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Mathematics Analysis and Approaches (AA) IA Exemplar: Optimum Solar… | RevisionDojo
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IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example
Maximising Efficiency of Solar Panels: Investigating the Optimum Tilt AngleHL
5
Official IB Result
12/20
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12/20
0
10
20
5.1·Weakness
Page 10• Click to view
The argument that −e−h/cos(α)=0 is “never happens” is correct, but it would be stronger with a brief justification rather than a bare statement. The student should explain that an exponential expression is always positive, so it cannot equal zero.
5.2·Suggestion
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The reflection on the optimum incidence angle would be stronger if the student added a brief comparison with the physical meaning of α=0. For example, explaining why this corresponds to perpendicular sunlight would make the conclusion feel more analytically integrated.
5.3·Suggestion
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The final tilt expression would be more persuasive if the student briefly checked it against the earlier maximum condition by substitution. A quick verification would help the reader see that the geometry and calculus conclusions are consistent with one another.
5.4·Weakness
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The claim that the average annual optimum tilt is equal to the latitude is presented as a result, but there is no explicit justification for why the averaging process leads to that outcome. The student should show the reasoning or at least a short verification step so the conclusion feels earned rather than asserted.
5.5·Suggestion
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The conclusion would be stronger if it distinguished between the mathematically derived optimum and the practical recommendation for a fixed domestic panel. That distinction would show deeper evaluation of what the model says versus what can realistically be implemented.
5.6·Weakness
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The limitations section identifies important real-world factors, but it remains descriptive rather than analytical. The student should explain how each limitation could change the result, for example whether cloud cover, orientation, or degradation would shift the optimum tilt or simply reduce overall output.
5.7·Suggestion
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The future exploration section would benefit from one or two concrete next steps with measurable aims, such as comparing the model with a solar calculator or testing sensitivity to latitude and season. Specific next steps would make the reflection feel more actionable.
Criteria A: Presentation
3/4
0
2
4
Criteria Strands
A.1Coherence and logical development
Good
A.2Organization and structure
Good
A.3Conciseness and relevance
Moderate
Criteria Feedback
Your exploration is logically developed and the main argument is easy to follow from the initial aim through to the final recommendation.
You use a clear structure with an introduction, development, and conclusion, which helps the reader navigate the work.
Most of the material stays relevant to the investigation and supports the central question.
A few parts of the writing interrupt the flow, including an incomplete statement and some reliance on an appendix for key algebra.
Some sections are longer than necessary and repeat ideas rather than moving the argument forward efficiently.
The layout of the long technical sections makes the work feel less streamlined than it could be.
1.1·Strength
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The methodology section gives the reader a useful roadmap by naming the main tools in advance: trigonometry, geometry, and calculus. That upfront signposting improves coherence because the subsequent sections feel purpose-driven rather than abrupt.
1.2·Suggestion
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The rationale is personally relevant, but it would be stronger if the student added a more specific research context, such as the intended location, roof constraints, or expected panel use. That would make the motivation feel more grounded in a real decision rather than a general interest in solar energy.
1.3·Strength
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The transition into the angle-relationship section is clear and logical. The student explains why a geometric diagram is needed before introducing the algebraic relationship, which helps the reader understand the purpose of the next steps.
1.4·Strength
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Using Desmos to plot declination across the year is a good organisational choice because it makes the seasonal pattern immediately visible. This supports the written explanation by converting a formula into an interpretable representation.
1.5·Suggestion
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The interpretation of the seasonal graph would be stronger if the student explicitly stated the key numerical trend, not just the general pattern. For example, identifying the approximate range between winter and summer tilt would make the discussion more concise and informative.
1.6·Strength
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The bibliography is clearly separated from the main exploration, which supports overall structure. Keeping references together in one place helps the reader distinguish between the mathematical argument and source material.
Criteria B: Mathematical Communication
3/4
0
2
4
Criteria Strands
B.1Mathematical language and notation
Good
B.2Multiple representations
Good
B.3Logical structure and clarity
Good
Criteria Feedback
You use mathematical language and notation mostly consistently, and your variables are usually defined before they are used.
You include several appropriate representations, including diagrams, graphs, and a table, which support the written explanation.
Your mathematics is organised in a logical sequence, so each step generally follows from the last.
Some derivative and algebra notation is difficult to read because symbols and factors are not always grouped clearly.
A few diagrams and graphs are more descriptive than analytical, so they could be used more strongly to justify results.
There are occasional formatting and phrasing issues that reduce clarity, especially in the longer worked sections.
2.1·Strength
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The variable definitions are explicit and helpful: the student defines I, I0, and α before using them in later derivations. This is good mathematical communication because it reduces ambiguity and allows the reader to follow the model independently.
2.2·Suggestion
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The constant of integration is identified, but the link to I0cos(α) would be clearer if the student wrote the substitution explicitly on the next line. A short sentence showing how the constant is absorbed into the physical model would improve transparency.
2.3·Weakness
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The derivative notation is still inconsistent here: writing dAMdI=−hI is clear, but later lines sometimes attach an extra I to the derivative symbol. The student should keep the left-hand side in a single standard form throughout to avoid suggesting multiplication where none is intended.
2.4·Weakness
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The simplification is broadly heading toward the right structure, but the notation remains difficult to parse because the exponent, sine factor, and denominator are not grouped consistently. Clear parentheses and a consistent exponent form would make the final derivative much easier to trust.
2.5·Strength
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The relationship α+β+θ=90∘ is communicated clearly and is exactly the kind of compact notation that supports the later derivation. This is effective mathematical communication because it summarises the geometric idea efficiently.
Criteria C: Personal Engagement
2/3
0
2
3
Criteria Strands
C.1Independent thinking
Good
C.2Personal approach
Good
C.3Creativity and initiative
Good
Criteria Feedback
Your motivation is clearly personal, especially in the way you connect the topic to your family’s interest in solar panels.
You show initiative by going beyond a purely symbolic answer and creating graphs and a seasonal summary to make the result more usable.
You combine different mathematical ideas independently to build your model rather than simply following one formula.
The personal context is introduced well, but it is not carried through strongly enough in the rest of the investigation.
The modelling choices are interesting, but they are not pushed far enough into tailored or highly original extensions.
There is room for more evidence that you shaped the investigation around a specific real-world situation rather than a general solar model.
3.1·Strength
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The student shows a personal stake in the topic by connecting the investigation to the family’s solar-panel plans. This makes the exploration feel purposeful, and it gives the modelling choices a clear practical reason beyond abstract mathematics.
3.2·Strength
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The student shows initiative by creating a graph of optimum tilt angle over the year rather than stopping at a symbolic formula. That extra step makes the investigation more personally useful and demonstrates independent follow-through.
Criteria D: Reflection
1/3
0
2
3
Criteria Strands
D.1Depth of reflection
Poor
D.2Critical analysis
Poor
D.3Evaluation of outcomes
Poor
Criteria Feedback
You do include some reflection on limitations, such as idealised weather assumptions and panel orientation constraints.
You briefly evaluate the usefulness of the results by relating them back to the practical goal of choosing a tilt angle.
You show some awareness that not all mathematical results transfer perfectly to real-life conditions.
The reflection is mostly descriptive rather than deeply analytical.
There is limited discussion of how strong the model is or how sensitive the result might be to the assumptions you made.
The implications of the final recommendation are mentioned, but not explored in enough detail to show comprehensive evaluation.
Criteria E: Use of Mathematics
3/6
0
3
6
Criteria Strands
E.1Relevance of mathematics
Good
E.2Level appropriateness
Good
E.3Understanding and accuracy
Moderate
Criteria Feedback
You use mathematics that is directly relevant to the problem, including trigonometric modelling, exponential decay, and calculus-based optimisation.
You show a solid understanding of the methods you use, especially when solving the differential equation and interpreting the stationary point.
You keep the mathematics focused on the central aim, so the techniques support the investigation rather than sitting beside it as separate exercises.
Some algebra and derivative working is hard to follow because the notation is not always presented cleanly.
A few statements in the explanation are incomplete or not fully checked, which weakens confidence in parts of the argument.
The later worked sections become visually dense, which makes it harder to verify the mathematics line by line.
4.1·Strength
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The student states a clear mathematical aim — finding the optimum tilt angle to maximise efficiency — and keeps the whole exploration directed toward that goal. This focus helps the later calculus, geometry, and seasonal analysis remain relevant to one another rather than becoming separate exercises.
4.2·Strength
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The student appropriately introduces Lambert’s Cosine Law as the starting point for the irradiance model. This is a mathematically relevant choice because it directly links the panel’s incidence angle to received energy.
4.3·Suggestion
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The air-mass definition would be stronger if the student briefly stated the modelling assumptions and limitations of AM=cos(α)1, especially for large incidence angles. That would show deeper awareness of when the approximation is reasonable.
4.4·Strength
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The separation of variables is handled correctly and shown step by step. This is a strong use of mathematics because the student demonstrates the mechanism behind the exponential attenuation model rather than only stating the result.
4.5·Weakness
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The product-rule working is difficult to verify because the derivative appears with inconsistent placement of symbols and factors. The student should present this step more cleanly, with one clear line for the rule and one line for the substituted derivatives, so the algebra can be checked more reliably.
4.6·Strength
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The student correctly identifies sin(α)=0 as the meaningful stationary condition and links it to α=0. That is a strong mathematical interpretation because it shows the student is not merely manipulating algebra but also reading the solution in context.
4.7·Strength
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The second-derivative check is used appropriately to confirm that the stationary point is a maximum. Even though the algebra is long, the student uses the test for its intended purpose, which strengthens the validity of the conclusion.
4.8·Strength
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The student successfully brings in the solar elevation formula to connect the geometric result to real seasonal variation. This is a strong modelling move because it turns a general tilt identity into something that can be evaluated for specific days.
4.9·Weakness
Page 18• Click to view
The seasonal table is useful, but the values should be checked for consistency with the stated latitude and the formulas used earlier. If the location is at 38.3458∘ latitude, the student should make sure the table headings and the resulting tilt values are clearly linked to the same assumption set.
