SACE General Mathematics subject outline — topics explained

Topic 3: Statistical models

Topic 3 has two subtopics. Subtopic 3.1, Bivariate statistics, works through the statistical investigation process: identifying the independent (explanatory) and dependent (response) variables, building a scatter plot, and describing the association by direction, form and strength. Pearson's correlation coefficient r and the coefficient of determination r-squared are found with technology and used to justify that description; outliers are identified visually and removed only with reasonable justification, and a strong correlation is never treated as proof of causation. The least squares line y = a + bx is fitted with technology and its slope read as a rate of change and its intercept as an initial value, both in context. Residuals and residual plots then decide whether the linear model was appropriate at all — a pattern or large residuals sends you to the exponential model y = a·b^x, where a is the initial value and b is the proportional rate of change expressed as a percentage. Predictions are made by interpolation or extrapolation and always qualified by reliability. Subtopic 3.2 covers the normal distribution: the parameters mu and sigma, the bell shape and symmetry, the 68:95:99.7% rule with and without technology, non-standard proportions, and inverse normal work where you are given an area and asked for the value.

What the exam asks

Marks are spread across the whole topic. The 2025 paper asked for Pearson's r on Adelaide median rental data, an interpretation of the slope 5.38 in context, two reasons a specific prediction could be considered reasonable, a residual calculated by hand and added to a printed residual plot, the meaning of the growth factor 1.065 in an exponential visitor model, and normal-distribution work on airport luggage weights that ran from a proportion between two weights to an expected count out of 7815 bags, an inverse normal for the lightest 2%, a standard deviation estimated from a printed distribution, and a three-mark question disproving a stated claim. The 2024 paper set the same shapes on childhood development ages and on cumulative vet costs, including choosing between a linear and an exponential model on r-squared, interpreting the a value, and finding and interpreting the point of intersection of the two models.

Where marks go missing

Answering the calculation and stopping. The 2025 Subject Assessment Advice lists the recurring losses precisely: quoting a prediction without commenting on the strength of the fit, confusing a scatter plot with a residual plot, interpreting a slope or growth factor without units or a per-unit statement, choosing the wrong tail or area in an inverse normal, and ignoring integer constraints when a proportion is converted to a number of people or bags. Interpretation parts need distinct, context-specific statements matched to the mark allocation, not one idea rephrased twice.

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Topic 4: Financial models

Topic 4 is annuity work done on the graphics calculator's financial solver. Subtopic 4.1, Models for saving, covers compound interest (finding FV, PV, n or I), future value annuities and the regular deposit needed to reach a target, the value of accumulating savings after a given period and the total interest earned, and the 'what if' investigations and limitations that go with them. It also covers the factors that decide between investments: interest counting as taxable income, the effect of inflation, institution and government charges, and the conversion of nominal or flat rates to an equivalent effective annual rate. Present value annuities close the subtopic, reversing the savings model so a lump sum or a superannuation balance provides a regular income. Subtopic 4.2, Models for borrowing, covers interest-only loans and sinking funds and then reducing-balance loans in detail — the repayment, the total interest paid, the outstanding balance after a given time — followed by the strategies that reduce interest (more frequent payments, larger payments, a shorter term, a lump sum off the principal, rate changes and offset accounts) and their reasonableness, and the calculation of comparison rates where a set-up fee is added to the present value and an ongoing charge is added to the payment.

What the exam asks

The financial questions are long and cumulative. The 2025 paper compared two accounts by effective annual rate, took interest earned in the third year alone and taxed it at a marginal rate, tested whether a savings balance beat an inflating purchase price, calculated a comparison rate for a loan carrying a $600 establishment fee and a $20 monthly service fee, found the outstanding balance of a 30-year mortgage after 10 years and the interest saved by refinancing, and ran a superannuation account from a $9810 opening balance through to a fortnightly retirement withdrawal, including the amount that could be drawn from interest alone. The 2024 paper showed a fortnightly superannuation contribution, found the time for an annuity balance to halve and asked why that is not halfway through the term, completed effective-rate tables for different compounding periods and for a flat rate, and traced an offset account across ten years to the total interest saved.

Where marks go missing

Calculator discipline. The 2025 advice names incorrect rates and compounding settings (P/Y and C/Y), an off-by-one error when isolating the interest in a specified interval, mishandling comparison-rate fees, sign errors on PV and PMT, and faulty logic in 'live off the interest' withdrawal questions. Two conventions matter: in the examination the number of compounding periods per year always equals the number of payments per year, and money answers are rounded to the nearest cent with the unrounded calculator value shown first. On a 'show that' part you must show the method — restating the printed value earns nothing, though you may use that value to continue.

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Topic 5: Discrete models

Topic 5 is the non-calculator topic. Subtopic 5.1, Critical path analysis, starts from a precedence table, draws the directed network (a bipartite ordering helps), and inserts dummy links where precedence cannot otherwise be shown correctly. A forward scan gives earliest starting times and the minimum completion time; a backward scan gives latest starting times; the critical path is the longest path through the network and there may be more than one. Slack time is calculated as latest completion time minus the sum of the task time and the earliest starting time, and you are expected to explain what shortening or lengthening a task does to the path and the completion time, and to state the assumptions and limitations of the model in the context given. Subtopic 5.2, Assignment problems, allocates tasks to minimise a cost in time, distance or money using the Hungarian algorithm: reduce rows then columns, cover the zeros with the minimum number of lines, subtract the smallest uncovered element and add it at the crossings, and repeat until an optimal assignment can be read off. Maximisation is handled by minimising the profit lost, non-square arrays are squared up with dummy rows or columns, and more than one optimal assignment may exist.

What the exam asks

Every recent paper carries two or three discrete questions. In 2025 students stated a critical path and a minimum completion time from a precedence table, drew a missing dummy link onto a printed network, explained why a six-day delay pushed the whole project out, justified a dummy link, completed forward and backward scans on a printed diagram, found slack for a named task and stated the assumption behind it, and then worked a 4x4 Hungarian array before a fifth operator forced a 5x4 array and a full set of algorithm steps. The 2024 paper ran the same machinery on a dog grooming roster — including why an extra column of zeros is required and which staff member ends up unallocated — on a house-move network with two dummy links, and on two rival ticketing networks compared after one task was underestimated by four weeks.

Where marks go missing

Minimum completion time is the project finish time, not the final task's earliest starting time. In 2025 this single error — giving 42 days instead of 43 by leaving off the last task's one-day duration — made part (b) of Question 1 the worst-answered part of the easiest question on the paper. The other repeat offenders the advice lists are omitting the arrowhead on a dummy link, failing to carry earliest starting times across dummy links, missing a second critical path after a change, and, in the Hungarian algorithm, forgetting the dummy column, reducing rows after inserting the zeros, covering the zeros with too many lines and not noticing that two optimal assignments exist.

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Topic 1: Modelling with linear relationships (school-assessed, not examined)

Topic 1 is compulsory teaching but is never examined. Subtopic 1.1, Simultaneous linear equations, extends the linear functions met in Stage 1 to problems solved by the intersection of two or more lines, set in familiar realistic contexts. Subtopic 1.2, Linear programming, is the major application: writing constraints as inequalities, drawing and reading the feasible region, and locating the optimal value of an objective function at a vertex. The outline deliberately asks that students meet these problems by trial and error before the algorithm, so the dynamic nature of a constrained optimisation is understood rather than memorised.

What the exam asks

Not on the examination. Topic 1 is assessed at school through Assessment Type 1: Skills and Applications Tasks or Assessment Type 2: Mathematical Investigation, and the outline requires that every non-examined topic be covered by at least one of them. The 2025 Subject Assessment Advice points to changing a constraint or a profit function as the standard way teachers add complexity to a linear programming task.

Where marks go missing

Spending revision time on linear programming because it feels like exam mathematics. It is not: the SACE Board examination is set on Topics 3, 4 and 5 only, and any practice paper that puts a feasible region in front of you is not modelling the real thing. The reverse mistake matters too — a school-assessed linear programming task carries the same 40% or 30% weight as any other, so it cannot be treated as optional.

Topic 2: Modelling with matrices (school-assessed, not examined)

Topic 2 continues the discrete mathematics begun in Stage 1 and is studied unless the school replaces it with Topic 6. Subtopic 2.1 applies matrices to network problems, using connectivity matrices to represent and count connections and paths in a network. Subtopic 2.2 applies matrices to transition problems, using a transition matrix and an initial state to model how a population or a market share moves between states over successive steps, and to examine long-run behaviour.

What the exam asks

Not on the examination. Like Topic 1 it is assessed at school through a skills and applications task or the mathematical investigation. The 2025 advice suggests, as a way of lifting complexity, asking students to complete only one or two rows of a connectivity or transition matrix and then to change the matrix, rather than filling one in mechanically.

Where marks go missing

Confusing the matrix networks of Topic 2 with the directed networks of Topic 5. They look similar on the page and only one of them is examined: Topic 5 networks are drawn from precedence tables and scanned for a critical path by hand, while Topic 2 networks are handled through connectivity matrices and stay inside school assessment.

Topic 6: Open topic (school-developed, not examined)

Topic 6 is an open topic that a school may develop for its own local context, and it replaces Topic 2: Modelling with matrices when it is undertaken. The outline sets conditions rather than content: the topic must let students meet the learning requirements alongside the other topics, must emphasise appropriate use of electronic technology in teaching, learning and assessment, and must be of a standard comparable to the other topics. The school writes its own key questions, key concepts and any subtopics, and is expected to build the topic around a problem-based approach.

What the exam asks

Not on the examination, and not standardised across schools — two students in different schools may study entirely different open topics. Where a school runs Topic 6, it must be assessed through a skills and applications task or, if it is not, it must be the focus of the mathematical investigation.

Where marks go missing

Assuming your school runs it. Most do not, and if yours does, the material is unique to your class: no past paper, no shared revision guide and no other school's notes cover it, so the only reliable sources are your teacher's key questions and the outline's stated conditions.