What the investigation does
The investigation translates a rugby conversion into an angle-maximization problem. After specifying pitch dimensions, the conversion line and simplifying assumptions, it derives the preferred distance using trigonometry and differentiation, then checks the result with a separate circle-geometry construction.
The solutions are compared through numerical tables and hyperbola graphs. The essay tests a simpler asymptotic placement rule, explores how including the crossbar changes the geometry, and closes by discussing the gap between a geometric optimum and a real kicker's success probability.
Why this example is worth reading
The model begins by identifying fixed quantities, controllable variables and assumptions about ball motion, spin, player ability and weather.
Two mathematical approaches lead to matching values, providing a useful internal check on the algebra and geometry.
The conclusion revisits assumptions and proposes meaningful extensions involving wind, a fixed flight path and observed conversion probabilities.
What to examine critically
Maximizing the apparent goal angle is a surrogate objective, not a demonstrated maximum of scoring probability. The essay itself acknowledges the roles of range, accuracy, curved flight and spin.
Check the domains and geometry of the crossbar extension and tabulated boundary cases carefully before reusing its formulas; agreement between two idealized methods does not validate their real-world assumptions.
Apply it to your own work
Turn a practical situation into a defined mathematical objective before selecting techniques.
Use an alternative derivation and numerical examples to test a result.
Evaluate whether optimizing the model also optimizes the real outcome of interest.
Essay & assessment material
Read the original material alongside the breakdown, or download it to keep.
The examiner mark sheet is embedded on PDF page 3 and records 32/36. No separate narrative examiner commentary is included with this selection; the teaching analysis is our own.
Source & assessment context
IB Mathematics English exemplars, May 2013. The source PDFs retain their original attribution. The analysis on this page is AsterDraft’s independent teaching commentary.
Examiner mark: 32/36 (legacy criteria). The examiner mark sheet is embedded on PDF page 3 and records 32/36. No separate narrative examiner commentary is included with this selection; the teaching analysis is our own.