# 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.

## Unit 2

### Unit 2 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.

### Unit 2 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.

### Unit 2 Content Descriptions

#### 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)

#### 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 $$Area=\frac12ab\sin C$$, 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)

#### 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 3 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.

### Unit 3 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.

### Unit 3 Content Descriptions

#### 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)

#### 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)

#### 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)