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

The title is clear and informative, effectively signaling the focus and scope of the investigation.

The personal narrative engages the reader, but the lengthy anecdote could be trimmed to improve conciseness and maintain relevance.

Citation of “Admin, 2023” is not sufficiently authoritative. Consider using a peer-reviewed physics text or journal for kinematics definitions.

The variables U and N are described as upper and lower limits but their physical meaning (“ground distance from release to hoop”) should be reiterated here.

The symbol table (Table 1) is helpful, but the layout is cramped. Consider splitting into two columns or using smaller font to improve readability.

Figure numbering is inconsistent: the text refers to Figure 1, but earlier it was called Graph 1. Use a single naming convention throughout.

The Euler substitution step is derived from an online source but lacks academic citation. Include full reference details for rigor.

Table 2’s colspan formatting makes reading column alignments difficult. Simplify the table structure or split into separate tables.

The term “quadratic formula” is misused when describing the projectile path; you mean quadratic function. Precise terminology is essential.

Equation (1) uses absolute-value notation around L without justification. Remove the bars or explain their necessity.

Graph 1’s axes labels are too small to read clearly; ensure all graphs have legible, consistently styled axis titles and units.

Differentiation notation in the derivation f′(x)=2×(−0.5652x)+2.2142 is awkward. Write it as f′(x)=−1.1304x+2.2142 directly.

The bold formatting around the constant term in s(t) is inconsistent. Present mathematical expressions with uniform styling.

The velocity function is written as v(t)=28.57116t2 – 31.94248t+10.50269 without caret notation. Use v(t)=28.57116t^2–31.94248t+10.50269 for clarity.

Your suggestion to apply the method to other sports demonstrates initiative and deepens the personal engagement in the exploration.

When calculating u-limits, the text should explicitly link each u-value to the corresponding x-limit to improve clarity.

The assumption of constant horizontal speed in scaling time to ground distance should be critically examined and justified.

Selecting a cubic model with R²≈0.952 is reasonable, but discuss why residuals or non-linearities do not warrant a higher-order fit.

The conclusion asserts that horizontal motion reduces net acceleration below g; this conflates vector components. Clarify that horizontal velocity does not alter vertical g.

Noting that a single free throw limits generalizability is a strong reflection which you could extend by proposing statistical sampling methods.

The methodology lacks detail on how GeoGebra point coordinates were converted from pixels to real-world meters. Clarify calibration procedure.

In Equation (2), “total ground distance” is not defined in symbols. Introduce a symbol (e.g. D) for clarity and consistency.

Regression coefficients are quoted without the R² value. Including goodness-of-fit metrics would strengthen justification of the quadratic model.

The domain is incorrectly stated as “0 ≤ x ≥ 3.232”. It should read “0 ≤ x ≤ 3.232”.

The derivation of x(u) by squaring should include a step showing cancellation of the 1.2778x² terms to confirm correctness.

The quotient-rule differentiation of du/dx is dense; break into smaller algebraic steps to avoid potential errors.

Relying on an online integral solver for the final arc-length integral undermines mathematical rigor. Attempt a numeric approximation or show partial analytical work.

The integration steps for computing v_avg and a_avg jump to numeric results. Include intermediate antiderivative values to validate the calculations.