IB Mathematics AA HL — Internal Assessment Grading Report

Title: Calculating the surface area and volume of a solid revolution of a copper pitcher Course: Mathematics: Analysis and Approaches (AA) Higher Level Session: May 2026 Word Count: 1570 Examiner Report Date: April 2026

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Summary Score Table

Criterion	Score	Max	Descriptor Level
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: 5 (range 12–14 on the IB 1–7 scale)

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Criterion A — Presentation: 3 / 4

What the student did well

The introduction is the strongest part of the presentation. The student establishes a clear and convincing personal rationale: the copper pitcher belonged to their grandmother in Lebanon, and they want to calculate the surface area to determine the cost of applying beeswax as a preservative. This context gives the exploration genuine direction and purpose. The exploration follows a logical sequence — introduction, methodology, volume calculation, surface area calculation, beeswax cost, and conclusion — and is easy to navigate. The word count of 1570 is appropriate and the writing is generally focused. The photographs of the physical object, annotated with measurements, are effective in grounding the mathematics.

What is missing or underdeveloped

The exploration lacks an explicitly stated aim. The purpose is implied throughout the introduction but is never set out as a formal aim (for example: "The aim of this exploration is to model the copper pitcher as a solid of revolution and calculate its volume and surface area"). An IB examiner expects a clearly distinguished aim statement. Additionally, there is no brief outline at the start of how the aim will be achieved; the transition to the Methodology section is abrupt. The conclusion restates results but does not fully circle back to evaluate whether the aim was achieved with the precision that was intended.

Suggestions for improvement

• Add a single, explicitly labelled aim statement at the end of the introduction (one to two sentences).
• Include a short roadmap paragraph after the aim, previewing the mathematical tools that will be used.
• In the conclusion, explicitly revisit the aim ("The aim was to... The model achieved this by...") before moving to limitations.

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Criterion B — Mathematical Communication: 3 / 4

What the student did well

The student uses correct integral notation for both the volume and surface area formulas, presenting them in display-math format:

$V = \pi \int_a^b [g(x)]^2 \, dx \qquad S = 2\pi \int_a^b f(x)\sqrt{1 + (f'(x))^2} \, dx$

All functions are given with explicit domain restrictions, which is essential for piecewise-defined models and is handled consistently throughout. Figure 1 is labelled and the axes are identified ($y$ in cm for radius, $x$ in cm for height). The use of multiple forms of representation — photographs, GeoGebra coordinate graphs, individual function expressions, and computed numerical results — demonstrates awareness of the communicative value of varied representations. The derivative computations for the surface area integrands are set out and are mostly legible.

What is missing or underdeveloped

A persistent and significant notation issue runs through the surface area section: functions are written using programming-style exponentiation ($x\hat{}(3)$, $x\hat{}(2)$) rather than standard mathematical superscript notation ($x^3$, $x^2$). For example:

$c_2(x) = 0.79403x\hat{}(3) - 3.27914x\hat{}(2) + 3.40489x + 1.58022$

This is GeoGebra output that has been pasted without reformatting. An IB exploration must use standard mathematical notation throughout; copied software output does not meet this requirement. In addition, the volume formula is annotated with units "$\text{cm}^2$" where "$\text{cm}^3$" is required. No summary table is provided for the individual section volumes or surface areas; such a table would greatly clarify the piecewise structure and make the results easier to audit. Some variables (e.g. $t_1$, $g_2$, $p(x)$) are introduced without explanation of the naming convention.

Suggestions for improvement

• Reformat all GeoGebra-generated functions into proper mathematical notation before including them in the exploration.
• Correct the volume unit to $\text{cm}^3$ in the formula and all subsequent results.
• Add a results table listing each section, its function type, its domain, and its individual volume and surface area contribution.
• Briefly define the naming scheme for functions ($g$, $h$, $t_1$, etc.) so the reader can follow the structure.

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Criterion C — Personal Engagement: 2 / 3

What the student did well

The personal motivation is authentic and present throughout the exploration, not merely in the introduction. The choice of object — a specific pitcher inherited from their grandmother — gives the work genuine ownership that is perceptible to the reader. The student makes independent mathematical decisions: choosing to segment the pitcher into six or more sections rather than forcing a single regression model, selecting a logistic function for the wide-belly section rather than defaulting to another polynomial, and applying the surface area result to a real purchasing calculation using a specific commercial beeswax product (DICTUM). These decisions reflect initiative rather than mechanical execution of a textbook procedure.

What is missing or underdeveloped

The exploration is largely linear: the student completes the stated task and stops. There is no evidence of mathematical wondering beyond the original aim — for example, the student does not ask whether a different segmentation would give a materially different total, does not compare the computed volume to the manufacturer's stated capacity, and does not test whether a single high-degree polynomial over the full domain would give a reasonable approximation. Making and testing a prediction ("I expect the volume to be close to the 1-litre specification often cited for pitchers of this size") would significantly raise the engagement score. The volume calculation for the liquid capacity (noting that the top section should be excluded) is mentioned but not fully separated mathematically from the outer-surface calculation.

Suggestions for improvement

• Compare the computed volume to a manufacturer-stated or physically measured capacity, and discuss any discrepancy.
• Test an alternative model (e.g., a single degree-7 polynomial over the full domain) and compare results to the piecewise model to justify the chosen approach.
• Explicitly address the excluded handle: even a rough geometric estimate (e.g., treating the handle as a half-torus or a rectangular prism) would show mathematical curiosity.

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Criterion D — Reflection: 2 / 3

What the student did well

The Conclusion and Evaluation section identifies the most significant sources of error in a specific and mathematically grounded way. The student notes that the pitcher is not perfectly symmetrical (visible from the hammered surface), that the handle was excluded, and that manual placement of points along the profile introduces positional error that propagates through the regression and integration. The suggestion to measure the internal contour directly (to avoid the wall-thickness error in the volume calculation) is a concrete and appropriate improvement. The results are connected to the original aim — the cost of the beeswax is calculated and the practicality of the result ($0.01) is briefly discussed.

What is missing or underdeveloped

Reflection is confined almost entirely to the conclusion. An IB exploration at the higher levels is expected to carry reflective commentary throughout — for instance, noting why a logistic function was more appropriate than a polynomial for a particular segment, or observing that the sharp curvature near the base forced a very short domain for $g(x)$. None of the individual regression fits is assessed for quality: no $R^2$ or residual discussion appears anywhere. The student does not compare the computed volume to real-world data (e.g., the pitcher is marketed as holding approximately 1 litre; 380 cm$^3 \approx 0.38$ L, which is substantially less than expected and deserves comment). The significance of the near-equal values of $V_{\text{tot}}$ and $S_{\text{total}}$ (numerically close but in different units) is not remarked upon.

Suggestions for improvement

• Add brief in-text reflective sentences each time a modelling choice is made (e.g., "A logistic model was chosen here because the profile increases rapidly and then plateaus, matching the sigmoidal form").
• Include $R^2$ values or residual plots for at least the key regression functions to validate the models.
• Compare the computed liquid volume to the pitcher's expected capacity and discuss the discrepancy.
• Reflect on what the near-equality of volume and surface area values implies (or does not imply) physically.

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Criterion E — Use of Mathematics: 4 / 6

What the student did well

The mathematical content is clearly at the AA HL level. The volume of a solid of revolution,

$V = \pi \int_a^b [f(x)]^2 \, dx$

and the surface area of a surface of revolution,

$S = 2\pi \int_a^b f(x)\sqrt{1 + [f'(x)]^2} \, dx$

are both AA HL topics, and the student applies both correctly in concept. The use of a logistic function,

$t_1 = \frac{3.4314}{1 + 1.0138e^{-1.3022x}}$

to model the pitcher's widest section — where a polynomial would not reflect the natural sigmoidal profile — is mathematically sophisticated and shows awareness of function types beyond the standard polynomial toolkit. The differentiation of the logistic function using the chain rule is carried out and substituted into the surface area integrand, demonstrating understanding of how the formula works rather than blind substitution. The piecewise approach across six or more segments with different function types (cubic, quartic, degree-5, logistic) reflects genuine mathematical thinking about the shape of the object.

Errors and weaknesses

Significant error — Volume summation: The total volume is stated as:

$V_{\text{tot}} = 0.013 + 2.734 + 122.914 + 44.173 + 113.387 + 22.67831 + 74.43622 = 380.33553 \text{ cm}^3$

The value $122.914$ does not correspond to any individually computed section volume presented in the exploration. The six labelled section volumes are $V_1 = 0.013$, $V_2 = 2.734$, $V_3 = 22.678$ ($t_1$), $V(\text{p}) = 74.436$ ($g_2$), $V_4 = 44.173$ ($q$), $V_5 = 113.387$ ($r$), and $V_6 = 116.543$ ($s$). The value $V_6 = 116.543$ appears to have been replaced by the unexplained $122.914$ in the final sum. The correct total using all stated individual values would be approximately $373.97 \text{ cm}^3$, not $380.34 \text{ cm}^3$. This is a non-trivial error in the central result of the exploration.

Unit error: The volume formula is annotated "$\text{cm}^2$" rather than "$\text{cm}^3$."

Lack of model validation: No goodness-of-fit measure ($R^2$, residual plot, or visual overlay of curve and data points) is provided for any of the thirteen regression functions. Without this, the accuracy of the entire modelling exercise cannot be assessed.

Technology dependence without justification: The polynomial degree for each segment was selected by "trial and error" in GeoGebra. For an HL exploration, the student should provide a mathematical rationale: the degree should match the number of inflection points visible in the profile, or be justified by comparing $R^2$ values for successive degrees.

Minor inconsistency in price reporting: The introduction states the DICTUM beeswax costs EUR 13.20 per 500 g; the cost calculation on the final page uses EUR 13.50. The final numerical answer changes slightly as a result.

Suggestions for improvement

• Audit the volume summation and identify where 122.914 originated; correct the total using $V_6 = 116.543 \text{ cm}^3$.
• Include $R^2$ values from GeoGebra for each fitted function.
• Justify the polynomial degree for each segment with a brief mathematical argument (number of turning points in the profile, or comparative $R^2$ across degrees).
• Correct all unit labelling to ensure consistency ($\text{cm}^3$ for volume).
• Reconcile the beeswax price stated in the introduction with the value used in the calculation.

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Overall Summary

This is a well-motivated and mathematically ambitious exploration. The personal context is compelling, the choice of mathematical tools (solids of revolution, surface area of revolution, logistic regression) is appropriate and challenging for AA HL, and the piecewise approach to modelling a non-standard shape shows genuine mathematical initiative. The two strongest aspects of the work are the authenticity of the personal engagement and the appropriate use of HL-level integration techniques including the surface area formula.

The two areas requiring most attention are mathematical rigour and communication precision. The error in the total volume summation undermines the reliability of the central quantitative result, and the absence of any goodness-of-fit analysis means the accuracy of the regression models is entirely unvalidated. On the communication side, the persistent use of programming notation in the surface area functions ($x\hat{}(2)$, $x\hat{}(3)$) is a recurring issue that must be corrected. Addressing these two areas — verifying the summation and providing at least a brief model-validation step — would move this exploration from a solid 5 into the 6 range.

Predicted IB grade: 5 (Total: 14 / 20)

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Report prepared using IB Mathematics IA assessment criteria (2021 specification onwards).