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WAMathematics Applications← Mathematics Applications syllabus

Unit 3

18 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

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 to recognise a planar graph, redraw it without crossings, count faces including the outer region, and apply Euler's formula linking vertices, edges and faces.

7 min read →

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 to identify an arithmetic sequence by its common difference, use the recursive and explicit term rules, sum the terms, and apply the model to simple interest and flat-rate depreciation.

7 min read →

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 to identify a geometric sequence by its common ratio, use the recursive and explicit term rules, and apply the model to percentage growth and decay such as populations and reducing-balance depreciation.

7 min read →

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 to choose response and explanatory variables, plot a scatterplot the correct way round, and describe the association by its direction, form, strength and any outliers.

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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 to model a compound interest investment with a recurrence relation, convert a nominal annual rate to the compounding-period rate, find any balance, and compare effective annual rates.

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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.

How to model reducing-balance depreciation with a recurrence relation, contrast it with flat-rate depreciation, find an asset's book value after n years, and work out when it reaches a scrap value.

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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.

How to separate association from causation, explain confounding and coincidental correlation, and work through the four-step statistical investigation process for bivariate data.

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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 to tell walks, trails, paths, cycles and circuits apart, apply the odd-vertex condition for Eulerian trails and circuits, and recognise Hamiltonian paths and cycles.

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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 to read and draw graphs and networks, use vertices, edges and adjacency matrices, trace Euler and Hamilton paths, and find minimum spanning trees and shortest paths.

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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 to set up recurrence relations for compound interest, depreciation, loans and annuities, and how to use the finance solver to find payments, balances and the number of periods.

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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 to fit the least-squares regression line with technology, interpret its slope and intercept in real units, predict with the equation, and judge interpolation against extrapolation.

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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 to turn a worded optimisation problem into an objective function and inequality constraints, graph the feasible region, and test corner points to maximise or minimise the objective.

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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 to calculate a residual as observed minus predicted, build a residual plot, and read it to decide whether a straight line fits or whether the data needs transforming.

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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 to read and use a first-order linear recurrence relation, generate terms step by step, and recognise when it produces linear, growing or decaying behaviour.

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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 to calculate Pearson's r with technology, read its sign and size, convert to the coefficient of determination, interpret the proportion of variation explained, and respect the limits of both measures.

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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 to add, multiply and invert matrices, solve matrix equations with the inverse, and use transition matrices and steady states to model populations and market share.

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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 to apply the squared, log and reciprocal transformations to straighten curved bivariate data, fit a line to the transformed variable, and back-substitute to predict in original units.

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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.

How to use the core graph vocabulary of vertices, edges, degree, loops and multiple edges, apply the handshake rule, and translate between a graph and its adjacency matrix.

7 min read →