IB Mathematics Internal Assessment — Grading Report

Course: Mathematics Analysis and Approaches — Standard Level (AA SL) Title: Optimizing the Thickness of Chocolate Coating on Spherical Cake-Pops for Minimization of Chocolate Usage Date of Assessment: 6 April 2026 Total Pages: 17

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Summary Score Table

Criterion	Description	Score Awarded	Maximum
A	Presentation	3	4
B	Mathematical Communication	2	4
C	Personal Engagement	2	3
D	Reflection	2	3
E	Use of Mathematics	3	6
	Total	12	20

Predicted Grade Boundary: 4–5

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Criterion A — Presentation

Score: 3 / 4

What Was Done Well

The exploration is logically structured and easy to follow. A numbered section hierarchy (Introduction, Background, Methodology, Results, Cost Analysis, Conclusion, Evaluation, Extension, References) gives the reader a clear map of the work. The cover page states the course, assessment type, title, and research question concisely. The introduction contains a genuine rationale, a well-defined aim, and an explanation of the approach to be taken (algebraic, numerical, and graphical methods). The conclusion returns explicitly to the aim and summarises the key finding (optimal thickness of 0.4–0.5 cm). The evaluation section (7) and extension section (8) show awareness of the need to look beyond the immediate results.

What Is Missing or Underdeveloped

The exploration is not fully concise. The observation that $V_s(t)$ is strictly increasing — and therefore has no internal minimum — is stated correctly in Section 2.4 but then restated, with essentially the same argument, in Section 4.4 and again in Section 5.5. This repetition adds length without adding new insight.

The figure numbering contains an error that affects the coherence of the document: two separate figures are labelled "Figure 3." The derivative graph (Section 4.3) is captioned "Figure 3: Graph of $\frac{dV}{dt}$ against coating thickness," and the cost graph (Section 5.4) is independently captioned "Figure 3: Graphical Representation of Cost per Cake Pop." A reader following the figure references is consequently misled. The reference list contains only a single entry — a BBC Bitesize page — cited as the source for the standard sphere volume formula $V = \frac{4}{3}\pi r^3$. This formula appears in any secondary-level mathematics textbook and does not require a citation of this kind; the single weak reference suggests a lack of engagement with more appropriate mathematical sources.

Suggestions for Improvement

• Eliminate the repeated monotonicity argument; state it once and refer back to it.
• Renumber all figures sequentially throughout the document.
• Replace or supplement the BBC Bitesize citation with a textbook or peer-reviewed source.
• Add a brief outline at the end of the introduction enumerating the sections to follow, distinct from the rationale paragraph.

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Criterion B — Mathematical Communication

Score: 2 / 4

What Was Done Well

The student defines all key variables in a clearly formatted reference table (Table 1), which establishes units and notation before they are used in calculations. Mathematical notation is applied consistently throughout the algebraic derivations: $r_c$, $t$, $R$, $V_c$, $V_s$, and $V_{total}$ are introduced and used correctly in the formulas. The cross-sectional diagram (Figure 1) is well labelled and communicates the geometric model clearly, with $r_c$, $t$, and $r_c + t$ all indicated. The data tables (Tables 2 and 3) are titled, headed, and include units. The use of multiple representational forms — algebraic expressions, numerical tables, and graphs — is a strength.

What Is Missing or Underdeveloped

The derivative graph (Section 4.3) contains significant communication failures. The domain plotted extends from approximately $t = -6$ to $t = +2$, which includes values of thickness that are physically meaningless (negative thickness has no interpretation in this context). Because the derivative $\frac{dV_s}{dt} = 4\pi(r_c + t)^2$ is a quadratic with minimum at $t = -r_c = -2$, the graph shows the curve descending toward zero and rising again — giving the visual impression that the derivative does reach zero, which directly contradicts the student's written claim that "the derivative is always positive; hence the graph lies entirely above the x-axis." The graph does NOT lie entirely above the x-axis in the plotted range, because the minimum of the parabola at $t = -2$ brings the curve to near-zero. The student's textual interpretation is correct for the domain $t > 0$, but the graph as presented contradicts it. This is a material error in mathematical communication: the representation and the explanation are inconsistent.

The axes of the derivative graph are not labelled. The notation "dV/dT" appears as a curve label within the GeoGebra plot but is not a proper axis label; the horizontal axis is unlabelled entirely, making it impossible for a reader to determine what variable is being plotted on the x-axis without inference from context.

The cost graph (Section 5.4) similarly lacks axis labels within the figure itself; the reader must infer the axes from the surrounding text.

The variable notation shifts between sections: the derivative is written as $\frac{dV}{dT} = 4\pi(r+t)^2$ in the text (using $V$ instead of $V_s$, $T$ instead of $t$, and $r$ instead of $r_c$), which is inconsistent with the notation defined in Table 1.

A typographical error ("desnity" instead of "density") appears in the cost derivation, which is a minor but visible lapse.

Suggestions for Improvement

• Restrict the domain of all graphs to $t > 0$ (or at minimum $t \geq 0.1$ to match the tabulated data). Never plot meaningless values of the independent variable without explicit acknowledgement and justification.
• Add clear axis labels (variable name and unit) to every graph.
• Maintain consistent notation throughout: choose one symbol set and apply it in every equation, graph label, and sentence.

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Criterion C — Personal Engagement

Score: 2 / 3

What Was Done Well

The personal motivation for this topic is genuine and convincingly articulated. The student explains that they were assigned to make cake pops for their cousin's baby shower — a specific, real situation — and identified a practical mathematical problem within that experience. The connection between personal interest in baking and curiosity about mathematical modelling is expressed clearly and early in the introduction. The cost analysis in Section 5 represents an extension of the basic model that goes beyond what was strictly required and reflects the student's own practical concern about material costs for a large batch of cake pops.

What Is Missing or Underdeveloped

The exploration largely follows a predictable modelling framework — define variables, write a formula, differentiate, tabulate, graph — without significant moments of independent mathematical curiosity or divergence from this template. There is no evidence of the student testing a prediction, being surprised by a result, or exploring an unexpected direction. The acknowledgement that $V_s(t)$ has no internal minimum (because the derivative is always positive) is presented as a straightforward mathematical fact, but it represents an opportunity the student does not fully exploit: a genuinely curious student might pause here to ask why this happens geometrically (it is because adding any layer to a sphere always increases the surface area, hence always increases volume), or explore what this says about spherical versus non-spherical shapes.

The exploration does not demonstrate the student looking at the problem from different mathematical perspectives. For instance, considering how the ratio $V_s / V_{total}$ (chocolate fraction) changes with $t$ — an efficiency metric the student could have invented — would represent genuine independent thinking. The range 0.4–0.5 cm is identified as practical but the reasoning for this range combines mathematical and aesthetic/practical criteria in a way that is not fully distinguished.

Suggestions for Improvement

• Develop a mathematical question that arose unexpectedly during the investigation and show how it was pursued.
• Consider the chocolate-to-cake volume ratio as an efficiency measure and investigate how it changes with $t$.
• Explore the problem for more than one value of $r_c$ and compare how the optimal thickness changes — this would show genuine inquiry and independent direction.

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Criterion D — Reflection

Score: 2 / 3

What Was Done Well

The exploration contains dedicated sections for conclusion, evaluation, and extension (Sections 6, 7, and 8), and these sections are substantive rather than perfunctory. The student correctly identifies and explains a mathematically significant result: the minimum is not a turning point but a boundary solution imposed by the real-world coverage constraint. This is a genuine piece of critical reflection on the nature of the optimisation, linking the mathematical behaviour of the function to the physical limitations of the model. The limitations listed in Section 7.2 are relevant — uniform coating assumption, perfect sphericity, omission of temperature and viscosity effects, scaling variability — and represent an honest assessment of where the model falls short. The extensions in Section 8 are specific and connected logically to the identified limitations (e.g., modelling ellipsoidal shapes addresses the perfect sphericity assumption; investigating chocolate type density addresses the single-density assumption).

What Is Missing or Underdeveloped

Reflection is largely concentrated in the dedicated evaluation section (Section 7) and does not appear consistently throughout the body of the exploration. A more accomplished exploration would include moments of critical reflection at each stage: after deriving $V_s(t)$, one might ask what the formula reveals about scaling behaviour; after seeing the table, one might note that the increments between rows are not constant, and explain why. These in-body reflective moments are largely absent.

The student does not reflect on whether $r_c = 2.0$ cm is the right choice or what the model would predict for different core sizes, despite the research question explicitly referencing "fixed core radii" (plural). Investigating the sensitivity of the optimal thickness to the choice of $r_c$ would constitute meaningful reflection on the robustness of the result. The claim that "a coating thickness of 0.4 cm is the minimum thickness possible" for full coverage is stated as a fact drawn from the table but is never critiqued — the student does not reflect on the fact that the 0.4 cm result is a consequence of the 0.1 cm table resolution and may not represent the true mathematical minimum.

The extension section lists possibilities but does not develop any of them, even partially. Showing even one short calculation or diagram for an extended scenario would elevate the reflection significantly.

Suggestions for Improvement

• Integrate reflective commentary at each major step, not only in the evaluation section.
• Reflect explicitly on the sensitivity of the result to the choice of $r_c$: repeat the analysis for one or two other values of $r_c$ and compare.
• Critically examine the 0.4 cm finding: acknowledge that this is table-resolution dependent and discuss what a more refined method would yield.
• Develop at least one extension scenario with an actual calculation rather than listing it as a bullet point.

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Criterion E — Use of Mathematics

Score: 3 / 6

What Was Done Well

The student correctly derives the central formula for the volume of the chocolate shell as a spherical shell:

$V_s(t) = \frac{4}{3}\pi\left[(r_c + t)^3 - r_c^3\right]$

The expansion of this expression is performed correctly:

$V_s(t) = 4\pi r_c^2 t + 4\pi r_c t^2 + \frac{4}{3}\pi t^3$

The differentiation $\frac{dV_s}{dt} = 4\pi(r_c + t)^2$ is correct, and the student correctly recognises that this derivative is strictly positive for all $t > 0$, establishing that $V_s(t)$ is monotonically increasing on the relevant domain. The conclusion — that the minimum occurs at the smallest feasible thickness, not at a turning point — is mathematically correct and represents the conceptually most important result of the exploration. The cost derivation in Section 5 is correctly structured: volume is calculated from mass and density, cost per unit volume is derived, and the cost per cake pop is computed by multiplication. The numerical values in Table 2 are consistent with the formula when checked.

Errors and Significant Weaknesses

1. The central optimisation argument has a fundamental logical gap.

The research question asks which thickness "optimises the total amount of chocolate required while fully coating the cake pop." Full coverage is the binding constraint, but the student never provides a mathematical definition of what constitutes full coverage — that is, the value of $t_{min}$ is never derived. The student states that "from the table, 0.4 cm is that value and therefore the minimum thickness possible," but offers no mathematical justification for why 0.4 cm achieves full coverage. The table simply lists volumes for thicknesses from 0.1 cm to 1.0 cm in steps of 0.1 cm; it contains no coverage criterion. Without a mathematical definition of $t_{min}$ — derived from physical considerations, literature values, or a geometric argument — the claim that $t = 0.4$ cm is optimal is an assertion, not a conclusion. This is the most significant mathematical weakness in the exploration.

2. The derivative graph contradicts the written argument.

The student claims: "The derivative is always positive; hence the graph lies entirely above the x-axis, proving that the volume of chocolate is continuously growing with thickness." The graph plotted in Figure 3 (the derivative graph), however, shows the curve descending toward zero near $t \approx -2$ — the minimum of the parabola $4\pi(r_c + t)^2$ at $t = -r_c$. Because the domain is not restricted to $t > 0$, the graph does not "lie entirely above the x-axis" within the plotted range. The student's verbal claim is correct for $t > 0$, but the graph as presented does not support it. Using a graph as evidence for a mathematical claim when the graph contradicts that claim is a failure of mathematical rigour.

3. Inaccuracy in the density calculation.

The student computes the volume of one bar of chocolate as:

$V = \frac{225 \text{ g}}{1.24 \text{ g/cm}^3} = 180 \text{ cm}^3$

The correct result is $225 \div 1.24 = 181.45 \text{ cm}^3$, which the student rounds to 180 cm³. While the error is small in absolute terms, the student does not acknowledge this rounding, and 180 cm³ is not a correctly rounded value to 3 significant figures (which would be 181 cm³). All subsequent cost calculations inherit this inaccuracy.

4. Limited mathematical sophistication for AA SL.

The calculus content is restricted to differentiating a single cubic polynomial and observing the sign of the result. While this is mathematically correct, it represents a minimal deployment of the calculus tools available at AA SL. There is no integration, no second derivative test, no multi-variable analysis, no exploration of related rates or other calculus concepts. The polynomial expansion of $(r_c + t)^3$ is carried out correctly but is a standard binomial expansion. For AA SL, examiners expect mathematics that is "commensurate with the course" — this exploration uses tools from Topic 5 (Calculus) and Topic 3 (Geometry), but at a depth that falls short of what a thorough AA SL treatment would demonstrate.

5. Minor notation inconsistency.

The derivative is written in the body text as $\frac{dV}{dT} = 4\pi(r+t)^2$, using $V$ (not $V_s$), $T$ (not $t$), and $r$ (not $r_c$) — all of which are inconsistent with the notation established in Table 1.

Suggestions for Improvement

• Define $t_{min}$ mathematically. One approach: cite a source for the minimum practical coating thickness for chocolate confectionery, or derive it from a physical argument about the grain size of chocolate or minimum viscous layer depth.
• Restrict the domain of all plotted functions to $t > 0$ and ensure graphical evidence is consistent with claims made in text.
• Carry out the density division to the appropriate number of significant figures and acknowledge any rounding applied.
• Consider investigating how $V_s / V_{total}$ (the chocolate fraction) varies with $t$, or exploring the model for multiple values of $r_c$ using a general formula rather than a single substituted case. Both would add mathematical depth.
• Use the second derivative $\frac{d^2V_s}{dt^2} = 8\pi(r_c + t)$ to confirm that the rate of increase is itself increasing (confirming convexity), which would strengthen the argument about the accelerating cost of thicker coatings.

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Overall Summary

The exploration addresses a well-chosen, personally meaningful topic and demonstrates a sound understanding of the core geometric model. The derivation of the spherical shell volume formula and the monotonicity argument are mathematically correct, and the student shows commendable insight in recognising that the minimum is a boundary solution rather than a turning point. The structure is clear and the use of multiple representational forms (diagrams, equations, tables, graphs) is appropriate.

However, the exploration is held back by three principal weaknesses. First, the central optimisation claim — that $t = 0.4$ cm is the optimal thickness — is never mathematically justified because no criterion for "full coverage" is defined; this is a fundamental gap in the rigour of the investigation. Second, the derivative graph plots a domain that includes physically meaningless negative values of $t$, and the student's interpretation of this graph contradicts what the graph actually shows, representing a failure of mathematical precision. Third, the depth of calculus employed is modest relative to what is expected of AA SL: differentiating a single cubic and reading a table is a limited use of the tools available.

A score of 12 out of 20 is awarded, corresponding to an estimated grade boundary of 4–5. To move to the 5–6 band, the student should focus on: (1) establishing a rigorous mathematical definition of $t_{min}$; (2) correcting and restricting the derivative graph; and (3) deepening the mathematical analysis, for example by exploring the model across a range of $r_c$ values or introducing the second derivative to discuss convexity.

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Assessed against the IB Mathematics Internal Assessment criteria (Criteria A–E) as specified in the IB Mathematics: Analysis and Approaches subject guide.