Mathematics 2006-2011
The VCE Mathematics study has been reviewed and reaccredited for the period 2006-2011.
Summary of Changes
The principal developments in the revised Study Design are:
- the revision of topics and their reconfiguration within General Mathematics Units 1 and 2 using the six areas of study ‘Arithmetic’, ‘Data analysis and simulation’, ‘Algebra’, ‘Graphs of linear and non-linear relations’, ‘Decision and business mathematics’ and ‘Geometry and trigonometry’. A new topic, Kinematics has been included in the ‘Graphs of linear and non-linear relations’ area of study
- the inclusion of a new Matrices module in the ‘Applications’ area of study for Further Mathematics
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the inclusion of Mathematical Methods (CAS) as a fully accredited study available to all schools from 2006
- a greater focus on the concept of discrete and continuous random variables in the ‘Probability’ area of study in Mathematical Methods
- a revised examination structure for Mathematical Methods, Mathematical Methods (CAS) and Specialist Mathematics. This includes the introduction of a common technology-free Examination 1 for Mathematical Methods and Mathematical Methods (CAS) and a technology-free Examination 1 for Specialist Mathematics.
The structure of School-assessed Coursework for VCE Mathematics has not changed.
Outline of Key Changes
Foundation Mathematics Units 1 & 2
Material related to maps and plans has been re-allocated from the current ‘Measurement and design’ area of study to the ‘Space and shape’ area of study, which have be re-named as ‘Measurement’ and ‘Space, shape and design’ respectively.
General Mathematics Units 1 & 2
The rules for construction of a course of study for General Mathematics Units 1 and 2 are unchanged. Topics have been revised and re-configured within the areas of study as:
Arithmetic
- Matrices
- Integer and rational number systems
- Real and complex number systems
- Sequences and series
Graphs of linear and nonlinear relations
- Linear graphs and modelling
- Sketching and interpreting linear and non-linear graphs
- Variation
- Kinematics (new topic, numerical and graphical emphasis)
Data analysis and simulation
- Univariate data
- Bivariate data
- Simulation
Decision and business mathematics
- Networks
- Linear programming
- Financial arithmetic
Algebra
- Linear relations and equations
- Non-linear relations and equations
- Algebra and logic
Geometry and trigonometry
- Shape and measurement
- Geometry in two and three dimensions
- Coordinate geometry
- Vectors
- Trigonometric ratios and their applications
These changes provide greater coherence in grouping like topics within an over-arching area of study.
In 'Arithmetic', work on number systems has been given a distinctive focus through the development of separate topics on integers and rational numbers, and real and complex numbers, respectively.
The work on sequences and series has been re-configured to emphasise mapping from the natural numbers to the real numbers, and the related use of technology in their representation, analysis and application.
In 'Data analysis and simulation', simulation has been given greater emphasis, with consideration of bernoulli trials, markov chains, simple queuing problems and multi-event problems.
A new topic, Kinematics, which takes a graphical and numerical approach to analysis of rectilinear motion of a single particle, has been included in the ' Graphs of linear and non-linear relations' area of study, to support implementations of General Mathematics Units 1 and 2 designed to lead to Specialist Mathematics Units 3 and 4.
The 'Geometry and trigonometry' area of study incorporates the former ' Trigonometry' area of study with the single topic, Trigonometric ratios and their applications, as a topic of the same name within the combined area of study.
Further Mathematics Units 3 & 4
Changes to the ‘Data analysis’ and existing modules are generally minor and intended to account for changing real world practices, including the use of technology, as well as refining mathematical content. The inclusion of the new Matrices module reflects the importance of matrices in discrete mathematics and related applications.
Data Analysis
The core material requires students to make quantitative and qualitative analysis of various types of data, and minor changes to content include:
- the use of standardised scores (z scores) to assist in comparisons of data sets of different size;
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explicit consideration of the relationship between the correlation coefficient and the least squares regression line of best fit where
in the equation y = a + bx;
- explicit consideration of forecasting using trend lines with time series.
Applications
Module 1: Number patterns
- Ratios, proportions and percentages has been deleted as a separate topic.
- The use of numerical and graphical approaches to finding solutions to difference equations has been emphasised.
- Fibonacci and related sequences and their applications have been included as examples of sequences that can be expressed using a simple second order linear difference equation.
Module 2: Geometry and trigonometry
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Consideration of surface area and volume of solids has been made explicit, independent of change in linear dimension.
The use of Heron’s formula for calculating the area of a triangle has been included. - The use traverse surveys has been removed following advice that it is no longer a significant component of surveying practice.
Module 3: Graphs and relations
- No changes have been made to this module.
Module 4: Business related mathematics
- The material in this module has been reorganised without major change to the content. The module addresses the key concepts of financial transactions, and the growth of assets and investments. It reflects the relative importance of compound interest in investments and loans with an appreciation that the TVM facility of approved calculators can be extensively used to support problem-solving and related calculations.
Module 5: Networks and decision mathematics
- Consideration of dominance and reachability in networks has been included to broaden the range of applications that can be used.
- The ‘activity as edge’ method for network construction in critical path analysis has been specified.
- Familiarity with the use of the Hungarian algorithm for optimal allocation is now required.
Module 6: Matrices
- This is a new module with content covering matrix representation, arithmetic and applications such as solving simultaneous equations and transition matrices.
- The intended approach is numerical, with the use of technology to assist computation.
Mathematical Methods Units 1-4
Specification of common key by-hand skills statements with Mathematical Methods (CAS) for Outcome 1 across Units 1-4. Description of common content with Mathematical Methods (CAS) has been aligned, as applicable and appropriate.
In Unit 1 ‘Functions and graphs’ graphs of quartic polynomial functions with rule given in factorised form, simple power functions and inverse relations have been included.
In Unit 2, the content on permutations and the applications of permutations to probability, and the informal treatment of examples involving binomial and hypergeometric probabilities has been deleted, and calculation of probabilities for combinations of successive non-independent events emphasised.
The graphical treatment of circular functions has been generalised to simple cases of the form af(bx) + c for sine, cosine and tangent functions.
In Units 3 and 4, the content in the ‘Functions and graphs’ area of study has been re-configured to provide a more general description of transformation and combination (sum, difference and product) of functions and their graphs, including an explicit treatment of composition of functions (where the required composite is defined). The modulus function has also been included as a basic function.
The 'Probability' area of study has been substantially revised to focus on the study of discrete and continuous random variables (numerical integration by graphics calculator technology for the latter), properties and applications, with the binomial and normal distributions used as specific examples. Simple markov chains are also included as an application of conditional probability (intended treatment by tree diagram).
The hypergeometric distribution and informal consideration of relationship between binomial, hypergeometric and normal distributions have been deleted.
Mathematical Methods (CAS) Units 1-4
Specification of common key by-hand skills statements with Mathematical Methods (CAS) for Outcome 1 across Units 1-4. Description of common content with Mathematical Methods (CAS) has been aligned, as applicable and appropriate.
The content for Units 1-4 is essentially unchanged with some minor refinement. The compound and double angle formulas for sine and cosine have been deleted from Unit 2.
For Units 3 and 4, the hypergeometric distribution and informal consideration of relationship between binomial, hypergeometric and normal distributions has been deleted from the ‘Probability’ area of study.
Specialist Mathematics Units 3 & 4
In general, only minor refinements and have been made to the areas of study, with more explicit description of some content. Consistent with the areas of study for Mathematical Methods and Mathematical Methods(CAS), topics have been grouped under the areas of study ‘Functions, relations and graphs’ (which combines the previously separate ‘Coordinate geometry’ and ‘Circular (trigonometric) functions’ areas of study), ‘Algebra’, ‘Calculus’, ‘Vectors’ and ‘Mechanics’.
The use of implicit differentiation has been made explicit, and the work on the relationship between the graph of a function and the graph of its anti-derivative function linked to direction (slope) fields of a differential equation. The numerical solution of certain simple differential equations has also been linked to numerical integration using technology.
Assessment
Further Mathematics Units 3 & 4
There are no changes to the examination structure for Further Mathematics Units 3 and 4.
Mathematical Methods Units 3 & 4 and Mathematical Methods (CAS) Units 3 & 4
For Mathematical Methods Units 3 and 4 and Mathematical Methods (CAS) Units 3 and 4 there will be a common one-hour technology-free Examination 1 where students are required to answer a collection of short-answer and some extended-response questions.
For each of Mathematical Methods Units 3 and 4 and Mathematical Methods (CAS) Units 3 and 4 there will be a separate two hour Examination 2 where students are required to answer a collection of multiple-choice questions and extended-response questions. Student access to an approved graphics calculator will be assumed for Mathematical Methods (the use of CAS is not permitted for examinations in Mathematical Methods) and student access to an approved CAS will be assumed for Mathematical Methods (CAS).
Specialist Mathematics Units 3 & 4
For Specialist Mathematics Units 3 and 4 there will be a one-hour technology-free Examination 1 where students are required to answer a collection of short-answer and some extended-response questions; and a two-hour Examination 2 where students are required to answer a collection of multiple-choice questions and extended-response questions. Student access to an approved graphics calculator or CAS will be assumed for Examination 2. This examination will be set so that it is technology active, but neutral with respect to graphics calculator/CAS functionality.
Calculators and Support Material
For Examination 1 of Mathematical Methods, Mathematical Methods (CAS) and Specialist Mathematics, no calculators, CAS or notes of any kind are permitted. A sheet of formulas will be provided with the examination.
For Examinations 1 and 2 of Further Mathematics, and Examination 2 of Mathematical Methods, Mathematical Methods (CAS) and Specialist Mathematics, student access to approved technology, as applicable to each study, will be assumed. Approved technology and one bound reference, text (which may be annotated) or lecture pad may be brought into the examination. A sheet of formulas will be provided with the examination.
The revised examination structure takes account of developments in graphics calculator and other technology in recent years - in particular the enhanced functionality and substantial memory capacity of recent models of graphics calculators, and the increasing convergence between CAS functionality and the functionality of graphics calculators with supplementary programs. These developments are likely to continue.
Accordingly, the VCAA foreshadows that it will direct examiners to assume student access to an approved CAS for Examination 2 in Mathematics Methods and Specialist Mathematics examinations for 2009. As with the earlier introduction of graphics calculators into VCE Mathematics examinations in the late 1990s, this locates the process of familiarisation for all at the end of an existing accreditation period rather than at the beginning of a new accreditation period.
Details on the use of approved technology for VCE Mathematics examinations are published annually in the VCAA Bulletin.
Enquiries
Enquiries about the content of the Study Design can be directed to
David Leigh-Lancaster, Manager, Mathematics
Telephone: (03) 9651 4537
Email: leigh-lancaster.david.d@edumail.vic.gov.au