Mathematics Curriculum for Adolescents Aged Fifteen to Eighteen Years.

Knowledge, skills and understanding

Typically, individuals will:

01.   Connect the compound interest formula to repeated applications of simple interest using appropriate digital technologies

Knowledge, skills and understanding

Typically, individuals will:

01.   Factorise algebraic expressions by taking out a common algebraic factor

02.    Simplify algebraic products and quotients using index laws

03.   Apply the four operations to simple algebraic fractions with numerical denominators

04.    Expand binomial products and factorise monic quadratic expressions using a variety of strategies 

05.   Substitute values into formulas to determine an unknown 

Knowledge, skills and understanding

Typically, individuals will:

01.  Solve problems involving linear equations, including those derived from formulas 

02.   Solve linear inequalities and graph their solutions on a number line 

03.   Solve linear simultaneous equations, using algebraic and graphical techniques, including using digital technology

04.    Solve problems involving parallel and perpendicular lines

05.   Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate 

06.   Solve linear equations involving simple algebraic fractions 

07.   Solve simple quadratic equations using a range of strategies 

08.   Work with number sentences and equations:

• using pronumerals in equations

• solving simple linear equations

• using substitution to solve equations

• balancing equations

• identifying equivalent equations

• performing the same operation on both sides to solve equations

• solving equations involving two operations

• deriving an equation from a word problem to solve the problem

Knowledge, skills and understanding

Typically, individuals will:

01. Work with different types of algebraic structure:

• operations and their order

• geometric and graphic models of operations

• laws of exponents for radicals

• distributive law: minus and parentheses

• identities

• multiplying, dividing, factoring

• inverses: opposites and reciprocals, inverse operations, inverse functions

  algebraic functions: equivalent fractions, lowest terms

• abstract algebra

Knowledge, skills and understanding

Typically, individuals will:

01. Work with equations in different ways:

• solving equations graphically

• solving equations by trial and error

• linear equations

• quadratic equations

• inequalities

• simultaneous equations

• finding the equations of a line

Knowledge, skills and understanding

Typically, individuals will:

.01 Study and apply knowledge of functions:

• input-output tables

• linear functions: gradient-intercept form, standard form, constant sum, constant difference

• quadratic functions: intercept form, standard form, vertex form

• other functions: exponential functions, constant products, rational functions, step functions, absolute value functions, iterating linear functions

 .02 definitions of function, domain, range

Knowledge, skills and understanding

Typically, individuals will:

01. Using units of measurement

02. Solve problems involving surface area and volume for a range of prisms, cylinders and composite solids 

03. Formulate proofs involving congruent triangles and angle properties 

04. Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes 

05. Investigate nets of solids and Euler’s rule for polyhedra 

Knowledge, skills and understanding

Typically, individuals will:

01. Solve right-angled triangle problems including those involving direction and angles of elevation and depression

02. Trigonometric relationships, identities, sine and cosine rule 

Knowledge, skills and understanding

Typically, individuals will:

01. Select appropriate operations to solve a variety of application problems using real numbers

02. Identify coordinates of a point in a plane or in space

03. Find the length and the midpoint of a segment in two or three dimensions to solve problems

04. Use definitions, properties and theorems of lines, angles and polygons to solve problems:

05. Recognize, identify and model regular and non-regular polyhedra and use coordinate geometry to confirm properties

06. Use formulae to solve problems related to:

• the perimeter of a geometric figure and circumference of a circle

• the area of a triangle, parallelogram, rhombus, trapezoid, square, rectangle, regular polygons, and circles

• arc lengths and the area of sectors of a circle

• the ratio of the perimeters, areas, and volumes of similar geometric figures

• the lateral area, surface area, and volume of a right prism, pyramid, right circular cylinder, cone, and sphere

.07 Use gradients to determine if two lines are parallel or perpendicular

.08 Write equation of a line parallel or perpendicular to a given line through a given point

.09 Transform (translate, reflect, rotate, dilate) polygons in the coordinate plane, describing the transformation in simple algebraic terms

Knowledge, skills and understanding

Typically, individuals will:

01. Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence 

02. Use the language of ‘if ....then, ‘given’, ‘of’, ‘knowing that’ to investigate conditional statements and identify common mistakes in interpreting such language  

Knowledge, skills and understanding

Typically, individuals will:

01. Determine quartiles and interquartile range

02. Construct and interpret box plots and use them to compare data sets 

03. Compare shapes of box plots to corresponding histograms and dot plots 

04. Use scatter plots to investigate and comment on relationships between two numerical variables 

05. Investigate and describe bivariate numerical data where the independent variable is time 

06. Evaluate statistical reports in the media and other places by linking claims to displays, statistics and representative data

Knowledge, skills and understanding

Typically, individuals will:

01. Determine quartiles and interquartile range

02. Construct and interpret box plots and use them to compare data sets 

03. Compare shapes of box plots to corresponding histograms and dot plots 

04. Use scatter plots to investigate and comment on relationships between two numerical variables 

05. Investigate and describe bivariate numerical data where the independent variable is time 

06. Evaluate statistical reports in the media and other places by linking claims to displays, statistics and representative data

Knowledge, skills and understanding

Typically, individuals will:

01. Percentage calculations:

• review calculating a percentage of a given amount

• review one amount expressed as a percentage of another.

02. Applications of percentages:

• determine the overall change in a quantity following repeated percentage changes; for example, an increase of 10% followed by a decrease of 10%

• calculate simple interest for different rates and periods.

Knowledge, skills and understanding

Typically, individuals will:

01. Ratios:

• demonstrate an understanding of the elementary ideas and notation of ratio

• understand the relationship between fractions and ratio

• express a ratio in simplest form

• find the ratio of two quantities

• divide a quantity in a given ratio

• use ratio to describe simple scales.

02. Rates:

• review identifying common usage of rates such as km/h

• convert between units for rates; for example, km/h to m/s, mL/min to L/h

• complete calculations with rates, including solving problems involving direct proportion in terms of rate

• use rates to make comparisons

• use rates to determine costs; for example, calculating the cost of a tradesman using rates per hour, call-out fees.

Knowledge, skills and understanding

Typically, individuals will:

01. Time:

• use units of time, conversions between units, fractional, digital and decimal representations

• represent time using 12-hour and 24-hour clocks

• calculate time intervals, such as time between, time ahead, time behind

• interpret timetables, such as bus, train and ferry timetables

• use several timetables and electronic technologies to plan the most time-efficient routes

• interpret complex timetables, such as tide charts, sunrise charts and moon phases

• compare the time taken to travel a specific distance with various modes of transport

02. Distance:

• use scales to find distances, such as on maps; for example, road maps, street maps, bushwalking maps, online maps and cadastral maps

• optimise distances through trial-and-error and systematic methods; for example, shortest path, routes to visit all towns, and routes to use all roads.

03. Speed:

• identify the appropriate units for different activities, such as walking, running, swimming and flying

• calculate speed, distance or time using the formula speed = distance/time

• calculate the time or costs for a journey from distances estimated from maps

• interpret distance-versus-time graphs

• calculate and interpret average speed; for example, a 4-hour trip covering 250 km.

Knowledge, skills and understanding

Typically, individuals will:

01. Location:

• locate positions on Earth’s surface given latitude and longitude using GPS, a globe, an atlas, and digital technologies

• find distances between two places on Earth on the same longitude

• find distances between two places on Earth using appropriate technology.

02. Time:

• understand the link between longitude and time

• solve problems involving time zones in Australia and in neighbouring nations, making any necessary allowances for daylight saving

• solve problems involving Greenwich Mean Time and the International Date Line

• find time differences between two places on Earth

• solve problems associated with time zones; for example, internet and phone usage

• solve problems relating to travelling east and west, incorporating time zone changes.

Knowledge, skills and understanding

Typically, individuals will study:

01. Compound interest:

• review the principles of simple interest

• understand the concept of compound interest as a recurrence relation

• consider similar problems involving compounding; for example, population growth

• use technology to calculate the future value of a compound interest loan or investment and the total interest paid or earned

• use technology to compare, numerically and graphically, the growth of simple interest and compound interest loans and investments

• use technology to investigate the effect of the interest rate and the number of compounding periods on the future value of a loan or investment.

02. Reducing balance loans (compound interest loans with periodic repayments):

• use technology and a recurrence relation to model a reducing balance loan

• investigate the effect of the interest rate and repayment amount on the time taken to repay a loan.

Knowledge, skills and understanding

Typically, individuals will study:

01. Applications of rates and percentages:

• review rates and percentages

• calculate weekly or monthly wage from an annual salary, 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

• 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; for example, 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 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.

02. Use of spreadsheets:

• use a spreadsheet to display examples of the above computations when multiple or repeated computations are required; for example, 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.

Knowledge, skills and understanding

Typically, individuals will study:

01. Compound interest loans and investments:

• use a recurrence relation to model a compound interest loan or investment, and investigate (numerically or graphically) the effect of the interest rate and the number of compounding periods on the future value of the loan or investment

• calculate the effective annual rate of interest and use the results to compare investment returns and cost of loans when interest is paid or charged daily, monthly, quarterly or six-monthly

• with the aid of a calculator or computer-based financial software, solve problems involving compound interest loans or investments; for example, determining 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.

02. Reducing balance loans (compound interest loans with periodic repayments):

• use a recurrence relation 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 a financial calculator or computer-based financial software, solve problems involving reducing balance loans; for example, determining the monthly repayments required to pay off a housing loan.

03. Annuities and perpetuities (compound interest investments with periodic payments made from the investment):

• use a recurrence relation 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

• with the aid of a financial calculator or computer-based financial software, solve problems involving annuities (including perpetuities as a special case); for example, determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount.

Knowledge, skills and understanding

Under development in accordance with Australian curriculum.

Knowledge, skills and understanding

Typically, individuals will study:

01. Single substitution:

• substitute numerical values into algebraic expressions; for example, substitute different values of xx to evaluate the expressions 3x5,5(2x−4)3x5,5(2x−4) 

 02. General Substitution:

• substitute given values for the other pronumerals in a mathematical formula to find the value of the subject of the formula. 

Knowledge, skills and understanding

Typically, individuals will study:

01. Linear and non-linear expressions:

• substitute numerical values into linear algebraic and simple non-linear algebraic expressions, and evaluate

• find the value of the subject of the formula, given the values of the other pronumerals in the formula

• 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; for example, a table displaying the body mass index (BMI) of people of different weights and heights.

02. Matrices and matrix arithmetic:

• use matrices for storing and displaying information that can be presented in rows and columns; for example, databases, links in social or road networks

• recognise different types of matrices (row, column, square, zero, identity) and determine their size

• perform matrix addition, subtraction, multiplication by a scalar, and matrix multiplication, including 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; for example, 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.

Knowledge, skills and understanding

Typically, individuals will study:

01. Linear equations:

• identify and solve linear equations

• develop a linear formula from a word description

02. Straight-line graphs and their applications:

• construct straight-line graphs 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; for example, modelling the cost of filling a fuel tank of a car against the number of litres of petrol required.

03. Simultaneous linear equations and their applications:

• solve a pair of simultaneous linear equations, using technology when appropriate

• solve practical problems that involve finding the point of intersection of two straight-line graphs; for example, determining the break-even point where cost and revenue are represented by linear equations.

04. Piece-wise linear graphs and step graphs:

• sketch piece-wise linear graphs and step graphs, using technology when appropriate

• interpret piece-wise linear and step graphs used to model practical situations; for example, the tax paid as income increases, the change in the level of water in a tank over time when water is drawn off at different intervals and for different periods of time, the charging scheme for sending parcels of different weights through the post.

Knowledge, skills and understanding

Typically, individuals will study:

01. 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

• deduce a rule for the nth term 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; for example, 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.

02. 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

• deduce a rule for the nth term 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 only) practical problems involving geometric growth and decay; for example, analysing a compound interest loan or investment, the growth of a bacterial population that doubles in size each hour, 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.

03. Sequences generated by first-order linear recurrence relations:

• use a general first-order linear recurrence relation to generate the terms of a sequence and to display it in both tabular and graphical form

• recognise that a sequence generated by a first-order linear recurrence relation can have a long term increasing, decreasing or steady-state solution

• use first-order linear recurrence relations to model and analyse (numerically or graphically only) practical problems; for example, investigating the growth of a trout population in a lake recorded at the end of each year and where limited recreational fishing is permitted, or the amount owing on a reducing balance loan after each payment is made.

Knowledge, skills and understanding

Typically, individuals will study:

01. Linear measure:

• use metric units of length, their abbreviations, conversions between them, and appropriate levels of accuracy and choice of units 

• estimate lengths 

• convert between metric units of length and other length units 

• calculate perimeters of familiar shapes, including triangles, squares, rectangles, and composites of these. 

02. Area measure:

• use metric units of area, their abbreviations, conversions between them, and appropriate choices of units 

• estimate the areas of different shapes 

• convert between metric units of area and other area units 

• calculate areas of rectangles and triangles. 

 03. Mass:

• use metric units of mass, their abbreviations, conversions between them, and appropriate choices of units 

• estimate the mass of different objects.

04. Volume and capacity:

• use metric units of volume, their abbreviations, conversions between them, and appropriate choices of units 

• understand the relationship between volume and capacity 

• estimate volume and capacity of various objects 

• calculate the volume of objects, such as cubes and rectangular and triangular prisms. 

05. Units of energy:

• use units of energy to describe consumption of electricity, such as kilowatt hours 

• use units of energy used for foods, including calories 

• use units of energy to describe the amount of energy in activity, such as kilojoules 

• convert from one unit of energy to another. 

Knowledge, skills and understanding

Typically, individuals will study:

01. Linear measure:

• review metric units of length, their abbreviations, conversions between them, estimation of lengths, and appropriate choices of units

• calculate perimeters of familiar shapes, including triangles, squares, rectangles, polygons, circles, arc lengths, and composites of these.

• find the area of irregular figures by decomposition into regular shapes

02. Area measure:

• review metric units of area, their abbreviations, and conversions between them

• use formulas to calculate areas of regular shapes, including triangles, squares, rectangles, parallelograms, trapeziums, circles and sectors

• find the surface area of familiar solids, including cubes, rectangular and triangular prisms, spheres and cylinders

• find the surface area of pyramids, such as rectangular- and triangular-based pyramids

• use addition of the area of the faces of solids to find the surface area of irregular solids.

03. Mass:

• review metric units of mass (and weight), their abbreviations, conversions between them, and appropriate choices of units

• recognise the need for milligrams

• convert between grams and milligrams.

04. Volume and capacity:

• review metric units of volume, their abbreviations, conversions between them, and appropriate choices of units

• recognise relations between volume and capacity, recognising that and

• use formulas to find the volume and capacity of regular objects such as cubes, rectangular and triangular prisms and cylinders

• use formulas to find the volume of pyramids and spheres.

Knowledge, skills and understanding

Typically, individuals will study:

01. Geometry:

• recognise the properties of common two-dimensional geometric shapes and three-dimensional solids

• interpret different forms of two-dimensional representations of three-dimensional objects, including nets and perspective diagrams

• use symbols and conventions for the representation of geometric information, for example, point, line, ray, angle, diagonal, edge, curve, face and vertex.

02. Interpret scale drawings:

• interpret commonly used symbols and abbreviations in scale drawings

• find actual measurements from scale drawings, such as lengths, perimeters and areas

• estimate and compare quantities, materials and costs using actual measurements from scale drawings; for example, using measurements for packaging, clothes, painting, bricklaying and landscaping.

03. Creating scale drawings:

• understand and apply drawing conventions of scale drawings, such as scales in ratio, clear indications of dimensions, and clear labelling

• construct scale drawings by hand and by using software packages.

04. Three dimensional objects:

• interpret plans and elevation views of models

• sketch elevation views of different models

• interpret diagrams of three-dimensional objects.

05. Right-angled triangles:

• apply Pythagoras’ theorem to solve problems

• apply the tangent ratio to find unknown angles and sides in right-angled triangles

• work with the concepts of angle of elevation and angle of depression

• apply the cosine and sine ratios to find unknown angles and sides in right-angled triangles

• solve problems involving bearings.

Knowledge, skills and understanding

Typically, individuals will study:

01. Cartesian plane:

• demonstrate familiarity with Cartesian coordinates in two dimensions by plotting points on the Cartesian plane

• generate tables of values for linear functions, including for negative values of

• graph linear functions for all values of with pencil and paper and with graphing software.

02. Using graphs:

• interpret and use graphs in practical situations, including travel graphs and conversion graphs

• draw graphs from given data to represent practical situations

• interpret the point of intersection and other important features of given graphs of two linear functions drawn from practical contexts; for example, the ‘break-even’ point.

Knowledge, skills and understanding

Typically, individuals will study:

.01 Pythagoras Theorem:

• review Pythagoras’ Theorem and use it to solve practical problems in two dimensions and for simple applications in three dimensions.

.02 Mensuration:

• solve practical problems requiring the calculation of perimeters and areas of circles, sectors of circles, triangles, rectangles, parallelograms and composites

• calculate the volumes of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the volume of water contained in a swimming pool

• calculate the surface areas of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the surface area of a cylindrical food container.

.03 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

• obtain a scale factor and use it to solve scaling problems involving the calculation of surface areas and volumes of similar solids.

Knowledge, skills and understanding

Typically, individuals will study:

01. 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, or given three sides by using Heron’s rule, and solve related practical problems

• solve problems involving non-right-angled triangles using the sine rule (ambiguous case excluded) and the cosine rule

• solve 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 bearings in navigation.

Knowledge, skills and understanding

Under development according to the Australian Curriculum

Knowledge, skills and understanding

Typically, individuals will study:

01. Reading and interpreting graphs:

• interpret information presented in graphs, such as conversion graphs, line graphs, step graphs, column graphs and picture graphs 

• interpret information presented in two-way tables 

• discuss and interpret graphs found in the media and in factual texts. 

 02. Drawing graphs:

• determine which type of graph is best used to display a dataset 

• use spreadsheets to tabulate and graph data 

• draw a line graph to represent any data that demonstrate a continuous change, such as hourly temperature. 

Knowledge, skills and understanding

Typically, individuals will study:

01. Classifying data:

• identify examples of categorical data

• identify examples of numerical data.

02. Data presentation and interpretation:

• display categorical data in tables and column graphs

• display numerical data as frequency distributions, dot plots, stem and leaf plots, and histograms

• recognise and identify outliers

• compare the suitability of different methods of data presentation in real-world contexts.

03. Summarising and interpreting data:

• identify the mode

• calculate measures of central tendency, the arithmetic mean and the median

• investigate the suitability of measures of central tendency in various real-world contexts

• investigate the effect of outliers on the mean and the median

• calculate and interpret quartiles, deciles and percentiles

• use informal ways of describing spread, such as spread out/dispersed, tightly packed, clusters, gaps, more/less dense regions, outliers

• calculate and interpret statistical measures of spread, such as the range, interquartile range and standard deviation

• investigate real-world examples from the media illustrating inappropriate uses, or misuses, of measures of central tendency and spread.

04. Comparing data sets:

• compare back-to-back stem plots for different data-sets

• complete a five number summary for different datasets

• construct box plots using a five number summary

• compare the characteristics of the shape of histograms using symmetry, skewness and bimodality.

Knowledge, skills and understanding

Typically, individuals will study:

01. Census:

• investigate the procedure for conducting a census

• investigate the advantages and disadvantages of conducting a census.

02. Surveys:

• understand the purpose of sampling to provide an estimate of population values when a census is not used

• investigate the different kinds of samples; for example, systematic samples, self-selected samples, simple random samples

• investigate the advantages and disadvantages of these kinds of samples; for example, comparing simple random samples with self-selected samples.

03. Simple survey procedure:

• identify the target population to be surveyed

• investigate questionnaire design principles; for example, simple language, unambiguous questions, consideration of number of choices, issues of privacy and ethics, and freedom from bias.

04. Sources of bias:

• describe the faults in the collection of data process

• describe sources of error in surveys; for example, sampling error and measurement error

• investigate the possible misrepresentation of the results of a survey due to misunderstanding the procedure, or misunderstanding the reliability of generalising the survey findings to the entire population

• investigate errors and misrepresentation in surveys, including examples of media misrepresentations of surveys.

05. Bivariate scatterplots:

• describe the patterns and features of bivariate data

• describe the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak).

06. Line of best fit:

• identify the dependent and independent variable

• find the line of best fit by eye

• use technology to find the line of best fit

• interpret relationships in terms of the variables

• use technology to find the correlation coefficient (an indicator of the strength of linear association)

• use the line of best fit to make predictions, both by interpolation and extrapolation

• recognise the dangers of extrapolation

• distinguish between causality and correlation through examples.

Knowledge, skills and understanding

Typically, individuals will study:

01. Probability expressions:

• interpret commonly used probability statements, including ‘possible’, ‘probable’, ‘likely’, ‘certain’

• describe ways of expressing probabilities formally using fractions, decimals, ratios, and percentages.

02. Simulations:

• perform simulations of experiments using technology

• recognise that the repetition of chance events is likely to produce different results

• identify relative frequency as probability

• identify factors that could complicate the simulation of real-world events.

03. Simple probabilities:

• construct a sample space for an experiment

• use a sample space to determine the probability of outcomes for an experiment

• use arrays or tree diagrams to determine the outcomes and the probabilities for experiments.

04. Probability applications:

• determine the probabilities associated with simple games

• determine the probabilities of occurrence of simple traffic-light problems.

Knowledge, skills and understanding

Typically, individuals will study:

01. The statistical investigation process:

• review the statistical investigation process; for example, identifying a problem and posing a statistical question, collecting or obtaining data, analysing the data, interpreting and communicating the results.

02. Making sense of data relating to a single statistical variable:

• classify a categorical variable as ordinal, such as income level (high, medium, low), or nominal, such as place of birth (Australia, overseas), and use tables and bar charts to organise and display the data

• classify a numerical variable as discrete, such as the number of rooms in a house, or continuous, such as the temperature in degrees Celsius

• with the aid of an appropriate graphical display (chosen from dot plot, stem plot, bar chart or histogram), describe the distribution of a numerical dataset in terms of modality (uni or multimodal), shape (symmetric versus positively or negatively skewed), location and spread and outliers, and interpret this information in the context of the data

• determine the mean and standard deviation of a dataset and use these statistics as measures of location and spread of a data distribution, being aware of their limitations.

03. Comparing data for a numerical variable across two or more groups:

• construct and use parallel box plots (including the use of the ‘Q1 – 1.5 x IQR’ and ‘Q3 + 1.5 x IQR’ criteria for identifying possible outliers) to compare groups in terms of location (median), spread (IQR and range) and outliers and to interpret and communicate the differences observed in the context of the data

• compare groups on a single numerical variable using medians, means, IQRs, ranges or standard deviations, as appropriate; interpret the differences observed in the context of the data; and report the findings in a systematic and concise manner

• implement the statistical investigation process to answer questions that involve comparing the data for a numerical variable across two or more groups; for example, are Year 11 students the fittest in the school?

Knowledge, skills and understanding

Typically, individuals will study:

01. The statistical investigation process:

• review the statistical investigation process; for example, identifying a problem and posing a statistical question, collecting or obtaining data, analysing the data, interpreting and communicating the results.

02. Identifying and describing associations between two categorical variables:

• 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

• describe 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.

03. Identifying and describing associations between two numerical variables:

• construct a scatterplot to identify patterns in the data suggesting the presence of an association

• describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak)

• calculate and interpret the correlation coefficient (r) to quantify the strength of a linear association.

 04. 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 the coefficient of determination 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

• write up the results of the above analysis in a systematic and concise manner.

05. Association and causation:

• recognise that an observed association between two variables does not necessarily mean that there is a causal relationship between them

• identify possible non-causal explanations for an association, including coincidence and confounding due to a common response to another variable, and communicate these explanations in a systematic and concise manner.

06. The data investigation process:

• implement the statistical investigation process to answer questions that involve identifying, analysing and describing associations between two categorical variables or between two numerical variables; for example, is there an association between attitude to capital punishment (agree with, no opinion, disagree with) and sex (male, female)? is there an association between height and foot length?

Knowledge, skills and understanding

Typically, individuals will study:

01. The definition of a graph and associated terminology:

• explain 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; for example, 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.

02. Planar graphs:

• explain the meaning of the terms: planar graph, and face

• apply Euler’s formula, to solve problems relating to planar graphs.

03. Paths and cycles:

• explain 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)

• explain 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; for example, the Königsberg Bridge problem, planning a garbage bin collection route

• explain the meaning of the terms: Hamiltonian graph and semi-Hamiltonian graph, and use these concepts to investigate and solve practical problems; for example, planning a sight-seeing tourist route around a city, the travelling-salesman problem (by trial-and-error methods only).

Knowledge, skills and understanding

Typically, individuals will study:

01. 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; for example, one-off unanticipated events.

02. 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.

03. The data investigation process:

• implement the statistical investigation process to answer questions that involve the analysis of time series data.

Knowledge, skills and understanding

Typically, individuals will study:

01. Trees and minimum connector problems:

• explain the meaning of the terms tree and spanning tree identify practical examples

• identify a minimum spanning tree in a weighted connected graph either by inspection or by using Prim’s algorithm

• use minimal spanning trees to solve minimal connector problems; for example, minimising the length of cable needed to provide power from a single power station to substations in several towns.

02. Project planning and scheduling using critical path analysis (CPA):

• construct a network to represent the durations and interdependencies of activities that must be completed during the project; for example, 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 ESTs 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.

.03 Flow networks:

• solve small-scale network flow problems including the use of the ‘maximum-flow minimum- cut’ theorem; for example, determining the maximum volume of oil that can flow through a network of pipes from an oil storage tank (the source) to a terminal (the sink).

.04 Assignment problems:

• use a bipartite graph and/or its tabular or matrix form to represent an assignment/ allocation problem; for example, 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), by inspection for small-scale problems, or by use of the Hungarian algorithm for larger problems.

Knowledge, skills and understanding

Under development in accordance with the Australian Curriculum