IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Exemplar: Mathematical Modeling of Rainfall Patterns and Predicting Flood Risk of Kinabatangan River in Sandakan, Sabah

The exploration lacks an explicit, focused research question. Clarify the precise question you intend to answer to improve coherence and logical development.

The introduction’s historical narrative is verbose and interrupts the mathematical focus. Consider condensing background information and focusing more directly on the research question for conciseness.

Your interpretation of skewness refers to Figure A2 in the appendix. Consider including a small version inline or preview so the reader doesn’t have to flip back and forth.

Table 1.1’s headings are cramped. Consider merging header cells or using multi‐line titles to improve readability.

Your conclusion “Hence,” leads into the table but offers no summary of the key insight. Add a brief sentence stating the numerical coefficient values to close the argument.

Figure 2’s caption is minimal. Use a descriptive caption (e.g., “Cubic regression fit with critical points marked”) to improve reader guidance.

Bibliography entries contain tracking parameters (e.g., ?form=MG0AV3) which clutter citations. Clean URLs to adhere to academic formatting.

The flow‐chart figure is useful but lacks clear axis labels and step descriptions. Add concise labels and a legend to improve interpretability.

The sentence ends abruptly with “in” and no continuation. Complete the thought or split into two sentences to maintain clarity.

The term “helow” is a typo in the description of the standard deviation formula. Correct spelling and grammar for precision.

When introducing $\bar{x}=\sum X_i/n$, reference Figure A1 before using its data for context. This links equations to the displayed data.

When introducing R², emphasize why a close‐to‐1 value risks overfitting. A brief note on cross‐validation here would strengthen logical clarity.

Figures 1.1 and 1.2 lack axis labels and units on both axes. Add these to ensure each graph is self‐contained and interpretable.

The matrix inversion approach for coefficient calculation is correct but introduces a 4×4 system without clarifying dimensions. Add a brief note on the dimension and structure of each matrix.

The description of x and y is clear and well‐explained, providing effective mathematical communication.

The DE solution steps in Equation (7) skip justification for the integrating factor. Include the explicit step $\mu (t)=e^{kt}$ application to clarify the derivation.

Table 2’s header uses “m/day” but sample uses mm/day conversion; ensure consistent unit definitions in header and calculations.

The assumption $k=0.03$ day⁻¹ from Wikipedia is reasonable, but consider seeking local flow‐rate data to justify this choice and deepen independent thinking.

Choosing December data and $\Delta t=1$ day is logical but not justified. Explain why daily release optimization is appropriate vs. monthly average.

Switching to $k=0.01$ to simulate drought is an innovative approach, showing creative exploration of extreme scenarios.

Reflection on data limitations and suggestions for Fourier or stochastic methods is insightful and shows strong personal engagement.

The correlation coefficient of 0.9999 is suspiciously high. Reflect on possible overfitting or data artifacts, and consider adding a caveat.

The optimization derivative sets $W_{new}=W_{safe}$ but omits discussion of boundaries and feasibility of D(t). Reflect on these conditions.

When calculating $W(31)$ for the drought scenario, the day count mismatches months—use consistent Δt to avoid confusion.

The conclusion repeats results but does not critically evaluate model validity against real flood events. Add discussion of model limitations.

The formula for standard deviation is correctly stated, but please include the unit (mm) in the final result to reinforce consistency of units.

You mention using MSE from Appendix B but no formula is given in the main text. State the MSE definition to improve mathematical completeness.

The coefficient vector substitution in Equation (4) is executed accurately; this demonstrates solid understanding of matrix algebra.

When defining $R(t)$ and $kW$, specify their units (e.g., m/day and day⁻¹) to maintain consistency throughout.

The expression for $R=$ monthly rainfall ÷ days is ambiguous about units. Clarify conversion from mm to m before dividing to ensure correct dimensional analysis.

The product‐rule derivation in Equation (7) assumes one term’s derivative is zero without explanation. Justify why $dR/dt=0$ clearly.