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Mathematics Applications & Interpretation (AI) IA Exemplar: Conical… | RevisionDojo
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IB Mathematics Applications & Interpretation (AI) HL Internal Assessment Example
Determining the volume of a conical flask through 3 different methods of modeling; using 3D shapes to estimate volume, finding volume through cylindrical shells, and employing volume of revolution to estimate volumeHL
6
Official IB Result
15/20
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Criteria A: Presentation
3/4
0
2
4
Criteria Strands
A.1Coherence and logical development
Good
A.2Organization and structure
Good
A.3Conciseness and relevance
Good
Criteria Feedback
Clear, descriptive title and well‐labelled subheadings allowing easy navigation
Good overall structure with introduction, body, and conclusion sections
Comparative summary table in the conclusion is concise and highly relevant
Inconsistent pagination between contents and actual page numbers
Some long algebraic blocks are not broken across lines, reducing readability
Verbose passages and occasional redundant explanations
1.1·Strength
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The title is clear, descriptive and immediately establishes the focus of the exploration. This supports reader engagement and frames the investigation effectively.
1.2·Weakness
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The table of contents displays incomplete placeholders (“…”) and lacks accurate page numbers. Complete each entry and ensure consistency between contents and actual pagination.
1.3·Weakness
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The direct quotation defining “frustum” is useful but the citation is informal. Adopt a consistent referencing style (e.g., include author, year, page) to improve presentation.
1.4·Suggestion
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The transition to the measurement table mentions “the following values” but the table appears on the next page. Move or reference the table immediately after this statement for better organization.
1.5·Weakness
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Tables 2 and 3 of plotted points contain empty cells and misaligned headers. Ensure each column is labeled and rows are complete for readability.
1.6·Strength
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Comparative summary table in the conclusion is concise, highly relevant and offers an excellent overview of percentage errors across methods.
1.7·Suggestion
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The conclusion text duplicates table numbering (“Table 10”) and description. Remove redundancy and focus on reflective insights and future improvements.
Criteria B: Mathematical Communication
3/4
0
2
4
Criteria Strands
B.1Mathematical language and notation
Good
B.2Multiple representations
Good
B.3Clarity and consistency
Good
Criteria Feedback
Accurate use of standard calculus notation (integrals, summation) in most places
Effective use of multiple representations (tables, graphs, diagrams, Desmos screenshots)
Generally clear explanations with consistent notation within sections
Some axes and tables lack proper labels or complete variable definitions
Minor errors in subscript formatting and omission of definitions reduce clarity
2.1·Suggestion
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The aim statement is clear but verbose. Consider revising to explicitly state the comparison metric (percentage error) and the three methods: 3D shapes, shells, and discs.
2.2·Suggestion
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Diagram 1 is a clear student-created visual but lacks axis labels and scales. Add dimensional labels on each axis to strengthen clarity and reproducibility.
2.3·Weakness
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The table of symbols omits the full description of r₂ (“Radius of the cylinder and…”). Complete and clarify each variable definition for consistency.
2.4·Weakness
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In the smaller flask table, the variable “r2” lacks subscript formatting. Maintain consistent mathematical notation (e.g., r₂) across all tables.
2.5·Strength
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The external diagram of cylindrical shells is well chosen and cited, providing an effective multiple representation of the method.
2.6·Suggestion
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The notation Δr is introduced but not defined prior to use. Provide a clear definition of Δr as the shell thickness when first mentioned.
2.7·Strength
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Good use of Desmos-generated graphs to compare function fits. Technology effectively supports the selection of appropriate models.
2.8·Weakness
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The logistic function exponent appears as e^{-(-1.84x)} creating a double negative. Simplify to e^{1.84x} or adjust notation for clarity.
2.9·Suggestion
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The disc method section is well explained conceptually. Consider adding a schematic of disc thickness Δx and radius f(x) to reinforce understanding.
2.10·Weakness
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The sine-based integral omits “dx” in the display. Add the differential element (dx) to the integral for correct mathematical communication.
Criteria C: Personal Engagement
2/3
0
2
3
Criteria Strands
C.1Independent thinking
Good
C.2Personal approach
Good
C.3Creativity and initiative
Good
Criteria Feedback
Clear evidence of independent thinking in selecting and comparing three distinct calculus approaches
Personal measurements and choice of two flask sizes demonstrate ownership of the exploration
Creative initiative shown by applying curve‐fitting to logistic and sine models beyond a single standard method
Techniques remain within conventional boundaries without novel extension
No entirely new mathematical result or substantially altered methodology
Personal approach, while clear, does not reach the ‘highly personal’ descriptor for band 3
3.1·Strength
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The introduction rationale is engaging and connects the exploration to personal biology experience, demonstrating ownership of the topic.
Criteria D: Reflection
2/3
0
2
3
Criteria Strands
D.1Depth of reflection
Good
D.2Critical analysis
Good
D.3Connection to understanding
Good
Criteria Feedback
Meaningful reflection on measurement errors, model limitations, and possible improvements
Relevant analysis of percentage errors and critique of assumptions (symmetry, photo angle)
Reflections clearly linked to understanding of calculus applications and modelling accuracy
Reflection is relatively brief and largely descriptive without deeper quantitative analysis
No formal error propagation or sensitivity analysis included
Limited probing of domain restrictions and statistical goodness-of-fit measures
4.1·Suggestion
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The explanation notes measurement error sources but omits quantitative error propagation. Include a brief sensitivity analysis to deepen critical reflection.
4.2·Suggestion
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The inverse function derivation is mathematically sound but omits domain discussion. Reflect on domain restrictions (y>0) to deepen critical analysis.
4.3·Strength
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Thoughtful reflection on methodological limitations and future refinements (e.g., multiple photo angles, automated point distribution) shows meaningful critical analysis.
Criteria E: Use of Mathematics
5/6
0
3
6
Criteria Strands
E.1Relevance and level
Good
E.2Accuracy and correctness
Good
E.3Knowledge and understanding
Good
Criteria Feedback
Use of three HL calculus approaches (shells, discs, regression to logistic and sine models) demonstrates breadth
Algebraic derivations and integral set-ups are mostly correct and concise
Solid conceptual understanding of frustum volume, shell integration, and model justification
Minor transcription slips (missing absolute values, one rounding inconsistency)
Regression justification lacks full R² comparison and formal parameter-estimation discussion
No explicit error propagation or deeper optimization analysis
5.1·Suggestion
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The derivation of the frustum volume skips an explicit definition of F before substitution. Introduce F symbolically, define terms, then substitute to promote mathematical rigor.
5.2·Weakness
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The percentage error for the smaller flask is approximated as 4.32% but recalculation of |287.59–301|/301×100 yields ≈4.46%. Review the arithmetic and rounding.
5.3·Suggestion
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The shell volume formula is introduced but the text could simplify earlier by noting πh(r₂²–r₁²)=πh(r₂–r₁)(r₂+r₁) before further manipulation.
5.4·Strength
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The algebraic manipulation leading to V=2πrhΔr is concise and demonstrates clear understanding of shell integration.
5.5·Strength
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The final shell integral V=∫2π x f(x) dx is correctly set up. This aligns well with standard shell method formulation.
5.6·Suggestion
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The choice between quartic and logistic functions is based on R² but only quartic R² values are cited. Include logistic R² to justify final selection rigorously.
