IB Mathematics Analysis and Approaches (AA) SL Extended Essay Example

The research question is clear, but the phrase “compare” is too broad. The student should define what comparison means here: target accuracy, final-score prediction, match result, or error against DLS. Without an operational measure, the investigation cannot fully answer its own question.

The title communicates a focused mathematical investigation and immediately signals the comparison between two methods. This clarity helps the reader understand the purpose from the outset and sets up a coherent line of inquiry.

The student clearly states the intended mathematical direction and links the modelling work to a real-world comparison. This makes the purpose visible and gives the reader a useful roadmap for the essay.

This claim is more rhetorical than methodologically precise. The student should explain how the models are actually being compared, rather than implying flexibility automatically makes them superior. A sharper framing would specify the prediction task and the error metric used.

The opening to the modelling section is well motivated and connects the mathematics to the sporting context. This helps justify why differential equations are being introduced instead of appearing as an isolated technique.

It would be helpful to separate cumulative runs from scoring rate more carefully throughout the essay. The student should keep $R(t)$ for cumulative score and reserve a different symbol for runs per over, since later sections sometimes call $R(t)$ the “scoring rate,” which creates confusion.

The mathematical relationship is stated, but the cricket interpretation is oversimplified. The student should justify why current score should drive future scoring rate, because that assumption is not obvious in cricket and strongly affects the validity of the exponential model.

The student’s interpretation of the constant is not fully correct. The factor from integration is a multiplicative constant, not automatically the initial score itself. The student should show the substitution step that produces $R_i$ and clarify when the model begins so the parameter meaning is mathematically valid.

The student describes the DLS formula, but the explanation stops short of showing how the published resource table is used in an actual calculation. Because DLS is central to the essay, the student should demonstrate one worked target calculation instead of only giving a general description.

This is a strong piece of subject knowledge because it identifies the two central DLS resources and explains their cricket meaning. That conceptual grounding helps the reader understand why DLS is fundamentally different from a score-only model.

The student treats the carrying capacity as if it is a fixed cricket maximum, but in a chase it is not a universal constant. It depends on wickets, overs, match state, and tactical context. The model would be stronger if the student justified how $K$ is chosen in each innings.

Choosing one ODI and one T20I shows sound awareness of format differences and gives the investigation a useful contrast. This strengthens the applicability of the mathematics because the student is not treating all limited-overs cricket as identical.

This is not a correct use of the DLS result. The student is scaling the winning margin linearly to 50 overs, but DLS does not work by proportional extension of runs. The student should either compute the official DLS comparison properly or remove this claim entirely.

The student makes a useful reality check, but the “theoretical maximum” is not convincingly derived from the match context. The calculation should account for balls actually remaining and wickets in hand using a transparent cricket-based argument. This would make the critique of the exponential model far more persuasive.

The student is clearly structuring the analysis into a derivation, application, and evaluation sequence. That organisation supports the argument because each model is introduced before being tested against match data.

A higher $R^2$ here only shows a better in-sample fit, not a better prediction. The student should be careful not to equate regression fit with forecasting accuracy. A stronger evaluation would compare predictions against held-out overs or a genuinely independent target.

What does the vertical shift mean in cricket terms? If the parameter improves $R^2$ but changes the model’s initial behaviour, the student should ask whether the statistical gain is worth the loss of interpretability.

This conclusion overstates what the graphs alone show. Better visual fit does not automatically mean more accurate prediction, especially when the models are extrapolated beyond the observed innings. The student should support this claim with explicit error comparisons.

The comparative summary is useful because it brings the main numerical results into one place. This makes it much easier for the reader to evaluate the models side by side and see the size of the differences.

This comparison is not internally consistent with the earlier DLS result. The student should make sure the same scoring basis is used throughout: DLS par score, projected 50-over total, and winning margin are not interchangeable quantities. Mixing them weakens the analysis.

This is a relevant idea, but the section becomes conceptual rather than analytical because the SDEs are never fitted, simulated with real parameters, or compared to the matches. The student should either apply the method or clearly scale back its role in the investigation.

The student correctly identifies a reduction in residual error, but the evaluation remains one-sided. The essay should also discuss whether the adjusted curves still make cricket sense, because a numerically better fit may produce a less meaningful model.

The student introduces Ito’s Lemma, but the later use is not rigorous enough to support the essay’s conclusion. If the lemma is included, the derivation should be correctly applied to a specific SDE and then used in the cricket context; otherwise it reads as an unsupported detour.

The conclusion does return to the research question and makes a clear evaluative judgment. This helps the essay feel complete because the student does not simply present results but also interprets them in relation to the original aim.

The student could strengthen the evaluation by quantifying the comparison. For example, include an absolute error or percentage error for each model against the observed interrupted-match outcome, then use those numbers to justify the conclusion.

The report is well structured and easy to navigate. The contents page helps the reader follow the progression from background theory to applications, modifications, and conclusion.

Including appendices for the resource table, raw data, calculations, and code is a strong structural choice. It keeps the main body focused while still making supporting material available for transparency.

The tabulated match data are useful, but the presentation would be more effective if the student explicitly labelled which values are used for modelling and which are included only as context. Clearer signposting would make the data easier to interpret.

The graphs would be more informative if they included clear axis labels, a legend, and a viewing window that highlights the relevant range of overs and runs. That would let the reader see the model comparison without needing to infer the scale from surrounding text.

A small improvement would be to add a short sentence immediately after this summary introducing what the table shows and how it should be interpreted. That would help the reader move from raw results to the comparison more smoothly.

This section is structurally out of proportion with its role in the argument. The student should either develop the stochastic modelling properly and use it in the comparison, or reduce it to a brief future extension so the main analysis stays focused.