IB Mathematics Analysis and Approaches (AA) SL Internal Assessment Exemplar: Investigation into the fractal dimensions through the coastline paradox

The research question is clear but could be refined to indicate why these two islands were chosen over others and what scale considerations apply. Consider specifying criteria for island selection.

The explanation of the coastline paradox is informative but overly verbose; consider summarizing main points and reducing repetitive details to improve conciseness and relevance.

The series of images illustrates resolution effects effectively, showing complexity at multiple scales; this visual integration strengthens the argument.

Citation placement interrupts sentence flow; integrate the reference more smoothly and standardize citation formatting throughout.

The introduction of the box counting method is clear, yet the phrasing 'hard otherwise in other methods' lacks precision; revise for clarity and conciseness.

The computed dimension of Hachijō-jima (1.03) differs from the later stated value of 1.06; reconcile these discrepancies to maintain coherence and logical structure.

The caption for Figure 1 lacks explicit mention of resolution values used; including exact pixel or scale data would enhance clarity of representation.

The derivation of $$n=s^D$$ is clear, but the base of the logarithm used in subsequent steps is not specified; stating whether natural or common logs improves mathematical precision.

The logarithmic transformation is correct, yet the student could include a brief justification of each algebraic step to reinforce logical clarity.

The caption for Figure 2 is truncated by OCR artifacts and lacks full reference details; ensure complete citation and explanation of the example displayed.

The list summarizing dimension values is concise, but formatting the mathematical expressions in separate lines could improve readability and communication.

Consistent notation for logarithms (capitalization of 'log') should be maintained; correct the mixture of 'log' and 'Log' for mathematical consistency.

Typographical error: the list item reads 'm = (1 D)'; it should read $$m=1-D$$. Correcting this will avoid confusion in later calculations.

The trial data table is comprehensive, yet units for ruler length (s) and coastline length (p) are omitted; including units (e.g., km or pixels) ensures full clarity.

The graph of log(ruler length) vs log(coastline length) effectively supports the Richardson method, but axis labels and units should be explicitly shown on the graph.

The fractal dimension of Himaka-jima is calculated as 1.26 here, yet the conclusion later states 1.24; ensure consistency of results throughout the exploration.

The student effectively frames a personal context regarding Japanese coastlines and curiosity, demonstrating clear personal engagement. To deepen this, they could connect personal experiences to specific aspects of fractal analysis.

The final personal reflection is insightful but repetitive; refine this section to highlight specific learning outcomes and how these skills may apply to future studies.

The stated range for real coastline dimensions is appropriate, though adding a brief source discussion would enhance critical depth of reflection on these values.

The evaluation compares deviation percentages effectively, but the calculation steps for percentage deviation are not shown; include these for transparency in critical analysis.

The conclusion interprets the fractal dimensions well, but the reasoning attributing complexity to man-made structures could be extended with quantitative evidence or literature support.

The limitation regarding image resolution is critical; propose specific image processing or higher-resolution data sources as concrete steps for future research.

The Richardson method description is accurate, but the relationship between $$L(s)$$ and coastline length could be illustrated with a diagram tailored to the coastline context.

The equation $$D=\frac{\Delta \log n}{\Delta \log s}$$ is presented correctly, but the student should discuss how least-squares regression would yield a more accurate gradient than manual point selection.

Using only four grid scales in the box counting trial may limit accuracy; consider adding intermediate scales to improve regression reliability.

The slope calculation for Britain is clear, but using only two data points can bias the result; apply regression analysis on all four points to validate the dimension.

The uncertainty propagation table presents clear bounds, but the method used to derive absolute uncertainties is not explained; include a brief outline of the calculation process.