General Mathematics syllabus, dot point by dot point

Summarise the distribution of a numerical variable with the mean and sample standard deviation or the median, interquartile range and range, choose measures suited to the shape and outliers, and use them to compare distributions in terms of centre and spread

How do you summarise the distribution of a numerical variable with measures of centre and spread, choose the right measures, and use them to compare distributions?

Construct and interpret frequency tables and two-way (contingency) frequency tables, percentage them by the explanatory variable, display them with percentage segmented bar charts, and use them to identify and describe associations between two categorical variables

How do you use frequency tables, two-way tables and percentage segmented bar charts to identify and describe an association between two categorical variables?

Model a simple perpetuity as a special case of an annuity that lasts indefinitely, in which the regular payment equals the interest earned each period so the balance never changes, and find the payment, the principal, the interest rate or the recurrence relation

What is a perpetuity, how does the balance stay constant when the payment exactly equals the interest, and how do you find the payment or the principal?

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), and interpretation and use of the equation of the least squares line fitted to the transformed data to make predictions

When a scatterplot is curved, how do the squared, log and reciprocal transformations straighten the data so a least-squares line can be fitted?

Seasonal adjustment including the use and interpretation of seasonal indices and their calculation using seasonal and yearly means, and modelling trend by fitting a least squares line to a time series with time as the explanatory variable (data de-seasonalised where necessary), with forecasts re-seasonalised where necessary

How do seasonal indices quantify a repeating pattern, and how do you deseasonalise and reseasonalise time series data?

Use a logarithmic (base 10) scale to display data ranging over several orders of magnitude, and interpret the display in terms of powers of ten

Why do we display data that ranges over several orders of magnitude on a logarithmic (base 10) scale, and how do we read and interpret such a display?

Use the normal model for bell-shaped distributions and the 68-95-99.7% rule to estimate percentages and give meaning to the standard deviation, and use standardised values (z-scores) to compare data values across distributions

How does the normal distribution describe data, and how do the 68-95-99.7 rule and standardised z-scores let you compare and find percentages?

Calculate residuals, construct a residual plot, and use residual analysis to test the assumption of linearity and check the quality of fit of a least squares line

What is a residual, how do we construct and read a residual plot, and how does a residual analysis tell us whether a linear model is appropriate?

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; the difference between nominal and effective interest rates and the use of effective interest rates to compare investment returns and the cost of loans; and the future value of a compound interest investment or loan after n compounding periods

How do simple and compound interest differ, how does each follow a recurrence relation, and how do you compare them with effective rates?

Qualitative features of time series plots (trend, seasonality, irregular fluctuations, possible outliers and structural change); numerical smoothing using moving means, with centring when appropriate, and graphical smoothing using moving medians over an odd number of points; and modelling trend by fitting a least squares line with time as the explanatory variable to make forecasts, considering the limitations of a linear model and of extending into the future

How do you display and describe a time series, and how do moving-mean and moving-median smoothing reveal the underlying trend?

Summary of the distributions of numerical variables: the five-number summary and boxplots (including the use of the lower fence (Q1 - 1.5 x IQR) and upper fence (Q3 + 1.5 x IQR) to identify and display possible outliers), and the use of parallel boxplots to compare distributions

How do you build a boxplot from a five-number summary, test for outliers, and compare two groups using parallel boxplots?

12 min read →

How is a reducing balance loan modelled with a recurrence relation and an amortisation table, and how do we use a finance solver to analyse repayments, interest and the effect of changing the rate or repayment?

Use a first-order linear recurrence relation and an amortisation table to model and analyse a reducing balance loan, and use technology to find the repayment, balance, term, final payment and total interest, including the impact of a change in interest rate or repayment

18 min read →

How do flat rate, unit cost and reducing balance depreciation reduce the value of an asset, and how are they modelled with a recurrence relation and a rule?

Use a first-order linear recurrence relation to model and compare flat rate, unit cost and reducing balance depreciation of an asset, and use the rule for the value of the asset after n depreciation periods

18 min read →

How do back-to-back stem plots, parallel dot plots and parallel boxplots show whether a numerical variable is associated with a categorical variable?

Use back-to-back stem plots, parallel dot plots and parallel boxplots to identify and describe associations between a numerical variable and a categorical variable, comparing centre, spread, shape and outliers

Scatterplots and their use in describing the association between two numerical variables in terms of direction, form and strength; the Pearson correlation coefficient r; the least squares line y = a + bx, with a and b found by technology or from b = r(s_y/s_x) and a = y-bar - b(x-bar); interpretation of the slope and intercept; prediction, being aware of the limitations of extrapolation; and the coefficient of determination r squared in terms of explained variation

How do you measure the association between two numerical variables and fit a least-squares line to model and predict?

Use a first-order linear recurrence relation and an amortisation table to model and analyse an annuity, and use technology to find the amount to invest, the regular payment, the number of payments or the balance

How does an annuity provide a regular income from a lump sum, and how do we model and analyse its balance with a recurrence relation, an amortisation table and a finance solver?

18 min read →

How do first-order linear recurrence relations generate sequences, and how do we recognise arithmetic, geometric and other patterns of growth and decay?

Use a first-order linear recurrence relation of the form u(0) = a, u(n+1) = R u(n) + d to generate the terms of a sequence, recognise arithmetic and geometric sequences, use the explicit rules, and describe the behaviour of the sequence numerically and graphically

Types of data (categorical nominal and ordinal, numerical discrete and continuous), frequency tables, bar charts, segmented bar charts, dot plots, stem plots and histograms, and describing and interpreting the distribution of a categorical or numerical variable to answer statistical questions

How do you classify data, choose the right display, and describe the shape, centre and spread of a distribution?

Compound interest investment with periodic and equal additions to the principal: use of a first-order linear recurrence relation to model and analyse (numerically and graphically) annuity investment, use of a table to analyse its growth step by step, and use of technology with financial modelling functionality to find the future value, 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

How do annuity investments and savings plans grow when regular payments are added to a compounding balance, and how is the finance solver set up for them?

Recognise and use a permutation matrix as a binary matrix with exactly one 1 in each row and column, apply it to reorder the entries of a state matrix, and identify the matrix that reverses or repeats the reordering

Unit 4 Matrices

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How does a permutation matrix rearrange the rows of a state matrix, and how are binary matrices used to encode and reorder information?

Identify trees and spanning trees, determine a minimum spanning tree of a weighted connected graph by inspection or by Prim's algorithm (from a network or a table), and use minimum spanning trees to solve minimal connector problems

What is a tree and a spanning tree, and how are minimum spanning trees found by inspection or by Prim's algorithm to solve minimal connector problems?

Construct a transition matrix from a transition diagram or written description, use the matrix recurrence relation S(n+1) = TS(n) to generate a sequence of state matrices, informally identify the equilibrium state of a regular transition matrix, and use S(n+1) = TS(n) + B to model populations with culling and restocking

How do transition diagrams and transition matrices model step-by-step change in a system, and how are future, past and equilibrium states found, including with culling and restocking?

Distinguish walks, trails, paths, cycles, Eulerian trails and circuits, and Hamiltonian paths and cycles, and use the number of odd-degree vertices to decide whether an Eulerian trail or circuit exists in a connected graph

What is the difference between Eulerian and Hamiltonian travel, and how do vertex degrees decide whether an Eulerian trail or circuit exists?

Model a directed capacitated network with a source and sink, find the maximum flow from source to sink, identify cuts and their capacities, and use the minimum cut to confirm the maximum flow

How do you find the greatest flow a network can carry from source to sink, and how does the minimum cut prove the maximum flow?

Construct an activity network from a precedence table, including dummy activities where necessary, use forward and backward scanning to find earliest and latest start times, identify the critical path and float times, and use crashing to reduce the completion time of a project

How is a project scheduled with critical path analysis, including dummy activities, float and crashing, and how does this connect to flow and allocation problems?

Set up a Leslie matrix from age-specific birth (fecundity) and survival rates, multiply it by an age-structure state matrix to project a population forward, and interpret the long-term growth and age distribution

How does a Leslie matrix model an age-structured population, and how do birth and survival rates project the population forward?

Determine the shortest path between two specified vertices in a graph, digraph or network by inspection, and use Dijkstra's algorithm to find the shortest path between a given vertex and each of the other vertices

How do we find the shortest path between two vertices of a weighted graph, and how does Dijkstra's algorithm find the shortest path from one vertex to every other vertex?

Build a binary communication or dominance matrix, square it to count two-step links, add the matrix and its square to combine one-step and two-step connections, and rank competitors by their total

How do communication and dominance matrices use a matrix of zeros and ones to count direct and indirect links, and how do you rank competitors?

Perform elementary matrix operations (sum, difference, multiplication by a scalar, product and power), interpret them in context, and find the determinant and inverse of a matrix, including the condition for a matrix to have an inverse

How are matrices added, subtracted, multiplied and raised to powers, what do the elements of a product mean in context, and when does a matrix have an inverse?

Use the order of a matrix, element notation and the types of matrices (row, column, square, diagonal, symmetric, triangular, zero, binary, permutation and identity) and the transpose, represent tabular information as a matrix, and construct a matrix from a rule for its ij element

What are the order, elements and types of a matrix, and how do we use matrices to store tabular information and construct a matrix from a rule for its elements?

Use graph and network terminology (vertices, edges, degree, connected, planar), apply Euler's formula and the handshake result, find a minimum spanning tree, and determine the shortest path through a weighted network

What are the key terms of graph theory, and how are spanning trees and shortest paths found in a weighted network?

Use a bipartite graph and its tabular or matrix form to represent a matching problem, and determine the optimum assignment of people or machines to tasks by inspection or by the Hungarian algorithm for larger problems

How is an allocation problem represented by a bipartite graph or cost table, and how does the Hungarian algorithm find the optimum assignment of people or machines to tasks?