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Year 12: QCE General Mathematics

Listed Mathematics syllabus sub-topics at Units 3 and 4, in the words of the Queensland Curriculum and Assessment Authority (QCAA), with the practice on this site that covers it. Checked against the source on 4 September 2026.

2025 syllabus, version 1.3. Queensland numbers its units and topics but gives the content itself no codes, so each entry here is a sub-topic and its text is the sub-topic's own bullet list. Mathematical notation is flattened out of the syllabus PDF.

  • 19syllabus sub-topics
  • 28skills mapped
  • 100%have practice

Coverage is the percentage of listed syllabus sub-topics with at least one linked practice skill; it does not measure how fully every part of an entry is practised. A skill can practise more than one syllabus sub-topic, so some appear more than once below.

Unit 3: Bivariate data and time series analysis, sequencesand Earth geometry

  1. Topic 1 · Identifying and describing associations between two categorical variablesUnderstand the meaning of bivariate data. Construct two-way frequency tables and determine the associated row and column sums and percentages. Use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association. 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.

  2. Topic 1 · Identifying and describing associations between two numerical variablesIdentify the explanatory variable and the response variable. 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). 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. Use the correlation coefficient, 𝑟, to determine the coefficient of determination, 𝑅2 , and vice versa.

  3. Topic 2 · Fitting a linear model to numerical dataModel 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 𝑠_𝑦/𝑠_𝑥 and 𝑐 = 𝑦̅ − 𝑚𝑥̅ to determine the equation of a least-squares line, where 𝑚 is slope (gradient), 𝑟 is correlation coefficient, 𝑠_𝑦 is (sample) standard deviation of 𝑦 values, 𝑠_𝑥 is (sample) standard deviation of 𝑥 values, 𝑐 is 𝑦-intercept, 𝑦̅ is mean of 𝑦 values and 𝑥̅ is mean of 𝑥 values. Construct a residual plot and use it to assess the appropriateness of fitting a linear model to the data. Interpret the 𝑦-intercept and slope (gradient) of the fitted line. Distinguish between interpolation and extrapolation. Use the equation of the least-squares line to make predictions. Recognise and explain the potential dangers of extrapolation.

  4. Topic 2 · Association and causationRecognise and explain that an observed association between two variables (categorical and/or numerical) does not necessarily mean that there is a causal relationship between them. Identify and communicate possible non-causal explanations for an association, including coincidence or the influence of another variable. Solve practical problems by identifying, analysing and describing associations between two variables (categorical and/or numerical).

  5. Topic 3 · Describing and interpreting patterns in time series dataConstruct and use time series plots. Describe time series plots by identifying features, including trend (long-term direction, e.g. increasing/decreasing), seasonality (systematic, calendar-related movements) and irregular fluctuations (unsystematic, short-term fluctuations).

  6. Topic 3 · Analysing time series dataSmooth 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. 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. Solve practical problems that involve the analysis of time series data.

  7. Topic 4 · The arithmetic sequenceUse recursion to generate an arithmetic sequence. 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 the rule for the 𝑛th term of an arithmetic sequence. 𝑡_𝑛 = 𝑡_1 + (𝑛 − 1)𝑑 where 𝑡_𝑛 is 𝑛th term, 𝑡_1 is first term, 𝑛 is term number and 𝑑 is common difference 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.

  8. Topic 4 · The geometric sequenceUse recursion to generate a geometric sequence. 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 the rule for the 𝑛th term of a geometric sequence. 𝑡_𝑛 = 𝑡_1 𝑟^(𝑛−1) where 𝑡_𝑛 is 𝑛th term, 𝑡_1 is first term, 𝑛 is term number and 𝑟 is common ratio 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.

  9. Topic 5 · Locations on the EarthUnderstand the meaning of great circles. 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. Locate positions on Earth’s surface given latitude and longitude, e.g. using a globe, map, GPS and other digital technologies. State latitude and longitude for positions on Earth’s surface, e.g. investigating a map of Australia and locating boundary positions for Aboriginal peoples’ and Torres Strait Islander peoples’ language groups, Australian landmarks or local land boundaries. 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 Solve practical problems involving latitude, longitude, angular distance and distance.

  10. Topic 5 · Time zonesUnderstand the meaning of Greenwich Mean Time (GMT), International Date Line and Coordinated Universal Time (UTC). Understand the link between longitude and time. Determine the number of degrees of longitude for a given time difference. Calculate time differences between two places on Earth. 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.

Unit 4: Investing and netw orking

  1. Topic 1 · Compound interest loans and investmentsUse 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 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.

  2. Topic 1 · Present value of ordinary annuitiesUse 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 ( 𝑖 ) where 𝐴𝑃𝑉 is total amount, 𝑑 is periodic payment, 𝑖 is interest rate per compounding period and 𝑛 is number of compounding periods 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.

  3. Topic 2 · Perpetuities and future value of ordinary annuitiesUse 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 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 the perpetuity formula, 𝐴 = 𝑑 𝑖 where 𝐴 is total amount, 𝑑 is periodic payment and 𝑖 is interest rate per compounding period. Solve practical problems involving perpetuities, including determining the total amount of the perpetuity, periodic payment and interest rate per compounding period.

  4. Topic 3 · Graphs, associated terminology and the adjacency matrixUnderstand 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. Construct a network diagram to represent practical situations, e.g. tracks 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. Construct a graph or digraph from a given adjacency matrix.

  5. Topic 3 · Planar graphs, paths and cyclesUnderstand the meaning of planar graph and face. 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 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. 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. Solve practical problems involving semi-Eulerian graphs and Eulerian graphs. Understand the meaning of Hamiltonian path, semi-Hamiltonian graph, Hamiltonian cycle and Hamiltonian graph. Solve practical problems involving semi-Hamiltonian graphs and Hamiltonian graphs (by trial-and-error methods only).

  6. Topic 4 · Trees and minimum connector problemsUnderstand the meaning of tree, spanning tree and minimum spanning tree. 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.

  7. Topic 4 · Project planning and scheduling using critical path analysis (CPA)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). Use forward and backward scanning to determine the earliest starting time (EST) and latest starting time (LST) for each activity in the project. Use ESTs and LSTs to locate the critical path/s for a project. Use the critical path to determine the minimum time for a project to be completed. Calculate float times for non-critical activities. Solve small-scale practical problems involving critical path analysis.

  8. Topic 5 · Flow networksUnderstand the meaning of source node, sink node, cut, minimum cut and maximum flow. Use a flow network diagram to identify a cut. 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.

  9. Topic 5 · Assigning order and the Hungarian algorithmUse a bipartite graph and its tabular or matrix form to represent possible assignments for an allocation problem. Determine the optimum (minimum and maximum) assignment/s for small-scale practical problems by inspection. 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.

Other Year 12 skills

Year 12 skills not currently linked to an entry on this page. These may include revision, extension practice, or skills still awaiting a curriculum mapping.

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