General Mathematics Study Notes

Area of Study 1 – Data analysis, probability and statistics. Students cover data types, representation and distribution of data, location, spread, association, correlation and causation, response and explanatory variables, linear regression, data transformation and goodness of fit, time series, seasonality, smoothing and prediction. Investigating data distributions. This topic includes: types of data; representation, display and description of the distributions of categorical variables: data tables, two-way frequency tables and their associated segmented bar charts; representation, display and description of the distributions of numerical variables: dot plots, stem plots, histograms; the use of a logarithmic (base 10) scale to display data ranging over several orders of magnitude and their interpretation in terms of powers of ten; use of the distribution(s) of one or more categorical or numerical variables to answer statistical questions; summary of the distributions of numerical variables; the five-number summary and boxplots (including the use of the lower fence (Q1 − 1.5 × IQR) and upper fence (Q3 + 1.5 × IQR) to identify and display possible outliers); the sample mean and standard deviation and their use in comparing data distributions in terms of centre and spread; the normal model for bell-shaped distributions and the use of the 68–95–99.7% rule to estimate percentages and to give meaning to the standard deviation; standardised values (z-scores) and their use in comparing data values across distributions. Investigating association between two variables. This topic includes: response and explanatory variables and their role in investigating associations between variables; contingency (two-way) frequency tables, their associated bar charts (including percentage segmented bar charts) and their use in identifying and describing associations between two categorical variables; back-to-back stem plots, parallel dot plots and boxplots and their use in identifying and describing associations between a numerical variable and a categorical variable; scatterplots and their use in identifying and qualitatively describing the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak); answering statistical questions that require a knowledge of the associations between pairs of variables; Pearson correlation coefficient, r, and its calculation and interpretation; cause and effect; the difference between observation and experimentation when collecting data and the need for experimentation to definitively determine cause and effect. Investigating and modelling linear associations. This topic includes: least squares line of best fit y = a + bx, where x represents the explanatory variable, and y represents the response variable; the determination of the coefficients a and b using technology, and the formulas b = r × (s_y / s_x) and a = ȳ − b x̄; modelling linear association between two numerical variables, including the identification of the explanatory and response variables, use of the least squares method to fit a linear model to the data, interpretation of the slope and intercepts of the least squares line in the context of the situation being modelled, use of the rule of the fitted line to make predictions being aware of the limitations of extrapolation, use of the coefficient of determination, r², to assess the strength of the association in terms of explained variation, and use of residual analysis to check quality of fit; data transformation and its use in transforming some forms of non-linear data to linearity using a square, logarithmic (base 10) or reciprocal transformation (applied to one axis only); interpretation and use of the equation of the least squares line fitted to the transformed data to make predictions. Investigating and modelling time series data. This topic includes: qualitative features of time series plots; recognition of features such as trend (long-term direction), seasonality (systematic, calendar related movements) and irregular fluctuations (unsystematic, short-term fluctuations); possible outliers and their sources, including one-off real-world events, and signs of structural change such as a discontinuity in the time series; numerical smoothing of time series data using moving means with consideration of the number of terms required (using centring when appropriate) to help identify trends in time series plot with large fluctuations; graphical smoothing of time series plots using moving medians (involving an odd number of points only) to help identify long-term trends in time series with large fluctuations; seasonal adjustment including the use and interpretation of seasonal indices and their calculation using seasonal and yearly means; modelling trend by fitting a least squares line to a time series with time as the explanatory variable (data de-seasonalised where necessary), and the use of the model to make forecasts (with re-seasonalisation where necessary) including consideration of the possible limitations of fitting a linear model and the limitations of extending into the future.

Area of Study 2 – Discrete mathematics. Students cover the use of first-order linear recurrence relations and the time value of money (TVM) to model and analyse a range of financial situations, and using technology to solve related problems involving interest, appreciation and depreciation, loans, annuities and perpetuities. Depreciation of assets. This topic includes: use of a first-order linear recurrence relation of the form u_0 = a, u_(n+1) = R × u_n + d, where a, R and d are constants, to generate the terms of a sequence; use of a recurrence relation to model and compare (numerically and graphically) flat rate, unit cost and reducing balance depreciation of the value of an asset with time, including the use of a recurrence relation to determine the depreciating value of an asset after n depreciation periods for the initial sequence; use of the rules for the future value of an asset after n depreciation periods for flat rate, unit cost and reducing balance depreciation and their application. Compound interest investments and loans. This topic includes: the concepts of simple and compound interest; use of a recurrence relation to model and analyse (numerically and graphically) a compound interest investment or loan, including the use of a recurrence relation to determine the value of the compound interest loan or investment after n compounding periods for an initial sequence from first principles; the difference between nominal and effective interest rates and the use of effective interest rates to compare investment returns and the cost of loans when interest is paid or charged, for example, daily, monthly, quarterly; the future value of a compound interest investment or loan after n compounding periods and its use to solve practical problems. Reducing balance loans. This topic includes: use of a first-order linear recurrence relation to model and analyse (numerically and graphically) the amortisation of a reducing balance loan, including the use of a recurrence relation to determine the value of the loan or investment after n payments for an initial sequence from first principles; use of a table to investigate and analyse the amortisation of a reducing balance loan on a step-by-step basis, the payment made, the amount of interest paid, the reduction in the principal and the balance of the loan; use of technology with financial modelling functionality to solve problems involving reducing balance loans, such as repaying a personal loan or a mortgage, including the impact of a change in interest rate on repayment amount, time to repay the loan, total interest paid and the total cost of the loan. Annuities and perpetuities. This topic includes: use of a first-order linear recurrence relation to model and analyse (numerically and graphically) the amortisation of an annuity, including the use of a recurrence relation to determine the value of the annuity after n payments for an initial sequence from first principles; use of a table to investigate and analyse the amortisation of an annuity on a step-by-step basis, the payment made, the interest earned, the reduction in the principal and the balance of the annuity; use of technology to solve problems involving annuities including determining the amount to be invested in an annuity to provide a regular income paid, for example, monthly, quarterly; simple perpetuity as a special case of an annuity that lasts indefinitely. Compound interest investment with periodic and equal additions to the principal. This topic includes: use of a first-order linear recurrence relation to model and analyse (numerically and graphically) annuity investment, including the use of a recurrence relation to determine the value of the investment after n payments have been made for an initial sequence from first principles; use of a table to investigate and analyse the growth of an annuity investment on a step-by-step basis after each payment is made, the payment made, the interest earned and the balance of the investment; use of technology with financial modelling functionality to solve problems involving annuity investments, including determining the future value of an investment after a number of compounding periods, the number of compounding periods for the investment to exceed a given value and the interest rate or payment amount needed for an investment to exceed a given value in a given time.

Area of Study 2 – Discrete mathematics. Students cover the definition of matrices, different types of matrices, matrix operations, transition matrices and the use of first-order linear matrix recurrence relations to model a range of situations and solve related problems. Matrices and their applications. This topic includes: matrix arithmetic: the order of a matrix, types of matrices (row, column, square, diagonal, symmetric, triangular, zero, binary and identity), the transpose of a matrix, and elementary matrix operations (sum, difference, multiplication of a scalar, product and power); inverse of a matrix, its determinant, and the condition for a matrix to have an inverse; use of matrices to represent numerical information presented in tabular form, and the use of a rule for the a_ij th element of a matrix to construct the matrix; binary and permutation matrices, and their properties and applications; communication and dominance matrices and their use in analysing communication systems and ranking players in round-robin tournaments. Transition matrices. This topic includes: use of the matrix recurrence relation S_0 = initial state matrix, S_(n+1) = T × S_n or S_(n+1) = L × S_n, where T is a transition matrix, L is a Leslie matrix, and S_n is a column state matrix, to generate a sequence of state matrices (assuming the next state only relies on the current state); informal identification of the equilibrium state matrix in the case of regular transition matrices (no noticeable change from one state matrix to the next state matrix); use of transition diagrams, their associated transition matrices and state matrices to model the transitions between states in discrete dynamical situations and their application to model and analyse practical situations such as the modelling and analysis of an insect population comprising eggs, juveniles and adults; use of the matrix recurrence relation S_0 = initial state matrix, S_(n+1) = T × S_n + B to extend modelling to populations that include culling and restocking.

Area of Study 2 – Discrete mathematics. Students cover the definition and representation of different kinds of undirected and directed graphs, Eulerian trails, Eulerian circuits, bridges, Hamiltonian paths and cycles, and the use of networks to model and solve problems involving travel, connection, flow, matching, allocation and scheduling. Graphs and networks. This topic includes: the concepts, conventions and terminology of graphs including planar graphs and Euler's rule, and directed (digraphs) and networks; use of matrices to represent graphs, digraphs and networks and their application. Exploring and travelling problems. This topic includes: the concepts, conventions and notations of walks, trails, paths, cycles and circuits; Eulerian trails and Eulerian circuits: the conditions for a graph to have a Eulerian trail or a Eulerian circuit, properties and applications; Hamiltonian paths and cycles: properties and applications. Trees and minimum connector problems. This topic includes: trees and spanning trees; minimum spanning trees in a weighed connected graph and their determination by inspection or by Prim's algorithm; use of minimal spanning trees to solve minimal connector problems. Flow problems. This topic includes: use of networks to model flow problems: capacity, sinks and sources; solution of small-scale network flow problems by inspection and the use of the 'maximum-flow minimum-cut' theorem to aid the solution of larger scale problems. Shortest path problems. This topic includes: determination of the shortest path between two specified vertices in a graph, digraph or network by inspection; Dijkstra's algorithm and its use to determine the shortest path between a given vertex and each of the other vertices in a weighted graph or network. Matching problems. This topic includes: use of a bipartite graph and its tabular or matrix form to represent a matching problem; determination of the optimum assignment(s) of people or machines to tasks by inspection or by use of the Hungarian algorithm for larger scale problems. Scheduling problems and critical path analysis. This topic includes: construction of an activity network from a precedence table (or equivalent) including the use of dummy activities where necessary; use of forward and backward scanning to determine the earliest starting times (EST) and latest starting times (LST) for each activity; use of earliest starting times and latest starting times to identify the critical path in the network and determine the float times for non-critical activities; use of crashing to reduce the completion time of the project or task being modelled.