General Mathematics Unit 3 Subject Matter

General Mathematics Unit 3: Bivariate Data and Time Series Analysis, Sequences and Earth Geometry

GM Unit 3 Bivariate Data Analysis 1

Sub-topic: Identifying and Describing Associations between Two Categorical Variables (5 hours)

• Understand the meaning of bivariate data.

• Draw two-way frequency tables and Determine the associated row and column sums and percentages.

• Use an appropriately percentaged 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 in a systematic and concise manner, and Interpret this in the context of the data.

Sub-topic: Identifying and Describing Associations between Two Numerical Variables (6 hours)

• Identify the explanatory variable and the response variable.

• Construct and use a scatterplot to identify the association between two numerical variables.

• Describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak).

• Calculate Pearson’s correlation coefficient, , from raw data Use technology, and interpret it to quantify the strength of a linear association.

• Calculate the coefficient of determination, , from raw data using technology, and interpret it to assess the strength of a linear association in terms of the explained variation.

• Use the correlation coefficient, , to determine the coefficient of determination, , and vice versa.

GM Unit 3 Topic 2: Bivariate Data Analysis 2

Sub-topic: Fitting a Linear Model to Numerical Data (6 hours)

• Model a linear relationship by using technology to fit a least-squares line to the data, in the form of where is slope (gradient) and is -intercept.

• Understand and use and to determine the equation of a least-squares line, where is slope (gradient), is correlation coefficient, is (sample) standard deviation of values, is (sample) standard deviation of values, is -intercept, is mean of values and is mean of values.

• Construct a residual plot and use it to assess the appropriateness of fitting a linear model to the data.

• Interpret the -intercept and slope (gradient) of the fitted line.

• Distinguish between interpolation and extrapolation.

• Use the equation of the least-squares line to make predictions.

• Recognise and Explain the potential dangers of extrapolation.

Sub-topic: Association and Causation (6 hours)

• Recognise and explain that an observed association between two variables (categorical and/or numerical) does not necessarily mean that there is a causal relationship between them.

• Identify and communicate possible non-causal explanations for an association, including coincidence or the influence of another variable.

• Solve practical problems by identifying, analysing and describing associations between two variables (categorical and/or numerical).

GM Unit 3 Topic 3: Time Series Analysis

Sub-topic: Describing and Interpreting Patterns in Time Series Data (3 hours)

• Construct and use time series plots.

• Describe time series plots by identifying features, including trend (long-term direction, e.g. increasing/decreasing), seasonality (systematic, calendar-related movements) and irregular fluctuations (unsystematic, short-term fluctuations).

Sub-topic: Analysing Time Series Data (8 hours)

• Smooth time series data by calculating a simple moving average using the mean or median for an odd number of data, including the use of spreadsheets.

• Deseasonalise a time series by calculating the seasonal indices using the average percentage method, including the use of spreadsheets.

• 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.

GM Unit 3 Topic 4: Growth and Decay in Sequences

Sub-topic: The Arithmetic Sequence (5 hours)

• Use recursion to generate an arithmetic sequence.

• 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.

• Use the rule for the term of an arithmetic sequence.

• where is term,  is first term, is term number and is common difference

• Use arithmetic sequences to model and analyse practical situations involving linear growth or decay, e.g. analysing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, calculating the value of an item using the straight-line method of depreciation.

Sub-topic: The Geometric Sequence (6 hours)

• Use recursion to generate a geometric sequence.

• 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.

• Use the rule for the term of a geometric sequence.

• where is term,  is first term, is term number and is common ratio

• Use geometric sequences to model and analyse practical situations involving geometric growth and decay (use of logarithms not required), e.g. modelling the growth of a bacterial population that doubles in size each hour, calculating the value of an item using the diminishing-value method of depreciation.

GM Unit 3 Topic 5: Earth Geometry and Time Zones

Sub-topic: Locations on the Earth (5 hours)

• Understand the meaning of great circles.

• Understand the meaning of angles of latitude and longitude (in decimal degrees, and degrees and minutes) in relation to the equator and the prime meridian respectively.

• Locate positions on Earth’s surface given latitude and longitude, e.g. using a globe, map, GPS and other digital technologies.

• State latitude and longitude for positions on Earth’s surface, e.g. investigating a map of Australia and locating boundary positions for Aboriginal peoples’ and Torres Strait Islander peoples’ language groups, Australian landmarks or local land boundaries.

• Calculate angular distance and distance between two places on Earth on the same meridian.

• where is distance in kilometres

• Calculate angular distance and distance between two places on Earth on the same parallel of latitude.

• where is distance in kilometres and  is latitude

• Solve practical problems involving latitude, longitude, angular distance and distance.

Sub-topic: Time Zones (5 hours)

• Understand the meaning of 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 given time difference.

• Calculate time differences between two places on Earth.

• Solve practical problems involving time zones, making allowances for daylight saving where necessary, e.g. seasonal time systems used by Aboriginal peoples and Torres Strait Islander peoples, making phone calls, broadcasting events, travelling, preparing an itinerary.