Specialist Mathematics Units 3 and 4 consist of the areas of study: “Functions and graphs”, “Algebra”, “Calculus”, “Vectors”, “Mechanics” and “Probability and statistics”. The development of course content should highlight mathematical structure, reasoning and applications across a range of modelling contexts with an appropriate selection of content for each of Unit 3 and Unit 4. The selection of content for Unit 3 and Unit 4 should be constructed so that there is a balanced and progressive development of knowledge and skills with connections among the areas of study being developed as appropriate across Unit 3 and Unit 4.

Specialist Mathematics Units 3 and 4 assumes familiarity with the key knowledge and skills from Mathematical Methods Units 1 and 2, the key knowledge and skills from Specialist Mathematics Units 1 and 2 topics ‘Number systems and recursion’ and “Geometry in the plane and proof”, and concurrent or previous study of Mathematical Methods Units 3 and 4. Together these cover the assumed knowledge and skills for Specialist Mathematics, which are drawn on as applicable in the development of content from the areas of study and key knowledge and skills for the outcomes.

In Unit 3 a study of Specialist Mathematics would typically include content from “Functions & graphs” and a selection of material from the “Algebra”, “Calculus” and “Vectors” areas of study.

In Unit 4 this selection would typically consist of the remaining content from the “Algebra”, “Calculus”, and ‘Vectors’ areas of study and the content from the “Mechanics” and “Probability & statistics” areas of study.

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, graphs, differentiation, anti-differentiation and integration and inference 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 areas of study for each unit. For each of Unit 3 and Unit 4 the outcomes apply to the content from the areas of study selected for that unit.

Outcome 1

On the completion of each unit the student should be able to define and explain key concepts as specified in the content from the areas of study, and apply a range of related mathematical routines and procedures.

To achieve this outcome the student will draw on knowledge and skills outlined in all the areas of study.

Key knowledge

Key skills

sketch graphs and describe behaviour of specified functions and relations with and without the assistance of technology, clearly identifying their key features and using the concepts of first and second derivatives
perform operations on complex numbers expressed in cartesian form or polar form and interpret them geometrically
represent curves on an argand diagram using complex relations
apply implicit differentiation, by hand in simple cases
use analytic techniques to find derivatives and anti-derivatives by pattern recognition, and apply anti-derivatives to evaluate definite integrals
set up and evaluate definite integrals to calculate arc lengths, areas and volumes
set up and solve differential equations of specified forms
represent and interpret differential equations by direction(slope) fields
perform operations on vectors and interpret them geometrically
apply vectors to motion of a particle and to geometric problems
solve kinematics problems using a variety of techniques
set up and solve problems involving Newton’s laws of motion
apply a range of analytical, graphical and numerical processes to obtain solutions (exact or approximate) to equations
set up and solve problems involving the distribution of sample means
construct approximate confidence intervals for sample means
undertake a hypothesis test for a mean of a sample from a normal distribution or a large sample.

Outcome 2

On the completion of each unit, the student should be able to apply mathematical processes, with an emphasis on general cases, in non-routine contexts, and analyse and discuss these applications of mathematics.

To achieve this outcome the student will draw on knowledge and skills outlined in one or more areas of study.

Key skills

specify the relevance of key mathematical content from one or more areas of study to the investigation of various questions related to a given context
give mathematical formulations of specific and general cases used to derive results for analysis within a given application context
develop functions as possible models for data presented in graphical form and apply a variety of techniques to decide which function provides an appropriate model
use a variety of techniques to verify results
establish proofs for general case results
make inferences from analysis and use these to draw valid conclusions related to a given application context
communicate conclusions using both mathematical expression and everyday language, in particular in relation to a given application context.

Outcome 3

On completion of each unit the student should be able to select and appropriately use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.

To achieve this outcome the student will draw on knowledge and related skills outlined in all the areas of study.

Key skills

distinguish between exact and approximate presentations of mathematical results produced by technology, and interpret these results to a specified degree of accuracy
use technology to carry out numerical, graphical and symbolic computation as applicable
produce results using a technology which identify examples or counter-examples for propositions
produce tables of values, symbolic expressions, families of graphs and collections of other results using technology, which support general analysis in problem-solving, investigative and modelling contexts
use appropriate domain and range specifications to illustrate key features of graphs of functions and relations
identify the relation between numerical, graphical and symbolic forms of information about functions and equations and the corresponding features of those functions and equations
specify the similarities and differences between formal mathematical expressions and their representation by technology, in particular, equivalent forms of symbolic expressions
select an appropriate functionality of technology in a variety of mathematical contexts, and provide a rationale for these selections
apply suitable constraints and conditions, as applicable, to carry out required computations
relate the results from a particular technology application to the nature of a particular mathematical task (investigative, problem solving or modelling) and verify these results
specify the process used to develop a solution to a problem using technology, and communicate the key stages of mathematical reasoning (formulation, solution, interpretation) used in this process.