Specialist Mathematics Units 1 and 2 provide a course of study for students who wish to undertake an in-depth study of mathematics, with an emphasis on concepts, skills and processes related to mathematical structure, modelling, problem-solving and reasoning. This study has a focus on interest in the discipline of mathematics in its own right and investigation of a broad range of applications, as well as development of a sound background for further studies in mathematics and mathematics-related fields.

Mathematical Methods Units 1 and 2 and Specialist Mathematics Units 1 and 2, taken in conjunction, provide comprehensive preparation for Specialist Mathematics Units 3 and 4. The areas of study for Units 1 and 2 of Specialist Mathematics are ‘Algebra and structure’, ‘Arithmetic and number’, ‘Discrete mathematics’, ‘Geometry, measurement and trigonometry’, ‘Graphs of linear and non-linear relations’ and ‘Statistics’. For Units 1 and 2, to suit the range of students entering the study, and cover the four prescribed topics, content must be selected from the six areas of study using the following rules:

for each unit, content covers four or more topics in their entirety, selected from at least three different areas of study
each unit must include two of the prescribed topics: Number systems and recursion; Vectors in the plane; Geometry in the plane and proof; and Graphs of non-linear relations
other topics can be selected from those included in the areas of study for Specialist Mathematics Units 1 and 2 and/or General Mathematics Units 1 and 2
courses intended as preparation for study at the Units 3 and 4 level should include a selection of content from areas of study that provide a suitable background for these studies
content from an area of study provides a clear progression in knowledge and skills from Unit 1 to Unit 2.

In undertaking these units, students are expected to be able to apply techniques, routines and processes involving rational, real and complex arithmetic, sets, lists and tables, diagrams and geometric constructions, algebraic manipulation, equations and graphs with and without the use of technology. They should have facility with relevant mental and by-hand approaches to estimation and computation. The use of numerical, graphical, geometric, symbolic and statistical functionality of technology for teaching and learning mathematics, for working mathematically, and in related assessment, is to be incorporated throughout each unit as applicable.

OUTCOMES

For each unit the student is required to demonstrate achievement of three outcomes. As a set these outcomes encompass all of the selected areas of study for each unit. For each of Unit 1 and Unit 2 the outcomes as a set apply to the content from the areas of study and topics selected for that unit.

Area of Study 1 - Algebra and structure

Logic and algebra

Key skills

represent and test the truth of propositions and validity of arguments using karnaugh maps and truth tables
develop proofs of propositions in natural language and mathematics
represent circuits using gates and simplify these circuits
use boolean operators for searches in databases and by search engines.

Transformations, trigonometry and matrices

Key skills

define and apply transformations to the plane and specify their effect on subsets of the plane
identify the set of points that are invariant under a given transformation
find and apply inverse transformations and composite transformations, and interpret their effects on subsets of the plane
prove trigonometric identities and apply them to solve problems.

Area of Study 2 - Arithmetic and number

Principles of counting

Key skills

use one-to-one correspondence to demonstrate the countability of certain subsets of R
solve problems which involve techniques of counting
use deductive reasoning to solve problems involving counting techniques, the pigeon-hole principle and Pascal’s triangle.

Number systems and recursion

Key skills

define and represent number in various structures and contexts such as integer, rational, real and complex number systems, ordered sets of numbers such as sequences and series
identify and determine special forms such as identity, inverse, conjugate, and limit value
perform exact and approximate computations and apply algorithms in various structures and contexts, including sequences and series, and interpret results
apply deductive reasoning, including mathematical induction, and use appropriate language in the construction of mathematical arguments and proofs involving number and algebra.

Area of Study 3 - Discrete mathematics

Graph theory

Key skills

construct graphs and use them to model situations
use algorithms to construct subsets of graphs according to conditions and solve related problems
develop and understand results on areas including planar graphs, trails and circuits
solve problems and prove theorems involving graphs.

Area of Study 4 - Geometry, measurement and trigonometry

Geometry in the plane and proof

Key skills

identify assumptions, give definitions and provide examples and counter-examples using appropriate mathematical language, diagrams and models
complete geometric constructions using compass and straight edge and dynamic geometry technology
model situations and solve problems involving geometry
prove theorems involving lines, polygons and circles.

Vectors in the plane

Key skills

use vectors to model situations involving direction and magnitude
apply vector operations of scalar multiples, addition, subtraction and scalar product
use vectors as an alternative geometric tool and apply to problems in describing position, displacement,velocity and force
use vectors to solve geometric problems and prove theorems.

Area of Study 5 - Graphs of linear and non-linear relations

Kinematics

Key skills

construct continuous position-graphs, velocity-time and acceleration-time graphs based on empirical data, and interpret these and given graphs in context
determine position, time, speed, displacement, distance travelled, velocity and acceleration in contexts involving rectilinear motion, and solve related problems
apply the formulas for rectilinear motion involving constant acceleration to solve problems.

Non-linear relations and functions

Key skills

construct graphs from empirical data and form continuous interpolations and extrapolations, and identify and interpret key features of these graphs and given graphs
construct the graph of a reciprocal function from the graph of the original function
use the distance formula and locus definitions to obtain the rule of a relation and draw the corresponding curve
graph non-linear relations in the plane from their cartesian, polar and parametric representations, and identify and interpret their key features.

Area of Study 6 - Statistics

Simulation, sampling and sampling distributions

Key skills

simulate the sampling process from a population
display the results of taking multiple samples of the same size from a fixed population
consider measures of central tendency and spread of the distribution of sample means and sample proportions.