Mathematical Methods Units 3 and 4 are completely prescribed and extend the introductory study of simple elementary functions of a single real variable, to include combinations of these functions, algebra, calculus, probability & statistics, and their applications in a variety of practical and theoretical contexts. Units 3 and 4 consist of the areas of study “Functions and graphs”, “Calculus”, “Algebra” and “Probability & statistics”, which must be covered in progression from Unit 3 to Unit 4, with an appropriate selection of content for each of Unit 3 and Unit 4. Assumed knowledge and skills for Mathematical Methods Units 3 and 4 are contained in Mathematical Methods Units 1 and 2, and will be drawn on, as applicable, in the development of related content from the areas of study, and key knowledge and skills for the outcomes of Mathematical Methods Units 3 and 4.

For Unit 3 a selection of content would typically include the areas of study ‘Functions and graphs’ and ‘Algebra’, and applications of derivatives and differentiation, and identifying and analysing key features of the functions and their graphs from the ‘Calculus’ area of study.

For Unit 4, this selection would typically consist of remaining content from the areas of study: “Functions and graphs”, “Calculus” and “Algebra”, and the study of random variables and discrete and continuous probability distributions and the distribution of sample proportions. For Unit 4, the content from the “Calculus” area of study would be likely to include the treatment of anti-differentiation, integration, the relation between integration and the area of regions specified by lines or curves described by the rules of functions, and simple applications of this content.

The selection of content from the areas of study should be constructed so that there is a development in the complexity and sophistication of problem types and mathematical processes used (modelling, transformations, graph sketching and equation solving) in application to contexts related to these areas of study. There should be a clear progression of skills and knowledge from Unit 3 to Unit 4 in each area of study.

In undertaking these units, students are expected to be able to apply techniques, routines and processes involving rational and real arithmetic, sets, lists and tables, diagrams and geometric constructions, algebraic manipulation, equations, graphs, differentiation, anti-differentiation, 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 selected areas of study for each unit. For each of Unit 3 and Unit 4 the outcomes as a set apply to the content from the areas of study covered in that unit.

Outcome 1

On 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 skills

identify key features and properties of the graph of a function or relation and draw the graphs of specified functions and relations, clearly identifying their key features and properties
describe the effect of transformations on the graphs of a function or relation
apply matrices to transformations of functions and their graphs
find the rule of an inverse function and give its domain and range
find the rule of a composite function and give its domain and range
apply a range of analytical, graphical and numerical processes, as appropriate, to obtain general and specific solutions (exact or approximate) to equations (including literal equations) over a given domain and be able to verify solutions to a particular equation or equations over a given domain
solve by hand equations of the form sin(ax + b) = c, cos(ax + b) = c and tan(ax + b) = c with exact value solutions over a given interval
apply algebraic, logarithmic and circular function properties to the simplification of expressions and the solution of equations
evaluate derivatives of basic, transformed and combined functions and apply differentiation to curve sketching and related optimisation problems
find derivatives of polynomial functions and power functions
apply the product, chain and quotient rules for differentiation to simple combinations of functions by hand
find derivatives of basic and more complicated functions and apply differentiation to curve sketching and optimisation problems
find anti-derivatives of polynomial functions and power functions,
evaluate rectangular area approximations to the area under a curve, find and verify anti-derivatives of specified functions and evaluate definite integrals
apply definite integrals to the evaluation of the area under a curve and between curves over a specified interval
analyse a probability mass function or probability density function and the shape of its graph in terms of the defining parameters for the probability distribution and the mean and variance of the probability distribution
calculate and interpret the probabilities of various events associated with a given probability distribution, by hand in cases where simple arithmetic computations can be carried out
apply probability distributions to modelling and solving related problems
simulate repeated random sampling and interpret the results, for a variety of population proportions and a range of sample sizes, to illustrate the distribution of sample proportions and variations in confidence intervals
calculate sample proportions and confidence intervals for population proportions.

Outcome 2

On completion of each unit the student should be able to apply mathematical processes in non-routine contexts, including situations requiring problem-solving, modelling or investigative techniques or approaches, 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 in a given context
develop mathematical formulations of specific and general cases used to derive results for analysis within a given context
use a variety of techniques to verify results
make inferences from analysis and use these to draw valid conclusions related to a given context
communicate conclusions using both mathematical expression and everyday language, in particular, the interpretation of mathematics with respect to the 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, 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.