11-12.ACMGM002 calculate weekly or monthly wage from an annual salary, wages from an hourly rate including situations involving overtime and other allowances and earnings based on commission or piecework
11-12.ACMGM003 calculate payments based on government allowances and pensions
11-12.ACMGM004 prepare a personal budget for a given income taking into account fixed and discretionary spending
11-12.ACMGM005 compare prices and values using the unit cost method
11-12.ACMGM006 apply percentage increase or decrease in various contexts; for example, determining the impact of inflation on costs and wages over time, calculating percentage mark-ups and discounts, calculating GST, calculating profit or loss in absolute and percentage terms, and calculating simple and compound interest
11-12.ACMGM007 use currency exchange rates to determine the cost in Australian dollars of purchasing a given amount of a foreign currency, such as US$1500, or the value of a given amount of foreign currency when converted to Australian dollars, such as the value of €2050 in Australian dollars
11-12.ACMGM008 calculate the dividend paid on a portfolio of shares, given the percentage dividend or dividend paid per share, for each share; and compare share values by calculating a price-to-earnings ratio.
11-12. Use of spreadsheets:
11-12.ACMGM009 use a spreadsheet to display examples of the above computations when multiple or repeated computations are required; for example, preparing a wage-sheet displaying the weekly earnings of workers in a fast food store where hours of employment and hourly rates of pay may differ, preparing a budget, or investigating the potential cost of owning and operating a car over a year.
11-12.1.2 Algebra and matrices
11-12. Linear and non-linear expressions:
11-12.ACMGM010 substitute numerical values into linear algebraic and simple non-linear algebraic expressions, and evaluate
11-12.ACMGM012 use a spreadsheet or an equivalent technology to construct a table of values from a formula, including two-by-two tables for formulas with two variable quantities; for example, a table displaying the body mass index (BMI) of people of different weights and heights.
11-12. Matrices and matrix arithmetic:
11-12.ACMGM013 use matrices for storing and displaying information that can be presented in rows and columns; for example, databases, links in social or road networks
11-12.ACMGM014 recognise different types of matrices (row, column, square, zero, identity) and determine their size
11-12.ACMGM015 perform matrix addition, subtraction, multiplication by a scalar, and matrix multiplication, including determining the power of a matrix using technology with matrix arithmetic capabilities when appropriate
11-12.ACMGM016 use matrices, including matrix products and powers of matrices, to model and solve problems; for example, costing or pricing problems, squaring a matrix to determine the number of ways pairs of people in a communication network can communicate with each other via a third person.
11-12.ACMGM018 solve practical problems requiring the calculation of perimeters and areas of circles, sectors of circles, triangles, rectangles, parallelograms and composites
11-12.ACMGM019 calculate the volumes of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the volume of water contained in a swimming pool
11-12.ACMGM020 calculate the surface areas of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the surface area of a cylindrical food container.
11-12.2.1 Univariate data analysis and the statistical investigation process
11-12. The statistical investigation process:
11-12.ACMGM026 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.
11-12. Making sense of data relating to a single statistical variable:
11-12.ACMGM027 classify a categorical variable as ordinal, such as income level (high, medium, low), or nominal, such as place of birth (Australia, overseas), and use tables and bar charts to organise and display the data
11-12.ACMGM028 classify a numerical variable as discrete, such as the number of rooms in a house, or continuous, such as the temperature in degrees Celsius
11-12.ACMGM029 with the aid of an appropriate graphical display (chosen from dot plot, stem plot, bar chart or histogram), describe the distribution of a numerical dataset in terms of modality (uni or multimodal), shape (symmetric versus positively or negatively skewed), location and spread and outliers, and interpret this information in the context of the data
11-12. Comparing data for a numerical variable across two or more groups:
11-12.ACMGM031 construct and use parallel box plots (including the use of the 'Q1 – 1.5 x IQR' and 'Q3 + 1.5 x IQR' criteria for identifying possible outliers) to compare groups in terms of location (median), spread (IQR and range) and outliers and to interpret and communicate the differences observed in the context of the data
11-12.ACMGM032 compare groups on a single numerical variable using medians, means, IQRs, ranges or standard deviations, as appropriate; interpret the differences observed in the context of the data; and report the findings in a systematic and concise manner
11-12.ACMGM033 implement the statistical investigation process to answer questions that involve comparing the data for a numerical variable across two or more groups; for example, are Year 11 students the fittest in the school?
11-12.2.2 Applications of trigonometry
11-12. Applications of trigonometry:
11-12.ACMGM034 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
11-12.ACMGM037 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.
11-12.2.3 Linear equations and their graphs
11-12. Linear equations:
11-12.ACMGM038 identify and solve linear equations
11-12.ACMGM043 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.
11-12.ACMGM045 solve practical problems that involve finding the point of intersection of two straight-line graphs; for example, determining the break-even point where cost and revenue are represented by linear equations.
11-12.ACMGM046 sketch piece-wise linear graphs and step graphs, using technology when appropriate
11-12.ACMGM047 interpret piece-wise linear and step graphs used to model practical situations; for example, the tax paid as income increases, the change in the level of water in a tank over time when water is drawn off at different intervals and for different periods of time, the charging scheme for sending parcels of different weights through the post.
11-12.3.1 Bivariate data analysis
11-12. The statistical investigation process:
11-12.ACMGM048 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.
11-12. Identifying and describing associations between two categorical variables:
11-12.ACMGM049 construct two-way frequency tables and determine the associated row and column sums and percentages
11-12.ACMGM058 use a residual plot to assess the appropriateness of fitting a linear model to the data
11-12.ACMGM059 interpret the intercept and slope of the fitted line
11-12.ACMGM060 use the coefficient of determination to assess the strength of a linear association in terms of the explained variation
11-12.ACMGM061 use the equation of a fitted line to make predictions
11-12.ACMGM062 distinguish between interpolation and extrapolation when using the fitted line to make predictions, recognising the potential dangers of extrapolation
11-12.ACMGM063 write up the results of the above analysis in a systematic and concise manner.
11-12. Association and causation:
11-12.ACMGM064 recognise that an observed association between two variables does not necessarily mean that there is a causal relationship between them
11-12.ACMGM065 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.
11-12. The data investigation process:
11-12.ACMGM066 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?
11-12.3.2 Growth and decay in sequences
11-12. The arithmetic sequence:
11-12.ACMGM067 use recursion to generate an arithmetic sequence
11-12.ACMGM068 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
11-12.ACMGM069 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
11-12.ACMGM070 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.
11-12. The geometric sequence:
11-12.ACMGM071 use recursion to generate a geometric sequence
11-12.ACMGM072 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
11-12.ACMGM073 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
11-12.ACMGM074 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.
11-12. Sequences generated by first-order linear recurrence relations:
11-12.ACMGM075 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
11-12.ACMGM076 recognise that a sequence generated by a first-order linear recurrence relation can have a long term increasing, decreasing or steady-state solution
11-12.ACMGM077 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.
11-12.3.3 Graphs and networks
11-12. The definition of a graph and associated terminology:
11-12.ACMGM078 explain the meanings of the terms: graph, edge, vertex, loop, degree of a vertex, subgraph, simple graph, complete graph, bipartite graph, directed graph (digraph), arc, weighted graph, and network
11-12.ACMGM079 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 roundrobin sporting competition
11-12.ACMGM080 construct an adjacency matrix from a given graph or digraph.
11-12. Planar graphs:
11-12.ACMGM081 explain the meaning of the terms: planar graph, and face
11-12.ACMGM082 apply Euler's formula, v + f – e = 2, to solve problems relating to planar graphs
11-12. Paths and cycles:
11-12.ACMGM083 explain the meaning of the terms: walk, trail, path, closed walk, closed trail, cycle, connected graph, and bridge
11-12.ACMGM084 investigate and solve practical problems to determine the shortest path between two vertices in a weighted graph (by trial-and-error methods only)
11-12.ACMGM085 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
11-12.ACMGM086 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).
11-12.4.1 Time series analysis
11-12. Describing and interpreting patterns in time series data:
11-12.ACMGM087 construct time series plots
11-12.ACMGM088 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.
11-12. Analysing time series data:
11-12.ACMGM089 smooth time series data by using a simple moving average, including the use of spreadsheets to implement this process
11-12.ACMGM090 calculate seasonal indices by using the average percentage method
11-12.ACMGM091 deseasonalise a time series by using a seasonal index, including the use of spreadsheets to implement this process
11-12.ACMGM092 fit a least-squares line to model long-term trends in time series data
11-12. The data investigation process:
11-12.ACMGM093 implement the statistical investigation process to answer questions that involve the analysis of time series data.
11-12.4.2 Loans, investments and annuities
11-12. Compound interest loans and investments:
11-12.ACMGM094 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
11-12.ACMGM095 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
11-12.ACMGM096 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.
11-12. Reducing balance loans (compound interest loans with periodic repayments):
11-12.ACMGM097 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
11-12.ACMGM098 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.
11-12. Annuities and perpetuities (compound interest investments with periodic payments made from the investment):
11-12.ACMGM099 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
11-12.ACMGM100 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.
11-12.4.3 Networks and decision mathematics
11-12. Trees and minimum connector problems:
11-12.ACMGM101 explain the meaning of the terms tree and spanning tree identify practical examples
11-12.ACMGM102 identify a minimum spanning tree in a weighted connected graph either by inspection or by using Prim's algorithm
11-12.ACMGM103 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
11-12. Project planning and scheduling using critical path analysis (CPA):
11-12.ACMGM104 construct a network to represent the durations and interdependencies of activities that must be completed during the project; for example, preparing a meal
11-12.ACMGM105 use forward and backward scanning to determine the earliest starting time (EST) and latest starting times (LST) for each activity in the project
11-12.ACMGM106 use ESTs and LSTs to locate the critical path(s) for the project
11-12.ACMGM107 use the critical path to determine the minimum time for a project to be completed
11-12.ACMGM108 calculate float times for non-critical activities.
11-12. Flow networks
11-12.ACMGM109 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).
11-12. Assignment problems
11-12.ACMGM110 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
11-12.ACMGM111 determine the optimum assignment(s), by inspection for small-scale problems, or by use of the Hungarian algorithm for larger problems.