosecx and cotx.

Reciprocal trig functions (Plus Only)

Solving trigonometric equations algebraically in a specified domain, including sinθ=a, cosθ=a, tanθ=a, equations with arguments of the form ax+b, and trigonometric quadratics using identities such as sin2θ+cos2θ=1.

Trig Equations (Plus Only)

The inverse trig functions arcsin, arccos and arctan, including their principal-value domains and ranges and the graphs as reflections of the restricted trig functions in y=x.

Inverse trig functions (Plus Only)

Trigonometric identities including sin2θ+cos2θ=1, sin2θ=2sinθcosθ, and the equivalent forms of cos2θ: cos2θ−sin2θ, 2cos2θ−1, and 1−2sin2θ.

Trigonometric Identities (Plus Only)

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Further trigonometric identities, including the compound angle formulas for sin(A±B), cos(A±B) and tan(A±B), the double angle identity tan2θ=1−tan2θ2tanθ, and the identities 1+tan2θ=sec2θ and 1+cot2θ=cosec2θ.

Further trigonometric identities (Plus Only)

Complex Numbers

Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).

Cartesian form

The complex conjugate: z=a+bi⟹z∗=a−bi, its properties, and using it to simplify fractions with complex denominators.

Complex conjugate

Equating real and imaginary parts to solve complex equations

Solving complex equations

Introduction, definition, and properties of the complex modulus

Complex Modulus (Plus Only)

rcisθ and eiθ

Complex Argument (Plus Only)

Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.

Polar Form

Using De Moivre's Theorem to find powers of complex numbers in polar form, z=reiθ, so that zn=rneinθ=rncis(nθ), and applying the argument rule arg(zw)=arg(z)+arg(w).

Powers of Complex Numbers (Plus Only)

Vectors

3D and 2D vector notation with base vectors and position vectors, including zero and negative vectors, scalar multiples, unit vectors, vector magnitude, addition and subtraction by components or triangle law, displacement vectors, and parallel vectors.

Vector arithmetic & geometry

The scalar (aka dot) product, and its connection to angles between vectors.

Scalar product

The vector (aka cross) product u×v and its connection to areas in 3D space.

Vector Product

Thumbnail for Equations of a vector line

Equations of a vector line in 3D using a point and direction vector, in vector form r=a+λb, parametric form x=x0+λl, y=y0+λm, z=z0+λn, Cartesian form lx−x0=my−y0=nz−z0, and the angle between lines from their direction vectors.

Equations of a vector line (Plus Only)

Thumbnail for Coincident, Parallel, Intersecting & Skew Lines

Classifying vector lines in 3D as coincident, parallel, intersecting or skew using direction vectors and point checks, with the angle between lines found by θ=cos−1(∣b1∣∣b2∣b1⋅b2).

Coincident, Parallel, Intersecting & Skew Lines (Plus Only)

Equations of a plane in 3D space using vector form r=a+λb+μc, scalar product form r⋅n=a⋅n, and Cartesian form n1x+n2y+n3z=d, with a point and two direction vectors or a normal vector.

Equations of a plane (Plus Only)

Thumbnail for Angles and intersections with planes

Finding angles between lines, between a line and a plane, and between two planes using direction vectors and normals, plus intersections of lines and planes and systems of three planes in 3D.

Angles and intersections with planes (Plus Only)

Differentiation

The basic idea of a limit x→alimf(x) from tables and graphs, slope as a limit, rate of change and gradient, using the derivative definition f′(x)=h→0limhf(x+h)−f(x), derivatives of xn, and linearity for sums and scalar multiples.

Limits and Derivatives

Derivatives of xr,ex,lnx,sinx and cosx. The chain, product and quotient rules, as well as the derivatives of reciprocal trig functionslogax, and ax.

Differentiation rules

Definitions of tangent and normal, finding tangent and normal lines to a function

Tangents and normals (Plus Only)

Thumbnail for Applications of the First Derivative

Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation

Applications of the First Derivative (Plus Only)

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Second derivatives f′′, using the sign of f′′ to determine concavity, locate inflexion points where f′′(x)=0 and changes sign, and classify stationary points with the second derivative test.

Second Derivatives and Applications (Plus Only)

Differentiating both sides of an equation with respect to x to find dxdy for relations not written as y=f(x), using the product and chain rules, and finding horizontal and vertical tangents and normals from the resulting derivative.

Implicit differentiation

Using implicit differentiation and the chain rule to relate changing quantities, including dtdy=dxdy⋅dtdx, and applying this to volume, distance, and angle rates.

Related Rates (Plus Only)

The nth derivative of a function, written f(n)(x) or dxndny, found by differentiating n times.

n^th Derivative (Plus Only)

Evaluating indeterminate limits of the form x→alimg(x)f(x) when both numerator and denominator approach 0 or ∞, by replacing them with x→alimg′(x)f′(x) and repeating if necessary.

L'Hôpital's rule (Plus Only)

Integration

Evaluating definite integrals ∫abf(x)dx=F(b)−F(a), using anti-derivatives ∫f(x)dx=F(x)+C and the power rule ∫xndx=n+1xn+1+C for n=−1, with linearity, splitting or combining intervals, boundary conditions, and finding areas under a curve, between a curve and the x-axis, or between two curves using ∣f(x)∣ and ∣f(x)−g(x)∣.

Definite Integrals, Areas, and Basic Anti-Derivatives

Anti-derivatives of sums and scalar multiples, using linearity together with standard forms such as ∫xndx=n+1xn+1+C, ∫sinxdx=−cosx+C, ∫cosxdx=sinx+C, ∫exdx=ex+C, ∫x1dx=ln∣x∣+C, and the usual trig and inverse trig results.

Anti-Derivative Rules (Plus Only)

Techniques for evaluating integrals, including substitution for f(ax+b) and compositions, handling definite integrals with changed bounds, partial fractions, integration by parts, repeated parts, and cyclical parts.

Techniques of Integration (Plus Only)

Displacement and distance, velocity and acceleration, speed as ∣v∣, average velocity and acceleration as Δs/Δt and Δv/Δt, and the relations v=ds/dt, a=dv/dt=d2s/dt2, ∫t1t2v(t)dt for change in displacement, and ∫t1t2∣v(t)∣dt for distance.

Kinematics (Plus Only)

Finding the volume of a solid revolved around a function or axis to create a 3D figure

Volumes of Revolution (Plus Only)

Differential Equations

Solving differential equations by direct integration, separable variables, homogeneous equations using the substitution y=vx, and linear first-order equations with an integrating factor, then using initial conditions to find particular solutions.

Solving Differential Equations

Euler's method as a numerical method for finding particular solutions.

Euler's Method (Plus Only)

Maclaurin

Maclaurin series express a function as a polynomial in powers of x using derivatives at 0, with general form f(x)=n=0∑∞n!f(n)(0)xn, and include standard expansions for ex, sinx, cosx, ln(x+1), arctanx, and the binomial extension for rational exponents.

Maclaurin Series Basics

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Operations on Maclaurin series including term-by-term differentiation and integration, adding and subtracting coefficients, multiplying and dividing series to find initial terms, using substitutions for composite functions, and applying series expansions to evaluate limits.

Operations on Maclaurin Series (Plus Only)

Probability

Theoretical and experimental probability, complementary events, expected number of outcomes, sample space

Probabilistic Events

Venn diagrams, intersection and union of events, conditional probability, independent events

Combined Events (Plus Only)

Tree diagrams, selection with/without replacement, dependence

Tree Diagrams (Plus Only)

Bayes' theorem for reversing conditional probabilities, using prior and posterior probabilities with two events or three mutually exclusive events, e.g. P(B∣A)=P(A∣B)P(B)+P(A∣B′)P(B′)P(A∣B)P(B) and its three-event form.

Bayes' Theorem (Plus Only)

Descriptive Statistics

Understand the basics of population, data collection, and sampling.

Population & Data

Learn different ways to measure the "center," or typical values, of a set of data: mean, median, and mode.

Measuring Center

Thumbnail for Quartiles and Box & Whisker Plots

Learn the concept of dispersion, range, IQR, outliers, and box and whisker plots.

Quartiles and Box & Whisker Plots (Plus Only)

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Variance as σ2=n∑fi(xi−μ)2 and standard deviation as σ=√σ2, including calculator Sx vs σx and how adding a constant changes the mean but not the standard deviation, while scaling multiplies both mean and standard deviation.

Standard Deviation and Variance (Plus Only)

Thumbnail for Frequency Tables, Histograms and cumulative frequency diagrams

In this lesson, we learn about different ways to visualize frequency data, including tables, histograms, and cumulative frequency diagrams.

Frequency Tables, Histograms and cumulative frequency diagrams (Plus Only)

Bivariate Statistics

Scatter diagrams and lines of best fit y=ax+b and x=ay+b, by eye or with GDC, the Pearson correlation coefficient r, and making predictions with awareness of limitations.

Linear Regression (Plus Only)

Distributions & Random Variables

Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution

Binomial Distribution (Plus Only)

The normal (bell-shaped) distribution X∼N(μ,σ2), its symmetry about μ, z-values z=σx−μ and standardization to Z∼N(0,1), the empirical rule, and calculator-based normal and inverse normal probability calculations.

Normal Distribution (Plus Only)

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Continuous random variables with pdfs, probabilities as areas under f(x), normalization to total area 1, and finding E(X), median, mode, variance, standard deviation, and the effects of linear transformations aX+b.

Continuous random variables (Plus Only)

Discrete random variables taking a finite set of values, with probability distributions in tables or as expressions where probabilities sum to 1, and finding E(X), Var(X), standard deviation, fair games, and linear transformations E(aX+b)=aE(X)+b, Var(aX+b)=a2Var(X).

Discrete random variables

Algebra Skills

$Thumbnail for Partial fractions$

Decomposing rational expressions into partial fractions, including linear factors such as x+3A+x−1B and more complex forms, by cross-multiplying and solving for the unknown constants.

Partial fractions

Solving systems of equations in three unknowns by substitution, elimination or calculator methods, and determining whether the system has no solution, a unique solution or infinitely many solutions.

3 by 3 Systems of equations

Concept Lessons

Access comprehensive explorations for all concepts in your objective, including problems, helpful visuals, and AI features.

100 Lessons Available

What you'll find in every lesson:

High-quality explanations: Each lesson is carefully crafted to help you truly understand the concept, not just memorize it.

Checkpoint exercises: Practice as you learn with interactive problems designed to reinforce your understanding.

AI-powered discussions: Engage with thought-provoking questions that help reveal your misconceptions.

What you'll find in every lesson:

High-quality explanations: Each lesson is carefully crafted to help you truly understand the concept, not just memorize it.

Checkpoint exercises: Practice as you learn with interactive problems designed to reinforce your understanding.

AI-powered discussions: Engage with thought-provoking questions that help reveal your misconceptions.

Cartesian plane & lines

Distance between two points, midpoint of two points, and finding gradient using two points

Distance, Midpoint, & Gradient

Gradient-intercept form, point-gradient form, vertical lines, horizontal lines, standard form of a line

Equations of a Line

Parallel and perpendicular lines, intersection of 2 straight lines, solving a two-variable system of equations

Line Intersections & Systems of Equations

Quadratics

Foundations of Quadratics

Applications of Quadratics

Exponents & Logarithms

Exponential Algebra

Square roots and nth roots, including n√a=an1, rational exponents anm, simplifying radicals, roots of negative numbers, and rationalizing denominators using roots or conjugates.

Radicals and Roots

Logarithm algebra

Exp & Log functions (Plus Only)

Function Theory

The concept of a function as a mathematical machine, the notation f(x)=…, and function tables / diagrams.

Functions and their properties

Graphing functions with a calculator, characteristics of a function, even and odd functions, x and y intercepts, horizontal and vertical asymptotes, maxima and minima

Function Graphs

The domain of a function is the set of possible inputs, and the range is the set of possible outputs.

Domain and range

Composing functions like f∘g

Function Composition (Plus Only)

Even functions satisfy f(−x)=f(x) and have symmetry in the y-axis, while odd functions satisfy f(−x)=−f(x) and have origin symmetry.

Even and odd functions

Inverse Functions

Rounding & Error

Rounding, significant figures, decimal places

Rounding Numbers

Converting numbers to and from scientific notation (a×10k)

Scientific Notation

Sequences & Series

Introduction to the definition, properties, and general term of an arithmetic sequence

Arithmetic Sequences

Introduction to definition, identification, and general term of geometric sequences

Geometric Sequences

Definition of series, formula for arithmetic series

Arithmetic Series (Plus Only)

Summation notation, properties of sums

Σ summation notation (Plus Only)

Finite geometric series with sum Sn=1−ru1(1−rn), and infinite geometric series convergence when ∣r∣<1 so that S∞=1−ru1.

Geometric Series (Plus Only)

Working with sequences on a calculator, computations and applications of a calculator

Sequence Mode (Calculator) (Plus Only)

Compound growth, depreciation, interest, inflation, real value

Compounding (Appreciation & Depreciation) (Plus Only)

Counting & Binomials

Pascal's triangle and binomial coefficients, using factorials and nCr=r!(n−r)!n! to find combinations and the coefficients in binomial expansions.

Pascal's Triangle and nCr

Expansion of (a+b)n using binomial coefficients, including the general term (nr)an−rbr and finding the coefficient of a specified term.

Binomial Theorem (Plus Only)

Expansion of (a+b)n using binomial coefficients, including the general term (nr)an−rbr and finding the coefficient of a specified term.

Counting (Plus Only)

Counting arrangements by splitting into subcases and using complementary counts, including treating items that must stay together as one block and placing items in separate slots to keep them apart.

Counting with Restrictions (Plus Only)

Proof and Reasoning

Proof by deduction

Proof by induction (Plus Only)

Contradiction and Counterexamples (Plus Only)

Transformations & asymptotes

Linear Transformations

Modulus & Inequalties (Plus Only)

Rational functions (Plus Only)

Graph transformations of functions including combined horizontal translations and stretches f(ax+b), and the effects of modulus, squaring and reciprocals: f(∣x∣), ∣f(x)∣, [f(x)]2, and f(x)1.

Further Transformations (Plus Only)

Polynomials

Polynomial Basics

Polynomial division, factors and remainders

Polynomial graphs

Polynomial roots

2D & 3D Geometry

Pythagoras's theorem, SOHCAHTOA, finding side length from angles

Right angled triangles

Area of non-right-angled triangles using A=21absinC, the sine rule sinAa=sinBb=sinCc, the cosine rule c2=a2+b2−2abcosC, and the ambiguous case when solving triangles.

Non-right-angled triangles (Plus Only)

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=rθ, A=21θr2, C=2πr, and A=πr2.

Circles: Radians, arcs and sectors (Plus Only)

Spheres, cylinders, prisms, right cones, right pyramids, and combinations of solids

3D space & solids (Plus Only)

Angles of elevation and depression, true bearings, projections and more

Applied triangle geometry

Trig equations & identities

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=rθ, A=21θr2, C=2πr, and A=πr2.

Circles: Radians, arcs and sectors (Plus Only)

The Unit Circle

Trigonometric Functions

Reciprocal trig functions secθ=cosθ1, cosecθ=sinθ1 and cotθ=tanθ1, including the graph, domain and range of secx, cosecx and cotx.

Reciprocal trig functions (Plus Only)

Trig Equations (Plus Only)

The inverse trig functions arcsin, arccos and arctan, including their principal-value domains and ranges and the graphs as reflections of the restricted trig functions in y=x.

Inverse trig functions (Plus Only)

Trigonometric identities including sin2θ+cos2θ=1, sin2θ=2sinθcosθ, and the equivalent forms of cos2θ: cos2θ−sin2θ, 2cos2θ−1, and 1−2sin2θ.

Trigonometric Identities (Plus Only)

Further trigonometric identities (Plus Only)

Complex Numbers

Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).

Cartesian form

The complex conjugate: z=a+bi⟹z∗=a−bi, its properties, and using it to simplify fractions with complex denominators.

Complex conjugate

Equating real and imaginary parts to solve complex equations

Solving complex equations

Introduction, definition, and properties of the complex modulus

Complex Modulus (Plus Only)

rcisθ and eiθ

Complex Argument (Plus Only)

Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.

Polar Form

Using De Moivre's Theorem to find powers of complex numbers in polar form, z=reiθ, so that zn=rneinθ=rncis(nθ), and applying the argument rule arg(zw)=arg(z)+arg(w).

Powers of Complex Numbers (Plus Only)

Vectors

Vector arithmetic & geometry

The scalar (aka dot) product, and its connection to angles between vectors.

Scalar product

The vector (aka cross) product u×v and its connection to areas in 3D space.

Vector Product

Equations of a vector line (Plus Only)

Classifying vector lines in 3D as coincident, parallel, intersecting or skew using direction vectors and point checks, with the angle between lines found by θ=cos−1(∣b1∣∣b2∣b1⋅b2).

Coincident, Parallel, Intersecting & Skew Lines (Plus Only)

Equations of a plane in 3D space using vector form r=a+λb+μc, scalar product form r⋅n=a⋅n, and Cartesian form n1x+n2y+n3z=d, with a point and two direction vectors or a normal vector.

Equations of a plane (Plus Only)

Finding angles between lines, between a line and a plane, and between two planes using direction vectors and normals, plus intersections of lines and planes and systems of three planes in 3D.

Angles and intersections with planes (Plus Only)

Differentiation

Limits and Derivatives

Derivatives of xr,ex,lnx,sinx and cosx. The chain, product and quotient rules, as well as the derivatives of reciprocal trig functionslogax, and ax.

Differentiation rules

Definitions of tangent and normal, finding tangent and normal lines to a function

Tangents and normals (Plus Only)

Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation

Applications of the First Derivative (Plus Only)

Second Derivatives and Applications (Plus Only)

Implicit differentiation

Using implicit differentiation and the chain rule to relate changing quantities, including dtdy=dxdy⋅dtdx, and applying this to volume, distance, and angle rates.

Related Rates (Plus Only)

The nth derivative of a function, written f(n)(x) or dxndny, found by differentiating n times.

n^th Derivative (Plus Only)

Evaluating indeterminate limits of the form x→alimg(x)f(x) when both numerator and denominator approach 0 or ∞, by replacing them with x→alimg′(x)f′(x) and repeating if necessary.

L'Hôpital's rule (Plus Only)

Integration

Definite Integrals, Areas, and Basic Anti-Derivatives

Anti-Derivative Rules (Plus Only)

Techniques of Integration (Plus Only)

Kinematics (Plus Only)

Finding the volume of a solid revolved around a function or axis to create a 3D figure

Volumes of Revolution (Plus Only)

Differential Equations

Solving Differential Equations

Euler's method as a numerical method for finding particular solutions.

Euler's Method (Plus Only)

Maclaurin

Maclaurin Series Basics

Operations on Maclaurin Series (Plus Only)

Probability

Theoretical and experimental probability, complementary events, expected number of outcomes, sample space

Probabilistic Events

Venn diagrams, intersection and union of events, conditional probability, independent events

Combined Events (Plus Only)

Tree diagrams, selection with/without replacement, dependence

Tree Diagrams (Plus Only)

Bayes' Theorem (Plus Only)

Descriptive Statistics

Understand the basics of population, data collection, and sampling.

Population & Data

Learn different ways to measure the "center," or typical values, of a set of data: mean, median, and mode.

Measuring Center

Learn the concept of dispersion, range, IQR, outliers, and box and whisker plots.

Quartiles and Box & Whisker Plots (Plus Only)

Standard Deviation and Variance (Plus Only)

In this lesson, we learn about different ways to visualize frequency data, including tables, histograms, and cumulative frequency diagrams.

Frequency Tables, Histograms and cumulative frequency diagrams (Plus Only)

Bivariate Statistics

Scatter diagrams and lines of best fit y=ax+b and x=ay+b, by eye or with GDC, the Pearson correlation coefficient r, and making predictions with awareness of limitations.

Linear Regression (Plus Only)

Distributions & Random Variables

Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution

Binomial Distribution (Plus Only)

Normal Distribution (Plus Only)

Continuous random variables (Plus Only)

Discrete random variables

Algebra Skills

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Decomposing rational expressions into partial fractions, including linear factors such as x+3A+x−1B and more complex forms, by cross-multiplying and solving for the unknown constants.

Partial fractions

Solving systems of equations in three unknowns by substitution, elimination or calculator methods, and determining whether the system has no solution, a unique solution or infinitely many solutions.

3 by 3 Systems of equations