IB Mathematics AA HL — Internal Assessment Grading Report

Title: Can the surface area of a Llama-shaped cake be calculated through mathematical modeling and integration based on a 2D photograph of a stuffed animal to estimate the amount of fondant required for decoration?

Course: Mathematics Analysis and Approaches — Higher Level (AA HL)

Word Count: 2102

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Summary Table

Criterion	Score	Max	Descriptor
A — Presentation	3	4	Good
B — Mathematical Communication	3	4	Good
C — Personal Engagement	2	3	Adequate
D — Reflection	2	3	Adequate
E — Use of Mathematics	4	6	Adequate–Good
Total	14	20

Predicted Grade Boundary: 6 (boundary range 14–16 / 20)

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Criterion A — Presentation: 3 / 4

What was done well

The exploration is organised into clearly labelled sections — Introduction, Aim, Background Information, Methodology, and Evaluation and Conclusion — and follows a logical sequence. The introduction provides relevant personal context (baking a birthday cake for a younger sister) and a rationale for why standard geometric formulas are insufficient for an irregular shape. The aim is stated precisely and remains the governing thread of the investigation. Figure captions are included for all four GeoGebra diagrams, and Table 1 is clearly labelled and referenced in the body. The word count of 2102 is appropriate, and the exploration does not contain obviously irrelevant or repetitive sections.

What is missing or underdeveloped

The Background Information and Methodology sections overlap to some extent: assumptions about cake height, flat top, and vertical sides are introduced in Background but are then restated implicitly in the calculations without being brought together in a single, clearly bounded list. The conclusion, while present and connected to the aim, does not provide a brief structured summary of each stage of the investigation before moving into evaluation — a reader meeting the work for the first time would benefit from a two- or three-sentence recap of the computational pathway. Additionally, the exploration would benefit from a short outline near the end of the introduction stating how the aim will be achieved.

Suggestions for improvement

• Add two to three sentences at the close of the introduction previewing the structure: what will be computed first (perimeter via arc length), then (base area via integration), and finally (fondant mass via the surface area to mass ratio).
• Consolidate all assumptions into one numbered list in the Background section and refer back to them explicitly in the Evaluation.
• Expand the conclusion to restate the key numerical results before moving to the evaluation of limitations.

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Criterion B — Mathematical Communication: 3 / 4

What was done well

The exploration uses mathematical notation consistently and appropriately throughout. Integral notation for arc length and area is correctly typeset. All 16 piecewise functions and their domains are presented in a well-structured table. Multiple forms of representation are employed effectively: algebraic formulas, a coordinate graph with the image overlay (Figure 1), coloured curve plots (Figure 2), and shaded area diagrams with colour coding for addition and subtraction (Figures 3 and 4). The use of distinct colours for above-axis (blue) and below-axis (green) regions, and areas to be subtracted (red), is a clear and purposeful communication choice. Subscript notation ($a_f$, $l_f$, etc.) is introduced consistently and makes the summation step easy to follow.

What is missing or underdeveloped

There is a notable typographic error: the integrand shown for $a_c$ on the area calculation page is the formula for $b(x)$ ($0.50219x^2 - 14.53189x + 107.61704$) rather than $c(x)$ ($-0.01121x^3 + 0.60691x^2 - 10.94037x + 67.64427$). Since the domain is different, the numerical result may have been computed using the correct function, but the written integrand is inconsistent with Table 1. An IB examiner will identify this as a mathematical communication failure. Additionally, the arc length antiderivative is written in non-standard notation:

$l_f = \left[\int \sqrt{1+(f'(x))^2}\,dx\right]_{x_n}^{x_m}$

The bracket-with-limits notation conventionally denotes evaluation of an antiderivative $[F(x)]_a^b$, not the indefinite integral itself. The standard form $\int_{x_n}^{x_m} \sqrt{1+(f'(x))^2}\,dx$ should be used throughout. Finally, the axes of the GeoGebra graphs show unlabelled tick marks for the y-axis in some figures; a unit label (cm) on both axes in each figure would complete the communication.

Suggestions for improvement

• Correct the integrand written for $a_c$ so that it matches $c(x)$ from Table 1.
• Replace the non-standard antiderivative bracket notation with the standard definite integral form.
• Add unit annotations (cm) to the axes of all GeoGebra figures.
• When introducing subscript area notation, include a brief sentence clarifying that each $a$ represents a signed area computed from a single segment of the piecewise outline.

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Criterion C — Personal Engagement: 2 / 3

What was done well

The personal connection to the topic is genuine and specific: the student bakes for family events, her sister's birthday is the motivation, and the choice of the llama plush is her sister's favourite toy. This gives the exploration a clear sense of ownership that goes beyond choosing an arbitrary context. The methodology itself — overlaying the photograph in GeoGebra, placing boundary points manually, fitting piecewise polynomials to the outline, and then extending to a fondant mass estimate using real-world baking data — reflects independent and creative thinking. The decision to source data from a baking reference to compute a surface-area-to-mass ratio, and to cross-validate it using both circular and square cakes, shows initiative that goes beyond the mechanical application of calculus.

What is missing or underdeveloped

The exploration does not make or test explicit mathematical predictions at any stage. For example, the student could have estimated the perimeter roughly by approximating the shape as a rectangle before performing the full arc length calculation, then comparing the rough estimate to the precise result to assess reasonableness. There is no moment in the work where the student expresses mathematical curiosity in a form that drives the investigation forward — for instance, questioning whether degree-3 versus degree-4 polynomials produce meaningfully different arc lengths, or exploring how sensitive the fondant estimate is to the choice of cake height. The exploration reads as systematic execution of a plan rather than active mathematical exploration.

Suggestions for improvement

• Include a rough preliminary estimate (e.g., approximating the llama outline as a rectangle or ellipse) and compare it to the arc length result, discussing whether the difference is expected.
• Add a short section exploring the sensitivity of the result to one key assumption — for example, recalculating SA if the height were 6 cm instead of 5 cm, and commenting on the practical implication.
• Frame at least one finding as a mathematical surprise or question that arose during the investigation, and describe briefly how it was addressed.

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Criterion D — Reflection: 2 / 3

What was done well

The Evaluation and Conclusion section engages meaningfully with the limitations of the model. The student identifies multiple specific sources of error: the accuracy of photograph scaling, the placement of boundary points, the precision of GeoGebra's polynomial fitting, the accuracy of numerical integration, the use of secondary fondant data (potentially rounded for recipe simplicity), and practical baking factors such as fondant thickness variation and cake-knife cutting straightness. The student then connects the ±59 g uncertainty back to the practical context, noting that in the scale of a 1433 g fondant preparation, this is a realistic tolerance. The conclusion links the result explicitly to the aim stated at the outset.

What is missing or underdeveloped

The reflection does not look backward at the mathematical choices made during the investigation. The student does not ask whether a different modelling approach — for example, using parametric curves, or increasing the number of boundary points for a higher-resolution fit — would have produced a meaningfully different or more accurate result. There is no connection to prior mathematical knowledge (for instance, noting that the arc length formula is an application of the Pythagorean theorem in the limit, which was derived in the body of the work, but not reflected upon in the conclusion). The single proposed extension — "actually making the cake and comparing the mass of fondant used" — is practical rather than mathematical; a mathematical extension (testing a different model or estimating approximation error) would strengthen this criterion.

Suggestions for improvement

• Add a paragraph reflecting on the mathematical modelling choices: why were polynomials of varying degrees used rather than a uniform degree? How might the accuracy change if all segments used degree-4 polynomials, or if more boundary points were added?
• Connect the arc length derivation to the broader Riemann sum concept, noting how the fundamental idea of approximating a curve by straight segments is the same principle used both in the arc length formula and in numerical integration.
• Propose a mathematical extension, such as estimating the error introduced by the polynomial fitting by computing the residuals at the boundary points in GeoGebra.

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Criterion E — Use of Mathematics: 4 / 6

What was done well

The mathematics is appropriate for AA HL. Arc length via integration, $L = \int_a^b \sqrt{1+(f'(x))^2}\,dx$, is an HL-specific topic not required at SL, and its application as the principal computational tool of the investigation is well-chosen. The derivation of the arc length formula from the Pythagorean theorem is presented logically: beginning with $l_\text{small} = \sqrt{(dx)^2 + (dy)^2}$, factoring out $dx$, and integrating to obtain the formula over a full interval. The piecewise approach — 16 segments with individually fitted polynomial functions of degrees 2, 3, and 4 — is methodologically sound and produces a perimeter of 92.909 cm. The area calculation correctly handles the sign of areas above and below the x-axis, and the colour-coded diagrams in Figures 3 and 4 make the logic of addition and subtraction of regions transparent. The uncertainty analysis, including the propagation formula $\frac{\Delta M}{M} = \frac{\Delta SA}{SA} + \frac{\Delta(sa/m)}{sa/m}$ and the derivation of $\Delta M = 59.35$ g, shows additional mathematical thinking beyond the minimum required.

The final result $SA = 383.3077 + (92.90938 \times 5) = 847.8546$ cm$^2$ and $M = 847.8546 \times 1.69 = 1432.87$ g are computed correctly from the student's intermediate values, and the chain of calculation is traceable from first principles to the final answer.

What is missing or underdeveloped

Mathematical error in communication: As noted under Criterion B, the written integrand for $a_c$ does not match $c(x)$ from Table 1. If the numerical result 16.582 cm$^2$ was computed using $b(x)$ rather than $c(x)$, this would propagate an error into the total area. If it was computed using the correct $c(x)$, then the written integrand is a communication error rather than a computational one. In either case, this inconsistency undermines the precision of the written work.

Justification of polynomial degree: The choice of polynomial degree for each segment (quadratic, cubic, or quartic) is determined by GeoGebra with no mathematical justification provided. An HL exploration should explain why a particular degree was sufficient or necessary — for example, by noting the number of fitted points or discussing the visual residuals of the fit. Without this, the reader cannot assess whether the chosen functions accurately represent the llama outline.

Reliance on numerical computation: All definite integrals are evaluated numerically by GeoGebra. No analytical antiderivative is computed. While this is not prohibited, it means the exploration provides limited evidence of the student's independent mastery of integration technique. A brief analytical computation — for example, evaluating one of the simpler arc length integrals (such as $l_u$ or $l_o$, where the derivative is constant) by hand — would demonstrate procedural fluency.

Sophistication: The work is competent and systematic, but the mathematical insight is limited to the setup and execution of a standard algorithm (arc length formula + area by integration). There is no evidence of the student encountering and resolving a genuine mathematical difficulty, no exploration of the sensitivity of the model to its parameters, and no use of mathematical techniques beyond those directly required by the algorithm. For 5 or 6 marks at HL, the exploration would need to demonstrate either a higher degree of mathematical sophistication in the methods chosen or a more analytically rigorous treatment of the results.

Suggestions for improvement

• Correct the integrand for $a_c$ and verify that the numerical result is consistent with $c(x)$.
• Include a paragraph explaining the basis on which each polynomial degree was chosen (e.g., by inspecting the number of turning points in each segment of the outline, or by comparing goodness-of-fit indicators in GeoGebra).
• Compute at least one arc length integral analytically, even a simple one, to demonstrate command of integration technique beyond GeoGebra.
• Consider a sensitivity analysis: recalculate the total perimeter using a coarser or finer set of boundary points on one segment and compare the result, to quantify the effect of point placement on the final answer.

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Overall Summary

This is a well-motivated and clearly structured exploration that applies HL calculus — specifically arc length integration and definite integration for enclosed area — to a genuinely irregular, real-world shape. The personal context is authentic, the methodology is independently devised, and the extension to a fondant mass estimate using secondary data shows initiative. The total score of 14 out of 20 places this exploration in the Grade 6 boundary.

Top two strengths:

1. The central mathematical tool (arc length formula applied to 16 piecewise polynomial segments) is appropriate for HL, correctly set up, and consistently applied throughout.
2. The exploration is well-structured with a clear aim, systematic methodology, and a conclusion that returns to the original research question with a specific, quantified result including uncertainty.

Top two areas for improvement:

1. The exploration needs greater mathematical rigour and depth at HL level: polynomial degree choices must be justified, at least one integral should be evaluated analytically, and a sensitivity analysis would demonstrate the mathematical sophistication expected for the highest marks.
2. Reflection must go beyond listing practical limitations and engage with the mathematical modelling choices themselves — asking what alternative approaches could have been used, how the model could be refined mathematically, and what the results reveal about the nature of approximation in calculus.

The topic and scope are appropriate for AA HL. Arc length is an HL-only topic and is used meaningfully, not superficially. With targeted improvements to mathematical rigour (Criterion E) and deeper reflection on mathematical choices (Criterion D), this exploration has the potential to reach the Grade 7 boundary.