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, and the areas of study are ‘Functions and graphs’, ‘Algebra’, ‘Calculus’ and ‘Probability and statistics’. At the end of Unit 1, students are expected to have covered the content outlined in each area of study, with the exception of ‘Algebra’ which extends across Units 1 and 2.

AREA 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.

AREA OF STUDY 2 - Algebra

This area of study supports students’ work in the ‘Functions and graphs’, ‘Calculus’ and ‘Probability and statistics’ areas of study, and content is to be distributed between Units 1 and 2. In Unit 1 the focus is on the algebra of polynomial functions of low degree and transformations of the plane.

AREA OF STUDY 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.

AREA OF STUDY 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

OUTCOME 1

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.

OUTCOME 2

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.

OUTCOME 3

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.

ASSESSMENT TASKS

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

• assignments

• tests

• summary or review notes.

Demonstration of achievement of Outcome 2 will 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 will be 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.

In Unit 2 students focus on the study of simple transcendental functions and the calculus of simple algebraic functions. The areas of study are ‘Functions and graphs’, ‘Algebra’, ‘Calculus’, and ‘Probability and statistics’. At the end of Unit 2, students are expected to have covered the material outlined in each area of study. Material from the ‘Functions and graphs’, ‘Algebra’, ‘Calculus’, and ‘Probability and statistics’ areas of study should be organised so that there is a clear progression of skills and knowledge from Unit 1 to Unit 2 in each area of study.

AREA OF STUDY 1 - Functions and Graphs

In this area of study students cover 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.

AREA OF STUDY 2 - Algebra

This area of study supports students’ work in the ‘Functions and graphs’, ‘Calculus’ and ‘Probability and statistics’ areas of study. In Unit 2 the focus is on the algebra of some simple transcendental functions and transformations of the plane. This area of study provides an opportunity for the revision, further development and application of content prescribed in Unit 1, as well as the study of additional algebra material introduced in the other areas of study in Unit 2 as follows: • use of inverse functions and transformations to solve equations of the form Af (bx) + c = k, where A, b, c, k ∈ R and A, b ≠ 0 and f is sine, cosine, tangent or ax , using exact or approximate values on a given domain • index (exponent) laws and logarithm laws, including their application to the solution of simple exponential equations • numerical approximation of roots of cubic polynomial functions using Newton’s method.

AREA OF STUDY 3 - CALCULUS

In this area of study students cover 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.

AREA OF STUDY 4 - Probability and Statistics

In this area of study students cover introductory counting principles and techniques and their application to probability and the law of total probability in the case of two events.

OUTCOME 1

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.

OUTCOME 2

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.

OUTCOME 3

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.

ASSESSMENT TASKS

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

• assignments

• tests

• summary or review notes.

Demonstration of achievement of Outcome 2 will 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 will be 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.

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

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 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, 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.

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.

OUTCOMES

1. On completion of this unit the student should be able to define and explain key concepts and apply related mathematical techniques and models as specified in Area of Study 1 in routine contexts. To achieve this outcome the student will draw on knowledge and skills outlined in Area of Study 1.
2. 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.
3. 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.

ASSESSMENT TASKS

• Applications Investigations
• Topic Tests