Further Mathematics consists of two areas of study, a compulsory Core area of study to be completed in Unit 3 and an Applications area of study to be completed in Unit 4. The Core comprises “Data analysis” and “Recursion and financial modelling”. The Applications comprises two modules to be completed in their entirety, from a selection of four possible modules: “Matrices”, “Networks and decision mathematics”, “Geometry and measurement” and “Graphs and relations”. “Data analysis” comprises 40% of the content to be covered, “Recursion and financial modelling” comprises 20% of the content to be covered, and each selected module comprises 20% of the content to be covered. Assumed knowledge and skills for the Core are contained in the General Mathematics Units 1 and 2 topics: “Computation and practical arithmetic”, “Investigating and comparing data distributions”, “Investigating relationships between two numerical variables”, “Linear graphs and modelling”, “Linear relations and equations”, and “Number patterns and recursion”. For each module, there are related topics in General Mathematics Units 1 and 2.

In undertaking these units, students are expected to be able to apply techniques, routines, and processes involving rational and real arithmetic, sets, lists and tables, diagrams and geometric constructions, algebraic manipulation, equations, and graphs. They should have a facility with relevant mental and by-hand approaches to estimation and computation. The use of numerical, graphical, geometric, symbolic, financial and statistical functionality of technology for teaching and learning mathematics, for working mathematically, and in related assessment, is to be incorporated throughout each unit as applicable.

OUTCOMES – Unit 3

For this unit the student is required to demonstrate achievement of three outcomes. As a set these outcomes encompass Area of Study 1.

Data analysis

Key skills

construct frequency tables and bar charts and use them to describe and interpret the distributions of categorical variables
answer statistical questions that require a knowledge of the distribution/s of one or more categorical variables
construct stem and dot plots, boxplots, histograms and appropriate summary statistics and use them to describe and interpret the distributions of numerical variables
answer statistical questions that require a knowledge of the distribution/s of one or more numerical variables
solve problems using the z-scores and the 68–95–99.7% rule
construct two-way tables and use them to identify and describe associations between two categorical variables
construct parallel boxplots and use them to identify and describe associations between a numerical variable and a categorical variable
construct scatterplots and use them to identify and describe associations between two numerical variables
calculate the correlation coefficient, r, and interpret it in the context of the data
answer statistical questions that require a knowledge of the associations between pairs of variables
determine the equation of the least-squares line giving the coefficients correct to a required number of decimal places or significant figures as specified
distinguish between correlation and causation
use the least squares line of best fit to model and analyse the linear association between two numerical variables and interpret the model in the context of the association being modelled
calculate the coefficient of determination, r2, and interpret in the context of the association being modelled and use the model to make predictions, being aware of the problem of extrapolation
construct a residual analysis to test the assumption of linearity and, in the case of clear non-linearity, transform the data to achieve linearity and repeat the modelling process using the transformed data
identify key qualitative features of a time series plot including trend (using smoothing if necessary), seasonality, irregular fluctuations and outliers, and interpret these in the context of the data
calculate, interpret and apply seasonal indices
model linear trends using the least squares line of best fit, interpret the model in the context of the trend being modelled, use the model to make forecasts being aware of the limitations of extending forecasts too far into the future.

Recursion and financial modelling

Key skills

use a given first-order linear recurrence relation to generate the terms of a sequence
model and analyse growth and decay in financial contexts using a first-order linear recurrence relation
demonstrate the use of a recurrence relation to determine the depreciating value of an asset or the future value of an investment or a loan after n time periods, including from first principles for n ≤ 5
use a rule for the future value of a compound interest investment or loan, or a depreciating asset, to solve practical problems
use a table to investigate and analyse on a step-by-step basis the amortisation of a reducing balance loan or an annuity, and interpret amortisation tables
with the aid of technology with financial mathematics capabilities, solve practical problems associated with compound interest investments and loans, reducing balance loans, annuities and perpetuities, and annuity investments.

OUTCOMES – Unit 4

For this unit the student is required to demonstrate achievement of three outcomes. As a set these outcomes encompass the two selected modules from Area of Study 2, Applications.

Matrices

Key skills

use the matrix recurrence relation: S0 = initial state matrix, Sn+1 = TSn to generate a sequence of state matrices, including an informal identification of the equilibrium or steady state matrix in the case of regular state matrices
construct a transition matrix from a transition diagram or a written description and vice versa
construct a transition matrix to model the transitions in a population with an equilibrium state
use the matrix recurrence relation S0 = initial state matrix, Sn+1 = TSn + B to extend the modelling to populations that include culling and restocking.

Networks and decision mathematics

Key skills

construct graphs, digraphs and networks and their matrix equivalents to model and analyse practical situations
recognise that a problem is an example of the exploring and travelling problem and to solve it by utilising the concepts of walks, trails, paths, eulerian trails and circuits, and hamiltonian paths and cycles
recognise that a problem is an example of the minimum connector problem and solve it by utilising the properties of trees, spanning trees and by determining a minimum spanning tree by inspection or using Prim’s algorithm for larger-scale problems
recognise that a problem is an example of the flow problem, use networks to model flow problems and
determine the minimum flow problem by inspection, or by using the minimum cut/maximum flow theorem for larger-scale problems
recognise that a problem is an example of the shortest path problem and solve it by inspection or using Dijkstra’s algorithm for larger-scale problems
recognise that a problem is an example of the matching problem and solve it by inspection or using the Hungarian algorithm for larger-scale problems
recognise that a problem is an example of the scheduling problem and solve it by using critical path analysis.

Geometry and measurement

Key skills

solve practical problems involving the calculation of the side lengths, angles and areas of triangles, including the construction of diagrams based on word descriptions
solve practical problems involving the calculation of the surface area and volume of spheres, cylinders, prisms and their composites
solve practical problems involving the use of a linear scale factor to scale lengths, areas and volumes of similar figures and shapes
use a sphere of radius 6400 km as a model of the earth to solve practical problems involving meridians and parallels and latitude and longitude (specified in decimal degrees)
use the concept of a great circle to find the shortest distance between two points on the earth’s surface with the same longitude
solve time zone problems.

Graphs and relations

Key skills

construct and interpret straight-line graphs, line segment graphs and step graphs used to model practical situations
construct from a table of values and interpret non-linear graphs used to model practical situations
solve practical problems involving finding the point of intersection of two straight-line graphs
solve graphically practical problems involving finding the point of intersection of a linear graph with a non-linear graph
interpret graphs used to model situations involving two independent variables
graph linear inequalities in one or two variables and interpret them when used in practical situations
formulate a linear programming problem with two decision variables and solve graphically
extend the linear programming method of solution to include only integer solutions where required.