IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example

The list of four attributes duplicates earlier game rules. Condense this list by referencing the previous section or presenting only variations that change in the two‐attribute model.

Figure 1’s table is cluttered with stray characters and misaligned columns, interrupting the reader’s flow. Reformat the table to remove artifacts and improve legibility.

The general game‐rules paragraph in Section 2 repeats content unnecessary for the two‐attribute investigation. Remove or shorten to maintain focus.

The three‐attribute deck image is oversized and interrupts flow. Consider replacing it with a summary table of key card counts and attributes.

Figure 11 mapping is small and difficult to read, undermining its explanatory role. Enlarge the graphic or simplify to essential connections.

Figure 12’s multi‐line plot is dense and lacks a clear legend. Use distinct line styles or colors and annotate the inflection region for clarity.

The Python code listings contain OCR artifacts (e.g., “\ddot{6}values”). Provide clean, tested code in an appendix or online repository for reproducibility.

Clear distinction between the game‐specific “set” and the mathematical set strengthens conceptual clarity. This precision in language supports logical development.

Provide formal symbolic definitions (e.g., using set notation) for terms like “set” and “variation” to reinforce mathematical rigor in the definitions section.

The rationale for generalising to n attributes is well motivated and demonstrates foresight in simplifying later derivations.

The complement calculation P(No Set)=6/7 is correct but introduced abruptly. Add a sentence explaining “1−P(Set)” to guide less experienced readers.

The impossibility proof for five cards relies on a verbal pigeonhole argument. Formalize this using combinatorial bounds or the pigeonhole principle to strengthen mathematical communication.

Introduce the simulation approach by explicitly linking it to computational infeasibility of analytical enumeration for six or more cards.

The final general formula in Equation 16 is dense. Add inline commentary or label terms (intersets, single‐set groups, denominator) to aid reader interpretation.

The introduction offers an engaging personal narrative but could be more concise. Consider summarizing key experiences and linking them directly to the mathematical investigation to maintain relevance.

The simplified two‐attribute attribute list is clearly presented, helping the reader transition to the reduced model.

Table 2 shows an unexpected probability jump at 19 cards, indicating sampling error. Discuss confidence intervals or increase trial count to justify reliability.

Insightful identification of a logarithmic‐type decrease in probability demonstrates thoughtful critical analysis of trends.

Future extension proposals are creative but generic. Frame specific testable hypotheses (e.g., effect of 4 values per attribute on inflection point) to deepen reflection.

Figures 2 and 3 effectively illustrate valid and invalid sets, aiding comprehension through visual representation.

When generalising P(Set)=1/(3^n–2), clarify that 3^n is the deck size and discuss restrictions on n to avoid negative denominators.

When deriving P(Set)=1/7, insert a brief combinatorial count (e.g., “there are 9×8/2=36 pairs and 36 possible third cards”) to show how the fraction emerges.

The derivation of P(No Set with Fourth Card)=1/2 glosses over how the denominator relates to 3^n−2 and 3^n−3. Include the general case for arbitrary n.

The general formula P(No Set with three cards)=(3^n–3)/(3^n–2) is accurately stated and sets up the four‐card analysis well.

The worked example on page 8 vividly demonstrates inevitability of a set in five cards; continue this visual approach to aid reader intuition.

After applying the general formula for n=3, include a brief enumeration or simulated check to confirm P(No Set)=24/25, reinforcing accuracy.

The combinatorial derivation of total sets (117) is sound. Clear explanation of permutation vs. combination shows strong mathematical communication.

Calculation of total intersets =27×C(13,2)=2106 is correctly executed and clearly supports the five‐card analysis.