Syllabus
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Unit 1: Money, measurement and relations
Topic 1: Consumer arithmetic
Applications of rates, percentages and use of spreadsheets
Unit 1: Money, measurement and relations > Topic 1: Consumer arithmetic > Applications of rates, percentages and use of spreadsheets
- Review definitions of rates and percentages
- Calculate weekly 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
- Calculate payments based on government allowances and pensions, such as youth allowances, unemployment, disability and study
- Prepare a personal budget for a given income, taking into account fixed and discretionary spending
- Compare prices and values using the unit cost method
- Apply percentage increase or decrease in various contexts, e.g. determining the impact of inflation on costs and wages over time, calculating percentage mark-ups and discounts, calculating GST, calculating profit or loss in absolute and percentage terms, and calculating simple and compound interest
- Use currency exchange rates to determine the cost in Australian dollars of purchasing a given amount of a foreign currency, such as US$1500, or the value of a given amount of foreign currency when converted to Australian dollars, such as the value of 2050 euros in Australian dollars
- Calculate the dividend paid on a portfolio of shares, given the percentage dividend or dividend paid per share, for each share; and compare share values by calculating a price-to-earnings ratio
- 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 a fast-food store where hours of employment and hourly rates of pay may differ, preparing a budget or investigating the potential cost of owning and operating a car over a year.
Topic 2: Shape and measurement
Pythagoras' theorem
Unit 1: Money, measurement and relations > Topic 2: Shape and measurement > Pythagoras' theorem
- Review Pythagoras' theorem and use it to solve practical problems in two dimensions and simple applications in three dimensions.
Mensuration
Unit 1: Money, measurement and relations > Topic 2: Shape and measurement > Mensuration
- Solve the problem regarding the calculation of perimeters and areas of circles, sectors of circles, triangles, rectangles, trapeziums, parallelograms and composites
- Calculate the volumes and capacities of standard three-dimensional objects, including spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations, such as the volume of water contained in a swimming pool
- Calculate the surface areas of standard three-dimensional objects, e.g. spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations, such as the surface area of a cylindrical food container.
Similar figures and scale factors
Unit 1: Money, measurement and relations > Topic 2: Shape and measurement > Similar figures and scale factors
- Review the conditions for similarity of two-dimensional figures, including similar triangles
- Use the scale factor for two similar figures to solve linear scaling problems
- Obtain measurements from scale drawings, such as maps or building plans, to solve problems
- Obtain a scale factor and use it to solve scaling problems involving the calculation of the areas of similar figures, including the use of shadow sticks, calculating the height of trees, use of a clinometer
- Obtain a scale factor and use it to solve scaling problems involving the calculation of surface areas and volumes of similar solids.
Topic 3: Linear equations and their graphs
Linear equations
Unit 1: Money, measurement and relations > Topic 3: Linear equations and their graphs > Linear equations
- Identify and solve linear equations, including variables on both sides, fractions, non-integer solutions
- Develop a linear equation from a description in words.
Straight-line graphs and their applications
Unit 1: Money, measurement and relations > Topic 3: Linear equations and their graphs > Straight-line graphs and their applications
- Construct straight-line graphs using \( y = a + bx \) both with and without the aid of technology
- Determine the slope and intercepts of a straight-line graph from both its equation and its plot
- Interpret, in context, the slope and intercept of a straight-line graph used to model and analyse a practical situation
- Construct and analyse a straight-line graph to model a given linear relationship, such as modelling the cost of filling a fuel tank of a car against the number of litres of petrol required.
Simultaneous linear equations and their applications
Unit 1: Money, measurement and relations > Topic 3: Linear equations and their graphs > Simultaneous linear equations and their applications
- Solve a pair of simultaneous linear equations in the format \( y = mx + c \), using technology when appropriate; they must solve equations algebraically, graphically, by substitution and by the elimination method
- Solve practical problems that involve finding the point of intersection of two straight-line graphs, such as determining the break-even point where cost and revenue are represented by linear equations.
Piece-wise linear graphs and step graphs
Unit 1: Money, measurement and relations > Topic 3: Linear equations and their graphs > Piece-wise linear graphs and step graphs
- Sketch piece-wise linear graphs and step graphs, using technology where appropriate
- Interpret piece-wise linear and step graphs used to model practical situations.
Unit 2: Applied trigonometry, algebra, matrices and univariate data
Topic 1: Applications of trigonometry
Applications of trigonometry
Unit 2: Applied trigonometry, algebra, matrices and univariate data > Topic 1: Applications of trigonometry > Applications of trigonometry
- Review the use of the trigonometric ratios to find the length of an unknown side or the size of an unknown angle in a right-angled triangle
- Determine the area of a triangle given two sides and an included angle by using the rule \( area=\frac{1}{2}bc \sin A \), or given three sides by using Heron's rule \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \frac{a+b+c}{2} \), and solve related practical problems
- Solve two-dimensional problems involving non-right-angled triangles using the sine rule (ambiguous case excluded) and the cosine rule
- 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.
Topic 2: Algebra and matrices
Linear and non-linear relationships
Unit 2: Applied trigonometry, algebra, matrices and univariate data > Topic 2: Algebra and matrices > Linear and non-linear relationships
- Substitute numerical values into linear algebraic and simple non-linear algebraic expressions, and evaluate, e.g. order two polynomials, proportional, inversely proportional
- Find the value of the subject of the formula, given the values of the other pronumerals in the formula
- Transpose linear equations and simple non-linear algebraic equations, e.g. order two polynomials, proportional, inversely proportional
- 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, e.g. a table displaying the body mass index (BMI) of people with different weights and heights.
Matrices and matrix arithmetic
Unit 2: Applied trigonometry, algebra, matrices and univariate data > Topic 2: Algebra and matrices > Matrices and matrix arithmetic
- 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
- Recognise different types of matrices (row matrix, column matrix (or vector matrix), square matrix, zero matrix, identity matrix) and determine the size of the matrix
- Perform matrix addition, subtraction, and multiplication by a scalar
- Perform matrix multiplication (manually up to a 3 x 3 but not limited to square matrices)
- Determining the power of a matrix using technology with matrix arithmetic capabilities when appropriate
- 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.
Topic 3: Univariate data analysis
Making sense of data relating to a single statistical variable
Unit 2: Applied trigonometry, algebra, matrices and univariate data > Topic 3: Univariate data analysis > Making sense of data relating to a single statistical variable
- Define univariate data
- Classify statistical variables as categorical or numerical
- 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); or nominal: place of birth (Australia, overseas)
- Classify a numerical variable as discrete or continuous, e.g. discrete: the number of rooms in a house; or continuous: the temperature in degrees Celsius
- 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 or histogram
- Describe the graphical displays 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
- Determine the mean, \(\bar{x}\), and standard deviation (using technology) of a dataset and use statistics as measures of location and spread of a data distribution, being aware of the significance of the size of the standard deviation.
Comparing data for a numerical variable across two or more groups
Unit 2: Applied trigonometry, algebra, matrices and univariate data > Topic 3: Univariate data analysis > Comparing data for a numerical variable across two or more groups
- Construct and use parallel box plots (including the use of the \(Q_1 - 1.5 \times IQR \leq x \leq Q_3 + 1.5 \times IQR\) criteria for identifying possible outliers) to compare datasets in terms of median, spread (IQR and range) and outliers to interpret and communicate the differences observed in the context of the data
- Compare datasets using medians, means, IQRs, ranges or standard deviations for a single numerical variable, interpret the differences observed in the context of the data and report the findings in a systematic and concise manner.
Unit 3: Bivariate data, sequences and change, and Earth geometry
view_agenda query_statsTopic 1: Bivariate data analysis
view_agenda query_statsIdentifying and describing associations between two categorical variables
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 1: Bivariate data analysis > Identifying and describing associations between two categorical variables
- Define 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.
Identifying and describing associations between two numerical variables
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 1: Bivariate data analysis > Identifying and describing associations between two numerical variables
- Construct a scatterplot to identify patterns in the data suggesting the presence of an association
- Understand an association between two numerical variables in terms of direction (positive/negative), form (linear) and strength (strong/moderate/weak)
- Calculate and interpret the correlation coefficient (r) to quantify the strength of a linear association using Pearson's correlation coefficient.
Fitting a linear model to numerical data
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 1: Bivariate data analysis > Fitting a linear model to numerical data
- Identify the response variable and the explanatory variable
- Use a scatterplot to identify the nature of the relationship between variables
- Model a linear relationship by fitting a least-squares line to the data
- Use a residual plot to assess the appropriateness of fitting a linear model to the data
- Interpret the intercept and slope of the fitted line
- Use, not calculate, the coefficient of determination (\( R^2 \)) to assess the strength of a linear association in terms of the explained variation
- Use the equation of a fitted line to make predictions
- Distinguish between interpolation and extrapolation when using the fitted line to make predictions, recognising the potential dangers of extrapolation.
Association and causation
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 1: Bivariate data analysis > Association and causation
- Recognise that an observed association between two variables does not necessarily mean that there is a causal relationship between them
- Identify and communicate possible non-causal explanations for an association, including coincidence and confounding due to a common response to another variable
- Solve practical problems by identifying, analysing and describing associations between two categorical variables or between two numerical variables.
Topic 2: Time series analysis
view_agenda query_statsDescribing and interpreting patterns in time series data
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 2: Time series analysis > Describing and interpreting patterns in time series data
- Construct time series plots
- Describe time series plots by identifying features such as trend (long-term direction), seasonality (systematic, calendar-related movements) and irregular fluctuations (unsystematic, short-term fluctuations), and recognise when there are outliers, e.g. one-off unanticipated events.
Analysing time series data
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 2: Time series analysis > Analysing time series data
- Smooth time series data by using a simple moving average, including the use of spreadsheets to implement this process
- Calculate seasonal indices by using the average percentage method
- Deseasonalise a time series by using a seasonal index, including the use of spreadsheets to implement this process
- Fit a least-squares line to model long-term trends in time series data, using appropriate technology
- Solve practical problems that involve the analysis of time series data.
Topic 3: Growth and decay in sequences
view_agenda query_statsThe arithmetic sequence
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 3: Growth and decay in sequences > The arithmetic sequence
- Use 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 using \( t_n = t_1 + (n - 1)d \), where \( t_n \) represents the nth term of the sequence, \( t_1 \) = first term, \( n \) = term number and \( d \) = common difference of a particular arithmetic sequence from the pattern of the terms in an arithmetic sequence, and use this rule to make predictions
- Use arithmetic sequences to model and analyse practical situations involving linear growth or decay, such as analysing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, or calculating the value of an office photocopier at the end of each year using the straight-line method or the unit cost method of depreciation.
The geometric sequence
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 3: Growth and decay in sequences > The geometric sequence
- Use 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 using \( t_n = t_1(r^{n-1}) \) where \( t_n \) represents the nth term of the sequence, \( t_1 \) = first term, \( n \) = term number and \( r \) = common ratio of a particular geometric sequence from the pattern of the terms in the sequence, and use this rule to make predictions
- Use geometric sequences to model and analyse (numerically or graphically) practical problems involving geometric growth and decay (logarithmic solutions not required), such as analysing a compound interest loan or investment, the growth of a bacterial population that doubles in size each hour or the decreasing height of the bounce of a ball at each bounce; or calculating the value of office furniture at the end of each year using the declining (reducing) balance method to depreciate.
Topic 4: Earth geometry and time zones
view_agenda query_statsLocations on the Earth
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 4: Earth geometry and time zones > Locations on the Earth
- Define the meaning of great circles
- Define the meaning of angles of latitude and longitude in relation to the equator and the prime meridian
- Locate positions on Earth's surface given latitude and longitude, e.g. using a globe, an atlas, GPS and other digital technologies
- State latitude and longitude for positions on Earth's surface and world maps (in degrees only)
- Use a local area map to state the position of a given place in degrees and minutes, e.g. investigating the map of Australia and locating boundary positions for Aboriginal language groups, such as the Three Sisters in the Blue Mountains or the local area's Aboriginal land and the positions of boundaries
- Calculate angular distance (in degrees and minutes) and distance (in kilometres) between two places on Earth on the same meridian using \( D = 111.2 \times \) angular distance
- Calculate angular distance (in degrees and minutes) and distance (in kilometres) between two places on Earth on the same parallel of latitude using \( D = 111.2 \cos \theta \times \) angular distance
- Calculate distances between two places on Earth, using appropriate technology.
Time zones
view_agenda query_statsUnit 3: Bivariate data, sequences and change, and Earth geometry > Topic 4: Earth geometry and time zones > Time zones
- Define 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 time difference of one hour
- Solve problems involving time zones in Australia and in neighbouring nations, making any necessary allowances for daylight saving, including seasonal time systems used by Aboriginal peoples and Torres Strait Islander peoples
- Solve problems involving GMT, International Date Line and UTC
- Calculate time differences between two places on Earth
- Solve problems associated with time zones, such as online purchasing, making phone calls overseas and broadcasting international events
- Solve problems relating to travelling east and west incorporating time zone changes, such as preparing an itinerary for an overseas holiday with corresponding times.
Unit 4: Investing and networking
view_agenda query_statsTopic 1: Loans, investments and annuities
view_agenda query_statsCompound interest loans and investments
view_agenda query_statsUnit 4: Investing and networking > Topic 1: Loans, investments and annuities > Compound interest loans and investments
- Use a recurrence relation \(A_{n+1} = rA_n\), to model a compound interest loan or investment, and investigate (numerically and graphically) the effect of the interest rate and the number of compounding periods on the future value of the loan or investment, e.g. payday loan
- Calculate the effective annual rate of interest and use the results to compare investment returns and the future value when interest is paid or charged daily, monthly, quarterly or six-monthly
- Solve problems involving compound interest loans or investments, e.g. determine the future value of a loan, the number of compounding periods for an investment to exceed a given value, the interest rate needed for an investment to exceed a given value.
Reducing balance loans (compound interest loans with periodic repayments)
view_agenda query_statsUnit 4: Investing and networking > Topic 1: Loans, investments and annuities > Reducing balance loans (compound interest loans with periodic repayments)
- Use a recurrence relation, \(A_{n+1} = rA_n - R\) (where \(R =\) monthly repayment) to model a reducing balance loan and investigate (numerically or graphically) the effect of the interest rate and repayment amount on the time taken to repay the loan
- With the aid of appropriate technology, solve problems involving reducing balance loans, e.g. determining the monthly repayments required to pay off a housing loan.
Annuities and perpetuities (compound interest investments with periodic payments made from the investment)
view_agenda query_statsUnit 4: Investing and networking > Topic 1: Loans, investments and annuities > Annuities and perpetuities (compound interest investments with periodic payments made from the investment)
- Use a recurrence relation \(A_{n+1} = rA_n + d\) to model an annuity and investigate (numerically or graphically) the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity
- Solve problems involving annuities, including perpetuities as a special case, e.g. determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount.
Topic 2: Graphs and networks
view_agenda query_statsGraphs, associated terminology and the adjacency matrix
view_agenda query_statsUnit 4: Investing and networking > Topic 2: Graphs and networks > Graphs, associated terminology and the adjacency matrix
- Understand the meanings of the terms graph, edge, vertex, loop, degree of a vertex, subgraph, simple graph, complete graph, bipartite graph, directed graph (digraph), arc, weighted graph and network
- Identify practical situations that can be represented by a network and construct such networks, e.g. trails connecting camp sites in a national park, a social network, a transport network with one-way streets, a food web, the results of a round-robin sporting competition
- Construct an adjacency matrix from a given graph or digraph.
Planar graphs, paths and cycles
view_agenda query_statsUnit 4: Investing and networking > Topic 2: Graphs and networks > Planar graphs, paths and cycles
- Understand the meaning of the terms planar graph and face
- Apply Euler's formula, \( v + f - e = 2 \), to solve problems relating to planar graphs
- Understand the meaning of the terms walk, trail, path, closed walk, closed trail, cycle, connected graph and bridge
- Investigate and 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 the terms Eulerian graph, Eulerian trail, semi-Eulerian graph, semi-Eulerian trail and the conditions for their existence, and use these concepts to investigate and solve practical problems, e.g. the Konigsberg bridge problem, planning a garbage bin collection route
- Understand the meaning of the terms Hamiltonian graph and semi-Hamiltonian graph and use these concepts to investigate and solve practical problems (by trial-and-error methods only), e.g. planning a sightseeing tourist route around a city, the travelling-salesman problem.
Topic 3: Networks and decision mathematics
view_agenda query_statsTrees and minimum connector problems
view_agenda query_statsUnit 4: Investing and networking > Topic 3: Networks and decision mathematics > Trees and minimum connector problems
- Understand the meaning of the terms tree and spanning tree
- Identify practical examples
- Identify a minimum spanning tree in a weighted connected graph, e.g. using Prim's algorithm
- Use minimal spanning trees to solve minimal connector problems, e.g. minimising the length of cable needed to provide power from a single power station to substations in several towns.
Project planning and scheduling using critical path analysis (CPA)
view_agenda query_statsUnit 4: Investing and networking > Topic 3: Networks and decision mathematics > Project planning and scheduling using critical path analysis (CPA)
- Construct a network diagram to represent the durations and interdependencies of activities that must be completed during the project, e.g. preparing a meal
- Use forward and backward scanning to determine the earliest starting time (EST) and latest starting times (LST) for each activity in the project
- Use EST and LSTs to locate the critical path/s for the project
- Use the critical path to determine the minimum time for a project to be completed
- Calculate float times for non-critical activities.
Flow networks
view_agenda query_statsUnit 4: Investing and networking > Topic 3: Networks and decision mathematics > Flow networks
- Solve small-scale network flow problems including the use of the maximum-flow minimum-cut theorem, e.g. determining the maximum volume of oil that can flow through a network of pipes from an oil storage tank to a terminal.
Assigning order and the Hungarian algorithm
view_agenda query_statsUnit 4: Investing and networking > Topic 3: Networks and decision mathematics > Assigning order and the Hungarian algorithm
- Use a bipartite graph and its tabular or matrix form to represent an assignment/allocation problem, e.g. assigning four swimmers to the four places in a medley relay team to maximise the team's chances of winning
- Determine the optimum assignment/s for small-scale problems by inspection, or by use of the Hungarian algorithm (3 x 3) for larger problems.