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Mathematics Analysis and Approaches (AA) IA Exemplar: Solar Panel… | RevisionDojo
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IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Exemplar: How can the tilt angle of a solar panel be
optimised in Kenya for a maximum efficiency during the
year?
How can the tilt angle of a solar panel be
optimised in Kenya for a maximum efficiency during the
year?
Well-structured sections with headings, numbered subsections and a conclusion
Clear title and research question effectively frame the exploration
Occasional jumps in derivations and unit conversions without explanation
Verbose personal anecdotes and repeated ideas reduce conciseness
Inconsistent citation formatting and long algebraic blocks lacking interleaved explanatory text
1.1·Strength
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The title and research question are clearly stated and prominently placed, framing the investigation effectively and orienting the reader to the IA’s focus.
1.2·Suggestion
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The extensive personal anecdote adds context but detracts from mathematical focus. Trim to essential motivation to maintain conciseness and relevance.
1.3·Suggestion
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The second paragraph reiterates the IA’s aim but repeats background. Consider consolidating to avoid redundancy and improve flow.
1.4·Suggestion
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The narrative in Section 2.2 is verbose with mixed clause lengths. Breaking it into shorter sentences would improve coherence.
1.5·Strength
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The derivation from sin(α)=sinϕsinδ+cosϕcosδ to α=arcsin(cosδ) follows logical development and is clearly shown.
1.6·Suggestion
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In Section 2.4, “what ensures maximum efficiency” is colloquial. Use precise mathematical phrasing (e.g., “which guarantees maximisation”).
1.7·Weakness
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The opening sentence of Section 3.1 ends mid-clause. Complete the thought to maintain coherence and ensure clarity.
1.8·Suggestion
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Throughout the long algebraic derivations (pp. 11–14), steps are condensed. Interleave brief explanatory text between formulas to guide readers.
1.9·Strength
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The heading for Section 3.3 is explicit and shows strong organizational structure, separating optimization neatly.
1.10·Suggestion
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The table on negative tilt angles is well organized, supporting comparison. Ensure descriptive text explicitly references “Table 1” before its appearance.
1.11·Strength
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The bibliography includes peer-reviewed articles and official data sources, indicating thorough research and credible grounding.
1.12·Suggestion
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Reference entries vary in format and lack consistency (e.g., missing publication years). Standardize citation style for professionalism and clarity.
Criteria B: Mathematical Communication
3/4
0
2
4
Criteria Strands
Good
Mathematical language and notation
Moderate
Multiple representations
Good
Logical structure and clarity
Criteria Feedback
Mostly consistent and correct use of mathematical notation and defined variables
Step-by-step derivations with accompanying explanations provide clear logical flow
Accurate citation of mathematical laws (e.g., linearity of integration, Lambert’s cosine law)
Limited use of multiple representations (no function graphs or plots)
Inconsistent differential notation in integrals and mixing of degrees/radians
Diagrams and outputs lack labels and units
2.1·Suggestion
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In the diagram on Kenyan solar trapping, axes and variables are undefined. Label each element to enhance clarity and mathematical communication.
2.2·Suggestion
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Units for energy terms are omitted in the equation. Include units (e.g., kWh, W/m²) to improve precision and reader understanding.
2.3·Strength
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The statement of Lambert’s cosine law is accurate and the block equation Ecollected=Eincidentcos(θtilt) is well formatted.
2.4·Strength
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The list of factors (irradiance, area, incidence angle) is clearly enumerated, aiding reader comprehension of model variables.
2.5·Suggestion
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The text refers to Python outputs but includes no graphical representation. Incorporate a plot of cos∣α(d)−β∣ versus d to enrich multiple representations.
2.6·Strength
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The linearity of integration property is properly cited with ∫abcf(x)dx=c∫abf(x)dx, reinforcing mathematical communication.
2.7·Suggestion
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The identity cos(α)=1−sin2(α) is invoked without specifying sign branch. Clarify that positive root matches physical context.
2.8·Weakness
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The integrand in the final expression uses dα instead of dd, causing an inconsistency in the integration variable. Correct this for accuracy.
Criteria C: Personal Engagement
2/3
0
2
3
Criteria Strands
Good
Independent thinking
Good
Personal approach
Good
Creativity and initiative
Criteria Feedback
Personal motivation grounded in a visit to Kenya is well integrated
Clear reflection on coding challenges and modelling choices
Use of Python for numerical integration shows initiative
Approach follows standard textbook methods more than offering novel insights
Limited originality in mathematical pathway
Creativity is good but not exceptional
3.1·Strength
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Personal experience visiting Kenya is well integrated, revealing genuine engagement and motivation behind the choice of topic.
3.2·Strength
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The description of personal interests in applying models to real problems is compelling and demonstrates clear personal engagement.
3.3·Strength
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The final reflective link of personal experience to mathematical study effectively demonstrates engagement and connection to real-world context.
Criteria D: Reflection
1/3
0
2
3
Criteria Strands
Poor
Depth of reflection
Poor
Critical analysis
Poor
Evaluation of outcomes
Criteria Feedback
Assumptions and limitations are acknowledged
Some descriptive comments on unexpected results
Reflection lacks depth and critical analysis
No exploration of quantitative impact of assumptions
Minimal evaluation of outcomes or future implications
4.1·Weakness
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Assumptions (constant irradiance, neglecting weather) are acknowledged but not critically examined. Reflect on their quantitative impact.
4.2·Suggestion
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The conclusion summarizes findings but offers minimal discussion of model limitations or future work. Expand reflection on improvements and real-world implementation.
Criteria E: Use of Mathematics
3/6
0
3
6
Criteria Strands
Good
Relevance of mathematics
Good
Level appropriateness
Moderate
Understanding and accuracy
Criteria Feedback
Relevant trigonometry, differentiation and numerical integration applied to drive the investigation
Appropriate use of calculus and solar geometry at course level
Effective integration of technology (Python/SciPy)
Errors in degree/radian conventions and unexplained intermediate values
Unjustified algebraic steps (square rooting, integral approximation)
Minor inaccuracies in notation and rounding
5.1·Strength
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The declination angle formula is correctly presented, grounding the mathematical background in established solar geometry.
5.2·Suggestion
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When simplifying solar altitude at noon, units for the hour angle and cosine assumption are not specified. Clarify degree vs radian convention.
5.3·Weakness
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The rationale for squaring θ=∣α−β∣ before differentiation is implied but not stated. Explicitly justify this step to validate the method.
5.4·Strength
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The optimisation steps are well-structured: squaring the absolute value, differentiating, setting to zero, and confirming with the second derivative.
5.5·Strength
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The integral model Eyear=∫03656Acos∣α(d)−β∣dd is relevant and appropriately applied to the research question.
5.6·Suggestion
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The transition from discrete daily sums to a continuous integral is used without justification. Explain why the integral approximates the sum.
5.7·Suggestion
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Selecting fixed angles of 0°, 15°, 30°, 45° is clear, but criteria for selecting negative angles could be further justified numerically.
5.8·Strength
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Describing use of Python and SciPy highlights effective technology integration in solving complex integrals.
5.9·Weakness
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Results state rounding to three significant figures; however, intermediate values (e.g., “241”) lack explanation. Clarify source of these figures.
5.10·Strength
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Efficiency ratios are computed accurately and concisely, demonstrating solid application of proportional reasoning.
