# General Mathematics

### Rationale

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe many aspects of the world in the twenty-first century.

### Links to Foundation to Year 10

The General Mathematics subject provides students with a breadth of mathematical and statistical experience that encompasses and builds on all three strands of the F-10 curriculum.

### Representation of General capabilities

The seven general capabilities of Literacy, Numeracy, Information and Communication Technology (ICT) capability, Critical and creative thinking, Personal and social capability, Ethical understanding, and Intercultural understanding are identified where they offer opportunities to add depth and richness to student learning.

### Structure of General Mathematics

General Mathematics is organised into four units. The topics in each unit broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving. The units provide a blending of algebraic, geometric and statistical thinking.

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## Unit 4

### Unit 4 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.

### Unit 4 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.

### Unit 4 Content Descriptions

#### 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)

#### 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)

#### 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)