IB Mathematics Analysis and Approaches (AA) SL Internal Assessment Exemplar: Modelling a beauty blender and finding its volume

The exploration opens with a clear introduction and stated personal context, which helps the reader understand the purpose early. This supports coherence because the aim is established before the modelling work begins.

The graph of the data points helps the reader see the overall shape before the calculus begins. That visual anchor supports the written modelling process and makes the exploration feel more organised.

The main body would be clearer if the student briefly summarised all piecewise functions and their intervals in one place, rather than sending most of that information to the appendices. A compact summary table would make the model easier to follow.

The sectioning is effective: the student separates modelling, volume, surface area, experimental testing, and evaluation into distinct parts. That structure makes the investigation easy to follow and shows clear logical progression.

Several steps are repeated in a way that adds length without adding new development. The student should condense repeated phrases such as 'I did the same process' and replace them with a brief explanation of what changes from one section to the next.

The total volume is stated clearly, but the model is not yet fully convincing because the section-by-section calculations are not sufficiently connected back to the full shape. The large difference from the experimental value should be used to test the logic of the model, not just reported after the fact.

The experimental section is placed after the theoretical work, which creates a sensible model-then-check sequence. This ordering strengthens the overall narrative because the reader can see the calculation being tested against real data.

The conclusion would be stronger if it explicitly answered the original aim with a final judgement about reliability, not just a restatement of the numerical results. The student should end by saying what the investigation proves and what remains uncertain.

The student uses appropriate mathematical language in context, linking the real object to modelling, calculus, and measurement. The terminology is suitable for an SL exploration and generally supports clear communication.

The plotted coordinate system and labeled points provide a useful representation of the object’s profile. This helps connect the numerical measurements to the model being built.

The notation becomes inconsistent when the student refers to 'g(x)' here but later works with 't(x)'. The same function names should be used consistently so the reader can track which expression is being integrated.

The expression $$t(x)=y=0.61x^3-1.53x^2+1.87x$$ is a rounded version of the fitted function, but the student should be clear that later calculations are based on this approximation. Making the approximation explicit helps avoid confusion about where small numerical differences come from.

The student correctly introduces the solid of revolution approach and uses integrals with bounds to represent the model mathematically. This shows a sound grasp of the relevant calculus language.

The integration work is readable, but the algebraic steps are presented without clearly stating where the calculus result comes from. The student should label the antiderivative step more explicitly so the reasoning is easier to follow.

The student includes both volume and surface area formulas, showing that the mathematics is being communicated in more than one form. That supports readability because the reader can see the method before the computation begins.

The volume formula is stated as $$V=\pi \times \int y^{2}dx$$, but elsewhere the text suggests confusion between $$\pi$$ and $$2\pi$$. The student should distinguish clearly between the volume formula for a solid of revolution and the surface area formula, since they are different.

It would improve communication if the student included a short summary table of each section's volume and the interval used. That would make the overall calculation easier to verify at a glance.

The derivative is written incorrectly as $$y^{\prime }=3\times 0.31x^{3-1}-2\times 1.53x^{2-1}+1\times 1.87x^{1-1}$$ because the coefficient rounding and final simplification are not matched cleanly. The student should write the derivative directly from the chosen polynomial to avoid ambiguity.

The experimental section uses the unit ratio $$300\frac{\text{ml}}{{\text{cm}}^3}$$, which is not appropriate notation for a volume measurement. The student should write the water volume simply in ml or cm${}^3$, not as both at once.

The equation for the size change is written as a multiplication of length and width differences, but it does not directly represent volume. The student should be careful to distinguish between a 2D change in footprint and a 3D volume measurement.

The student shows clear independent motivation by linking the topic to personal interest in makeup and daily life. This personal context gives the investigation purpose and helps justify why the modelling is worth doing, rather than appearing as a purely abstract calculus exercise.

The student demonstrates initiative by taking their own measurements and cutting the beauty blender to measure its dimensions directly. This makes the exploration genuinely self-generated and supports a more authentic modelling process.

The student could strengthen the independent thinking further by explaining why a polynomial model was preferred over other possibilities, such as splines or a simple geometric solid. Justifying the modelling choice would show more deliberate decision-making.

What made the student confident that rotating the profile around the axis would give a realistic model of the beauty blender, given that the object is compressible and not perfectly symmetric?

The experimental water-displacement test is a strong sign of initiative because it adds an independent check on the model rather than relying only on the fitted functions and GDC output.

The uncertainty diagram shows that the student is thinking about the reliability of the fitted model, not just the final answer. That awareness of error sources is an important part of independent mathematical investigation.

The student could deepen creativity by testing an improved version of the model after noticing the discrepancy between modelled and experimental volume. For example, adjusting the curve segments or comparing with a simpler geometric approximation would show more iterative thinking.

How reliable is the water-displacement method for a porous sponge that can absorb liquid and change shape when squeezed? That issue should be considered before treating the experimental value as a direct measure of volume.

The reflection would be stronger if the student linked each limitation to a specific consequence on the final volume or surface area. For example, they could explain whether point placement error would increase or decrease the estimated volume.

The student does well to identify multiple limitations, including measurement uncertainty and imperfect curve fitting. This is a good starting point for reflection because it shows awareness that the model is only an approximation.

The conclusion evaluates the result only in a broad way and does not really reconcile the large gap between the calculated and experimental volumes. A stronger evaluation would explain what the difference means for the validity of the model.

The student appropriately suggests future improvements such as using more than one beauty blender and improving measurement precision. These are realistic follow-up steps because they target the main weaknesses in the data collection.

The final statement about taking the mean of the theoretical and experimental values is not justified. A mean does not automatically produce a more accurate result, especially when the two methods measure different things and disagree by such a large amount.

The expanded square is written in a way that is hard to verify manually and includes repeated terms before simplification. The student should present the simplified result more directly, especially when technology is later used for the integration.

The choice of solids of revolution is highly relevant to the aim of estimating the beauty blender’s volume. This is appropriate modelling mathematics because it turns the profile into a measurable three-dimensional object.

The student uses technology-assisted polynomial fitting to build a workable model from measured points. That is a suitable SL-level method and shows good use of mathematics to represent a real shape.

The exploration applies calculus effectively across multiple sections and then combines the partial results into a total volume. This shows that the mathematics is being used for the central purpose of the task rather than as a side calculation.

If the modelled volume is about $$18.01{\text{cm}}^3$$ but the experimental value is about $$38{\text{cm}}^3$$, what does that suggest about the accuracy of the curve fit or the assumptions behind the model?

A brief note explaining that the GDC is being used to evaluate the definite integrals would improve transparency. This would make it clearer which parts of the process are hand-calculated and which are technology-assisted.

The surface area result is reported as $$3.91{\text{cm}}^2$$, but for a surface area calculation the student should be especially careful to show how the integrand and bounds produce that value. More working would make the result mathematically more convincing.

The surface area integrand is not fully explained before evaluation, so the reader has to infer how $$2\pi f(x)\sqrt{1+(f^{\prime }(x){)}^2}$$ was assembled. More explicit setup would show stronger understanding of the formula’s components.

The experimental calculation is connected to the mathematical aim of the exploration, so the volume estimate is not isolated from the investigation. That keeps the mathematics focused and relevant to the task.

The percentage change in dimensions is noted qualitatively, but the effect on the volume estimate is not quantified. The student should connect the change in length and width to the modelling error more directly.

The derivative here is written as $$y^{\prime }=-0.28+0.82$$, which omits the variable term. It should be $$y^{\prime }=-0.28x+0.82$$; otherwise the subsequent squared expression is not justified correctly.