Short Descriptions
Algebra Group
1.

Algebra and Indices

Methods Unit 1, Topic 1

Introduces algebra as the language of mathematics and briefly discusses using indices as an example how effective teaching happens when things make sense. For more on Algebra see the Algebra supplement.

2.

Quadratic equations, Completing the square

Methods Unit 1, Topic 1

The history of quadratic equations, and their solutions through geometric construction including the geometrical understanding of the method of completing the square. For more on Algebra see the Algebra supplement.

3.

Quadratics: Factors and Formula

Methods Unit 1, Topic 1

Shows how the general quadratic equation can be factorized and how this process naturally leads to quadratic formula.

4.

Coordinate Geometry, Basic ideas

Methods Unit 1, Topic 1

Covers basic ideas in coordinate geometry including midpoint and ratio division, and equations of lines.

Relations Group
5.

Mappings

Methods Unit 1, Topic 1

The language around functions and relations and includes terms such as domain and range, mappings and inverses.

6.

Continuity and Smoothness

Methods Unit 1, Topic 1

Concepts around continuity and smoothness of a function.

7.

Equivalence and other relations

Methods Unit 1, Topic 1

Equivalence relations and the three properties of relations (reflexive, transitive and symmetric) are explained using a simple example.

Trigonometry Group
8.

Common Ratios

Methods Unit 1, Topic 2

Establishes the basic trigonometric ratios and show how these ratios together with Pythagoras’ theorem can be used to solve for sides of triangles and real world problems.

9.

Angles and applications

Methods Unit 1, Topic 2

Continues with work around the basic ratios and uses them to solve for angles in right triangles, showing how this knowledge can be practically applied.

10.

Radians and applications

Methods Unit 1, Topic 2

Defines the radian, arc length and sector area, and
provides an interesting application involving milliradians, mils, and the
estimate of distances.

11.

The Sine Rule

Methods Unit 1, Topic 2

Provides an interesting proof of the Sine Rule in terms of a triangle’s circumscribed circle and shows how the rule can be applied to triangles.

12.

The Cosine Rule

Methods Unit 1, Topic 2

Provides an historic development of the Cosine rule in terms of Ptolemy’s discovery and shows how the Cosine rule can be applied to triangles.

13.

The Area of a Triangle Rule

Methods Unit 1, Topic 2

Establishes the Area of the triangle rule and shows how the rule can be applied to Kepler’s 2nd law.

14.

Angles of Any Magnitude: Definitions

Methods Unit 1, Topic 2

Redefines the basic ratios in terms of the unit circle, and provides insights into how the unit circle is related to the graphs of the trigonometric functions.

15.

Angles of any Magnitude: Ratios and Angles

Methods Unit 1, Topic 2
Specialist Unit 2, Topic 1 
Introduces the reciprocal ratios and shows them geometrically expressed on the unit circle. This allows for the geometrical formulation of the Pythagorean results. Also includes examples of the solutions of trigonometric equations with restricted domains.

16.

Compound Angles

Specialist Unit 2, Topic 1

Establishes the compound angle formulae for sine, cosine and tan, providing insightful approaches to the proofs.

17.

Functions and Modelling: Part 1

Specialist Unit 2, Topic 1

Shows the effect of the constants a, b, c, and d on asin(bx+c) + d.

18.

Functions and Modelling: Part 2

Specialist Unit 2, Topic 1

Models the movement of the London Eye using an appropriate trigonometric function and introduces the horizon formula showing distances to the horizon from any height, h, above the surface of the Earth.

The Vectors Group
19.

History and Arrows

Specialist Unit 1, Topic 2

Provides an interesting historical perspective to the idea of Vectors, in particular the work of Sir William Rowan Hamilton on Quaternions. Goes on to define the arrow as an entity having both magnitude and direction.

20.

Algebra

Specialist Unit 1, Topic 2

Carefully discusses and defines the properties of vectors including addition, scalar multiplication, uniqueness of inverses and identities, and finishes with a real life application.

21.

Linear Independence

Specialist Unit 1, Topic 2

Establishes the concept of linear independence and defines basis vectors as linear combinations of other vectors. It also demonstrates how to form vector representations of systems of linear equations.

22.

Plane Geometry Proofs

Specialist Unit 1, Topic 2

Carefully explains how vectors can be harnessed to prove results in plane geometry, including “The diagonals of a parallelogram bisect each other” and “the medians of a triangle divide each other in the ratio 1:2” and other interesting examples.

23.

Basis and Scalar Product

Specialist Unit 1, Topic 2

Explores the idea of an ndimensional vector space, defining length, angles, the scalar product and orthogonality.

24.

The Vector or Cross Product

Specialist Unit 3, Topic 3

Defines the vector or cross product, and explains its application to orthonormal basis vectors, leading to the coordinate definition of the cross product, and establishing formulae for the area of a parallelogram and the volume of a parallelepiped.

25.

Projections and Functions

Specialist Unit 3, Topic 3

Shows that Pythagoras’ theorem holds for an ndimesional vector space, and establishes the idea of a projection of one vector onto another, and onto a subspace. Establishes the concept of a vector valued function and explores their application to velocity and acceleration.

The Probability Group
26.

Probability, Basics

Methods Unit 1, Topic 3

Introduces the concept of probability as a theoretical construct and as notion that emerges from the consideration of relative frequency. Discusses the gamblers ruin, and examines two probing examples.

27.

Probability, Two Events: Part 1

Methods Unit 1, Topic 3

Establishes for two events the concepts of mutual exclusivity and independence in relation to two events. Introduces the addition rule and the idea of conditional probability.

28.

Probability, Two events: Part 2

Methods Unit 1, Topic 3

Introduces the idea of tree diagrams through an example, and discusses conditional probability and independence in relation to them. Also introduces Karnaugh tables and establishes Bayes Theorem.

The Counting and Combinatorics Group
29.

Counting

Methods Unit 1, Topic 3

Introduces the fundamental counting principle and then establishes the concepts of factorials, arrangements and combinations developing formulae along the way using examples where necessary.

30.

The Binomial

Methods Unit 1, Topic 3

Develops Pascal’s Triangle from binomial products and establishes the binomial formula. Shows how the coefficients of that formula relates to arrangements, and then goes on to generalize binomial expansions.

31.

The InclusionExclusion Principle

Specialist Unit 1, Topic 1

Using examples the Inclusion/Exclusion Principle (IEP) described as one of the most useful principles of mathematical enumeration.

32.

Derangements

Specialist Unit 1, Topic 1

Develops the idea of a derangement and carefully establishes the associated formula.

33.

The Pigeon Hole Principle

Specialist Unit 1, Topic 1

Known by the French as the Principle of the drawers of Dirichlet the Pigeon Hole Principle is a simple but versatile tool for solving a range of realworld problems. The idea is established along with a number of examples.

The Plane Geometry Group
34.

Euclid’s Elements

Specialist Unit 1, Topic 3

Discusses the Euclidean Elements  origins of plane geometry, and their role in developing the mindsets of the emerging scientific revolution of the 17th Century. Establishes plane geometry as a welldefined axiomatic system and offers proposition 1 of book 1 of the Elements as an example.

35.

Circles 1

Specialist Unit 1, Topic 3

Focuses on circles and the use of congruency and similarity to prove certain theorems, including theorems around chords, angles and tangents.

36.

Circles 2

Specialist Unit 1, Topic 3

Continues with the discussion on circles, and addresses the case of secants intersecting inside and outside of a circle. It also discusses the case of a tangent and secant intersection and shows how the results can be used to develop an alternative horizon formula.

37.

Quadrilaterals

Specialist Unit 1, Topic 3

Generally discusses quadrilaterals and goes on to establish theorems on quadrilaterals, including Ptolemy’s theorem for cyclic quadrilaterals and other theorems and properties.

The Other Functions Group
38.

Rational Functions

Specialist Unit 3, Topic 2

A Rational function takes the form P(x)/Q(x) where P(x) and Q(x) are polynomials. Using a strategy known as ‘roots and poles’ many of these functions can be easily sketched. The technique also has ramifications for polynomial sketching as well.

39.

Exponential Functions

Methods Unit 2, Topic 1

Describes the exponential function using a general base and outlines the procedure for changing bases and explores modeling with the exponential function using the example of Moore’s Law.

40.

Logarithms, Introduction

Methods Unit 4, Topic 1

Before logarithms there were quarter square tables. These are examined as an introduction to logs. The idea of logs as codes for multiplication and division is explained, and includes a discussion on how Briggs constructed his common logarithms. For the rules of logarithms see the Algebra supplement.

41.

Logarithms, Functions

Methods Unit 4, Topic 1

The logarithmic function is defined and intuitively explained. Transformations on the function are discussed and the inverse function is established. Included is a discussion around when the log function base b is mutually tangential to its inverse exponential function. Kepler’s 3rd law and sound intensity are examined as two applications of log functions.

The AP and GP Group
42.

Sequences and Series

Methods Unit 2, Topic 2

Sequences are defined including explicit and recursive formulae and some interesting examples are examined including the Fibonacci sequence, The Tower of Hanoi, John Conway’s Morris sequence and the Pizzacut sequence.

43.

Arithmetic and Geometric Sequences

Methods Unit 2, Topic 2

Focuses in on Arithmetic and Geometric sequences and compares one with the other. Also covers the means of the two sequences and provides a geometric proof from antiquity that the GM <= AM.

44.

Arithmetic and Geometric Series

Methods Unit 2, Topic 2

Focuses in on Arithmetic and Geometric series establishing the formulae for Sn and the limiting sum for a GP.

45.

Geometric Progression, Applications 1

Methods Unit 2, Topic 2

Includes two interesting applications. The first provides a proof, using a geometric series, that the area under the curve y=x^2 between x=0 and x=a is 1/3 a^3. The second is a discussion about the rule of 72 – an old banking rule of thumb that estimates the time required for an investment to double.

46.

Geometric Progression, Applications 2

Methods Unit 2, Topic 2

A few examples are discussed. The first shows the maximum amount of interest that can be earned at r% per annum given any number of compounding periods within the term. The second derives a formula for the future value of an ordinary annuity. The third demonstrates how repeating decimals can be shown to be rational.

Matrices
47.

Basics

Specialist Unit 2, Topic 2

Traces the origins of matrices, and establishes the matrix rules of operation, in particular develops the ideas around multiplication, identities, determinants and inverses.

48.

Solving Systems

Specialist Unit 2, Topic 2

Carefully establishes the idea of row reduction to solve systems of linear equations. Shows how row reduction leads naturally to Cramer’s rule. Discusses linear independence, rowechelon form, and shows a row reduction example for a 3 by 3 system.

49.

Transformations 1

Specialist Unit 2, Topic 2

Establishes the concept of a linear transformation, and in particular what actually makes the transformation linear. Explains the terms linear independence, basis, and span using examples and summarizes the six most common transformations in the plane.

50.

Transformations 2

Specialist Unit 2, Topic 2

Shows how linear transformations can be applied to relations, in particular focusing on translating lines and rotating parabolas and ellipses. It finishes with a discussion of combined transformations and establishes the formula for reflection y=tan(theta) in the line and as an example reflects y=1/x in the line y=root(3) x.

51.

Transformations 3

Specialist Unit 2, Topic 2

Introduces the idea of a general transformation on the unit square to a parallelogram whose area is given by the determinant of the transformation matrix. Discusses inverse transformations and establishes the concept of eigenvalues and eigenvectors, and the characteristic equation. It provides a geometrical interpretation of eigenvectors in terms of the transformational matrix A.

The Calculus Group
52.

Rates of Change

Methods Unit 2, Topic 3

The rainbow is a beautiful example to use in any discussion about rates of change. The rainbow can be thought of as a physical manifestation of a minimum turning point. The mathematics is examined and the understanding of rates of change is graphically demonstrated. In the second investigation, instantaneous velocity is shown as a limiting value for the average velocity between t and delta t.

53.

Differentiation and Integration

Builds the idea of a rate of change leading into the formal definition of the derivative. An historical perspective is given to the concept of instantaneous rates of change and this is followed by a discussion around velocity and displacement as functions of time. Finally a formal definition of limits and continuity is established.

Methods Unit 2, Topic 3
Methods Unit 3, Topic 2 Specialist Unit 3, Topic 2 
54.

The Fundamental Theorem

Carefully builds an exceptionally clear explanation of the fundamental theorem that underpins integration. The establishment of the theorem is accompanied by clarifying illustrations.

Methods Unit 2, Topic 3
Methods Unit 3, Topic 2 Specialist Unit 3, Topic 2 
55.

The Derivative

Methods Unit 3, Topic 1

Establishes the first and second derivative as tools for curve sketching. Establishes the rule of differentiation and then rules for products, quotients, and the composition of functions. Finishes with a discussion about notation from an historical perspective and provides examples using different notations.

56.

Trigonometric Functions

Methods Unit 3, Topic 1

Carefully establishes some results leading to the first principles development of the derivatives for the trigonometric functions. Describes implicit differentiation and the derivatives of the inverse functions in particular the inverse trigonometric functions. Ends with a rationale for why these functions are so important.

57.

Logarithm and Exponential Functions

Methods Unit 3, Topic 1

Starting with a function L(x) with certain properties the derivative of L is deduced as 1/x and the result establishes that the integral of 1/x is ln x + c . Consideration of the inverse of y = ln x leads naturally to the derivative of e^x, and these results are finally generalized.

The Integration Techniques Group
58.

Integration Techniques: 1

Specialist Unit 4, Topic 1

Establishes the context for these techniques and goes on to describe them. The techniques include integration by substitution or change of variable, using compound angle formulae, and integration by parts.

59.

Integration Techniques: 2

Specialist Unit 4, Topic 1

Integration continues with explanations of the techniques given as integration using partial fractions, integration using half angle tformulae, and integration leading to reduction formulae using the parts process. Finishes with a short discussion on approximation strategies to determine areas under curves, including the midordinate rule and Simpson’s rule.

The Complex Numbers Group
60.

The Story of the Cubic

Specialist Unit 2, Topic 3

Takes an authentic approach to the solution of the cubic equation, and provides in detail the steps required to solve it. It shows how Rafael Bombelli in 1572 solved the irreducible case, and gives a geometric interpretation of the classification of cubic polynomials in terms of their roots.

61.

The Basics

Specialist Unit 2, Topic 3

Defines the complex number as z = a+bi and as the ordered pair (a,b) and the operations on them. Defines and describes algebraically the inverse of z, the conjugate of z and the polar form of z and the operations in polar form. Also introduces de Moivre’s theorem.

62.

Geometry

Specialist Unit 3, Topic 1

Establishes the number i as a rotational operator on z, gives a geometrical description of the conjugate and the inverse and establishes the method for finding square roots. It also derives an equation that is useful for evaluating digits of pi.

63.

Polynomials

Specialist Unit 3, Topic 1

Establishes the method for finding the nth roots of a complex number z using de Moivre’s theorem, outlines the conjugate root theorem and shows examples of its use and finishes with a description of Descartes Rule of Signs used to determine the nature of the roots of a polynomial.

64.

Solutions and Subsets

Specialist Unit 3, Topic 1

Describes how polynomial equations can be factorized over the complex field. Establishes a rationale for looking at subsets of the complex plane by looking at an example of a complex function of a complex variable showing how elements in the zplane are mapped to the wplane. Examines, using examples, subsets of the complex plane.

The Differential Equations Group
65.

First Order

Specialist Unit 4, Topic 2

Defines differential equations and introduces the terms first order, linear, ordinary and separable equations. Establishes the rationale for separation of variables and gives examples including the logistic equation and discusses the technique of integrating factors and the Bernoulli equation.

66.

Homogeneous

Specialist Unit 4, Topic 2

Establishes the concept of a homogeneous equation and demonstrates the technique of solution for equations of degree 0. Describes some examples and illustrates some solution families on the coordinate axes.

67.

Direction Fields and the Logistic equation

Specialist Unit 4, Topic 2

Motivates and builds the concept of a direction field with examples, and explores and establishes the general logistic growth equation.

68.

Euler’s method

Specialist Unit 4, Topic 2

Carefully establishes the Euler method as an effective numerical method for solving a differential equation.

69.

Euler’s method

Specialist Unit 4, Topic 2

Defines rectilinear motion establishing expressions for displacement, velocity and acceleration, and the difference between average speed and average velocity. Establishes the various forms of acceleration and gives a comprehensive example linking all the forms.

70.

Forces 1

Specialist Unit 4, Topic 2

Newton’s three laws are discussed and defined, and the concepts of kinetic and potential energy and work are explored in some detail.

71.

Forces 2

Specialist Unit 4, Topic 2

The concept of impulse is established leading to the law of conservation of momentum. Variable forces are discussed with examples including the problem of determining escape velocity. Concurrent forces are discussed along with examples.

72.

Simple Harmonic Motion

Specialist Unit 4, Topic 2

Through the example of the pendulum, its historical significance and its mathematical formulation, simple harmonic motion is established, along with a discussion of the history around the ‘seconds’ pendulum.

73.

Projectile Motion

Specialist Unit 4, Topic 2

Outlines a history of the thinking about projectiles and then establishes the equations of motion of a projectile and uses these to derive formulae for the height and range of a projectile. Establishes the conservation of mechanical energy as it applies to projectile motion and finishes with a discussion of particles projected up an inclined plane.

Statistics
74.

Random Variables and Expected Value

Methods Unit 3, Topic 3

Establishes the concept of a sample space, and the axioms of probability. Carefully explains the meaning of a random variable, and the probability distribution associated with it, using a clear example. Defines the expected value and variance of a random variable. Establishes the fact that E[ax+b]=aE[x]+b and Var[aX+b}=a^2Var[X].

75.

Discrete Distributions

Methods Unit 3, Topic 3

Outlines the idea of a sampling distribution and establishes the formula for the expected value of the sample variance.

76.

Continuous Distributions

Methods Unit 4, Topic 2

Discusses the Normal, Uniform and Exponential probability density functions and their parameters, particularly the expected value and variance of each.

Sampling
77.

Introduction

Specialist Unit 4, Topic 3

Introduces the concept of a random sample and explores the idea of a statistic as an estimate of a population parameter. Using a simple example validates the sample mean and variance as unbiased estimators of the population mean and variance, and establishes the idea of the sampling distribution leading to the Central Limit Theorem.

78.

Distributions

Specialist Unit 4, Topic 3

Continues with discussion of the sampling distribution. Introduces the studentt distribution for small samples and shows how it can be used to find various probabilities.

79.

Confidence Intervals

Specialist Unit 4, Topic 3

Introduces the concept of (1alpha)100% confidence intervals on the mean. It demonstrates the idea with an example, and gives a method of determining a minimum sample size given a
(1alpha)100% confidence requirement. 
80.

Proportions

Methods Unit 4, Unit 3
Specialist Unit 4, Topic 3 
Carefully establishes bounds on the sample proportion and gives an example showing how the statistic is used. Also gives a minimum sample size given a predetermined (1alpha)100% confidence requirement.
