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Stage 2 Specialist Mathematics - AT2 - Topic 5 - Integration Techniques and Applications
Page 1 of 6
Ref: A500041 (revised December 2021)
SACE Board of South Australia 2021
Stage 2 Specialist Mathematics
Assessment Type 2: Mathematical Investigation
Topic 5 – Integration Techniques and Applications
Mathematics can be used to model the shapes of objects. In the first part of this investigation the cross
section of the bowl of a wine glass is modelled with the aim to mathematically obtain a reasonable
volume of the glass. The second part of the investigation allows different objects to be modelled to find
their volume.
bowl
stem
base
Wine glass
Ensure the following points are addressed in this investigation.
Wine Glass:
Mathematically model the shape of the cross-section of the bowl of a chosen wine glass.
You may use your knowledge of inverse functions to find a 1-1 function to rotate about the
xaxis, or otherwise, to find the volume of the glass.
Investigate adjusting your model (see the flow chart on page 2) to improve the accuracy of the
volume calculated compared to the actual volume of the chosen wine glass.
Discuss the reasonableness of the results.
Another Object:
Consider another object and find its actual volume discussing the process used.
Develop a mathematical model to find an approximate volume. Use the flow chart on page 2 to
adjust the model to improve the answer you have found mathematically.
Compare the actual volume and calculated volume and discuss the reasonableness of the
results.
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Stage 2 Specialist Mathematics - AT2 - Topic 5 - Integration Techniques and Applications
Page 2 of 6
Ref: A500041 (revised December 2021)
SACE Board of South Australia 2021
The format of the investigation report may be written or multimodal. The report should include the
following:
an outline of the problem and context
the method required to find a solution, in terms of the mathematical model or strategy used
the application of the mathematical model or strategy, including
o relevant data and/or information o mathematical calculations and results, using
appropriate representations o the analysis and interpretation of results, including
consideration of the reasonableness and limitations of the results
the results and conclusions in the context of the problem.
A bibliography and appendices, as appropriate, may be used.
The investigation report, excluding bibliography and appendices if used, must be a maximum of 15
A4 pages if written, or the equivalent in multimodal form. The maximum page limit is for single-sided
A4 pages with minimum font size 10. Page reduction, such as 2 A4 pages reduced to fit on 1 A4
page, is not acceptable. Conclusions, interpretations and/or arguments that are required for the
assessment must be presented in the report, and not in an appendix. Appendices are used only to
support the report, and do not form part of the assessment decision.
Test Model and
Reflect
Real World Pathway
to Model
Initial model
Adjust model
providing
explanations/reasons
Final model -
reflection and
extensions
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Stage 2 Specialist Mathematics - AT2 - Topic 5 - Integration Techniques and Applications
Page 3 of 6
Ref: A500041 (revised December 2021)
SACE Board of South Australia 2021
Mathematical Report
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Stage 2 Specialist Mathematics - AT2 - Topic 5 - Integration Techniques and Applications
Page 4 of 6
Ref: A500041 (revised December 2021)
SACE Board of South Australia 2021
Appropriate use of electronic technology to find accurate solutions. Reasonable graphical interpretation are
needed.
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Stage 2 Specialist Mathematics - AT2 - Topic 5 - Integration Techniques and Applications
Page 5 of 6
Ref: A500041 (revised December 2021)
SACE Board of South Australia 2021
Performance Standards for Stage 2 Specialist Mathematics
Concepts and Techniques
Reasoning and Communication
A
Comprehensive knowledge and understanding of concepts
and relationships.
Highly effective selection and application of mathematical
techniques and algorithms to find efficient and accurate
solutions to routine and complex problems in a variety of
contexts.
Successful development and application of mathematical
models to find concise and accurate solutions.
Appropriate and effective use of electronic technology to find
accurate solutions to routine and complex problems.
Comprehensive interpretation of mathematical results in the
context of the problem.
Drawing logical conclusions from mathematical results, with a
comprehensive understanding of their reasonableness and
limitations.
Proficient and accurate use of appropriate mathematical notation,
representations, and terminology.
Highly effective communication of mathematical ideas and
reasoning to develop logical and concise arguments.
Effective development and testing of valid conjectures, with
proof.
B
Some depth of knowledge and understanding of concepts
and relationships.
Mostly effective selection and application of mathematical
techniques and algorithms to find mostly accurate solutions
to routine and some complex problems in a variety of
contexts.
Some development and successful application of
mathematical models to find mostly accurate solutions.
Mostly appropriate and effective use of electronic
technology to find mostly accurate solutions to routine and
some complex problems.
Mostly appropriate interpretation of mathematical results in the
context of the problem.
Drawing mostly logical conclusions from mathematical results,
with some depth of understanding of their reasonableness and
limitations.
Mostly accurate use of appropriate mathematical notation,
representations, and terminology.
Mostly effective communication of mathematical ideas and
reasoning to develop mostly logical arguments.
Mostly effective development and testing of valid conjectures,
with substantial attempt at proof.
C
Generally competent knowledge and understanding of
concepts and relationships.
Generally effective selection and application of
mathematical techniques and algorithms to find mostly
accurate solutions to routine problems in a variety of
contexts.
Successful application of mathematical models to find
generally accurate solutions.
Generally appropriate and effective use of electronic
technology to find mostly accurate solutions to routine
problems.
Generally appropriate interpretation of mathematical results in
the context of the problem.
Drawing some logical conclusions from mathematical results, with
some understanding of their reasonableness and limitations.
Generally appropriate use of mathematical notation,
representations, and terminology, with reasonable accuracy.
Generally effective communication of mathematical ideas and
reasoning to develop some logical arguments.
Development and testing of generally valid conjectures, with some
attempt at proof.
D
Basic knowledge and some understanding of concepts and
relationships.
Some selection and application of mathematical techniques
and algorithms to find some accurate solutions to routine
problems in some contexts.
Some application of mathematical models to find some
accurate or partially accurate solutions.
Some appropriate use of electronic technology to find some
accurate solutions to routine problems.
Some interpretation of mathematical results.
Drawing some conclusions from mathematical results, with some
awareness of their reasonableness or limitations.
Some appropriate use of mathematical notation, representations,
and terminology, with some accuracy.
Some communication of mathematical ideas, with attempted
reasoning and/or arguments.
Attempted development or testing of a reasonable conjecture.
E
Limited knowledge or understanding of concepts and
relationships.
Attempted selection and limited application of mathematical
techniques or algorithms, with limited accuracy in solving
routine problems.
Attempted application of mathematical models, with limited
accuracy.
Attempted use of electronic technology, with limited
accuracy in solving routine problems.
Limited interpretation of mathematical results.
Limited understanding of the meaning of mathematical results,
their reasonableness, or limitations.
Limited use of appropriate mathematical notation,
representations, or terminology, with limited accuracy.
Attempted communication of mathematical ideas, with limited
reasoning.
Limited attempt to develop or test a conjecture.
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Stage 2 Specialist Mathematics - AT2 - Topic 5 - Integration Techniques and Applications
Page 6 of 6
Ref: A500041 (revised December 2021)
SACE Board of South Australia 2021