The statements in this section, taken from documents endorsed by Education Ministers as the agreed and common base for course development, are to be used to define expectations for the meaning (nature, scope and level of demand) of relevant aspects of the sections in this document setting out course requirements, learning outcomes, the course content and standards in the assessment.

Unit 1 – Topic 2: Algebra and matrices

**Linear and non-linear expressions:**

- substitute numerical values into linear algebraic and simple non-linear algebraic expressions, and evaluate (ACMGM010)

Unit 1 – Topic 3: Shape and Measurement

**Pythagoras’ theorem:**

- review Pythagoras’ Theorem and use it to solve practical problems in two dimensions and for simple applications in three dimensions (ACMGM017)

Unit 2 – Topic 2: Applications of Trigonometry

**Applications of trigonometry:**

- review the use of the trigonometric ratios to find the length of an unknown side or the size of an unknown angle in a right-angled triangle (ACMGM034)
- determine the area of a triangle given two sides and an included angle by using the rule `Area = 1/2 ab sin C`, or given three sides by using Heron’s rule, and solve related practical problems. (ACMGM035)
- solve problems involving non-right-angled triangles using the sine rule (ambiguous case excluded) and the cosine rule (ACMGM036)
- solve practical problems involving the trigonometry of right-angled and non-right-angled triangles, including problems involving angles of elevation and depression and the use of bearings in navigation (ACMGM037)

Unit 2 – Topic 3 Linear Equations and their graphs

**Linear equations:**

- identify and solve linear equations (ACMGM038)
- develop a linear formula from a word description. (ACMGM039)

**Straight-line graphs and their applications:**

- construct straight-line graphs both with and without technology (ACMGM040)
- determine the slope and intercepts of a straight-line graph from both its equation and its plot (ACMGM041)
- interpret, in context, the slope and intercept of a straight-line graph used to model and analyse a practical situation (ACMGM042)
- construct and analyse a straight-line graph to model a given linear relationship; for example, modelling the cost of filling a fuel tank of a car against the number of litres of petrol required (ACMGM043)

Unit 3 – Topic 1: Bivariate Data Analysis

**The statistical investigation process:**

- review the statistical investigation process; for example, identifying a problem and posing a statistical question, collecting or obtaining data, analysing the data, interpreting and communicating the results (ACMGM048)

**Identifying and describing associations between two categorical variables:**

- construct two-way frequency tables and determine the associated row and column sums and percentages (ACMGM049)
- use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association (ACMGM050)
- describe an association in terms of differences observed in percentages across categories in a systematic and concise manner, and interpret this in the context of the data (ACMGM051)

**Identifying and describing associations between two numerical variables:**

- construct a scatterplot to identify patterns in the data suggesting the presence of an association (ACMGM052)
- describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak) (ACMGM053)
- calculate and interpret the correlation coefficient (`r`) to quantify the strength of a linear association (ACMGM054)

**Fitting a linear model to numerical data:**

- identify the response variable and the explanatory variable (ACMGM055)
- use a scatterplot to identify the nature of the relationship between variables (ACMGM056)
- model a linear relationship by fitting a least-squares line to the data (ACMGM057)
- use a residual plot to assess the appropriateness of fitting a linear model to the data (ACMGM058)
- interpret the intercept and slope of the fitted line (ACMGM059)
- use the coefficient of determination to assess the strength of a linear association in terms of the explained variation (ACMGM060)
- use the equation of a fitted line to make predictions (ACMGM061)
- distinguish between interpolation and extrapolation when using the fitted line to make predictions, recognising the potential dangers of extrapolation (ACMGM062)
- write up the results of the above analysis in a systematic and concise manner (ACMGM063)

**Association and causation:**

- recognise that an observed association between two variables does not necessarily mean that there is a causal relationship between them (ACMGM064)
- identify possible non-causal explanations for an association, including coincidence and confounding due to a common response to another variable, and communicate these explanations in a systematic and concise manner (ACMGM065)

**The data investigation process:**

- implement the statistical investigation process to answer questions that involve identifying, analysing and describing associations between two categorical variables or between two numerical variables; for example, is there an association between attitude to capital punishment (agree with, no opinion, disagree with) and sex (male, female)? Is there an association between height and foot length? (ACMGM066)

Unit 3 – Topic 2: Growth and Decay in Sequences

**The arithmetic sequence:**

- use recursion to generate an arithmetic sequence (ACMGM067)
- display the terms of an arithmetic sequence in both tabular and graphical form and demonstrate that arithmetic sequences can be used to model linear growth and decay in discrete situations (ACMGM068)
- deduce a rule for the nth term of a particular arithmetic sequence from the pattern of the terms in an arithmetic sequence, and use this rule to make predictions (ACMGM069)
- use arithmetic sequences to model and analyse practical situations involving linear growth or decay; for example, analysing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, or calculating the value of an office photocopier at the end of each year using the straight-line method or the unit cost method of depreciation (ACMGM070)

**The geometric sequence:**

- use recursion to generate a geometric sequence (ACMGM071)
- display the terms of a geometric sequence in both tabular and graphical form and demonstrate that geometric sequences can be used to model exponential growth and decay in discrete situations (ACMGM072)
- deduce a rule for the nth term of a particular geometric sequence from the pattern of the terms in the sequence, and use this rule to make predictions (ACMGM073)
- use geometric sequences to model and analyse (numerically, or graphically only) practical problems involving geometric growth and decay; for example, analysing a compound interest loan or investment, the growth of a bacterial population that doubles in size each hour, the decreasing height of the bounce of a ball at each bounce; or calculating the value of office furniture at the end of each year using the declining (reducing) balance method to depreciate (ACMGM074)

**Sequences generated by first-order linear recurrence relations:**

- use a general first-order linear recurrence relation to generate the terms of a sequence and to display it in both tabular and graphical form (ACMGM075)
- recognise that a sequence generated by a first-order linear recurrence relation can have a long term increasing, decreasing or steady-state solution (ACMGM076)
- use first-order linear recurrence relations to model and analyse (numerically or graphically only) practical problems; for example, investigating the growth of a trout population in a lake recorded at the end of each year and where limited recreational fishing is permitted, or the amount owing on a reducing balance loan after each payment is made (ACMGM077)

Unit 3 – Topic 3: Graphs and networks

**The definition of a graph and associated terminology:**

- explain the meanings of the terms: graph, edge, vertex, loop, degree of a vertex, simple graph, complete graph, bipartite graph, directed graph (digraph), arc, weighted graph, and network (ACMGM078)
- identify practical situations that can be represented by a network, and construct such networks; for example, trails connecting camp sites in a National Park, a social network, a transport network with one-way streets, a food web, the results of a round-robin sporting competition (ACMGM079)

**Planar graphs:**

- explain the meaning of the terms: planar graph, and face (ACMGM081)
- apply Euler’s formula, `v + f- e = 2`, to solve problems relating to planar graphs (ACMGM082)

**Paths and cycles:**

- explain the meaning of the terms: walk, trail, path, closed walk, closed trail, cycle, connected graph, and bridge
- investigate and solve practical problems to determine the shortest path between two vertices in a weighted graph (by trial and-error methods only) (ACMGM084)
- explain the meaning of the terms: Eulerian graph, Eulerian trail, semi-Eulerian graph, semi-Eulerian trail and the conditions for their existence, and use these concepts to investigate and solve practical problems; for example, the Königsberg Bridge problem, planning a garbage bin collection route (ACMGM085)
- explain the meaning of the terms: Hamiltonian graph and semi-Hamiltonian graph, and use these concepts to investigate and solve practical problems; for example, planning a sight-seeing tourist route around a city, the travelling-salesman problem (by trial-and-error methods only) (ACMGM086)

Unit 4 – Topic 1: Time Series Analysis

**Describing and interpreting patterns in time series data:**

- construct time series plots (ACMGM087)
- describe time series plots by identifying features such as trend (long term direction), seasonality (systematic, calendar-related movements), and irregular fluctuations (unsystematic, short term fluctuations), and recognise when there are outliers; for example, one-off unanticipated events (ACMGM088)

**Analysing time series data:**

- smooth time series data by using a simple moving average, including the use of spreadsheets to implement this process (ACMGM089)
- calculate seasonal indices by using the average percentage method (ACMGM090)
- deseasonalise a time series by using a seasonal index, including the use of spreadsheets to implement this process (ACMGM091)
- fit a least-squares line to model long-term trends in time series data (ACMGM092)

**The data investigation process:**

- implement the statistical investigation process to answer questions that involve the analysis of time series data (ACMGM093)

Unit 4 – Topic 2: Loans, Investments and Annuities

**Compound interest loans and investments:**

- use a recurrence relation to model a compound interest loan or investment, and investigate (numerically or graphically) the effect of the interest rate and the number of compounding periods on the future value of the loan or investment (ACMGM094)
- calculate the effective annual rate of interest and use the results to compare investment returns and cost of loans when interest is paid or charged daily, monthly, quarterly or six-monthly (ACMGM095)
- with the aid of a calculator or computer-based financial software, solve problems involving compound interest loans or investments; for example, determining the future value of a loan, the number of compounding periods for an investment to exceed a given value, the interest rate needed for an investment to exceed a given value (ACMGM096)

**Reducing balance loans (compound interest loans with periodic repayments):**

- use a recurrence relation to model a reducing balance loan and investigate (numerically or graphically) the effect of the interest rate and repayment amount on the time taken to repay the loan (ACMGM097)
- with the aid of a financial calculator or computer-based financial software, solve problems involving reducing balance loans; for example, determining the monthly repayments required to pay off a housing loan (ACMGM098)

**Annuities and perpetuities (compound interest investments with periodic payments made from the investment):**

- use a recurrence relation to model an annuity, and investigate (numerically or graphically) the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity (ACMGM099)
- with the aid of a financial calculator or computer-based financial software, solve problems involving annuities (including perpetuities as a special case); for example, determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount (ACMGM100)

Unit 4 – Topic 3: Networks and Decision Mathematics

**Trees and minimum connector problems:**

- explain the meaning of the terms tree and spanning tree and identify practical examples (ACMGM101)
- identify a minimum spanning tree in a weighted connected graph either by inspection or by using Prim’s algorithm (ACMGM102)
- use minimal spanning trees to solve minimal connector problems; for example, minimising the length of cable needed to provide power from a single power station to substations in several towns (ACMGM103)

**Project planning and scheduling using critical path analysis (CPA):**

- construct a network to represent the durations and interdependencies of activities that must be completed during the project; for example, preparing a meal (ACMGM104)
- use forward and backward scanning to determine the earliest starting time (EST) and latest starting times (LST) for each activity in the project (ACMGM105)
- use ESTs and LSTs to locate the critical path(s) for the project (ACMGM106)
- use the critical path to determine the minimum time for a project to be completed (ACMGM107)
- calculate float times for non-critical activities (ACMGM108)

**Flow networks**

- solve small-scale network flow problems including the use of the ‘maximum-flow minimum- cut’ theorem; for example, determining the maximum volume of oil that can flow through a network of pipes from an oil storage tank (the source) to a terminal (the sink) (ACMGM109)

**Assignment problems**

- use a bipartite graph and/or its tabular or matrix form to represent an assignment/allocation problem; for example, assigning four swimmers to the four places in a medley relay team to maximise the team’s chances of winning (ACMGM110)
- determine the optimum assignment(s), by inspection for small-scale problems, or by use of the Hungarian algorithm for larger problems (ACMGM111)