IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example

The exploration is clearly staged through section headers and a logical progression from data collection to coefficient calculation, analysis, and evaluation. That structure helps the reader follow the modelling process without losing sight of the research aim.

The student places the regression discussion close to the graph and uses it as the bridge into Fourier coefficient calculation. That close pairing of visual evidence and method improves the flow of the exploration and makes the modelling purpose easier to follow.

The repeated insertion of long calculator output interrupts the argument more than it advances it. The student should summarise the key steps and final coefficient result in the main text, then move the detailed antiderivative to the appendix to keep the exploration focused and easier to read.

The strengths list is useful, but it reads more like a postscript than an integrated evaluation. The student would improve the structure by connecting each strength directly to a result in the main body, so the conclusion feels like a synthesis rather than a separate list.

The variable definitions are clear and mathematically helpful, especially the explanation of amplitude, phase shift, and output variable. This makes the later modelling steps easier to interpret and shows stronger communication than in the previous draft.

The notation for the Fourier series is still not accurate, and that affects mathematical communication as well as correctness. The student should rewrite the formula using the standard additive structure and then carry that same structure into equation 28 so the final model matches the stated theory.

The processed data table is well chosen because it converts the raw season list into a usable mathematical dataset. That supports the later regression and coefficient calculations by making the variables explicit and consistently indexed.

The coefficient table condenses repeated results efficiently and improves readability. This is a good communication choice because it prevents the main body from being overloaded with nearly identical integration steps.

The definition of $R^2$ is stated correctly, but the notation should be tightened by distinguishing the observed value $y_i$ from the fitted value ${\hat{y}}_i$ consistently throughout. Using consistent subscript notation would make the statistical explanation more precise.

If the model is already known to need a later transformation, what does the reported $R^2$ actually measure: the original Fourier approximation, or the adjusted curve fit? Clarifying this distinction would make the statistical interpretation much more rigorous.

The topic has a clear personal motivation, and the student connects the exploration to both football and mathematics rather than choosing a detached textbook example. That personal investment helps explain why the dataset and method were selected.

The student shows initiative by acknowledging that football performance is not perfectly regular and by using that insight to motivate a periodic model. This shows more than routine application because the modelling idea is tied to the behaviour of the real data.

The adjustment to the model is still driven mainly by trial and error, so the independent thinking is not fully sustained through the mathematics. The student should test a few model variants systematically and compare them, rather than relying on visual tuning alone.

Consider computing one or two coefficients manually as a cross-check against the calculator outputs. That would show ownership of the method and reduce dependence on software while strengthening confidence in the final values.

The proposed extension to compare summation with integration is a genuine sign of initiative because it identifies a more natural approach for discrete season data. That is a good direction for future development because it emerges from the limitations of the current method.

The coefficient formulas still mix conventions: the interval is written as ${\int }_0^T$ in the general form, but later the model is evaluated on $[1,18]$ with a shifted variable. The student should explicitly state one consistent period convention and show how the shift changes the limits and trigonometric arguments.

The Fourier series is still written incorrectly as $f(x)\approx \frac{1}{2}{a}_0\sum \cdots$. The constant term should appear separately as $\frac{a_0}{2}+{\sum }_{n=1}^{\infty }(\cdots )$. This structural error affects the scaling of every later coefficient and should be corrected in the model statement and final equation.

The student justifies the 13th-degree regression by linking it to the best fit quality rather than simply choosing a convenient formula. That is a stronger modeling choice than before because it shows an attempt to optimize the regression stage before using it in the Fourier calculations.

The student now makes the horizontal shift explicit in the coefficient integrals using $x-1$. This is an improvement because it shows the periodic model is being aligned to the chosen season indexing rather than leaving the transformation implicit.

The analysis correctly recognises that the initial Fourier curve does not align well with the data, which is an important diagnostic observation. Noticing the mismatch shows that the student is evaluating the model rather than blindly accepting the first output.

What feature of the original Fourier model specifically suggested that a dilation and shift were needed, rather than revisiting the coefficient definition or period choice? Answering this would strengthen the mathematical diagnosis behind the adjustment.

The transformation $g(x)=\left(\frac{f(\frac{x}{1.078}+18.48296)}{38}\right)+45$ is introduced as a trial-and-error fix, but no mathematical criterion is given for why these exact constants are appropriate. The student should justify the scaling and translation quantitatively, otherwise this step reads as curve-warping rather than model refinement.

The interpretation of the transformed curve relies too heavily on the fact that the model now shows three visible peaks. The student should connect those peaks back to the actual seasons and explain why that shape is mathematically meaningful rather than just visually similar.