IB Mathematics Applications & Interpretation (AI) SL Internal Assessment Example

The overview of the investigation is concise and signposted well. The student previews the stages of the method, which creates a clear roadmap and supports the coherent flow that follows.

The student opens with a clear personal motivation, linking the investigation to gym nutrition and protein estimation. This makes the exploration purposeful and helps the reader immediately understand why the model matters beyond pure mathematics.

This explanation becomes repetitive: the discussion of full polynomials and the Lagrange method restates ideas already introduced without adding new mathematical purpose. The student should trim repeated definitions and use the space for comparison, validation, or reflection on why the chosen method is preferable.

What does a total volume of $291{\text{cm}}^3$ imply when compared with a typical egg, and how should that affect the interpretation of the rest of the investigation? A strong exploration would use this comparison to challenge the model rather than simply accept the number.

The statement that the protein content “seems inline” is not convincing because it is not checked against a realistic egg value. The student should compare the result to a known benchmark and explain any mismatch, since this is essential to judging whether the exploration’s narrative and mathematics remain aligned.

The conclusion would be stronger if it synthesised the main modelling decisions and their consequences more explicitly. The student could briefly state which function choices worked best, where the model was weakest, and what that means for the reliability of the final estimate.

The student uses standard mathematical notation appropriately in the Lagrange formula, which makes the modelling method identifiable and precise. Clear symbolic definitions like this strengthen communication because the reader can track the structure of the interpolation without ambiguity.

The graph annotation would be more informative if the key points, axes, and scale were summarised in the caption or accompanying text. That would make the visual representation easier to interpret and more useful for comparing model fit.

The use of both $1.1667$ and $1.17$ creates avoidable inconsistency in the same model. The student should either keep a consistent number of decimal places or justify why a rounded form is used, because small precision shifts matter when later squaring and integrating the function.

The table of plotted points is helpful, but the communication would be stronger if it were linked more explicitly to the equation being fitted. The student could state which listed points determine the model and how each one supports continuity between sections.

The final piecewise definition is a strong communication feature because it brings the three local models together in one place. This makes the overall profile easy to reference and shows that the student understands how the section models connect.

The integration formula is written with reversed bounds as $V=\pi {\int }_b^a(f(x){)}^{2}dx$, which is inconsistent with standard convention and can confuse the reader. The student should write the limits in ascending order and keep this notation consistent throughout the volume calculations.

The expansion of the squared quadratic is not correct as written: the signs and coefficients do not match the square of $-0.133x^2+1.1667x+1.2$. The student should re-expand the expression carefully, because this error propagates into the integrated result and weakens the reliability of the whole section.

The antiderivative for Section 3 is not communicated clearly enough to verify the calculation, especially around the logarithmic term and the handling of $\pi$. The student should show each integration step explicitly so the reader can follow how $\int \left(\frac{64}{(x-11.8{)}^2}+\frac{80}{x-11.8}+25\right)dx$ is obtained.

The photographic scaling method would be more convincing if the student explained how distortion was checked or minimised beyond the camera distance. A short note on calibration, viewpoint control, or reference measurement would make the personal data-collection approach feel more rigorous.

Which of the candidate models was actually tested against a measurable criterion, and which were selected mainly because they looked plausible? Reflecting on that distinction would show a deeper level of independent decision-making.

The student shows independent thinking by comparing a quadratic model with a square-root alternative rather than settling on the first fit. This demonstrates active model testing, which is a genuine sign of personal judgement in the investigation.

The iterative adjustment of the hyperbolic model shows initiative: the student did not simply copy a standard function, but tried to adapt parameters to the shape. That willingness to refine the model is valuable personal engagement, even though the method could be made more transparent.

The hyperbolic function appears to have been chosen mainly by visual tuning, but the exploration does not show a clear, reproducible method for selecting $8$ and $11.8$. The student should explain how these parameters were determined or tested so the choice is defensible rather than impressionistic.

The personal investigation could be strengthened by using an independent measurement of the egg’s mass or volume to compare with the model. That would turn the topic from a personal context into a more authentic inquiry with stronger ownership of the evidence.

The student reflects meaningfully on why a quadratic model is appropriate for the lower section, showing awareness that the function should match the local geometry. This kind of reasoning demonstrates more than description; it explains the modelling choice.

The comment that the quadratic “doesn't align precisely” is useful because it shows the student is not treating a visual fit as automatically perfect. Recognising mismatch is an important step toward critical modelling reflection.

The statement that the scaling factor was chosen “after several adjustments” is too vague to count as deep reflection. The student should explain what changed between attempts and why the final choice was preferred, so the reflective process becomes visible rather than just asserted.

If the model is described as effective, what evidence supports that claim beyond the calculation itself? Asking this would help the student distinguish between completing a method and critically evaluating whether it worked.

The reflection on Lagrange interpolation is accurate in principle, but it does not examine whether the chosen points caused the unrealistic final result. The student should connect method choice to outcome quality, since this is the clearest way to show mathematical self-evaluation.

The limitations section identifies the symmetry assumption, but it stops short of analysing how much that assumption could affect the final volume. The student should move from naming limitations to estimating their impact, because that is where reflection becomes critical rather than descriptive.

The student applies Lagrange interpolation correctly in principle to construct a quadratic from three points. This is relevant AI SL mathematics used for a clear modelling purpose, rather than as decorative algebra.

The algebra in the quadratic interpolation needs checking, because the coefficients later used in the model are not transparently derived from the substituted expression. The student should verify each term before simplifying, since any arithmetic slip here affects the fitted curve and the subsequent volume.

The integrated polynomial is written with incorrect coefficient processing, especially in the higher-power terms. The student should show the expanded expression and then integrate it term by term with exact fractions or clearly justified decimals, because the current result cannot be confidently checked.

The Section 1 integration is handled correctly and clearly: the student simplifies ${\left(\sqrt{4.8x}\right)}^2$ and evaluates the definite integral logically. This shows secure use of standard volume-of-revolution techniques.

What would the Section 2 contribution look like if the polynomial were recomputed with exact coefficients instead of rounded decimals? Checking this would reveal whether the final volume is sensitive to the precision used in the fit.

The final protein estimate inherits the earlier modelling error, so the mathematical chain loses credibility here. The student should avoid presenting $37.97\text{g}$ as a meaningful protein result until the volume estimate has been independently checked and corrected.

The total volume is a major point of concern: $291{\text{cm}}^3$ is implausibly large for the egg dimensions given earlier. The student should investigate whether the profile function, integration, or scaling is incorrect, because an unchallenged unrealistic total weakens the mathematical conclusion.

A brief reasonableness check would strengthen the mathematics considerably. The student could compare the final volume to an estimate from displacement or to a typical egg volume range, then use that comparison to identify where the model may be overestimating.

The conversion from volume to mass is mathematically simple but only valid if the volume is realistic. Because the volume estimate is likely inflated, the mass and protein values are also unreliable and should not be presented as dependable conclusions without validation.

The stated value $V_3=141{\text{cm}}^3$ is difficult to trust because the preceding antiderivative is not fully justified and the scale of the result is not checked against the dimensions. The student should verify this numerically or by substitution, since the final volume depends heavily on this section.