General Mathematics · QCAA
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▶1 — Money, measurement, algebra and linear equations30 LOs
▶Consumer arithmetic9 LOs
▶Applications of rates, percentages and use of spreadsheets9 LOs
●Apply percentage increase or decrease in various contexts, e.g. inflation of costs and wages, percentage mark-ups and discounts, percentage profit and loss, GST, simple interest. 𝐼 = 𝑃𝑖𝑛 where 𝐼 is simple interest, 𝑃 is principal, 𝑖 is interest rate per year and 𝑛 is number of years●Calculate income support payments based on government allowances and pensions.●Calculate the dividend paid on a portfolio of shares, given the dividend yield or dividend paid per share, and compare share values by calculating a price-to-earnings (P/E) ratio. dividend yield = dividend share price × 100 P/E ratio = market price per share annual earnings per share●Calculate weekly, fortnightly or monthly wages from an annual salary, and wages from an hourly rate, including situations involving overtime and other allowances and earnings based on commission or piecework.●Compare prices and values using the unit cost method.●Prepare a personal budget for a given income, taking into account fixed and discretionary spending.●Understand the meaning of rates and percentages.●Use a spreadsheet to display examples of the above computations when multiple or repeated computations are required, e.g. preparing a wage sheet displaying the weekly earnings of workers in an organisation, preparing a budget, investigating the potential cost of owning and operating a car over a year.●Use currency exchange rates to convert between the Australian dollar and foreign currencies.
▶Shape and measurement6 LOs
▶Pythagoras’ theorem1 LO
▶Mensuration5 LOs
●Calculate areas, 𝐴, of standard two-dimensional objects in practical situations, including circles, sectors of circles, triangles, rectangles, parallelograms, trapeziums and composites. circle: 𝐴 = 𝜋𝑟2 where 𝑟 is radius sector: 𝐴 = 𝜃 360 𝜋𝑟2 where 𝜃 is central angle and 𝑟 is radius triangle: 𝐴 = 1 2 𝑏ℎ where 𝑏 is base length and ℎ is perpendicular height parallelogram: 𝐴 = 𝑏ℎ where 𝑏 is base length and ℎ is perpendicular height trapezium: 𝐴 = 1 2 (𝑎 + 𝑏)ℎ where 𝑎 and 𝑏 are parallel lengths and ℎ is perpendicular height●Calculate perimeters, 𝑃, of standard two-dimensional objects in practical situations, including circles, sectors, triangles, rectangles, trapeziums, parallelograms and composites. circle: 𝐶 = 2𝜋𝑟 where 𝐶 is circumference and 𝑟 is radius sector: 𝑃 = 2𝑟 + 𝜃 180 𝜋𝑟 where 𝜃 is central angle and 𝑟 is radius General Mathematics 2025 v1.3●Calculate surface areas, 𝑆, of standard three-dimensional objects in practical situations, including rectangular prisms, cylinders, pyramids, cones, spheres and composites. cylinder: 𝑆 = 2𝜋𝑟ℎ + 2𝜋𝑟2 where 𝑟 is radius and ℎ is perpendicular height cone: 𝑆 = 𝜋𝑟𝑠 + 𝜋𝑟2 where 𝑟 is radius and 𝑠 is slant height sphere: 𝑆 = 4𝜋𝑟2 where 𝑟 is radius●Calculate volumes, 𝑉, and capacities of standard three-dimensional objects in practical situations, including rectangular prisms, cylinders, pyramids, cones, spheres and composites. prism: 𝑉 = 𝐴ℎ where 𝐴 is base area and ℎ is perpendicular height cylinder: 𝑉 = 𝜋𝑟2 ℎ where 𝑟 is radius and ℎ is perpendicular height pyramid: 𝑉 = 1 3 𝐴ℎ where 𝐴 is base area and ℎ is perpendicular height cone: 𝑉 = 1 3 𝜋𝑟2 ℎ where 𝑟 is radius and ℎ is perpendicular height sphere: 𝑉 = 4 3 𝜋𝑟3 where 𝑟 is radius●Solve practical problems involving shape and measurement. General Mathematics 2025 v1.3
▶Similarity and scale4 LOs
▶Similar figures and scale factors4 LOs
●Determine a scale factor and use it to solve scaling problems, e.g. calculating lengths and areas of similar figures; and calculating surface areas, volumes and capacit ies of similar solids.●Determine measurements from scale drawings (e.g. maps and building plans) to solve problems.●Understand the conditions for similarity of two-dimensional figures, including similar triangles.●Use the scale factor for two similar figures to solve linear scaling problems.
▶Algebra3 LOs
▶Linear and non-linear relationships3 LOs
●Find the value of a pronumeral in linear and simple non-linear equations given the values of the other pronumerals, transposing equations where necessary.●Substitute numerical values into linear and simple non-linear algebraic expressions, and evaluate.●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.
▶Linear equations and their graphs8 LOs
▶Linear equations3 LOs
▶Straight-line graphs5 LOs
●Construct a straight-line graph using a linear function of the form, 𝑦 = 𝑚𝑥 + 𝑐 where 𝑚 is slope (gradient) and 𝑐 is 𝑦-intercept.●Construct and analyse a straight-line graph to model a given linear relationship, e.g. modelling the cost of filling a fuel tank of a car against the number of litres of petrol required.●Determine the slope (gradient), 𝑥-intercept and 𝑦-intercept of a straight line from both its equation and its graph.●Interpret, in context, the slope (gradient) and intercept of a linear function used to model and analyse a practical situation.●Understand and use the slope-intercept form of a linear function, 𝑦 = 𝑚𝑥 + 𝑐 where 𝑚 is slope (gradient) and 𝑐 is 𝑦-intercept.
▶2 — Applications of linear equations and trigonometry, matrices and univariate data analysis24 LOs
▶Applications of linear equations and their graphs4 LOs
▶Applications of trigonometry4 LOs
▶Applications of trigonometry4 LOs
●Calculate the area of a non-right-angled triangle, △𝐴𝐵𝐶, and solve related practical problems. area = 1 2 𝑏𝑐 sin 𝐴, given two sides, 𝑏 and 𝑐, and an included angle, 𝐴 Heron’s rule: area = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where 𝑠 = 𝑎+𝑏+𝑐 2, given three sides, 𝑎, 𝑏 and 𝑐●Solve two-dimensional 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 true bearings. General Mathematics 2025 v1.3●Solve two-dimensional problems involving a non-right-angled triangle, △𝐴𝐵𝐶, with sides, 𝑎, 𝑏 and 𝑐, and corresponding angles, 𝐴, 𝐵 and 𝐶. sine rule: 𝑎 sin 𝐴 = 𝑏 sin𝐵 = 𝑐 sin 𝐶 (ambiguous case excluded) cosine rule: 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶●Understand and use the trigonometric ratios to find the size of an unknown angle, 𝜃, or the length of an unknown side in a right-angled triangle. cos 𝜃 = adjacent hypotenuse sin 𝜃 = opposite hypotenuse tan 𝜃 = opposite adjacent
▶Matrices5 LOs
▶Matrices and matrix arithmetic5 LOs
●Determine the power of a matrix using technology with matrix arithmetic capabilities when appropriate.●Perform matrix addition, subtraction and multiplication by a scalar.●Perform matrix multiplication manually up to 3 × 3 matrices but not limited to square matrices.●Use matrices for storing and displaying information that can be presented in rows and columns, e.g. tables, databases, links in social or road networks.●Use matrices, including matrix products and powers of matrices, to model and solve problems, e.g. 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.
▶Univariate data analysis 19 LOs
▶Making sense of data relating to a single statistical variable9 LOs
●Classify a categorical variable as ordinal or nominal and use tables and pie, bar and column charts to organise and display the data, e.g. ordinal: income level (high, medium, low); nominal: place of birth (Australia, overseas).●Classify a numerical variable as discrete or continuous, e.g. discrete: the number of people in a room; continuous: the temperature in degrees Celsius.●Classify a statistical variable as categorical or numerical.●Describe a graphical display in terms of the number of modes, shape (symmetric versus positively or negatively skewed), measures of centre and spread, and outliers, and interpret this information in the context of the data.●Select, construct and justify an appropriate graphical display to describe the distribution of a numerical dataset, including dot plot, stem-and-leaf plot, column chart and histogram.●Understand and calculate the (sample) standard deviation, 𝑠𝑥, of a dataset, using technology only.●Understand and calculate the mean, median, mode, range and interquartile range (IQR) of a dataset, with and without technology. mean: 𝑥̅ = ∑ 𝑥 𝑛 median: (𝑛+1 2)th data value●Understand the meaning of univariate data.●Use statistics as measures of centre and spread of a data distribution, being aware of their limitations. General Mathematics 2025 v1.3
▶Univariate data analysis 22 LOs
▶Comparing data for a single numerical variable across two or more groups2 LOs
●Compare datasets in terms of mean, median, range, IQR and standard deviation, interpret the differences observed in the context of the data, and report the findings in a systematic and concise manner.●Construct and use parallel box plots, including identifying possible outliers, to compare datasets in terms of median, spread (range and IQR) and outliers to interpret and communicate the differences observed in the context of the data. outliers (identifying): Q1 − 1.5 × IQR ≤ 𝑥 ≤ Q3 + 1.5 × IQR where Q1 is lower quartile and Q3 is upper quartile
▶3 — Bivariate data and time series analysis, sequences and Earth geometry44 LOs
▶Bivariate data analysis 110 LOs
▶Identifying and describing associations between two categorical variables4 LOs
●Construct two-way frequency tables and determine the associated row and column sums and percentages.●Understand 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.●Understand the meaning of bivariate data.●Use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association.
▶Identifying and describing associations between two numerical variables6 LOs
●Calculate Pearson’s correlation coefficient, 𝑟, from raw data using technology, and interpret it to quantify the strength of a linear association.●Calculate the coefficient of determination, 𝑅2, from raw data using technology, and interpret it to assess the strength of a linear association in terms of the explained variation.●Construct and use a scatterplot to identify the association between two numerical variables.●Describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak).●Identify the explanatory variable and the response variable.●Use the correlation coefficient, 𝑟, to determine the coefficient of determination, 𝑅2, and vice versa.
▶Bivariate data analysis 28 LOs
▶Fitting a linear model to numerical data6 LOs
●Construct a residual plot and use it to assess the appropriateness of fitting a linear model to the data.●Distinguish between interpolation and extrapolation.●Interpret the 𝑦-intercept and slope (gradient) of the fitted line.●Model a linear relationship by using technology to fit a least-squares line to the data, in the form of 𝑦 = 𝑚𝑥 + 𝑐 where 𝑚 is slope (gradient) and 𝑐 is 𝑦-intercept.●Understand and use 𝑚 = 𝑟 𝑠𝑦 𝑠𝑥 and 𝑐 = 𝑦 − 𝑚𝑥 to determine the equation of a least-squares line, where 𝑚 is slope (gradient), 𝑟 is correlation coefficient, 𝑠𝑦 is (sample) standard deviation of 𝑦 values, 𝑠𝑥 is (sample) standard deviation of 𝑥 values, 𝑐 is 𝑦-intercept, 𝑦 is mean of 𝑦 values and 𝑥 is mean of 𝑥 values.●Use the equation of the least-squares line to make predictions.
▶Time series analysis6 LOs
▶Analysing time series data4 LOs
●Deseasonalise a time series by calculating the seasonal indices using the average percentage method, including the use of spreadsheets.●Fit a least-squares line to model long-term trends in time series data.●Smooth time series data by calculating a simple moving average using the mean or median for an odd number of data, including the use of spreadsheets.●Solve practical problems that involve the analysis of time series data.
▶Growth and decay in sequences8 LOs
▶The arithmetic sequence4 LOs
●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.●Use arithmetic sequences to model and analyse practical situations involving linear growth or decay, e.g. analysing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, calculating the value of an item using the straight -line method of depreciation. General Mathematics 2025 v1.3●Use recursion to generate an arithmetic sequence.●Use the rule for the 𝑛th term of an arithmetic sequence. 𝑡𝑛 = 𝑡1 + (𝑛 − 1)𝑑 where 𝑡𝑛 is 𝑛th term, 𝑡1 is first term, 𝑛 is term number and 𝑑 is common difference
▶The geometric sequence4 LOs
●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.●Use geometric sequences to model and analyse practical situations involving geometric growth and decay (use of logarithms not required), e.g. modelling the growth of a bacterial population that doubles in size each hour, calculating the value of an item using the diminishing-value method of depreciation.●Use recursion to generate a geometric sequence.●Use the rule for the 𝑛th term of a geometric sequence. 𝑡𝑛 = 𝑡1 𝑟(𝑛−1) where 𝑡𝑛 is 𝑛th term, 𝑡1 is first term, 𝑛 is term number and 𝑟 is common ratio
▶Earth geometry and time zones12 LOs
▶Locations on the Earth7 LOs
●Calculate angular distance and distance between two places on Earth on the same meridian. 𝐷 = 111.2 × angular distance where 𝐷 is distance in kilometres●Calculate angular distance and distance between two places on Earth on the same parallel of latitude. 𝐷 = 111.2 cos 𝜃 × angular distance where 𝐷 is distance in kilometres and 𝜃 is latitude●Locate positions on Earth’s surface given latitude and longitude, e.g. using a globe, map, GPS and other digital technologies.●Solve practical problems involving latitude, longitude, angular distance and distance.●State latitude and longitude for positions on Earth’s surface, e.g. investigating a map of Australia and locating boundary positions for Aboriginal peoples’ an d Torres Strait Islander peoples’ language groups, Australian landmarks or local land boundaries.●Understand the meaning of angles of latitude and longitude (in decimal degrees, and degrees and minutes) in relation to the equator and the prime meridian respectively.●Understand the meaning of great circles.
▶Time zones5 LOs
●Calculate time differences between two places on Earth.●Determine the number of degrees of longitude for a given time difference.●Solve practical problems involving time zones, making allowances for daylight saving where necessary, e.g. seasonal time systems used by Aboriginal peoples and Torres Strait Islander peoples, making phone calls, broadcasting events, travelling, preparing an itinerary.●Understand the link between longitude and time.●Understand the meaning of Greenwich Mean Time (GMT), International Date Line and Coordinated Universal Time (UTC).
▶4 — Investing and netw orking40 LOs
▶Loans, investments and annuities 17 LOs
▶Compound interest loans and investments4 LOs
●Calculate the effective annual rate of interest, 𝑖effective, and use the results to compare interest on loans or investments when interest is paid or charged for different compounding periods, including daily, monthly, quarterly and six-monthly. 𝑖effective = (1 + 𝑖)𝑘 − 1 where 𝑖 is interest rate per compounding period and 𝑘 is number of compounding periods per year●Solve practical problems involving compound interest loans or investments, including determining the total amount of the loan or investment, total interest, principal, interest rate per year and per compounding period, and the effect of the interest rate and number of compounding periods on the total amount.●Use a recurrence relation to model a compound interest loan or investment. 𝐴𝑛+1 = 𝑟𝐴𝑛 where 𝐴𝑛+1 is total amount at the beginning of the (𝑛 + 1)th period, 𝐴𝑛 is total amount at the beginning of the 𝑛th period, and 𝑟 = 1 + 𝑖 where 𝑖 is interest rate per compounding period●Use the compound interest formula to model a compound interest loan or investment. 𝐴 = 𝑃(1 + 𝑖)𝑛 where 𝐴 is total amount, 𝑃 is principal, 𝑖 is interest rate per compounding period and 𝑛 is number of compounding periods
▶Present value of ordinary annuities3 LOs
●Solve practical problems involving the present value of an ordinary annuity, including determining the total amount of the annuity, periodic payment, total payments and total interest. General Mathematics 2025 v1.3●Use a recurrence relation to model the present value of an ordinary annuity, e.g. reducing balance loan or retirement pension with periodic payments where interest is calculated before the periodic payment is made. 𝐴𝑛+1 = 𝑟𝐴𝑛 − 𝑑 where 𝐴𝑛+1 is total amount at the beginning of the (𝑛 + 1)th period, 𝐴𝑛 is total amount at the beginning of the 𝑛th period, 𝑑 is periodic payment, and 𝑟 = 1 + 𝑖 where 𝑖 is interest rate per compounding period●Use the present value annuity formula to model the present value of an ordinary annuity, e.g. reducing balance loan or retirement pension with periodic payments where interest is calculated before the periodic payment is made. 𝐴𝑃𝑉 = 𝑑 (1−(1+𝑖)−𝑛 𝑖) where 𝐴𝑃𝑉 is total amount, 𝑑 is periodic payment, 𝑖 is interest rate per compounding period and 𝑛 is number of compounding periods
▶Loans, investments and annuities 25 LOs
▶Perpetuities and future value of ordinary annuities5 LOs
●Solve practical problems involving perpetuities, including determining the total amount of the perpetuity, periodic payment and interest rate per compounding period. General Mathematics 2025 v1.3●Solve practical problems involving the future value of an ordinary annuity, including determining the total amount of the annuity, periodic payment, total payments and total interest.●Use a recurrence relation to model the future value of an ordinary annuity, e.g. compound interest investment with periodic payments where interest is calculated before the periodic payment is made. 𝐴𝑛+1 = 𝑟𝐴𝑛 + 𝑑 where 𝐴𝑛+1 is total amount at the beginning of the (𝑛 + 1)th period, 𝐴𝑛 is total amount at the beginning of the 𝑛th period, 𝑑 is periodic payment and 𝑟 = 1 + 𝑖 where 𝑖 is interest rate per compounding period●Use the future value annuity formula to model the future value of an ordinary annuity, e.g. compound interest investment with periodic payments where interest is calculated before the periodic payment is made. 𝐴𝐹𝑉 = 𝑑 ((1+𝑖)𝑛−1 𝑖) where 𝐴𝐹𝑉 is total amount, 𝑑 is periodic payment, 𝑖 is interest rate per compounding period and 𝑛 is number of compounding periods●Use the perpetuity formula, 𝐴 = 𝑑 𝑖 where 𝐴 is total amount, 𝑑 is periodic payment and 𝑖 is interest rate per compounding period.
▶Graphs and networks12 LOs
▶Graphs, associated terminology and the adjacency matrix4 LOs
●Construct a graph or digraph from a given adjacency matrix.●Construct a network diagram to represent practical situations, e.g. tra cks connecting camp sites in a national park, a social network, a transport network with one -way streets, the results of a round-robin sporting competition.●Construct an adjacency matrix from a given graph or digraph.●Understand the meaning of graph, vertex (node), edge (arc), loop, degree of a vertex, subgraph, simple graph, complete graph, bipartite graph, directed graph (digraph), weighted graph and network.
▶Planar graphs, paths and cycles8 LOs
●Apply Euler’s formula to solve problems relating to planar graphs. 𝑣 + 𝑓 − 𝑒 = 2 where 𝑣 is number of vertices, 𝑓 is number of faces and 𝑒 is number of edges●Solve practical problems involving semi-Eulerian graphs and Eulerian graphs.●Solve practical problems involving semi-Hamiltonian graphs and Hamiltonian graphs (by trial-and-error methods only). General Mathematics 2025 v1.3●Solve practical problems to determine the shortest path between two vertices in a weighted graph (by trial-and-error methods only).●Understand the meaning of Eulerian trail, semi-Eulerian graph, Eulerian circuit and Eulerian graph, and the conditions for their existence.●Understand the meaning of Hamiltonian path, semi-Hamiltonian graph, Hamiltonian cycle and Hamiltonian graph.●Understand the meaning of planar graph and face.●Understand the meaning of walk, trail, path, open walk, open trail, open path, closed walk, closed trail (circuit), closed path (cycle), connected graph and bridge.
▶Networks and decision mathematics 19 LOs
▶Trees and minimum connector problems3 LOs
●Determine a minimum spanning tree in a weighted connected graph.●Solve practical problems involving minimum spanning trees, e.g. minimising the length of cable needed to provide power from a single power station to substations in several towns.●Understand the meaning of tree, spanning tree and minimum spanning tree.
▶Project planning and scheduling using critical path analysis (CPA)6 LOs
●Calculate float times for non-critical activities.●Construct a project network diagram (activity on arc) to represent the durations and interdependencies of activities that must be completed during the project (excluding dummy activities).●Solve small-scale practical problems involving critical path analysis.●Use ESTs and LSTs to locate the critical path/s for a project.●Use forward and backward scanning to determine the earliest starting time (EST) and latest starting time (LST) for each activity in the project.●Use the critical path to determine the minimum time for a project to be completed.
▶Networks and decision mathematics 27 LOs
▶Flow networks4 LOs
●Determine the capacity of a cut.●Solve small-scale practical problems involving flow networks (up to 8 possible cuts), including determining the minimum cut and the maximum flow.●Understand the meaning of source node, sink node, cut, minimum cut and maximum flow.●Use a flow network diagram to identify a cut.
▶Assigning order and the Hungarian algorithm3 LOs
●Determine the optimum (minimum and maximum) assignment/s for small-scale practical problems by inspection.●Use a bipartite graph and its tabular or matrix form to represent possible assignments for an allocation problem.●Use the Hungarian algorithm (3 × 3 up to 5 × 5 square matrices) to determine the optimum (minimum and maximum) assignment/s for larger practical problems.
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