Planning

Area of study: Algebra, number and structure

Topic: Proof and number

Weeks 1–3

• The structure, arithmetic and properties of the natural numbers N, integers Z and rational numbers Q.
• Complex numbers will be studied in Unit 2 but some initial mention can be made in the context of solution of equations.
• For the non-negative integers, divisibility properties, prime numbers, representation as products of powers of primes.
• Highest common factor and lowest common multiple and the relationship between the two and with products of powers of prime factors.

Set notation and its use in representing information in different areas of mathematics including the solution of `logic problems’.

For rational number Q:,

• fraction and decimal representation and their equivalence. Proof that is irrational and similar proofs.
• The algebra of surds, and the introduction of such concepts as conjugates, additive and multiplicative inverses.

The structure of the set of all numbers of the form where a, b are rational numbers with the usual operations can be used as an introduction to the complex numbers.

Introduction to important ideas of proof such as implication, equivalence, negation, converse and contrapositive, along with the classical case for proofs of the infinitude of primes.

Introduction to proof by mathematical induction including partial sums, divisibility and inequalities. AMSI – Proofs by induction

Sample learning activity: number proofs by induction

Possible assessment 1, Investigation: A collection of proofs and thoughts on proofs including:

• Results that can be proved by more than one technique n³ – n is divisible by 6. Prove by factorisation and induction.
• What are some possible mistakes in induction proofs? (See AMSI Induction linked above.)
• Interesting non-algebraic proofs involving induction such as Tower of Hanoi. (See AMSI Induction linked above.)

Possible assessment 2, Investigation: Investigation of aspects of the relationship between the mediant of two fractions, continued fractions, Farey sequences, Ford circles and the Stern-Brocot sequence. Construct algorithm to generate Farey and Stern-Brocot sequences (it is advantageous to use lists for the numerators and denominators and possibly the integer part function). See the linked file Pseudocode for examples and conventions of pseudocode.

Topic: Logic and algorithms

Weeks 4–6

Boolean operations, propositions, logical connectives, truth values, Karnaugh maps and truth tables.

Tautologies and using truth tables to check the validity of arguments.
AMSI – Logic and Boolean algebra

Boolean algebra, the algebra of sets, propositional logic and other models, applications to electronic gates and circuits and circuit simplification.

Fundamental constructs needed to describe algorithms: decision (selection, choice, if...then blocks) and repetition (iteration and loops). Using pseudocode to describe these.

Sample learning activity: Writing pseudocode to investigate solution of equations in the integers. For example, linear Diophantine equations. See the linked file Pseudocode for examples and conventions of pseudocode.

Possible assessment 3, Investigation: Writing pseudocode and implementing code with a device to investigate Pythagorean triples and related n-tuples and their properties. See the linked file Pseudocode for examples and conventions of pseudocode.

Possible assessment 4, Investigation: Squares of integers

Writing pseudocode to investigate particular partial sums and use induction to prove results about these sums. See the Pseudocode for examples and conventions of pseudocode.

Area of study: Discrete mathematics

Topic: Sequences and series

Weeks 7–9

Arithmetic and geometric sequences and their partial sums. ‘Infinite’ geometric sequences and their limiting sums.

Solution of first order linear recurrence relations of the form t_{n + 1} = a t_n + b, a ≠ 0 with constant coefficients. Applications of these to financial and population problems.

In Units 3 and 4, sequence and sum notation can appear in Logic and proof and in describing algorithms and in Euler’s method for obtaining approximate solutions differential equations.

AMSI – Sequences and series

Sample learning activity: Investigate the properties of Fibonacci and related sequences. Establish results by proof by induction. Develop algorithms to generate these sequences.

Possible assessment 5, Investigation: Apply induction to prove some results concerning sequences generated by recurrence relations.

Possible assessment 6, Investigation: Linear recurrence relations

Topic: Combinatorics

Weeks 10–12

Pigeon-hole and inclusion-exclusion principles, related applications and proofs.

Permutations and combinations, applications, problems involving restrictions, Pascal’s triangle and associated identities.

In Units 3 and 4 the ideas from this topic can appear in Logic and proof.

Sample learning activity: Investigate the possible number of paths through grids moving from ‘left to right’ and ‘up’.

Possible assessment 7, Investigation: A selection of applications and / or proofs involving counting principles.

Topic: Matrices

Weeks 13–15

Introduction to matrix algebra and matrix equations. Matrices will be used in the topics Graph Theory, Transformations and possibly Complex numbers. There are many results in matrices provable by induction and accessible to students who have undertaken this course.

Sample learning activity: Using matrices to determine sequences of numbers. For example, Fibonacci numbers.

Sample learning activity: Show how a particular family of 2 by 2 matrices can be used to construct an algebraic system which is the complex numbers.

Possible assessment 8, Investigation: Use of matrices in solutions of systems of linear equations.

Area of study: Algebra, number and structure

Topic: Graph theory

Weeks 16–18

This topic builds on the material in the Area of study: Algebra, number and structure and in the topic Combinatorics.

Vertices and edges for undirected graphs and their representation using lists, diagrams and matrices (including multiple edges and loops) with examples from a range of contexts.

Simple graphs, sub-graphs, connectedness, complete graphs and the complement of a graph, and isomorphism of graphs. 

Bi-partite graphs, trees, regular graphs (including the platonic graphs), planar graphs and related proofs and applications.

Walks, trails, paths, and circuits, Euler circuits and Euler trails, Hamiltonian cycles and paths. 

Proofs of results in graph theory are established and the results applied. Types of problems in graph theory including existence problems, construction problems, counting problems and optimisation problems. Questions and proofs such as:

• How many edges in a complete graph with n vertices? (Using combinatorics ideas)
• How many simple graphs with four vertices are there?
• Prove that a connected graph has a Euler circuit if and only if the degree of every vertex is even.
• Prove that if all the vertices of a connected graph have even degree then every Euler trail in the graph is a Euler circuit.
• Show that the cycle graph C₈ is bipartite. AMSI – Graph theory

Possible learning activity: Use of graphs to solve problems such as puzzles such as stacking coloured cubes problems or to describe social networks.

Sample learning activity: Graph theory constructions and proofs

Possible assessment 9, Investigation: Use of Eulerian and Hamiltonian graphs to discuss Knights tour problem.

Area of study: Space and measurement

Topic: Trigonometry

Weeks 1–3

The material in this topic is used extensively in the topics of Unit 2 and the topics of Units 3 and 4. For example, trigonometric identities, inverse trigonometric functions and reciprocal trigonometric functions are required for integration.

AMSI – Further trigonometry

Sine rule and cosine rule. Arc length, area of sectors and segments.

Proof and use of Pythagorean identities, angle sum, difference, double angle and other trigonometric identities.

Possible learning activity: Areas and perimeters in overlapping circles.

Possible learning activity: Induction proofs in trigonometry.

Possible assessment 1, Investigation: Three-dimensional trigonometry problems with a consideration of error.

Possible assessment 2, Investigation: Approximating trigonometric functions with polynomials. Use of coding to help evaluate approximating polynomials and the errors involved for different domains.

Topic: Transformations

Weeks 4–5

AMSI - Transformations

Points in the Cartesian plane, coordinates and column vectors. 

This topic extends the study of transformations undertaken in Mathematical Methods. Matrices are used extensively.

Linear transformations, matrices, dilations rotations about the origin, reflections in lines through the origin. Inverse and composite transformations. 

Effects of transformations, inverse transformations and composite transformations on subsets of the plane (points, lines, shapes and graphs). Invariance of properties under transformation, determinant of a transformation matrix and the effect of a transformation on area.

Use of matrix multiplication to establish results such as angle formulas, equivalence of transformations. 

Possible learning activity: Tessellations of the plane. Translating polygons to form tessellations.

Possible assessment 3, Investigation: Rotating conics.

Possible assessment 4, Investigation: Consideration of isometries and composition of reflections to obtain the other isometries.

Topic: Vectors

Weeks 6–9

The study of vectors is continued in Specialist Mathematics Units 3 and 4. The subject progresses to topics such as vector equations of lines and planes and to vector calculus.

Plane vectors, magnitude and direction, unit vectors, zero vector, position, displacement and velocity. Geometric representation of addition, subtraction, scalar multiple and linear combinations. 

Algebraic representation of vectors as ordered pairs, matrices and i, j form and the relationship between these and direction cosines.

Vector algebra of representation of addition, subtraction, scalar multiple and linear combinations. 

Scalar product, perpendicular and parallel vectors, vector projection and angle between two vectors. 

Application of vectors, including geometric proofs, navigation, statics, relative velocity.

Possible learning activity: Use of vector proofs in three-dimensions.

Possible learning activity: Investigation of sums of velocity vectors and their application to relative velocity.

Possible assessment 5, Investigation: Vector proofs

Area of study: Data analysis, probability and statistics

Topic: Simulation, sampling and sampling distributions

Weeks 10–12

AMSI – Random sampling
AMSI – Inference for means

Simulation of random experiments, events and event spaces, simulation and random samples.

Simple random sampling from a finite population, probability of obtaining a particular sample, introduction to random variables for discrete distributions.

Population parameters and a sample statistic, and use of the sample means and proportions to estimate corresponding population parameters.

Concept of a sampling distribution and its random variable, distribution of sample means and proportions considered empirically, including comparing the distributions of different size samples from the same population, in terms of centre and spread.

Display of variation in sample proportions and means, consideration of the mean and standard deviation of both the distribution of sample means and the distribution of sample proportions and consideration of the effect of taking larger samples.

This topic provides valuable background for the statistics topics of Specialist Mathematics Units 3 and 4.

Possible learning activity: Distribution of sample means

Area of Study: Algebra, number and structure

Topic: Complex numbers

Weeks 13–15

The study of Complex numbers is continued in Specialist Mathematics Units 3 and 4.
The structure, arithmetic and properties of the complex numbers, including modulus, and the argand diagram model. General solution in C of quadratic equations with real coefficients. 
Representation of subsets of the complex plane including lines and circles.

Possible assessment: Test involving the solution of quadratic equations using various techniques over Q, R and C.

Possible assessment 6, Investigation: Curves in the complex plane

Possible learning activity: Complex partial fractions, Adventures with complex numbers

Area of study: Functions, relations and graphs

Topic: Functions, relations and graphs

Weeks 16–18

Knowledge of the graphs of the inverse trigonometric functions and the trigonometric reciprocal functions will be useful in Specialist Mathematics Units 3 and 4 where they will appear in various topics.

Graphs of rational functions are studied in Units 3 and 4 and this follows on from the study of simple reciprocal functions.

The study of Parametric graphs leads to the study of vector functions in Units 3 and 4.

Graphs of simple reciprocal functions, including those for sine, cosine and tangent.

Graphs of the inverse trigonometric functions.

Locus definition and construction in the plane of lines, parabolas, circles, ellipses and hyperbolas, and Cartesian, polar and parametric graphs of these relations.

Polar and/or parametric graphs of other relations in the plane.

Possible assessment: A test involving interpretation of key features of a graph of empirical data, and the production of various graphs using relations drawn from the types specified.

Possible assessment 7, Investigation: Rotating conics is suitable here.

Area of study: Discrete mathematics Logic and proof – 2 weeks

This involves the basic constructs and techniques of proof. It is expected that students will be able to apply the techniques of these constructs and techniques to other areas. Proofs involve concepts from topics such as: divisibility, inequalities, graph theory, combinatorics, sequences and series including partial sums and partial products and related notations, complex numbers, matrices, vectors and calculus. Examples may include:

• Let n be an integer. Then n is divisible by 3 if and only if n² is divisible by 3.
• For all real numbers x and y prove that x² + 5y² ≥ 2xy.
• Every graph has an even number of vertices of odd degree.
• Pascal’s rule, ⁿC_r = ^n-1C_r–1 +  ^n-1C_r
• Induction proofs involving sequences, partial sums and products.
• Induction proof of De Moivre’s theorem.
• Induction proofs involving matrices e.g.
• Geometric proofs involving vectors.
• Reduction formula in integration (see example in integration).

Use of pseudocode and coding to help investigate properties and make conjectures.

See the linked file Pseudocode for examples and conventions of pseudocode.

Other proof by induction resources:

AMSI – Proofs by induction
Wolfram Alpha

Area of study: Space and measurement Vectors – 3 weeks

Vector arithmetic and algebra, linear dependence and independence of a set of vectors, geometric representation and interpretation.

Magnitude, direction, unit vectors, orthogonal vectors, i, j, k vectors, rectangular components, scalar product, parallel and perpendicular vectors, projections.

Vector proofs of simple geometric results.

Area of study: Space and measurement Vectors and Cartesian equations – 2 weeks

Vector and parametric equations in two and three dimensions.

Consider information necessary to define a line.

Three different forms of equations of lines: vector, Cartesian and parametric.

Consider information necessary to define a plane.

It is useful to understand and use three different forms of equations of planes: vector, Cartesian and parametric.

Parallel, perpendicular, and skew lines. Distance of a point to a line, line segments, intersection of lines, concurrence of lines, angles between two lines, distance between skew lines.

Vector product and the geometric interpretation.

Normal vector to a plane, distance between parallel planes, intersections of planes.

Wolfram – Equation of a plane
Wolfram – Shortest distance between two skew lines

Area of study: Algebra, number and structure Complex numbers and algebra – 2 weeks

This topic follows on from the study of Complex numbers in Specialist Mathematics Units 1 and 2.

Arithmetic and algebra of complex numbers in Cartesian and polar forms and representations in the complex plane (argand diagram), modulus and argument, basic identities.

De Moivre’s theorem, integer powers of complex numbers, nth roots of unity and other complex numbers, representation in the complex plane.

Factorisation of polynomial functions with integer coefficients over C, conjugate root theorem.

Fundamental theorem of algebra and simple cases of factorisation of polynomial functions with complex coefficients.

Application task – 2 weeks

This could be held earlier or later depending on coverage of relevant concepts, skills and processes with respect to content and context for the application task. See VCAA Bulletin Supplements, December: 2002, 2003, 2004.

It is useful to understand and use three different forms of equations of lines: vector, Cartesian and parametric.

Sample application task: Cubic and quartic polynomials
Sample application task: Curves in the complex plane
Sample application task: Graphs of rational functions with cubic polynomial numerators and denominators
Sample application task: Farey sequences
Sample application task: Graphs of rational functions

Area of study: Functions, relations and graphs Functions, relations and graphs – 2 weeks

This topic follows on from the topic: Functions, relations and graphs in Specialist Mathematics Units 1 and 2. This topic could be taught concurrently with the differentiation topics in Calculus.

Partial fractions are introduced here for use in integration.

Graphs of rational functions of low degree, including reciprocal functions of polynomial functions.

The absolute value function was introduced in Specialist Mathematics, Units 1 and 2.
Determining points of inflection and its inclusion in sketching graphs.

Possible learning activity: Quadratic reciprocal and rational functions

Area of study: Calculus Differential calculus (and integral calculus) – 2 weeks

This topic can be taught concurrently with Functions, relations and graphs.

Inverse circular functions were introduced in Specialist Mathematics Units 1 and 2

The chain rule – related rates, derivatives of inverse circular functions and implicit differentiation.

Implicit differentiation and its applications.

The second derivative and its application to analysing graphs of functions, points of inflection, concavity quotient functions and their graphs.

Area of study: Calculus (Differential calculus) and Integral calculus – 3 weeks

The relationship between differentiation and anti-differentiation as inverse processes, and graphs of the family of anti-derivatives of a function.

Compound and double angle formulas, identities, partial fractions of rational functions.

Anti-derivatives involving , derivatives of inverse circular functions, identities, substitution and partial fractions. Trigonometric identities, the reciprocal trigonometric functions and inverse trigonometric functions were introduced in Specialist Mathematics Units 1 and 2. The absolute value function will be required in integration and this was also introduced in Specialist Mathematics Units 1 and 2.

Integration by parts and reduction formula. For example:

• Integrate x sin (x) using integration by parts
• Let . Show that  and work iteratively with this formula to determine definite integrals such as .

Anti-differentiation and numerical and symbolic integration using technology. Reimann sums may be used to approximate the integrals of this section.

Application of integration to arc lengths for parametrically defined curves, areas, volumes of revolution about either coordinate axis and areas of a surface of revolution about either coordinate axis.

For surface area parametrically defined curves, and integration with respect to either variable when the function is one-to-one.

Possible learning activity: Arc lengths of sections of curves

Sample learning activity: Areas of a surface of revolution.

Area of study: Calculus Differential equations and kinematics – 3 weeks

Formulation of differential equations, representation and analysis using slope fields, verification of differential equations.

Analytic techniques for solving differential equations, numerical solutions using Euler’s method, use of technology to solve differential equations. Using pseudocode to describe the algorithm for Euler’s method.

Use of separation of variables to solve differential equations.

Kinematics, application of calculus to the analysis of rectilinear motion different forms of acceleration, velocity-time graphs.

Modelling or problem-solving task 1–2 weeks

Calculus context

Sample modelling or problem-solving task: Geese

Sample modelling or problem-solving task: Double Ferris wheel

Area of study: Space and measurement Vector calculus – 2 weeks

Position, velocity and acceleration vectors in two and three dimensions as functions of time.

Differentiation  and integration of vector functions.

Cartesian and parametric forms of paths in the plane, including circles, ellipses and hyperbolas.

The study of circles, ellipses and hyperbolas was undertaken in Specialist Mathematics Units 1 and 2.

Area of study: Data analysis, probability and statistics Data analysis, probability and statistics – 2 weeks

Linear combinations of random variables, simulation of repeated sampling, sample means and approximate confidence intervals for means.

Hypothesis testing for the mean (one and two tailed), p values for hypothesis testing related to the mean, errors in hypothesis testing.

Modelling or problem-solving task 1–2 weeks

Probability and statistics context.

Sample modelling or problem-solving task: Hypothesis testing from a known population

Sample modelling or problem-solving task: Sums of Bernoulli random variables