§-Mathematics Applications syllabus

WA · SCSA← Mathematics Applications

Mathematics Applications syllabus, dot point by dot point

Every dot point in the WA Mathematics Applications syllabus, in syllabus order, with a focused answer for each. Open any dot point for the answer, exam-style questions and links to related points.

Unit 3

Unit overview →

When can a graph be drawn without edges crossing, and what relation links its parts?

Identify planar graphs, count vertices, edges and faces, and verify and apply Euler's formula v minus e plus f equals 2.

How do we model situations that change by a constant amount each step?

Recognise arithmetic sequences, use the recursive and explicit rules, and apply them to simple interest and linear depreciation.

How do we model situations that change by a constant percentage each step?

Recognise geometric sequences, use the recursive and explicit rules with the common ratio, and apply them to growth and decay.

How do we display two numerical variables together and describe the association we see?

Identify response and explanatory variables, construct a scatterplot, and describe the association in terms of direction, form, strength and outliers.

How does recursion model a compound interest investment that grows each period?

Model compound interest with a recurrence relation, convert nominal annual rates to the period rate, and find balances and effective rates.

How do we model an asset that loses a fixed percentage of its value each year?

Model reducing-balance depreciation with a recurrence relation, compare it with flat-rate depreciation, and find book value and scrap-value timing.

Why does a strong association not prove that one variable causes the other?

Distinguish association from causation, identify confounding and coincidence, and place bivariate analysis within the statistical investigation process.

What are the different kinds of route through a graph, and when do special ones exist?

Distinguish walks, trails, paths, cycles and circuits, and determine when Eulerian and Hamiltonian routes exist.

How can diagrams of dots and lines model and solve real connection problems?

Represent situations with graphs and networks, use terminology and matrices, and solve shortest path, minimum spanning tree and connection problems.

How does money grow or shrink over time under interest and repayments?

Model and solve problems involving compound interest, depreciation, annuities, loans and investments using recursion and the financial solver.

How do we fit the best straight line to bivariate data and read meaning from it?

Fit a least-squares line using technology, interpret the slope and intercept in context, and predict while distinguishing interpolation from extrapolation.

How do we find the best decision when choices are limited by constraints?

Formulate linear programming problems, graph feasible regions, and locate the optimal solution at a vertex of the feasible region.

How do residuals tell us whether a straight line was the right model?

Calculate residuals, construct and interpret a residual plot, and use it to judge whether a linear model is appropriate.

How does a recurrence relation generate a sequence one term at a time?

Use first-order linear recurrence relations to generate sequences and recognise the patterns of growth and decay they produce.

How do we put a number on the strength and direction of a linear association?

Calculate and interpret Pearson's correlation coefficient r and the coefficient of determination r squared, and state their limitations.

How can a grid of numbers store data and model change over time?

Perform matrix operations, find determinants and inverses of 2x2 matrices, solve matrix equations, and apply transition matrices to model systems.

How can we straighten a curved relationship so a linear model works?

Apply squared, logarithmic and reciprocal transformations to linearise data, fit a least-squares line to the transformed data and use it to predict.

How do we describe a graph precisely and record it as a matrix?

Use vertex, edge, degree, loop and multiple-edge terminology, apply the handshake rule, and represent a graph with an adjacency matrix.

Unit 4

Unit overview →

How do we model quantities that grow or shrink by a constant ratio?

Recognise geometric growth and decay, use recurrence relations and the explicit rule for geometric sequences, and model compound and reducing situations.

How do regular payments build up or draw down a fund over time?

Model annuities and annuity-investments with recurrence relations and find balances, payments and the time to exhaust or reach a target.

How much can flow from a source to a sink through a capacitated network?

Model flow in a directed network, find the maximum flow, and use the maximum-flow minimum-cut relationship.

How does a loan reduce to zero when interest is charged and regular repayments are made?

Model a reducing-balance loan with a recurrence relation, build an amortisation table, and find balances, repayments and total interest.

How do we allocate workers to tasks for the lowest total cost?

Model an assignment problem as a bipartite graph and solve it with the Hungarian algorithm to minimise total cost.

How do we display data measured over time and identify the patterns within it?

Construct a time series plot and identify trend, seasonal, cyclic and irregular components.

How can we measure and use the relationship between two numerical variables?

Construct and interpret scatterplots, calculate the correlation coefficient and least-squares regression line, and use the line to make predictions.

How do we schedule dependent tasks to finish a project in the shortest time?

Construct an activity network, compute earliest and latest starting times and float, and identify the critical path and minimum completion time.

How can a fund pay out forever without ever running down?

Model a perpetuity, find the payment that keeps the balance constant, and relate it to the interest earned each period.

How do we find the route of least total weight between two points in a network?

Find the shortest path between two vertices in a weighted network and interpret it in context.

How do we measure a seasonal effect and remove it to compare periods fairly?

Calculate seasonal indices, deseasonalise and reseasonalise a time series, and interpret seasonal indices in context.

How do we describe a pattern over time and use it to forecast?

Plot and describe time series, smooth with moving averages, deseasonalise with seasonal indices, fit a trend line and forecast future values.

How do we fit a trend line to a time series and forecast future values?

Fit a least-squares trend line to a time series, forecast future values, and reseasonalise forecasts for seasonal data.

How do we connect every site in a network at the least total cost?

Solve connector problems by finding a minimum spanning tree using Prim's algorithm and interpret its total weight.

How do we smooth out the noise in a time series to reveal its trend?

Smooth a time series using moving averages, including centred even-order averages, and using median smoothing.

How do we describe a normal population and estimate it from a sample?

Use the normal distribution and the 68-95-99.7 rule, standardise to z-scores, and construct and interpret sample proportions and confidence intervals.