IB Mathematics Analysis and Approaches (AA) SL Internal Assessment Exemplar: Modelling of a 500ml water bottle to find surface area and optimal shape.

The contextual motivation is relevant, but it could be shortened slightly so the exploration reaches the mathematical modelling more quickly. Consider keeping only the statistics that directly justify the research question.

The student has a clear investigative aim that is sustained throughout the exploration. Choosing a specific bottle and linking it to the broader goal of reducing plastic use gives the work a coherent direction and helps the reader understand why the mathematics matters.

The goal is clear, but the wording is mathematically imprecise: a bottle with volume should be described as $$500{\text{ cm}}^3$$, not $$500{\text{ cm}}^2$$. Tightening this terminology would improve coherence and prevent confusion about what is being optimised.

The student gives a clear method for turning a real object into a mathematical model. This sentence shows a logical modelling pathway from measurement, to coordinates, to functions, which supports the overall structure of the investigation.

This transition is helpful, but the reason for switching methods should be made more analytical. The volume check exposes a mismatch in the first model, so the student should explain whether the new method is being used as a validation step, a replacement, or a refinement of the original model.

What evidence would be needed to show that packaging or user comfort is actually the main reason for the larger surface area, rather than just a plausible explanation? Developing that distinction would strengthen the logical development of the conclusion.

This conclusion is not supported by the evidence: a volume of 381.7 cm³ compared with 500 cm³ is not a minor error. The student should revise this interpretation and explain what the discrepancy implies about the reliability of the first model.

The evaluation identifies a realistic next step that would improve the investigation. By proposing weight, density, and comparisons with other bottles, the student shows awareness that the model could be extended beyond surface area alone.

The student correctly identifies perspective as a limitation, but the explanation remains too general. It would be stronger to state how the perspective affects the measured coordinates or the fitted functions, and which part of the surface area is most sensitive to that distortion.

The student is attentive to consistency in precision, which is an important part of mathematical communication. Linking the number of decimal places in the model to the measurement accuracy helps make the presentation more disciplined and transparent.

The communication would be clearer if the student used more precise mathematical wording, such as 'corresponding function' rather than 'responding function'. Small terminology errors can weaken clarity because they make the modelling process harder to follow.

The notation is mathematically acceptable, but the justification is not clearly communicated. The student should state explicitly why a normal-distribution curve is appropriate for this section of the bottle, rather than relying on a visual fit alone.

This table is a strong communication device because it brings together the model, the supporting points, and the domain of each piece. That structure makes it much easier to verify how the contour was assembled.

Consider defining the variable names more carefully here. If the formula uses $$r(x)$$, then the explanation should consistently refer to that same function rather than switching to $$g(x)$$, because consistency in notation is essential for mathematical clarity.

The explanation of the total surface area is incomplete because the separate top and base areas are not shown explicitly at this stage. The student should present the component areas in full so that the final total can be checked and reproduced.

The student shows good communication by breaking the total into separate components before combining them. This makes the calculation easier to audit and shows awareness that mathematical results should be traceable.

The communication becomes confusing here because the reliability claim does not match the numerical evidence: the basic-shapes model has the far better volume check. The student should separate surface-area similarity from model reliability and explain each claim using the evidence from both methods.

This is a memorable comparison, but it would be clearer if the student stated exactly how the football-pitch equivalence was calculated. Adding the conversion would make the claim more convincing and mathematically transparent.

The student shows a personal link between subjects by drawing on biology to justify the modelling choice. That interdisciplinary move is a good sign of independent thinking because it shows the model is not purely template-driven.

The student’s modelling choice is creative, but the parameter selection is too informal. Saying the values were found by 'tweaking' leaves the model vulnerable, so the student should justify the chosen parameters with a measurable fitting criterion or comparison.

The student shows initiative by developing a second modelling method instead of relying on the first one alone. Using a separate decomposition to check the result is a strong sign of independent thinking.

What would change if the comparison were based on actual material mass rather than surface area alone? This question matters because personal engagement is stronger when the student tests whether the chosen measure really captures environmental impact.

The exploration goes beyond routine calculation by trying to connect the mathematics to commercial decision-making. Even though the claim needs tighter evidence, it shows the student is trying to build a personal argument around the model.

This is a sensible extension, but it remains quite general. To show stronger initiative, the student could specify exactly how those measurements would be used to refine the model, rather than simply suggesting them as future possibilities.

The student computes the error correctly, but the reflection does not go far enough in interpreting its significance. A 23.7% discrepancy should prompt analysis of which assumptions caused the error and whether the model is valid in its current form.

This is a meaningful reflective move because the student notices a problem in the first model and responds by changing approach. Reflection is strongest when it leads to action, and that happens here.

This is a useful checkpoint because it shows the student is evaluating the model against the target volume rather than assuming the construction is correct. That comparison provides a clear basis for later reflection on accuracy.

The reflection here is too generous given the evidence. The student should distinguish between 'calculation done correctly' and 'model close enough to be useful'; the first model’s volume error is too large to describe as minor.

How does this comparison change if the student reflects on production efficiency, wall thickness, and material density as well as surface area? Thinking about those factors would deepen the evaluation of the environmental implication.

The student identifies a genuine limitation in the method and links it to the difference between the two surface-area results. That connection is important because reflection should explain where the uncertainty came from, not just list flaws.

This limitation is relevant, but the reflection would be stronger if the student explained the direction of the effect. For example, does the cap overlap increase the measured area, and does the imperfect base cause the model to overestimate or underestimate the true surface area?

The student uses highly relevant mathematics for the modelling task: a surface-area integral for a surface of revolution is exactly the right tool here. This is a strong choice at SL because it is applied directly to the real-world shape being studied.

Although the formula is relevant, the student should align the notation more carefully with the chosen functions. If the integral is written in terms of $$r(x)$$, then the explanation should not refer to $$g(x)$$ without clarifying the relationship between the two.

This is a good use of mathematics because the student applies disc integration to test whether the model preserves the target volume. That makes the mathematics functional rather than decorative.

The mathematics here is relevant, but the justification is weak because the approximation method is not analysed rigorously. The student should explain why these shapes are a reasonable representation and what error they introduce.

The student makes effective use of a quantitative error calculation, which is exactly the kind of mathematics needed to evaluate a model. This helps distinguish a numerical result from a validated result.

The mathematical interpretation is not convincing because the reliability claim does not follow from the evidence presented. The student should separate surface area comparison from volume validation, since a close surface area does not make a model more reliable if the volume is inaccurate.

The student applies the area calculation appropriately, but one crucial component appears to be missing from the mathematical reasoning: the formula for combining top and base areas. Since this directly affects the result, it should be shown explicitly and checked for correctness.

The conclusion is plausible, but the working is too compressed to verify. The student should show the derivative calculation more clearly so that the relationship between the critical point and the minimum is mathematically transparent.

The optimisation step uses appropriate course-level mathematics: algebraic substitution, differentiation, and a second-derivative test. This is a strong example of mathematics being used to solve a constrained optimisation problem.