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QLDGeneral Mathematics

Unit 3: Bivariate data, sequences and change, and Earth geometry

9 dot points across 9 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

Topic 1: Bivariate data analysis - how do we describe and measure the relationship between two numerical variables?

Construct a scatterplot, describe the association between two numerical variables in terms of direction, form and strength, calculate and interpret Pearson's correlation coefficient $r$ and the coefficient of determination $r^2$ , and recognise that correlation does not imply causation
A focused answer to the QCE General Mathematics Unit 3 dot point on bivariate data. Covers scatterplots, describing association by direction, form and strength, Pearson's correlation coefficient $r$ , the coefficient of determination $r^2$ , and the difference between correlation and causation, with CAS-supported worked examples for IA2 and the external assessment.
6 min answer →

Topic 1: Bivariate data analysis - how do we straighten a curved relationship so a least-squares line can be fitted?

Apply a square, logarithmic or reciprocal transformation to one variable to linearise a non-linear association, fit a least-squares line to the transformed data, use the transformed equation to predict, and choose the transformation that best straightens the scatter
A focused answer to the QCE General Mathematics Unit 3 dot point on data transformation. Covers when to transform, the square, log and reciprocal transformations, how to fit and use a least-squares line on transformed data, and how to predict by back-substituting, with arithmetic-verified worked examples for IA2 and the external assessment.
6 min answer →

Topic 2: Growth and decay in sequences - how do we model the loss in value of an asset over time?

Model depreciation of an asset using flat-rate (straight-line), reducing-balance and unit-cost methods with recurrence relations and rules, compute book value and scrap value, and compare the methods over the life of the asset
A focused answer to the QCE General Mathematics Unit 3 dot point on depreciation. Covers flat-rate (straight-line), reducing-balance and unit-cost depreciation as recurrence relations and rules, book value and scrap value, and how the three methods differ over the life of an asset, with arithmetic-verified worked examples for IA2 and the external assessment.
6 min answer →

Topic 3: Earth geometry - how do we locate points on the Earth and measure distances along its surface?

Locate a position on the Earth using latitude and longitude, define great circles and small circles, calculate the distance along a meridian or the equator using the angular separation, and convert between nautical miles, minutes of arc and kilometres
A focused answer to the QCE General Mathematics Unit 3 dot point on Earth geometry positions and distances. Covers latitude and longitude coordinates, great and small circles, distance along meridians and the equator using angular separation, and conversions between minutes of arc, nautical miles and kilometres, with arithmetic-verified worked examples for IA2 and the external assessment.
6 min answer →

Topic 1: Bivariate data analysis - how do we model a linear relationship and use it to make predictions?

Fit a least-squares regression line to bivariate data, interpret the slope and intercept in context, use the line to make predictions through interpolation and extrapolation, and assess the fit using a residual plot and the coefficient of determination
A focused answer to the QCE General Mathematics Unit 3 dot point on least-squares regression. Covers fitting the regression line with CAS, interpreting slope and intercept in context, interpolation versus extrapolation, residuals and residual plots, and using the coefficient of determination to judge fit, with worked CAS examples for IA2 and the external assessment.
6 min answer →

Topic 1: Bivariate data analysis - how do we use residuals to judge whether a straight line is the right model?

Calculate residuals for a least-squares line, construct and interpret a residual plot, use the pattern in the residual plot to decide whether a linear model is appropriate, and identify when a transformation is needed because the residuals show curvature
A focused answer to the QCE General Mathematics Unit 3 dot point on residual analysis. Covers what a residual is, how to calculate residuals from a least-squares line, how to build and read a residual plot, and how a random scatter versus a clear pattern tells you whether a linear model fits, with arithmetic-verified worked examples for IA2 and the external assessment.
6 min answer →

Topic 2: Growth and decay in sequences - how do we model quantities that change by a constant amount or a constant ratio?

Use arithmetic and geometric sequences and first-order recurrence relations to model linear growth or decay and geometric growth or decay, find the nth term and partial sums, and apply these models to simple interest, reducing-balance depreciation and compound contexts
A focused answer to the QCE General Mathematics Unit 3 dot point on sequences and change. Covers arithmetic and geometric sequences, first-order recurrence relations, nth-term and sum formulas, and modelling linear versus geometric growth and decay including simple interest and depreciation, with arithmetic-verified worked examples for IA2 and the external assessment.
6 min answer →

Topic 1: Time series analysis - how do we describe patterns over time and smooth out short-term fluctuation?

Construct and interpret time series plots, identify trend, seasonality, cyclical and irregular variation, smooth a series using moving averages, calculate and apply seasonal indices to deseasonalise data, and fit a trend line to forecast future values
A focused answer to the QCE General Mathematics Unit 3 dot point on time series. Covers time series plots, the four components of variation, moving-average smoothing including centred averages, seasonal indices and deseasonalising, and fitting a trend line to forecast, with arithmetic-verified worked examples for IA2 and the external assessment.
6 min answer →

Topic 3: Earth geometry - how do we relate longitude to local time and work out time differences between places?

Relate longitude to local time using the rule that the Earth turns 15 degrees of longitude per hour, calculate the time difference between two locations from their longitudes, apply the conventions of east being ahead and west being behind, and combine this with flight times
A focused answer to the QCE General Mathematics Unit 3 dot point on time zones. Covers the 15-degrees-per-hour rotation rule, calculating the time difference between two longitudes, the east-ahead west-behind convention, the International Date Line, and combining time zones with flight times, with arithmetic-verified worked examples for IA2 and the external assessment.
6 min answer →