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Unit 1

Unit Description

This unit has three topics: ‘Consumer arithmetic’, ‘Algebra and matrices’, and ‘Shape and measurement’.

‘Consumer arithmetic’ reviews the concepts of rate and percentage change in the context of earning and managing money, and provides a fertile ground for the use of spreadsheets.

‘Algebra and matrices’ continues the F-10 study of algebra and introduces the new topic of matrices.

‘Shape and measurement’ builds on and extends the knowledge and skills students developed in the F-10 curriculum with the concept of similarity and associated calculations involving simple and compound geometric shapes. The emphasis in this topic is on applying these skills in a range of practical contexts, including those involving three-dimensional shapes.

Classroom access to the technology necessary to support the computational aspects of the topics in this unit is assumed.

Learning Outcomes

By the end of this unit, students:

  • understand the concepts and techniques introduced in consumer arithmetic, algebra and matrices, and shape and measurement
  • apply reasoning skills and solve practical problems arising in consumer arithmetic, algebra and matrices, and shape and measurement
  • communicate their arguments and strategies, when solving problems, using appropriate mathematical language
  • interpret mathematical information, and ascertain the reasonableness of their solutions to problems
  • choose and use technology appropriately and efficiently.

Content Descriptions

Topic 1: Consumer arithmetic

Applications of rates and percentages:

review rates and percentages (ACMGM001)

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 (ACMGM002)

calculate payments based on government allowances and pensions (ACMGM003)

prepare a personal budget for a given income taking into account fixed and discretionary spending (ACMGM004)

compare prices and values using the unit cost method (ACMGM005)

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 (ACMGM006)

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 (ACMGM007)

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. (ACMGM008)

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. (ACMGM009)

Topic 2: Algebra and matrices

Linear and non-linear expressions:

substitute numerical values into linear algebraic and simple non-linear algebraic expressions, and evaluate (ACMGM010)

find the value of the subject of the formula, given the values of the other pronumerals in the formula (ACMGM011)

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. (ACMGM012)

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 (ACMGM013)

recognise different types of matrices (row, column, square, zero, identity) and determine their size (ACMGM014)

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 (ACMGM015)

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. (ACMGM016)

Topic 3: Shape and measurement

Pythagoras’ Theorem:

review Pythagoras’ Theorem and use it to solve practical problems in two dimensions and for simple applications in three dimensions. (ACMGM017)

Mensuration:

solve practical problems requiring the calculation of perimeters and areas of circles, sectors of circles, triangles, rectangles, parallelograms and composites (ACMGM018)

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 (ACMGM019)

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. (ACMGM020)

Similar figures and scale factors:

review the conditions for similarity of two-dimensional figures including similar triangles (ACMGM021)

use the scale factor for two similar figures to solve linear scaling problems (ACMGM022)

obtain measurements from scale drawings, such as maps or building plans, to solve problems (ACMGM023)

obtain a scale factor and use it to solve scaling problems involving the calculation of the areas of similar figures (ACMGM024)

obtain a scale factor and use it to solve scaling problems involving the calculation of surface areas and volumes of similar solids. (ACMGM025)

 

Unit 2

Unit Description

This unit has three topics: ‘Univariate data analysis and the statistical investigation process’, ‘Linear equations and their graphs’; and ‘Applications of trigonometry’.

‘ Univariate data analysis and the statistical investigation process’ develops students’ ability to organise and summarise univariate data in the context of conducting a statistical investigation.

‘Linear equations and their graphs’ uses linear equations and straight-line graphs, as well as linear-piecewise and step graphs, to model and analyse practical situations.

‘Applications of trigonometry’ extends students’ knowledge of trigonometry to solve practical problems involving non-right-angled triangles in both two and three dimensions, including problems involving the use of angles of elevation and depression and bearings in navigation.

Classroom access to the technology necessary to support the graphical, computational and statistical aspects of this unit is assumed.

Learning Outcomes

By the end of this unit, students:

  • understand the concepts and techniques in univariate data analysis and the statistical investigation process, linear equations and their graphs, and applications of trigonometry
  • apply reasoning skills and solve practical problems in univariate data analysis and the statistical investigation process, linear equations and their graphs, and the applications of trigonometry
  • implement the statistical investigation process in contexts requiring the analysis of univariate data
  • communicate their arguments and strategies, when solving mathematical and statistical problems, using appropriate mathematical or statistical language
  • interpret mathematical and statistical information, and ascertain the reasonableness of their solutions to problems and their answers to statistical questions
  • choose and use technology appropriately and efficiently.

Content Descriptions

Topic 1: Univariate data analysis and the statistical investigation process

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. (ACMGM026)

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 (ACMGM027)

classify a numerical variable as discrete, such as the number of rooms in a house, or continuous, such as the temperature in degrees Celsius (ACMGM028)

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 (ACMGM029)

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. (ACMGM030)

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 (ACMGM031)

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 (ACMGM032)

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? (ACMGM033)

Topic 2: 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 (ACMGM034)

determine the area of a triangle given two sides and an included angle by using the rule Area=12absin⁡C, or given three sides by using Heron’s rule, and solve related practical problems (ACMGM035)

solve problems involving non-right-angled triangles using the sine rule (ambiguous case excluded) and the cosine rule (ACMGM036)

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. (ACMGM037)

Topic 3: Linear equations and their graphs

Linear equations:

identify and solve linear equations (ACMGM038)

develop a linear formula from a word description (ACMGM039)

Straight-line graphs and their applications:

construct straight-line graphs both with and without the aid of technology (ACMGM040)

determine the slope and intercepts of a straight-line graph from both its equation and its plot (ACMGM041)

interpret, in context, the slope and intercept of a straight-line graph used to model and analyse a practical situation (ACMGM042)

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. (ACMGM043)

Simultaneous linear equations and their applications:

solve a pair of simultaneous linear equations, using technology when appropriate (ACMGM044)

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. (ACMGM045)

Piece-wise linear graphs and step graphs:

sketch piece-wise linear graphs and step graphs, using technology when appropriate (ACMGM046)

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. (ACMGM047)

 

Unit 3

Unit Description

This unit has three topics: ‘Bivariate data analysis’, ‘Growth and decay in sequences’ and ‘Graphs and networks’.

‘Bivariate data analysis’ introduces students to some methods for identifying, analysing and describing associations between pairs of variables, including the use of the least-squares method as a tool for modelling and analysing linear associations. The content is to be taught within the framework of the statistical investigation process.

‘Growth and decay in sequences’ employs recursion to generate sequences that can be used to model and investigate patterns of growth and decay in discrete situations. These sequences find application in a wide range of practical situations, including modelling the growth of a compound interest investment, the growth of a bacterial population, or the decrease in the value of a car over time. Sequences are also essential to understanding the patterns of growth and decay in loans and investments that are studied in detail in Unit 4.

‘Graphs and networks’ introduces students to the language of graphs and the ways in which graphs, represented as a collection of points and interconnecting lines, can be used to model and analyse everyday situations such as a rail or social network.

Classroom access to technology to support the graphical and computational aspects of these topics is assumed.

Learning Outcomes

By the end of this unit, students:

  • understand the concepts and techniques in bivariate data analysis, growth and decay in sequences, and graphs and networks
  • apply reasoning skills and solve practical problems in bivariate data analysis, growth and decay in sequences, and graphs and networks
  • implement the statistical investigation process in contexts requiring the analysis of bivariate data
  • communicate their arguments and strategies, when solving mathematical and statistical problems, using appropriate mathematical or statistical language
  • interpret mathematical and statistical information, and ascertain the reasonableness of their solutions to problems and their answers to statistical questions
  • choose and use technology appropriately and efficiently.

Content Descriptions

Topic 1: Bivariate data analysis

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. (ACMGM048)

Identifying and describing associations between two categorical variables:

construct two-way frequency tables and determine the associated row and column sums and percentages (ACMGM049)

use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association (ACMGM050)

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. (ACMGM051)

Identifying and describing associations between two numerical variables:

construct a scatterplot to identify patterns in the data suggesting the presence of an association (ACMGM052)

describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak) (ACMGM053)

calculate and interpret the correlation coefficient (r) to quantify the strength of a linear association. (ACMGM054)

Fitting a linear model to numerical data:

identify the response variable and the explanatory variable (ACMGM055)

use a scatterplot to identify the nature of the relationship between variables (ACMGM056)

model a linear relationship by fitting a least-squares line to the data (ACMGM057)

use a residual plot to assess the appropriateness of fitting a linear model to the data (ACMGM058)

interpret the intercept and slope of the fitted line (ACMGM059)

use the coefficient of determination to assess the strength of a linear association in terms of the explained variation (ACMGM060)

use the equation of a fitted line to make predictions (ACMGM061)

distinguish between interpolation and extrapolation when using the fitted line to make predictions, recognising the potential dangers of extrapolation (ACMGM062)

write up the results of the above analysis in a systematic and concise manner. (ACMGM063)

Association and causation:

recognise that an observed association between two variables does not necessarily mean that there is a causal relationship between them (ACMGM064)

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. (ACMGM065)

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? (ACMGM066)

Topic 2: Growth and decay in sequences

The arithmetic sequence:

use recursion to generate an arithmetic sequence (ACMGM067)

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 (ACMGM068)

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 (ACMGM069)

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. (ACMGM070)

The geometric sequence:

use recursion to generate a geometric sequence (ACMGM071)

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 (ACMGM072)

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 (ACMGM073)

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. (ACMGM074)

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 (ACMGM075)

recognise that a sequence generated by a first-order linear recurrence relation can have a long term increasing, decreasing or steady-state solution (ACMGM076)

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. (ACMGM077)

Topic 3: Graphs and networks

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 (ACMGM078)

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 (ACMGM079)

construct an adjacency matrix from a given graph or digraph. (ACMGM080)

Planar graphs:

explain the meaning of the terms: planar graph, and face (ACMGM081)

apply Euler’s formula, v+f-e=2 , to solve problems relating to planar graphs. (ACMGM082)

Paths and cycles:

explain the meaning of the terms: walk, trail, path, closed walk, closed trail, cycle, connected graph, and bridge (ACMGM083)

investigate and solve practical problems to determine the shortest path between two vertices in a weighted graph (by trial-and-error methods only) (ACMGM084)

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 (ACMGM085)

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). (ACMGM086)

 

Unit 4

Unit Description

This unit has three topics: ‘Time series analysis’; ‘ Loans, investments and annuities’ and ‘Networks and decision mathematics’.

‘Time series analysis’ continues students’ study of statistics by introducing them to the concepts and techniques of time series analysis. The content is to be taught within the framework of the statistical investigation process.

‘Loans and investments and annuities’ aims to provide students with sufficient knowledge of financial mathematics to solve practical problems associated with taking out or refinancing a mortgage and making investments.

‘Networks and decision mathematics’ uses networks to model and aid decision making in practical situations.

Classroom access to the technology necessary to support the graphical, computational and statistical aspects of this unit is assumed.

Learning Outcomes

By the end of this unit, students:

  • understand the concepts and techniques in time series analysis; loans, investments and annuities; and networks and decision mathematics
  • apply reasoning skills and solve practical problems in time series analysis; loans, investments and annuities; and networks and decision mathematics
  • implement the statistical investigation process in contexts requiring the analysis of time series data
  • communicate their arguments and strategies, when solving mathematical and statistical problems, using appropriate mathematical or statistical language
  • interpret mathematical and statistical information, and ascertain the reasonableness of their solutions to problems and their answers to statistical questions
  • choose and use technology appropriately and efficiently.

Content Descriptions

Topic 1: Time series analysis

Describing and interpreting patterns in time series data:

construct time series plots (ACMGM087)

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. (ACMGM088)

Analysing time series data:

smooth time series data by using a simple moving average, including the use of spreadsheets to implement this process (ACMGM089)

calculate seasonal indices by using the average percentage method (ACMGM090)

deseasonalise a time series by using a seasonal index, including the use of spreadsheets to implement this process (ACMGM091)

fit a least-squares line to model long-term trends in time series data. (ACMGM092)

The data investigation process:

implement the statistical investigation process to answer questions that involve the analysis of time series data. (ACMGM093)

Topic 2: Loans, investments and annuities

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 (ACMGM094)

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 (ACMGM095)

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. (ACMGM096)

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 (ACMGM097)

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. (ACMGM098)

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 (ACMGM099)

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. (ACMGM100)

Topic 3: Networks and decision mathematics

Trees and minimum connector problems:

explain the meaning of the terms tree and spanning tree identify practical examples (ACMGM101)

identify a minimum spanning tree in a weighted connected graph either by inspection or by using Prim’s algorithm (ACMGM102)

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. (ACMGM103)

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 (ACMGM104)

use forward and backward scanning to determine the earliest starting time (EST) and latest starting times (LST) for each activity in the project (ACMGM105)

use ESTs and LSTs to locate the critical path(s) for the project (ACMGM106)

use the critical path to determine the minimum time for a project to be completed (ACMGM107)

calculate float times for non-critical activities. (ACMGM108)

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). (ACMGM109)

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 (ACMGM110)

determine the optimum assignment(s), by inspection for small-scale problems, or by use of the Hungarian algorithm for larger problems. (ACMGM111)

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