IB Mathematics Applications & Interpretation (AI) HL Internal Assessment Example

The student establishes a clear roadmap early on, which helps the reader understand how the exploration will develop. This is good organization because the methods and the central comparison are named before the detailed mathematics begins.

This opening to the mathematical section gives a logical starting point for the balancing work. It is effective because it connects the chemistry principle to the algebraic method that follows.

The student should be careful here: skipping the working weakens the logical development because the reader cannot check how the coefficients were obtained. Even a brief summary of the key row operations would make the argument much more transparent.

This sentence clearly summarises the progression from chemical species to vectors and then to a solvable system. It supports coherence because the student explicitly explains how the mathematical representation was built.

The comparison is introduced clearly, but the student should ensure that the plotted functions are stated with complete precision before discussion begins. Because the exact second-order expression is inconsistent later on, the reader cannot fully trust the comparison as written.

This is a useful interpretive sentence, but it would be stronger if it were immediately linked to the exact numerical evidence used to produce the graph. The reader needs the reasoning chain from formula to graph to conclusion to remain fully coherent.

The student could tighten the structure by condensing the early balancing demonstrations and moving more quickly to the core ethanol model. That would make the overall progression feel more seamless and keep the central investigation in focus.

A brief concluding link back to the original research question would improve closure. The student already evaluates the method well; one final sentence showing how the investigation answers the aim would strengthen the section’s ending.

The student uses appropriate linear-algebra notation and defines the objects in the system, which helps the reader follow the balancing method. This is effective communication because the symbols are not left unexplained.

The student should use more precise terminology here: “substrate” is not the usual term in this context, and the rate should be expressed with consistent differential notation. Clearer terminology would make the mathematical communication more accurate.

The student could improve clarity by stating the proportional-rate relationship in a compact symbolic form as well as in words. That would make the notation easier to reuse later in the exploration.

This is a clear and mathematically appropriate explanation of what an integrated rate law does. The student’s notation is mostly effective here because the meaning of each concentration quantity is made explicit.

This explanation would be stronger if the student stated the rate law with explicit notation and variable definitions before using it. The reader is left to infer too much, which reduces clarity in a mathematically dense section.

The graph is useful, but the communication would be clearer if the key features were referenced directly in the accompanying text, such as the intercept and decay behaviour. That would help the reader interpret the representation rather than simply see it.

The notation here becomes difficult to follow because the integrated expression is not written in a consistent mathematical form. The student should rewrite the result cleanly with clear brackets and standard notation so that the intended model can be checked unambiguously.

The student should keep notation consistent when referring to the variable $t$ and when reporting numerical values. Presenting the values with cleaner mathematical notation would make the comparison much easier to verify.

The graph is introduced with enough context for the reader to understand what is being compared. This is good mathematical communication because the notation used in the figure is tied back to the meaning of the two models.

The student shows a genuine personal connection to the topic, which helps explain why the investigation is sustained and motivated. This kind of rationale is effective because it gives the exploration a clear purpose beyond simply completing a task.

This is a strong indication of independent initiative: the student is not just applying a formula, but deliberately combining two mathematical fields in one context. That choice gives the exploration a clearer identity and shows ownership of the direction of the work.

The student states the transition clearly, but the reflection on why this mathematical route is the best choice is still limited. To show stronger independent thinking, the student should explain why matrices are preferable here beyond convenience, and what insight they give that trial-and-error balancing does not.

The student could strengthen personal engagement by briefly explaining why these particular conditions or concentrations were selected. A short justification of why this dataset is suitable for the model would make the mathematical choices feel more deliberately owned.

The student could push the personal and analytical originality further by quantifying this difference at several time points, not just describing it qualitatively. Adding a small table of percentage error would make the comparison more distinctive and mathematically purposeful.

What extension would best reflect the student’s own mathematical interests: a numerical simulation, a sensitivity analysis, or a different oxidation reaction? A more specific next step would show clearer initiative than a general comment about future software use.

The student presents the integrated law as settled, but does not verify whether the stoichiometric factor from the $1:3$ ratio has been carried through correctly. Stronger reflection would check the algebra before moving on, because later conclusions depend on this exact expression.

The student makes a reasonable limiting claim, but the conclusion is not tested against the earlier modelling assumptions or parameter values. A deeper reflection would ask whether the limit is physically meaningful in the presence of reactant depletion, and whether the exact and approximate models remain comparable for all times.

The student does more than describe the two curves: they interpret what the gap means chemically. This is effective reflection because it links the model comparison to the consequences of the approximation, not just to the shapes of the graphs.

What would change in the conclusion if the pseudo-first-order assumption were not justified by the concentration ratio? This is worth considering because reflection should test the model’s dependence on its assumptions, not only describe the method used.

The evaluation would be stronger if the student tied this extension to a concrete limitation already encountered, such as the algebraic inconsistency in the exact-model derivation or the lack of uncertainty analysis. That would make the reflection feel more analytical and less generic.

This is a strong evaluative comment because it acknowledges a realistic limitation of the model. The student shows awareness that mathematical accuracy alone is not enough; the validity of conclusions also depends on measurement uncertainty and modelling assumptions.

The student uses the atom counts correctly to justify the balancing process, which shows relevant mathematical reasoning rather than rote manipulation. This is a good example of linking chemical structure to algebraic setup.

This is a sound mathematical-chemical constraint to state explicitly. It shows understanding that the algebraic solution must be scaled into chemically meaningful coefficients.

The student could strengthen the mathematics by briefly showing one full worked row-reduction step or one nontrivial algebraic check. That would make the use of matrix methods more transparent and demonstrate greater command of the method.

The student could make the modelling assumption more rigorous by explaining why the excess is large enough to justify treating one concentration as effectively constant. A brief estimate or threshold would make the HL modelling stronger.

The student correctly encodes the stoichiometric $1:3$ relationship in the concentration substitutions, which is essential for the later modelling. This is a strong example of using the chemistry to inform the mathematics.

The final integrated law is introduced as if it were fully verified, but the subsequent analysis depends on getting the stoichiometric factor exactly right. The student should check the derivation carefully and confirm the algebra before using the expression in the graph.

How much of this difference is due to the modelling assumption, and how much might come from the algebraic inconsistency in the exact second-order formula? A reflection on that distinction would show much deeper mathematical rigour.

This plotted expression is mathematically important, but it appears to inherit the earlier inconsistency in the second-order derivation. The student should verify that the formula matches the stoichiometric coefficients and initial conditions before relying on the graph for conclusions.

The limit statement is reasonable, but it is presented without checking whether the model’s assumptions still hold as time increases. The student should connect the mathematics back to the physical chemistry more carefully, especially for long-time behaviour.