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

The introduction effectively contextualizes the investigation by linking Deepseek’s LLM to the mathematical reconstruction task, engaging the reader immediately.

The methodology list is clear, but clarifying the sequence of software and manual steps would improve coherence and user reproducibility.

Table 1’s screenshot effectively consolidates arc equations and sample points; the image is slightly blurred—export at higher resolution for clarity.

The descriptive graphs on this page aid understanding, but they lack captions explaining key features—add concise labels to each subplot.

The Gauss–Jordan procedure is explained clearly, but direct quoting from PennState could be paraphrased to demonstrate deeper understanding.

The elimination steps are clearly numbered, but formatting irregularities in step 4 (‘A, R₄’) should be corrected for professional presentation.

Fig. 3 effectively shows the matrix after initial row operations; consider shading pivot columns to direct the reader’s attention.

The final compilation of regression curves illustrates the investigation’s scope; add brief captions for each to clarify which function corresponds to each segment.

Including the logo image provides a clear visual reference; ensure the figure is high resolution and numbered consistently throughout.

The circle equation is stated correctly using the Pythagorean theorem in $(x-a{)}^2+(y-b{)}^2=r^2$. Consider annotating each parameter’s geometric meaning directly in the text.

The perpendicular bisector slope is found correctly; adding the exact fraction before decimal approximation would improve mathematical precision.

The simultaneous solve for the circle center is concise but omits key algebraic steps; include intermediate transformations to maintain clarity.

Domain restriction via $y\le -0.506x-2.285$ is well applied; verify that the inequality sign is consistent when graphing to avoid misrepresenting the arc.

Multiple graphical representations are used effectively; consider referencing each representation explicitly in the narrative to guide the reader.

The criteria for reduced row echelon form are stated well; adding a small example of a non-compliant matrix would illustrate common pitfalls.

General form $a{x}^3+b{x}^2+cx+d$ is presented correctly; maintain consistent spacing around operators throughout to improve readability.

The partial matrix in Fig. 5 still has nonzero off-pivot entries; a fully reduced echelon form requires clearing these for consistent logical structure.

The personal motivation is noted, but the student could deepen engagement by reflecting on why the whale logo resonates personally and how it influenced methodological choices.

The rationale linking circle insufficiency to logo complexity is insightful, demonstrating personal recognition of mathematical limits.

The decision to combine polynomial and circular functions is logical; adding deeper reflection on alternative functional forms would strengthen critical analysis.

The radius calculation is thorough; to deepen critical reflection, discuss how rounding to three significant figures may affect the arc’s fit.

Explanation for choosing x=3.5 as a midpoint is pragmatic, but evaluate how this approximation affects the circle fit quantitatively (e.g., measure residual).

The discussion on polynomial degree assumptions is insightful; including a brief discussion of overfitting risks when increasing degree would deepen critical reflection.

Comparing manual and Desmos results is excellent practice; include specific R² or residual statistics to quantitatively validate curve fits.

Specification of the logo’s scaling on Desmos is useful; consider explaining why width=10 and height=7.4 were chosen to justify coordinate ranges.

Providing three sample points is essential, but clarify the point-selection method and ensure they lie precisely on the curve to avoid fitting inaccuracies.

Midpoint calculation is executed correctly; the student shows command of basic geometry. Ensure all values are carried consistently to the final equation.

The rationale for using polynomial regression is well connected to local extrema; explicitly state the number of maxima/minima to justify degree choices.

Check that the matrix entries in Table 2 match the previously derived equations; minor OCR errors (e.g. -0.137 vs -0.737) can lead to incorrect solutions.

The advanced row operations in Fig. 4 are well documented; verify each numeric result against manual computation to guard against transcription errors.

The final polynomial is clearly stated, but there is a minor sign typo: the cubic coefficient should be $-0.130086x^3$. Double-check your row-reduction arithmetic.

The final RREF in Fig. 6 cleanly isolates each coefficient, demonstrating mastery of Gauss–Jordan elimination.