Planning
Accreditation period Units 1-4 from 2023
Introduction
The VCE General Mathematics Study Design 2023 support materials provides teaching and learning advice for Units 1 to 4, and assessment advice for school-based assessment in Units 3 and 4.
The program developed and delivered to students must be in accordance with the VCE General Mathematics Study Design 2023.
Sample course plan Units 1 and 2
The following sample course plan is based on the assumption of 18 teaching weeks per semester including time for review and assessment. A table of topics is given, followed by a detailed possible implementation sequence for these topics for the sample course.
Sample course overview
| Unit 1 | Unit 2 | ||
|---|---|---|---|
| Area of study | Topic | Area of study | Topic |
| Data analysis, probability and statistics | Investigating and comparing data distributions | Data analysis, probability and statistics | Investigating relationships between two numerical variables |
| Algebra, number and structure | Arithmetic and geometric sequences, first-order linear recurrence relations and financial mathematics | Discrete mathematics | Graphs and networks |
| Functions, relations and graphs | Linear functions, graphs, equations and models | Functions, relations and graphs | Variation |
| Discrete mathematics | Matrices | Space and measurement | Space, measurement and applications of trigonometry |
| 4 areas of study, 4 topics | 4 areas of study, 4 topics | ||
| Mathematical investigation | Mathematical investigation | ||
The following is a detailed possible implementation sequence for this selection of topics.
Sample course plan: Unit 1
| Assumed knowledge |
|---|
Topic: Computation and practical arithmetic Weeks 1–2 Review of integer, rational (fraction and decimal) and real arithmetic. Estimation, exact and approximate answers, scientific notation, significant figures, rounding and accuracy. Applications to mental, written and technology-based computation. 'Show' questions. Units of measure and order of magnitude, logarithm to base 10 scales. Ratio, proportion, percentage and percentage change, applications to solving practical problems including the unitary method. |
| Area of Study: Data analysis, probability and statistics |
Topic: Investigating and comparing data distributions Categorical (nominal or ordinal) and numerical (discrete or continuous, interval, ratio) data, and their display and description of the distribution of data. The mode and its interpretation. Display of numerical data using dot plot, stem plots and histogram as applicable. Description of the distribution of data in terms of shape, centre, and spread. Sample summary statistics (mean, median, range, interquartile range, standard deviation) and their use to describe numerical data distributions. Consideration of the range of distributions (symmetrical, asymmetrical), boxplots and the five-number summary, lower and upper fences and possible outliers. The use of back-to-back stem plots and parallel boxplots as applicable to compare distributions of a single numerical variable across two or more groups. Interpretations of similarities and differences in context. Sample investigation task: data investigation |
| Area of Study: Algebra, number and structure |
Topic: Arithmetic and geometric sequences, first-order linear recurrence relations and financial mathematics Sequences as function with domain N. Sequences generated by function rules and by recurrence relations. Tables of values and graphs for sequences. Arithmetic sequences and geometric sequences defined by rule and recurrence relations. Tables of values and graphs. Applications to practical problems. The use of a first-order linear recurrence relation to generate the values of an arithmetic or geometric sequence or to model and analyse practical situations of growth and decay including financial contexts. Percentage increase and decrease, marks-ups, discounts, GST, inflation, spending power and purchase options. Solving practical problems involving percentages and finance. |
| Area of Study: Functions, relations and graphs |
Topic: Linear functions, graphs, equations and models Review of linear functions and graphs. Linear models and their specifications, domain and range, interpretation of gradient and intercept in context. Using linear models to make predictions. Linear relations and function rule form, substitution and evaluation, transposition of formulas. Developing formulas word descriptions, tables of values and graphs. Sample learning activity: lines and graphs – Applications of piecewise defined line segment graphs. Solving linear equations and simultaneous linear equations numerically, graphically and algebraically. Applications of linear relations, equations and simultaneous linear equations. |
| Area of Study: Discrete mathematics |
Topic: Matrices Sets, arrays and matrices, elements and order of matrices, types of matrices, uses for storing and displaying data. Matrix arithmetic (sum, difference, scalar multiple, product and power); applications of matrices to model situations and solve problems. Matrix inverse, determinants, application to solving systems of simultaneous linear equations; communication matrices and regular transition matrices. |
Sample course plan: Unit 2
| Area of Study: Data analysis, probability and statistics |
|---|
Topic: Investigating relationships between two numerical variables Data involving response and explanatory variables. Scatterplots, identification and qualitative description of association in terms of direction, form and strength. Informal interpretation of association and causation. Modelling observed linear association with line of good fit by eye, interpretation of slope and intercept. Using the model to make predictions, interpolation and limitations of extrapolation. Sample learning activity: variation in random samples |
| Area of Study: Discrete mathematics |
Topic: Graphs and networks Introduction to the language and properties and types of graphs, including edge, face, loop, vertex, the degree of a vertex, isomorphic and connected graphs, and the adjacency matrix. Euler's formula for planar graphs. Walks, trails, paths, circuits, bridges and cycles in the context of traversing a graph, graphs and networks, and the shortest path problem. Trees and minimum spanning trees and the concept of a greedy algorithm in solving practical problems. Sample learning activity: Networks and types of graph |
| Area of Study: Functions, relations and graphs |
Topic: Variation Numerical, graphical and algebraic representation of direct and inverse variation. Transformation of data to linearity by plotting y against f(x), constant variation. Transformation of data to linearity, for example y and x2, y and Modelling of non-linear data using y = k f(x)+ C where f(x) is a sutable transformation Use a logarithmic (base 10) scale to represent quantities magnitude and to solve variation problems. Orders of magnitude including the use of log to base 10 scales. Sample mathematical investigation: Globalisation, education and wealth |
| Area of Study: Space and measurement |
Topic: Space, measurement and applications of trigonometry Units of measurement of length, angle, area, volume and capacity. Exact and approximate answers, scientific notation, significant figures and rounding. Application of trigonometry to right-angled triangles (Pythagoras' theorem and trigonometric ratios), including angles of elevation and depression and true bearings, sines and cosines of angles up to 180°. Review of mensuration and solving problems using practical applications of trigonometry involving perimeter and area of both standard and composite two-dimensional shapes including arc lengths and sectors. Practical situations involving quadrilaterals, circles and composite shapes including arcs and sectors. Length, surface area and volume problems involving three-dimensional objects and their composites. Applications of Pythagoras' theorem in three dimensions. Application of trigonometry to non-right-angled triangles including length (sine rule including the ambiguous case, cosine rule), area (three standard rules), surface area and volume and practical problems involving three-dimensional objects and their composites (includes Pythagoras in three dimensions). Similarity of shapes and solids and applications, the effect of linear scale factor on lengths, areas and volumes. Sample learning activity: Planning a house and land package Sample mathematical investigation: Tent size and guy rope length |
, y and log10(x).
Sample course plan Units 3 and 4
The following sample course is based on the assumption of 27 teaching weeks including time for review and assessment. A table of the topics is given, followed by a detailed possible implementation sequence for these topics.
Sample course overview
| Unit 3 | Unit 4 | ||
|---|---|---|---|
| Area of study | Topic | Area of study | Topic |
| Data analysis, probability and statistics | Data analysis | Discrete mathematics | Matrices |
| Discrete mathematics | Recursion and financial mathematics | Discrete mathematics | Graphs and networks |
| 2 Areas of study, 2 topics | 1 Area of study, 2 topics | ||
| School-Assessed Coursework | School-Assessed Coursework | ||
|
| ||
The following is a detailed possible implementation sequence for Units 3 and 4.
Sample course plan: Unit 3
| Area of Study: Data analysis, probability and statistics |
|---|
Topic: Investigating data distributions Review of data types and their representations. Distributions of categorical variables (data tables, two-way frequency tables, segmented bar charts) and numerical variables (dot plots, stem plots, histograms). Use of log scales as applicable for numerical data. Summary of distributions of numerical variables (five-number summary, Boxplots, upper and lower fences, possible outliers). Measures of centre and spread. Sample mean and standard deviation and their use in comparing distributions. Description of the distribution of data in terms of shape, centre, and spread (symmetrical, asymmetrical) Use of distributions of one or more categorical or numerical variables to answer statistical questions. Normal model for bell-shaped distributions, 68-95-99.7 rule, z-scores, comparing data across distributions. |
Investigating association between two variables Weeks 3–4 Response and explanatory variables and association between them. Association between categorical variables, two-way (contingency) frequency tables and segmented bar charts. Association between a categorical variable and a numerical variable: back-to-back stem-plots, parallel dot plots and boxplots. Association between numerical variables: scatterplots, qualitative description of association in terms of direction, form and strength. Pearson correlation coefficient, calculation and interpretation. Statistical questions about association between variables. Cause and effect, observation and experimentation. Non-causal explanations for association: common response, confounding and coincidence. Discussion and explanation in context. |
Investigating and modelling linear associations Weeks 5–6 Least squares regression line, y = a + bx, explanatory and response variables, determination of coefficients by technology and the relating summary statistics for these variables to the coefficients. Modelling linear association between two numerical variables. Interpretation of slope and intercepts, make predictions, interpolation and extrapolation in context. Use of coefficient of determination as a measure of strength of association in terms of explained variation, and use of residual plots to check quality of fit. Data transformation of non-linear to linearity by square, logarithmic (base 10) or reciprocal transformation of one axis only. Use of least squares regression line fitted to transformed data to make predictions. |
Investigating and modelling time series data Weeks 7–8 Time series, data: tables and graphs, features such as: trend, seasonality, irregular fluctuations, outliers, discontinuity, and their source. Numerical smoothing of time series data using moving means (centring as applicable) to identify long-term trends. Graphical smoothing of time series data using moving medians (for an odd number of points) to identify long-term trends. Seasonal adjustment, including seasonal indices, seasonal and yearly means, and their interpretation. Modelling trend in times series using least square regression lines (with de-seasonalisation, of data as applicable), forecasts, and consideration of limitations of the model. |
Week 9 – Application task This could be held earlier or later depending on coverage of relevant concepts, skills and processes with respect to content and context for the application task. Data analysis sample application task - exchange rates Data analysis sample application task - measles Data analysis sample application task - stock exchange Data analysis sample application task - longevity and prosperity Data analysis sample application task - agriculture Data sets for statistical investigations can be obtained from various sources, such as the Australian Bureau of Statistics, The Reserve Bank and Melbourne Water. |
| Topic: Recursion and financial mathematics |
| Area of Study: Discrete mathematics |
Depreciation of assets Week 10 Use of first order linear recurrence relations and their sequences to model and compare (numerically and graphically) flat rate, unit cost and reducing balance depreciation of an asset with time, including the use of a recurrence relation to determine the depreciating value of an asset after n depreciation periods for the initial sequence. Use of the rules for the future value of an asset after n depreciation periods for flat rate, unit cost and reducing balance depreciation and their application. |
Compound interest investments and loans Week 11 Simple and compound interest and their comparison. Use of a recurrence relation and their sequences to model and analyse (numerically and graphically) compound interest loan/investment after n compounding period for an initial sequence from first principles. Nominal and effective interest rates and how to calculate effective interest rates, use effective rates to compare returns/costs on a variety of loan periods. The future value of a compound interest investment/loan and its use to solve practical problems. |
Reducing balance loans Week 12 Use of first order linear recurrence relations to model and analyse (numerically and graphically) the amortisation of reducing balance investments/loans value at any time, including the use of a recurrence relation to determine the value of the loan or investment after n payments for an initial sequence from first principles. Use of technology to solve problems involving reducing balance loans, applications. Effect of change of interest rate. |
Annuities and perpetuities Use of first order linear recurrence relations to model and analyse (numerically and graphically) the amortisation of an annuity at any time, including the use of a recurrence relation to determine the value of the annuity after n payments for an initial sequence from first principles. Use of tables for step-by-step analysis of an annuity. Use of technology to solve problems involving annuities, applications. Perpetuities as annuities that last indefinitely. |
Compound interest investment with periodic and equal additions to the principal Week 14 Use of first order linear recurrence relations to model and analyse (numerically and graphically) annuity investments value at any time, including the use of a recurrence relation to determine the value of the investment after n payments have been made for an initial sequence from first principles. Use of tables for step-by-step analysis of an annuity investment. Use of technology to solve problems involving annuity investments, including future value, time or number of compounding periods to exceed a given value and interest rate or payment amount required to exceed a given value in a given time. |
Week 15 – Modelling or problem-solving task Recursion and financial modelling sample modelling or problem-solving task Recursion and financial modelling sample modelling or problem-solving task – superannuation Recursion and financial modelling sample modelling or problem-solving task – investments |
Sample course plan: Unit 4
| Area of Study: Discrete mathematics |
|---|
| Topic: Matrices |
Matrices and their applications Matrix definition, order and elements of a matrix, representation, structure and features, matrices as a list of lists and a rectangular array. Types of matrices, using matrices to represent information. Matrix arithmetic, the transpose of a matrix. Matrix basics. Matrix equality, operations (sum, difference, scalar multiple, product, power), and determinant and multiplicative inverse of a square matrix and the conditions for a matrix to have an inverse. Generating matrices based on rules for elements. Binary and permutation matrices, communication and dominance matrices, and their properties and applications. |
Transition matrices The use of the matrix recurrence relation: S0 = initial state matrix, Sn+1 = TSn or Sn+1 = LSn where T is a transition matrix (demonstration of Transition matrices of Markov chains), L is a Leslie matrix (demonstration of a Leslie matrix), and Sn is a column state matrix, to generate a sequence of state matrices (assuming the next state only relies on the current state). Sample Learning Activity: Leslie Matrices Construction of transition matrices from a transition diagram or a written description and vice versa. Transition diagrams, associated transition matrices and state matrices to model and analyse practical situations. Informal consideration of steady state (no noticeable change in state matrix). Modelling and analysing practical situations using transition matrices. Use of matrix recurrence relations to extend modelling to situations involving culling / restocking of populations (or similar situations). |
Modelling or problem-solving task Week 6 Matrices sample modelling or problem-solving Matrices sample modelling or problem-solving –trophic cascades |
| Area of Study: Discrete mathematics |
| Topic: Networks and decision mathematics |
Concepts, conventions and representations, and Exploring and travelling problems Week 1 Planar graphs and Euler's rule and directed graphs (digraphs) and networks. Use of matrices to represent, graphs, digraphs and networks and their applications. Construction of graphs and networks Concepts, conventions and notations of walks, trails, paths, cycles and circuits, Eulerian trails and Eulerian circuits, Hamiltonian paths and cycles, and related conditions, properties and applications. |
Trees and minimum connector problems, and shortest path problems Week 2 Concepts of tree and spanning tree, minimal spanning trees in a weighted connected graph, determination by inspection and by using Prim's algorithm. Application to minimal connector problems. Finding the shortest path between two specified vertices in a graph, digraph or network by inspection. Use Dijkstra's algorithm to find the shortest path between a given vertex and each of the other vertices in a weighted graph or network. |
Flow problems Week 3 Use of networks to model flow problems: capacity, sinks and sources. Solution of small-scale network flow problems by inspection and the 'maximum-flow minimum-cut' theorem for larger scale problems. |
Matching problems Week 4 Use of a bipartite graph and its tabular or matrix form to represent a matching problem. Finding the optimum assignment(s) of people or machines to tasks by inspection and use of the Hungarian algorithm for larger scale problems. |
Scheduling problems and critical path analysis Week 5 Construction of an activity network from a precedence table (or equivalent) with dummy activities where necessary, use of forward / backward scanning to find the earliest starting times (EST) and latest starting times (LST) for each activity. Use ESTs and LSTs to identify the critical path and find float times for non-critical activities, crashing to reduce completion time. |
Modelling or problem-solving task Week 6 Networks and decision mathematics sample modelling or problem-solving task –planning a campaign Networks and decision mathematics sample modelling or problem-solving task – distribution centres |
Aboriginal and Torres Strait Islander Perspectives in the VCE
Aboriginal and Torres Strait Islander Perspectives in the VCE
On-demand video recordings, presented with the Victorian Aboriginal Education Association Inc. (VAEAI) and the Department of Education (DE) Koorie Outcomes Division, for VCE teachers and leaders as part of the Aboriginal and Torres Strait Islander Perspectives in the VCE webinar program held in 2023.
Employability skills
Units 1 and 2
General Mathematics Units 1 and 2 provide students with the opportunity to engage in a range of learning activities. In addition to demonstrating their understanding and mastery of the content and skills specific to the study, students may also develop employability skills through their learning activities.
The nationally agreed employability skills* are: Communication; Planning and organising; Teamwork; Problem solving; Self-management; Initiative and enterprise; Technology; and Learning.
Each employability skill contains a number of facets that have a broad coverage of all employment contexts and designed to describe all employees. The table below links those facets that may be understood and applied in a school or non-employment related setting, to the types of assessment commonly undertaken within the VCE study.
Students undertaking the following types of assessment, in addition to demonstrating their understanding and mastery of the study, typically demonstrate the following key competencies and employability skills.
| Assessment task | Employability skills - selected facets | ||
|---|---|---|---|
| Assignments | Use of information and communications technology | ||
| Tests | Self management, use of information and communications technology | ||
| Summary or review notes | Self management | ||
| Mathematical investigations | Communication, team work, self management, planning and organisation, use of information and communications technology, initiative and enterprise | ||
| Short written responses | Communication, problem solving | ||
| Problem-solving tasks | Communication, problem solving, team work, use of information and communications technology | ||
| Modelling tasks | Problem solving, planning and organisation, use of information and communications technology |
Units 3 and 4
Students undertaking the following types of assessment, in addition to demonstrating their understanding and mastery of the content of the study, typically demonstrate the following key competencies and employability skills.
| Assessment task | Employability skills - selected facets | ||
|---|---|---|---|
| Modelling or problem-solving task | Planning and organising, solving problems, using mathematical ideas and techniques (written) communication, use of information and communications technology, self management | ||
| Application task | Planning and organising, solving problems, using mathematical ideas and techniques (written) communication, use of information and communications technology, self management |
*The employability skills are derived from the Employability Skills Framework (Employability Skills for the Future, 2002), developed by the Australian Chamber of Commerce and Industry and the Business Council of Australia, and published by the (former) Commonwealth Department of Education, Science and Training.
Resources
The following websites provide resources that include various approaches, activities and other materials which can be used to assist teachers in the implementation of General Mathematics Units 1- 4.
American Statistical Association
Australian Bureau of Statistics (ABS)
Hewlett – Packard Prime Links and Resources
Mathematical Association of Victoria (MAV) – Supporting implementation of VCE Mathematics
Mathematical Modelling (International Mathematical Modelling Challenge)
Wolfram Research – Demonstrations Project
Implementation videos
VCE General Mathematics (2023) implementation videos
Online video presentations which provide teachers with information about the new VCE General Mathematics Study Design for implementation in 2023.