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

The student’s title is clear and concise, effectively introducing the investigation’s focus on cookie volume changes and setting a coherent tone for the exploration.

The detailed camera-setup description is insightful but interrupts the mathematical flow. Consider placing this in an appendix to maintain conciseness.

The objective outlines the aim but could be refined by specifying the regression methods and modelling strategy for height and width functions to improve clarity.

The domain for w is mis-specified as “−w≤w”; this should read “0≤w”. Correct domain definitions are essential to avoid misinterpretation of variables.

The interpretation of Graph 1 identifies concavity changes and links them to ingredient behavior. This effective use of multiple representations enhances understanding.

The linearization of the width model is executed clearly and supported by an R² comparison, demonstrating effective use of multiple representations.

The derivative setup for V(t) is dense; consider adding intermediate steps or annotations to enhance logical clarity in the presentation.

Eliminating extraneous roots is logically sound and strengthens the argument that t=675 s is the valid maximum for V(t).

The graphical confirmation of V(t) peaking at (675,144) effectively corroborates the analytical result, demonstrating consistency across representations.

The rationale vividly connects personal baking experiences with mathematical inquiry, demonstrating genuine engagement and clear motivation for addressing the oven temperature problem.

The extension comparing baking soda vs. baking powder shows strong personal engagement and extends the investigation creatively beyond standard IA scope.

The scaling method is clever, but discuss potential sources of error in film-to-real measurements to deepen critical reflection.

Solving the cubic system manually increases risk of algebraic errors. Consider using technology to solve the simultaneous equations more efficiently.

The discussion of cubic model limitations is valuable. Extend this by quantifying potential error bounds to strengthen the evaluation.

The reflection on the cubic model’s negative height domain is perceptive and evaluates the model’s real-world applicability effectively.

Reflection identifies model limitations well; extend by suggesting how improved camera calibration or repeated trials could reduce scaling errors.

Assuming the cookie is a perfect partial sphere oversimplifies uneven expansion in the oven; discuss how deviations from spherical shape may impact volume calculations.

Several algebraic steps in the volume derivation contain sign-handling inconsistencies. Revisit the simplification from the integral evaluation to ensure full mathematical accuracy.

The use of the Pythagorean theorem to express r in terms of w and h is well executed and neatly integrated into the derivation.

The final expression $V=\frac{1}{6}\pi h(3w^2+h^2)$ is elegantly presented. Highlight its significance before proceeding to data modelling.

The derivative calculation for h(t) appears mis-typed (cubic in derivative). Recompute to ensure correctness of the model’s critical points.

The derivation of a = e^(−1.96) is correct and concise, but explicitly note the link to ln(a) from the regression intercept for mathematical clarity.

The choice of a rational function with a slant asymptote is well justified by data behavior; this demonstrates appropriate level-commensurate mathematics selection.

The quotient rule simplification for w′(t) contains typographical misalignments. Recompute and clearly present numerator and denominator expressions.

The scale factor computation omits the explicit formula and exponent notation (e.g. scale factor³). Include the full equation for transparency and reproducibility.