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

The exploration opens with a clear progression from context to purpose, and the student’s personal role as a shooting guard immediately helps the reader understand why the investigation matters. This gives the introduction a strong sense of direction and supports the overall coherence of the report.

The aim would be stronger if the student defined the three variables more precisely. Terms such as “apparent size” and “probability of shot success” need measurable definitions, otherwise the later analysis can only address them indirectly.

The background section connects the geometry of viewing the hoop with the investigation’s central idea, which helps justify why trajectory and entry angle matter. This is a relevant conceptual bridge and improves the logical development of the exploration.

The assumption about ideal projectile motion is useful, but it should be tightened mathematically. “Negligence of air resistance or spin” is imprecise phrasing; the student should state exactly which forces are ignored and explain how that simplification could bias the model.

Consider trimming repeated set-up sentences and moving straight into the comparison between manual and Desmos regression. That would make the section more concise and keep the focus on the mathematical outcome rather than the process description.

The paragraph repeats the procedural idea of calculating regression manually without yet showing why this adds value beyond the technology output. The student should state the purpose of the manual check more explicitly, such as verifying the digital regression or examining consistency between methods.

The discussion at this point begins to forecast later conclusions about improvement and consistency, but it does not yet tie those claims back to the specific vertex values just calculated. The student should use the computed numbers more directly to support the transition into the comparison of made and missed shots.

The Part 1 reflection effectively summarises the mathematical meaning of the vertex results and links them back to shot quality. This shows a coherent wrap-up of the section rather than a simple repetition of calculations.

The reflection would be stronger if the student explicitly compared the boxplot outcomes to the earlier vertex and angle results in one short synthesis. Bringing the three investigation parts together would make the final structure feel more integrated.

The conclusion states that successful shots had a narrower range and a higher entry angle, but it still relies on broad claims rather than a fully developed mathematical link to the research question. The student should connect these findings to the earlier calculations more explicitly and avoid leaving the argument at a descriptive level.

The references are separated clearly from the main body and the appendix follows afterwards, which makes the overall structure easy to navigate. This is good academic organisation and supports readability.

The student uses appropriate mathematical vocabulary such as “vertex calculations,” “projectile motion,” and “quadratic regression,” which helps establish the mathematical frame of the investigation. This language is clear and relevant to the topic.

The regression formula notation is unconventional and difficult to verify as written. In particular, the symbols should be defined unambiguously so the reader can tell whether the student means sums of products, sums of squares, or centered sums. Clearer notation would make the method much more trustworthy.

The notation here appears inconsistent with standard regression notation, which makes the derivation harder to follow. If the student is using centered sums, those should be written consistently and linked clearly to the definitions given below.

The graph is a useful representation, but it would be even more effective if the key regression quality information were cited directly in the main text, such as how well the curve fits the tracked points. That would strengthen the communication of why the model is reliable.

There is still a sign inconsistency in the regression parameters: the printed Desmos output shows one sign pattern, but the later listed values and substituted equation do not match it consistently. The student should reconcile the parameter signs before using the model in later calculations.

The standard quadratic form is clearly introduced and the variables are defined in context, which makes the later derivative work easier to follow. This is a solid example of effective mathematical communication.

The derivative step is correct in principle, but the student should preserve notation more consistently by using either $h(x)$ and $h^{\prime }(x)$ or $y$ and $\frac{dy}{dx}$ throughout. Mixing notation is not wrong, but it can make the chain of reasoning less clear.

If the angle of entry is being treated as a downward angle, what convention justifies taking the absolute value after obtaining a negative arctan result? Making that sign convention explicit would prevent ambiguity in the interpretation.

The label here should be median rather than IQR. The calculation is correct for the median of the even-sized data set, but the notation is misleading, so the student should correct the label to avoid confusing the reader.

The quartile calculation is shown step by step, which makes the statistical method transparent. This is a strong use of mathematical communication because the reader can reproduce the calculation from the working.

The missed-shot summary appears to contain a likely inconsistency: the median is listed as 32.675 while Q1 is 33.92 and Q3 is 44.83. Since the median should lie between Q1 and Q3, the student should re-check the calculation and confirm the ordered data.

The boxplot summary table communicates the five-number summary clearly and supports the later comparison between made and missed shots. This is an effective multiple-representation choice because it compresses the data into an interpretable statistical form.

The personal context is authentic and relevant: the student explains the investigation through their own role as a shooting guard. This gives the project clear ownership and shows that the question arises from genuine experience rather than a generic prompt.

The choice to use raw Logger Pro coordinates rather than relying only on a published or textbook example shows initiative. The student is making methodological choices based on the data collection process, which strengthens independence.

The personal approach would be even more distinctive if the student explained why this manual verification mattered for their own investigation. A brief comment about confidence in the model or sensitivity to tracking choices would make the independence more visible.

Comparing the manual regression with Desmos demonstrates independent checking rather than passive acceptance of technology output. This is a good example of mathematical initiative because the student is validating the model from two directions.

The reflection is grounded in the student’s own results and makes a personal judgement about what the vertex comparison means for future shooting choices. This is more than description; it shows the student using the mathematics to inform practice.

The statement that steeper arcs increase the apparent size of the hoop is interesting, but it is still asserted rather than modelled. To move this into stronger personal initiative, the student should attempt a geometric or probabilistic definition of “apparent size” instead of leaving it as an intuition.

Breaking the investigation into separate mathematical lenses — trajectory, angle, and distribution — shows thoughtful initiative in designing the exploration. This multi-part structure goes beyond a single routine calculation and reflects deliberate planning.

The student identifies a real limitation in the sample design and suggests extending the dataset in future work. Recognising this boundary shows mature ownership of the inquiry and a willingness to improve the investigation.

What would change in the investigation if side-rim misses were included rather than excluded? Considering that question could help the student decide whether the sample design is aligned with the research question.

The student extends beyond a simple black-box regression by showing an attempt to derive the quadratic model manually before checking it with technology. That demonstrates a solid grasp of how the mathematics is being generated.

The inconsistency between the regression coefficients shown here and the later substituted equation means the model is not yet fully reliable. The student should verify the sign of $b$ and ensure the same equation is used consistently in all later calculations.

The transformed equation should be checked carefully against the stated coefficient values. If $a=-0.2368$ and $b=-1.102$, the substituted model should reflect that sign exactly; any mismatch will affect the derivative and vertex results that follow.

The use of differentiation to find the vertex is mathematically appropriate and well chosen for this exploration. It shows the student applying course-level calculus in a meaningful way rather than simply quoting formulae.

The calculation of the vertex height should be checked for consistency with the substituted quadratic. Small arithmetic or sign errors here would propagate into the later averages, so the student should verify the numerical substitution carefully.

The trigonometric step is correctly selected: using $\theta =\arctan \left(\frac{dy}{dx}\right)$ is a valid way to convert slope into an angle of entry. This shows appropriate mathematical method choice for the investigation.

The slope calculation appears to use a coefficient set that does not match the earlier stated regression equation. The student should make sure the exact same model is being differentiated, otherwise the angle-of-entry results are not dependable.

The IQR method is an appropriate way to compare the spread of made and missed angles, and it is applied correctly for the made-shot data shown here. This is a sensible use of statistics to interpret consistency.

The missed-shot quartile summary looks inconsistent with the ordered-data logic used for the made shots. The student should re-order the dataset and recalculate Q1, median, and Q3 before drawing conclusions from the boxplot comparison.

The conclusion effectively uses the numerical patterns to support a performance claim about successful shots. This is a strong sign that the mathematical results are being used meaningfully in context.

A stronger final evaluation would include at least one quantified comparison of uncertainty or model fit, such as a spread measure, residual pattern, or error estimate. That would make the outcome analysis more robust without requiring highly advanced mathematics.