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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.

OFFICIAL

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.

OFFICIAL

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.

OFFICIAL

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