# General Mathematics

Like Maths A, this subject is designed for students who want to extend their mathematical skills beyond Year 10 but whose future studies or employment pathways do not require calculus.

Students will develop a mathematical mindset that can be used to solve complex problems in algebra, geometry, statistics, networks and matrices.

## What do you learn in Grade 11 General Mathematics?

### Money, Measurement and Relations

Question? More specifically, students will:

##### Consumer Arithmetic
Rates, Percentages and Spreadsheets
• calculate weekly or monthly wages from an annual salary, and wages from an hourly rate, including situations involving overtime and other allowances and earnings based on commission or piecework
• calculate payments based on government allowances and pensions, such as youth allowances, unemployment, disability and study
• prepare a personal budget for a given income, taking into account fixed and discretionary spending
• compare prices and values using the unit cost method
• determine the impact of inflation on costs and wages over time
• calculate percentage mark-ups and discounts
• calculate GST
• calculating profit or loss in absolute and percentage terms
• calculate simple and compound interest
• use currency exchange rates
• calculate the dividend paid on a portfolio of shares,
• compare share values by calculating a price-to-earnings ratio
• prepare 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
• investigate the potential cost of owning and operating a car over a year
##### Shape and Measurement
Pythagoras’ Theorem
• solve practical problems in two dimensions
• simple applications in three dimensions
Mensuration
• calculate perimeters and areas of circles, sectors of circles, triangles, rectangles, trapeziums, parallelograms and composites
• calculate volumes and capacities of standard three-dimensional objects, including spheres, rectangular prisms, cylinders, cones, pyramids and composites
• calculate volume of water contained in a swimming pool
• calculate surface areas of standard three-dimensional objects, e.g. spheres, rectangular prisms, cylinders, cones, pyramids and composites
• calculate surface area of a cylindrical food container
Similar Figures and Scale Factors
• review the conditions for similarity of two-dimensional figures
• 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
• use shadow sticks, tree height and clinometers to obtain scale factor and use it to solve scaling problems
• calculate the surface areas and volumes of similar solids
##### Linear Equations and Graphs
Linear Equations
• identify and solve linear equations, including variables on both sides, fractions, non-integer solutions
• develop a linear equation from a description in words
Straight Line Graphs
• solve a pair of simultaneous linear equations in the format y = mx + c
• solve equations algebraically, graphically, by substitution and by the elimination method
• find the point of intersection of two straight-line graphs
• determine the break-even point where cost and revenue are represented by linear equations
Piece-Wise Linear Graphs and Step Graphs
• sketch piece-wise linear graphs and step graphs, using technology
• interpret piece-wise linear and step graphs used to model practical situations

### Applied Trigonometry, Algebra, Matrices and Univariate Data

Question? More specifically, students will:

##### Applications of Trigonometry
Trigonometry
• find the length of an unknown side or the size of an unknown angle in a right-angled triangle using trigonometric ratios
• determine the area of a triangle given two sides and an angle
• determine the area of a triangle given three sides
• solve two-dimensional problems using the sine rule and cosine rule
• solve problems involving angles of elevation and depression and the use of true bearings
##### Algebra and Matrices
Linear and Non-Linear Relationships
• substitute numerical values into expressions and evaluate
• order two polynomials, proportional, inversely proportional
• find the value of the subject of the formula, given the values of the other pronumerals
• transpose linear equations and simple non-linear algebraic equations
• use technology to construct a table of values from a formula
• construct a table displaying the body mass index (BMI) of people with different weights and heights
Matrices and Matrix Arithmetic
• use matrices for storing and displaying information
• recognise row matrix, column matrix (or vector matrix), square matrix, zero matrix, identity matrix
• determine the size of the matrix
• perform matrix addition, subtraction, and multiplication by a scalar
• perform matrix multiplication (manually up to a 3 x 3)
• determine the power of a matrix using technology
• use matrix products and powers of matrices to model and solve pricing problems
• square a matrix to determine the number of ways pairs of people in a communication network can communicate with each other via a third person
##### Univariate Data Analysis
Single Statistical Variables
• classify statistical variables as categorical or numerical
• classify a categorical variable as ordinal or nominal
• use tables and pie, bar and column charts to organise and display data
• classify a numerical variable as discrete or continuous
• construct and justify the use of dot plots, stem-and-leaf plots, column charts and  histograms to graphically display data
• describe number of modes, shape (symmetric versus positively or negatively skewed), measures of centre and spread, and outliers of graphical displays
• determine the mean and standard deviation of a dataset
• use statistics as measures of location and spread of a data distribution
• explain significance of standard deviation
Numerical Variables Across Groups
• construct and use parallel box plots to compare datasets in terms of median, spread (IQR and range) and outliers
• interpret and communicate the differences observed between datasets
• use IQR to identify possible outliers
• compare datasets using medians, means, IQRs, ranges or standard deviations
• interpret the differences observed in the context of the data and report the findings in a systematic and concise manner

## What do you learn in Grade 12 General Mathematics?

### Bivariate Data, Sequences and Change, and Earth Geometry

Question? More specifically, students will:

##### Bivariate Data Analysis
Associations Between Two Categorical Variables
• construct two-way frequency tables and determine row and column sums and percentages
• use a two-way frequency table to identify patterns that suggest the presence of an association
• understand an association in terms of differences observed in percentages across categories
• interpret this in the context of the data
• construct scatterplots to identify the presence of an association
• understand an association between two numerical variables in terms of direction (positive/negative), form (linear) and strength (strong/moderate/weak)
• calculate and interpret the correlation coefficient (r) to quantify the strength of a linear association using Pearson’s correlation coefficient
Fitting a Linear Model to Numerical Data
• 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
• recognise the potential dangers of extrapolation
Association and Causation
• distinguish observed associations and causal relationships
• communicate possible non-causal explanations for an association (coincidence and confounding due to a common response to another variable)
• analyse associations between two categorical variables or between two numerical variables
##### Time Series Analysis
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)
• recognise when there are outliers or one-off unanticipated events
Analysing Time Series Data
• smooth time series data by using a simple moving average
• calculate seasonal indices by using the average percentage method
• deseasonalise a time series by using a seasonal index
• fit a least-squares line to model long-term trends in time series data
• solve practical problems that involve the analysis of time series data
##### Growth and Decay in Sequences
Arithmetic Sequence
• use recursion to generate an arithmetic sequence
• display the terms of an arithmetic sequence in both tabular and graphical form
• demonstrate that arithmetic sequences can be used to model linear growth and decay in discrete situations
• analyse a simple interest loan or investment using arithmetic sequences
• calculate a taxi fare based on the flag fall and the charge per kilometre using arithmetic sequences
• calculate the value of an office photocopier at the end of each year using the straight-line method or the unit cost method of depreciation
Geometric Sequence
• use recursion to generate a geometric sequence
• display the terms of a geometric sequence in both tabular and graphical form
• demonstrate that geometric sequences can be used to model exponential growth and decay in discrete situations
• use geometric sequences to model and analyse practical problems involving geometric growth and decay , such as
• analyse a compound interest loan or investment using geometric sequences
• analyse the growth of a bacterial population that doubles in size each hour using geometric sequences
• he decreasing height of the bounce of a ball at each bounce using geometric sequences
• calculating the value of office furniture at the end of each year using the declining (reducing) balance method to depreciate
##### Earth Geometry and Time Zones
Locations on the Earth
• define the meaning of great circles
• define the meaning of angles of latitude and longitude in relation to the equator and the prime meridian
• locate positions on Earth’s surface given latitude and longitude
• state latitude and longitude for positions on Earth’s surface and world maps
• use a local area map to state the position of a given place in degrees and minutes
• investigate the map of Australia and locate boundary positions for Aboriginal language groups
• calculate angular distance (in degrees and minutes) and distance (in kilometres) between two places on Earth on the same meridian
• calculate angular distance (in degrees and minutes) and distance (in kilometres) between two places on Earth on the same parallel of latitude
• calculate distances between two places on Earth
Time Zones
• define Greenwich Mean Time (GMT), International Date Line and Coordinated Universal Time (UTC)
• understand the link between longitude and time
• determine the number of degrees of longitude for a time difference of one hour
• solve problems involving time zones in Australia and in neighbouring nations, making any necessary allowances for daylight saving
• solve problems involving GMT, International Date Line and UTC
• calculate time differences between two places on Earth
• solve problems associated with time zones, such as online purchasing, making phone calls overseas and broadcasting international events
• solve problems relating to travelling east and west incorporating time zone changes, such as preparing an itinerary for an overseas holiday with corresponding times

### Investing and Networking

Question? More specifically, students will:

##### Loans, Investments and Annuities
Compound Interest Loans and Investments
• model a compound interest loan or investment
• investigate (numerically and graphically) the effect of the interest rate and the number of compounding periods on the future value of the loan or investment
• 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, etc
• determine the future value of a loan
• determine the number of compounding periods for an investment to exceed a given value
• determine the interest rate needed for an investment to exceed a given value
Reducing Balance Loans
• use 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 the loan
• determine the monthly repayments required to pay off a housing loan
Annuities and Perpetuities
• use a recurrence relation to model an annuity
• investigate the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity
• solve problems involving annuities
• determine the amount to be invested in an annuity to provide a regular monthly income of a certain amount
##### Graphs and Networks
Graphs and the Adjacency Matrix
• explain 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
• construct networks for trails connecting camp sites in a national park, social networks, transport networks with one-way streets, a food web, the results of a round-robin sporting competition
• construct an adjacency matrix from a given graph or digraph
Planar Graphs, Paths and Cycles
• define planar graph and face
• apply Euler’s formula to solve planar graph problems
• understand the meaning of the terms walk, trail, path, closed walk, closed trail, cycle, connected graph and bridge
• determine the shortest path between two vertices in a weighted graph
• understand the meaning of the terms Eulerian graph, Eulerian trail, semi-Eulerian graph, semi-Eulerian trail and the conditions for their existence
• use these concepts to investigate and solve the Königsberg bridge problem and planning a garbage bin collection route
• understand the meaning of the terms Hamiltonian graph and semi-Hamiltonian graph
• plan a sightseeing tourist route around a city
##### Networks and Decision Mathematics
Trees and Minimum Connector Problems
• understand the meaning of the terms tree and spanning tree
• identify a minimum spanning tree in a weighted connected graph using Prim’s algorithm
• use minimal spanning trees to solve minimal connector problems,
• minimise the length of cable needed to provide power from a single power station to substations in several towns

## How do you achieve an A+ in General Mathematics? Assessment Criteria

The key to high grades in any subject is strict adherence to the task and its criteria. To achieve an A+ level result for assessment in General Mathematics, students must:

• demonstrate a comprehensive knowledge and understanding of the subject matter
• recognise, recall and use facts, rules, definitions and procedures
• comprehend and apply mathematical concepts and techniques to solve problems drawn from:
• Number and algebra
• Measurement and geometry
• Statistics
• Networks and matrices in simple familiar, complex familiar and complex unfamiliar situations
• explain mathematical reasoning to justify procedures and decisions
• evaluate the reasonableness of solutions
• communicate using mathematical, statistical and everyday language and conventions
• uses technology to solve problems in simple familiar, complex familiar and complex unfamiliar situations