08

## Mathematics

Modified June 9, 2023
23.5 min
Modified June 9, 2023

### Foundation Mathematics

Prerequisites: Year 10 Standard Level Maths

#### Foundation Mathematics Units 1/2

Foundation Mathematics Units 1 and 2 focus on providing students with the mathematical knowledge, skills, understanding and dispositions to solve problems in real contexts for a range of workplace, personal, further learning, and community settings relevant to contemporary society.

##### Unit 1: Semester 1

Unit 1 involves the study of algebra, number and structure, including fractions, decimals, percentages, rates and approximations; data analysis, probability and statistics, including collection and representation of data, construction of charts, tables and graphs, and interpretation of data; financial and consumer mathematics, including, personal financial services and information, income calculations and taxation; space and measurement, including standard metric units, reading and interpretation of scales, estimation and approximation strategies, and time and duration conventions, schedules and timetables.

On completion of this unit students should be able to: use and apply a range of mathematical concepts, skills and procedures from selected areas of study to solve practical problems based on a range of everyday and real-life contexts; apply mathematical processes in non-routine practical contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in practical situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework:  60% (inclusive of a mathematical investigation)
2. Examination: 40%

##### Unit 2: Semester2

Unit 2 involves the study of algebra, number and structure, including construction, use and interpretation of formulas, manipulation of symbolic expressions, and estimation, approximation and reasonableness of calculations and results; data analysis, probability and statistics, including measure of central tendency and simple measure of spread, and interpretation, summary and comparison of related data sets; financial and consumer mathematics, including products and services, managing money and financial and economic data trends over time; space and measurement, including simple and composite shapes, two-dimensional plans, location, maps, routes and itineraries.

On completion of this unit students should be able to: use and apply a range of mathematical concepts, skills and procedures from selected areas of study to solve practical problems based on a range of everyday and real-life contexts; apply mathematical processes in non-routine practical contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in practical situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework: 60% (inclusive of a mathematical investigation)
2. Examination: 40%

#### Foundation Mathematics Units 3/4

Foundation Mathematics Units 3 and 4 focus on providing students with the mathematical knowledge, skills and understanding to solve problems in real contexts for a range of workplace, personal, further learning, community, and global settings relevant to contemporary society.

There are 4 areas of study to be completed over the two units, two areas of study per unit. The areas of study are Algebra, number and structure, including estimation, the use and application of different forms of numbers and calculations, algorithmic and computational thinking, and the representation of formal mathematical expressions and processes including formulas and other algebraic expressions to solve practical problems in community, business and industry contexts; data analysis, probability and statistics, including collection, presentation and analysis of gathered and provided data from community, work, recreation and media contexts, including consideration of suitable forms of representation and summaries. This area of study incorporates the ability to critically reflect on statistical data and results, and to be able to communicate and report on the outcomes and any implications; discrete mathematics including, the use and application of different forms of numbers and calculations, relationships and formulae, and their application in relation to the analysis of, and critical reflection on, personal, local, national and global financial, consumer and global matters; and space and measurement, including the use and application of the metric system and related measurement in a variety of domestic, societal, industrial and commercial contexts, including consideration of accuracy, precision and error.

On completion of these units students should be able to: use and apply a range of mathematical concepts, skills and procedures from selected areas of study to solve practical problems based on a range of everyday and real-life contexts; apply mathematical processes in non-routine practical contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in practical situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Unit 3 School-assessed Coursework: 40% (two mathematical investigations)
2. Unit 4 School-assessed Coursework: 20% (a mathematical investigation)
3. Units 3 and 4 Examination: 40%

### General Mathematics

Prerequisites

Units 1 & 2 – Year 10 Standard Level Maths

Units 3 & 4 – Unit 1 & 2 General Mathematics or for accelerating students, Year 10 Algebra at least a B average.

### General Mathematics Units 1/2

General Mathematics Units 1 and 2 cater for a range of student interests, provide preparation for the study of VCE General Mathematics at the Units 3 and 4 level and contain assumed knowledge and skills for these units.

#### Unit 1: Semester 1

Unit 1 involves the study of investigating and comparing data distributions, including types of data, displaying data, summarising data, five number summary and calculation of outliers, back-to-back stem plots and parallel box plots; arithmetic and geometric sequences, first order linear recurrence relations and financial mathematics, including percentage increase and decrease, inflation and comparison of purchase options; linear functions, graphs, equations and models, including interpreting and graphing linear functions, solving simultaneous equations and piecewise functions; matrices, including use of matrices to store and display information, matrices arithmetic, inverse matrices and transition matrices.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework: 40% (inclusive of a mathematical investigation)
2. Examination 1: 30%
3. Examination 2: 30%

#### Unit 2: Semester 2

Unit 2 involves the study of investigating relationships between two numerical variables, including response and explanatory variables, scatterplots, informal interpretation of association and strength, fitting a line of best fit and interpretation of the line of best fit; graphs and networks, including notations, conventions, and representations of graphs, planar, connected and weighted graphs, and trees and minimum spanning trees; variation, including numerical, graphical and algebraic approaches, transformation of data to linearity, and modelling of non-linear data; space, measurement and applications of trigonometry,  including units of measure, exact and approximate answers, similar shapes and objects, perimeter, area, volume and surface area, trigonometric ratios and Pythagoras’ Theorem, and the sine and cosine rules.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework: 40% (inclusive of a Mathematical Investigation)
2. Examination 1: 30%
3. Examination 2: 30%

### General Mathematics – Units 3 and 4

General Mathematics Units 3 and 4 focus on real-life application of mathematics and consist of the areas of study ‘Data analysis, probability and statistics’ and ‘Discrete mathematics’.

#### Unit 3: Semester 1

Unit 3 involves the study of data analysis, including data types, representation and distribution of data, location, spread, association, correlation and causation, response and explanatory variables, linear regression, data transformation and goodness of fit, times series, seasonality, smoothing and prediction; and recursion and financial modelling, including the use of first-order linear recurrence relations and the time value of money (TVM) to model and analyse a range of financial situations, and using technology to solve related problems involving interest, appreciation and depreciation, loans, annuities and perpetuities.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

#### Unit 4: Semester 2

Unit 4 involves the study of matrices, including the definition of matrices, different types of matrices, matrix operations, transition matrices and the use of first-order linear matrix recurrence relations to model a range of situations and solve related problems; and networks and decision mathematics, including the definition and representation of different kinds of undirected and directed graphs, Eulerian trails, Eulerian circuits, bridges, Hamiltonian paths and cycles, and the use of networks to model and solve problems involving travel, connection, flow, matching, allocation and scheduling.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Unit 3 School-assessed Coursework: 24% (an application task and a problem-solving task)
2. Unit 4 School-assessed Coursework: 16% (two problem-solving tasks)
3. Units 3 and 4 Examination 1: 30%
4. Units 3 and 4 Examination 2: 30%

### Mathematical Methods (CAS)

Prerequisites

Units 1 & 2 – Year 10 Algebra at least a C+ average

Units 3 & 4 – Maths Methods (CAS) (Units 1 & 2)

#### Course Description

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.

##### Unit 1: Semester 1

Unit 1 involves the study of functions, relations and graphs, including 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; algebra, number and structure, including the algebra of polynomial functions of low degree and transformations of the plane; calculus, including 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; data analysis, probability and statistics, including the concepts of experiment (trial), outcome, event, frequency, probability and representation of finite sample spaces and events using various forms such as lists, grids, Venn diagrams and tables.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework: 40% (inclusive of a mathematical investigation)
2. Examination 1: 20% (technology free)
3. Examination 2: 40%

##### Unit 2: Semester 2

Unit 2 involves the study of functions, relations and graphs, including graphical representation of circular, exponential and logarithmic 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; algebra, number and structure, including the algebra of some simple transcendental functions and transformations of the plane; calculus, including differentiation and anti-differentiation of polynomial functions by rule, different notations, and related applications including the analysis of graphs; data analysis, probability and statistics, including the use of lists, tables and diagrams to calculate probabilities, including consideration of complementary, mutually exclusive, conditional and independent events involving one, two or three events (as applicable), including rules for computation of probabilities for compound events.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework: 40% (inclusive of a mathematical investigation)
2. Examination 1: 20% (technology free)
3. Examination 2: 40%

##### Mathematical Methods (CAS) – Units 3 and 4

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.

##### Unit 3: Semester 1

Unit 3 follows directly on from Mathematical Methods (CAS) Units 1 and 2 and assumes knowledge normally acquired in Unit 2. It involves the study of functions, relations and graphs; algebra, number and structure; Calculus, including applications of derivatives and differentiation, and identifying and analysing key features of functions and their graphs; and data analysis, probability and statistics, including the study of random variables, discrete and continuous probability distributions, and the distribution of sample proportions.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

##### Unit 4: Semester 2

Unit 4 involves the study of calculus, including 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, including to probability distributions of continuous random variables.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Unit 3 School-assessed Coursework: 20% (an application task)
2. Unit 4 School-assessed Coursework: 20% (two modelling or problem-solving tasks)
2. Unit 3 and 4 Examination 1: 20% (technology free)
3. Unit 3 and 4 Examination 2: 40%

### Specialist Mathematics

Prerequisites

Units 1 & 2 – Year 10 Higher Level Maths, must also be enrolled in Maths Methods Unit 1 & 2 or Units 3 & 4.

Units 3 & 4 – Specialist Maths Units 1 & 2 and must also be enrolled in Maths Methods Units 3 & 4 or have already completed Maths Methods Units 3 & 4.

#### Course Description

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, reasoning and proof.

##### Unit 1: Semester 1

Unit 1 involves the study of: algebra, number and structure, including the development of formal mathematical notation, definition, reasoning and proof applied to number systems, graph theory, sets, logic, and Boolean algebra, and the development of algorithms to solve problems; discrete mathematics, including the study of sequences, series, and first-order linear difference equations, combinatorics, including the pigeon-hole principle, the inclusion-exclusion principle, permutations and combinations, combinatorial identities, and matrices.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework: 40% (inclusive of a mathematical investigation)
2. Examination 1: 20% (technology free)
3. Examination 2: 40%

##### Unit 2: Semester 2

Unit 2 involves the study of: data analysis, probability and statistics including the study of linear combinations of random variables and the distribution of sample means of a population, with the use of technology to explore variability of sample means; space and measurement, including trigonometry and identities, rotation and reflection transformations of the plane and vectors for working with position, shape, direction and movement in the plane and related applications; algebra, number and structure, including the arithmetic and algebra of complex numbers, including polar form, regions and curves in the complex plane and introduction to factorisation of quadratic functions over the complex field; functions, relations and graphs, including an introduction to partial fractions; reciprocal and inverse circular functions and their graphs and simple transformations of these graphs; locus definitions of lines, parabolas, circles, ellipses and hyperbolas and the cartesian, parametric and polar forms of these relations.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Coursework: 40% (inclusive of a mathematical investigation)
2. Examination 1: 20% (technology free)
3. Examination 2: 40%

##### Specialist Mathematics – Units 3 and 4

Specialist Mathematics Units 3 and 4 assumes familiarity with the key knowledge and key skills from Mathematical Methods Units 1 and 2; the key knowledge and key skills from Specialist Mathematics Units 1 and 2; and concurrent study or previous completion of Mathematical Methods Units 3 and 4.

##### Unit 3: Semester 1

This unit involves the study of discrete mathematics; including logic; functions, relations and graphs, including, rational functions and other simple quotient functions, curve sketching of these functions and relations, and the analysis of key features of their graphs including intercepts, asymptotic behaviour and the nature and location of stationary points and points of inflection and symmetry; algebra, number and structure, including the algebra of complex numbers, including polar form, factorisation of polynomial functions over the complex field and an informal treatment of the fundamental theorem of algebra; space and measurement, including arithmetic and algebra of vectors; linear dependence and independence of a set of vectors and proof of geometric results using vectors; and calculus, including the advanced calculus techniques for analytical and numerical differentiation and integration of a broad range of functions, and combinations of functions; and their application in a variety of theoretical and practical situations, including curve sketching and evaluation of arc length, area and volume.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

##### Unit 4: Semester 2

This unit involves the study of discrete mathematics; including, proof techniques including mathematical induction; space and measurement, including vector representation of curves in the plane and their parametric and cartesian equations; vector kinematics in one, two and three dimensions; vector, parametric and cartesian equations of lines and planes; calculus, including differential equations and kinematics, and modelling with differential equations drawing from a variety of fields such as biology, economics and science; and data analysis, probability and statistics, including the study of linear combinations of random variables and introductory statistical inference with respect to the mean of a single population, the determination of confidence intervals, and hypothesis testing for the mean using the distribution of sample means.

On completion of this unit students should be able to: define and explain key concepts and apply a range of related mathematical routines and procedures; apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics; and apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches.

ASSESSMENT
1. Unit 3 School-assessed Coursework: 20% (an application task)
2. Unit 4 School-assessed Coursework: 20% (two modelling or problem-solving tasks)
3. Unit 3 and 4 Examination 1: 20% (technology free)
4. Unit 3 and 4 Examination 2: 40%