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

The aim is clear and focused: the student states exactly what the investigation is trying to solve and connects it to a real decision about tea timing. That gives the exploration a strong purpose and helps every later section stay anchored to a single question.

The plan of action is logically sequenced from modelling to comparison, which makes the overall argument easy to follow. The student has improved the structure so the reader can see how the data collection leads into the modelling choices.

The statement that the test was repeated "to ensure reliability and consistency" is useful, but it would be stronger if the student briefly explained what reliability meant in this context and how three trials would reduce random variation. That would make the methodology more analytical rather than descriptive.

The raw data table is presented clearly and is directly relevant to the investigation. Listing time and temperature in a consistent format makes the later averaging and modelling steps much easier to verify.

The discussion of the asymptote is relevant, but the student could tighten the wording by linking the limitation directly to the tea data: the model is not just "continuously decreasing," it is assuming a fixed surrounding temperature. Explaining that connection would make the logic more precise.

The explanation of the graph is a little repetitive and stays at the level of "matches to a certain extent." The student should identify a specific feature, such as where the model overestimates or underestimates the mean data, so the discussion advances the argument instead of restating general fit.

This is a good place to compare the Newton model more explicitly with the other two models. A brief sentence explaining whether its deviation is larger at early times, later times, or both would improve the logical flow into the comparison section.

The transition into model comparison is effective because the student moves from qualitative observation to a quantitative decision rule. That shift strengthens the structure of the exploration and shows clear progression toward the final recommendation.

The evaluation begins well, but the reflection is still fairly general. To deepen the presentation, the student should connect each limitation to a specific stage of the investigation, rather than summarizing all uncertainties in broad terms.

The mean table is an effective representation because it summarizes the repeated trials into a single dataset for modelling. That reduction is mathematically meaningful and supports the later regression work.

The mean calculation is shown clearly and the notation is understandable. This makes the method reproducible and helps the reader follow how the combined data set was produced from the three trials.

The exponential model is written in a standard form and the variable meanings are listed immediately after it. This is good mathematical communication because the reader can interpret each symbol without needing to infer it.

The description of $k$ as "constant proportionality" is too informal for precise mathematical writing. The student should use a clearer phrase such as "rate constant" or explain exactly what proportional relationship it represents.

The phrase "to isolate and remove the k constant" is not mathematically accurate because setting $t=0$ isolates $C$, not $k$. Replacing this with a precise explanation would improve both notation and conceptual clarity.

The step "To remove the \ln we multiply it by e" is imprecise. The student should instead state that exponentiating both sides with base $e$ undoes the natural logarithm, because that describes the transformation correctly.

The statement that $e^C$ "can be taken as the temperature of the surrounding" is not correct. $e^C$ is just a constant factor from integration, while the surrounding temperature is represented separately by $T_s$; keeping these symbols distinct would strengthen the communication.

The MAE table communicates the comparison well, but the student could improve clarity by stating the units or meaning of the error values in the sentence that follows. That would make the interpretation of the numbers more immediate.

The final calculation to find the cooling time is written in a logical sequence and uses the chosen model correctly. The algebra is easy to follow, which is important because this is the step that answers the original practical question.

The student has converted the time into minutes and seconds, which is helpful. A short reminder of which model produced this value would make the final result easier to interpret at a glance.

The opening shows genuine personal motivation: the student links the investigation to sleep, studying, and exam performance. That personal context is meaningful because it gives the mathematics a real decision-making purpose rather than treating the task as abstract modelling.

The student makes independent choices about controlling variables such as time, volume, and surrounding conditions. This shows that the experiment was designed thoughtfully rather than copied mechanically from a template.

What would change in the model if the surrounding temperature were measured repeatedly instead of assumed to be 30°C? Considering that question could help the student evaluate how personal the experimental setup really is.

The explanation of why the negative sign is included shows active mathematical decision-making. The student is not only applying a formula, but also trying to interpret how the sign affects the shape of the model.

Using calculator regression to obtain the line of best fit shows initiative in selecting a practical tool for parameter estimation. That choice supports the exploration’s modelling aim and shows the student is engaging with the data independently.

The personal decision about which values to use for solving $k$ could be explained more explicitly. A brief justification for choosing these two points, rather than another pair, would make the student’s independent thinking more visible.

The worked MAE example shows initiative because the student does not just quote the error metric; they demonstrate how it is calculated. That helps show ownership of the comparison method.

The student states that values are substituted in, but does not explain why the point at $t=5$ is a good choice for finding $k$. Reflection would be stronger if the student discussed whether another time point might give a more stable estimate.

The student begins to reflect on real-world effects by noting that the temperature change may not follow the model exactly. This is a useful starting point because it shows awareness that assumptions have consequences for the fit.

The claim that the gradient and intercept can simply be reassigned as $k$ and $C$ is too procedural. The student should reflect on why the linearised line corresponds to the exponential parameters, because that would show genuine understanding rather than just substitution.

The evaluation acknowledges that Newton’s law depends on conditions and that anomalies may affect discrepancy. That shows the student is beginning to critique model validity rather than treating the equation as automatically correct.

The MAE calculation is shown, but the reflection stops short of interpreting what the size of the error means in context. The student should explain whether the differences are practically important for deciding when to drink the tea, not just numerically smaller.

The use of MAE to justify the choice of linear decay is a meaningful reflective step because it connects the numerical comparison back to the modelling decision. This helps the reader see how the conclusion was reached.

The conclusion would be stronger if it briefly revisited the main assumption that ambient temperature stays fixed at 30°C. That would connect the final result to the limits of the mathematical model and show more reflective maturity.

The evaluation names an improvement, but it does not explain how much better data collection would affect the model or final brewing time. The student should quantify the benefit where possible, because critical reflection needs to show impact as well as suggestion.