Senior Subject Selection

MATHEMATICAL METHODS

UNITS 1 & 2

Mathematical Methods Units 1 and 2 provide an introductory study of simple elementary functions of a single real variable, algebra, calculus, probability and statistics and their applications in a variety of practical and theoretical contexts. The Units are designed as preparation for Mathematical Methods Units 3 and 4 and contain assumed knowledge and skills for these units.

The focus of Unit 1 is the study of simple algebraic functions. The focus of unit 2 is the study of simple transcendental functions and the calculus of simple algebraic functions.

Areas of Study

1 - Functions and Graphs

In this area of study students cover

the graphical representation of simple algebraic functions (polynomial and power functions) of a single real variable and the key features of functions and their graphs such as axis intercepts, domain (including the concept of maximal, natural or implied domain), co-domain and range, stationary points, asymptotic behaviour and symmetry. The behaviour of functions and their graphs is explored in a variety of modelling contexts and theoretical investigations.
graphical representation of functions of a single real variable and the key features of graphs of functions such as axis intercepts, domain (including maximal, natural or implied domain), co-domain and range, asymptotic behaviour, periodicity and symmetry.

2 - Algebra

This area of study supports students’ work in the ‘Functions and graphs’, ‘Calculus’ and ‘Probability and statistics’ areas of study.

algebra of polynomial functions of low degree and transformations of the plane.
use of inverse functions and transformations to solve equations involving trigonometric, exponential or logarithmic functions
index (exponent) laws and logarithm laws, including their application to the solution of simple exponential equations
numerical approximation of roots using Newton’s method.

3 - CALCULUS

In this area of study students cover

constant and average rates of change and an introduction to instantaneous rate of change of a function in familiar contexts, including graphical and numerical approaches to estimating and approximating these rates of change.
first principles approach to differentiation, differentiation and anti-differentiation of polynomial functions and power functions by rule, and related applications including the analysis of graphs.

4 - PROBABILITY AND STATISTICS

In this area of study students cover

the concepts of event, frequency, probability and representation of finite sample spaces and events using various forms such as lists, grids, Venn diagrams, Karnaugh maps, tables and tree diagrams. This includes consideration of impossible, certain, complementary, mutually exclusive, conditional and independent events involving one, two or three events (as applicable), including rules for computation of probabilities for compound events
introductory counting principles and techniques and their application to probability and the law of total probability in the case of two events.

Outcomes

The student is required to demonstrate achievement of three outcomes. As a set these outcomes encompass all of the areas of study for the unit.

On completion of this 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.
On completion of this 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.
On completion of this unit the student should be able to select and 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 skills outlined in all the areas of study.

Assessment

The award of satisfactory completion for a unit is based on the set of outcomes specified for the unit.

Demonstration of achievement of Outcome 1 should be based on the student’s performance on a selection of the following assessment tasks:

topic tests & semester examinations
assignments
summary or review notes.

Demonstration of achievement of Outcome 2 should be based on the student’s performance on a selection of the following assessment tasks:

modelling tasks
problem-solving tasks
mathematical investigations.

Demonstration of achievement of Outcome 3 is based on the student’s performance on aspects of tasks completed in demonstrating achievement of Outcomes 1 and 2 that incorporate opportunity for the effective and appropriate use of technology.

UNITS 3 & 4

Mathematical Methods Units 3&4 is only available to students who have successfully completed Units 1&2 Mathematical Methods.

Mathematical Methods Units 3 and 4 extend the introductory study of simple elementary functions of a single real variable, to include combinations of these functions, algebra, calculus, probability and 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 and statistics’, which will be covered in progression from Unit 3 to 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.

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.

Areas of Study

1 - Functions and Graphs

In this area of study students cover transformations of the plane and the behaviour of some elementary functions of a single real variable, including key features of their graphs such as axis intercepts, stationary points, points of inflection, domain (including maximal, implied or natural domain), co-domain and range, asymptotic behaviour and symmetry. The behaviour of these functions and their graphs is to be linked to applications in practical situations.

2- Algebra

In this area of study students cover the algebra of functions, including composition of functions, simple functional relations, inverse functions and the solution of equations. They also study the identification of appropriate solution processes for solving equations, and systems of simultaneous equations, presented in various forms. Students also cover recognition of equations and systems of equations that are solvable using inverse operations or factorisation, and the use of graphical and numerical approaches for problems involving equations where exact value solutions are not required or which are not solvable by other methods. This content is to be incorporated as applicable to the other areas of study.

2 - Calculus

In this area of study students cover graphical treatment of limits, continuity and differentiability of functions of a single real variable, and differentiation, anti-differentiation and integration of these functions. This material is to be linked to applications in practical situations.

4 - Probability and statistics

In this area of study students cover discrete and continuous random variables, their representation using tables, probability functions (specified by rule and defining parameters as appropriate); the calculation and interpretation of central measures and measures of spread; and statistical inference for sample proportions. The focus is on understanding the notion of a random variable, related parameters, properties and application and interpretation in context for a given probability distribution.

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.

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
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.
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.

Assessment

The award of satisfactory completion for a unit is based on the set of outcomes specified for the unit. This includes

topic tests & semester examinations
assignments
summary or review notes

The student’s level of achievement for Units 3 and 4 will be determined by School-assessed Coursework (SACs). The SACs are

Unit 3
- Application Task
Unit 4
- Problem Solving or Modelling Task 1
- Problem Solving or Modelling Task 2

For more information about Maths Methods see Simon Tyler or anyone in the Maths team.

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