STAGE 2 MATHEMATICAL APPLICATIONS
FOLIO - INVESTIGATION 1
Purpose
To demonstrate your ability to:
 use a range of mathematical skills from the topic of Matrices to investigate a problem set in a context
 apply your knowledge, skills, and problem-solving strategies to develop a model from which you can make predictions.
Description of assessment
Use your knowledge of transition matrices to investigate the level of support that the fast-food outlets in your regional city could expect to have in the future. Using a justified sampling method and a survey of your design, collect information on the fast food habits of a sample of residents from the local area. Make predictions using transition matrix methods about how many of your sample will purchase their next fast food meal at each of the fast food outlets being investigated. Discuss the long term prospects for each of the fast food outlets in the regional city, including a discussion of the reliability of your answers and any assumptions that have been made. Discuss the limitations of extrapolating the model that you have created to make predictions about the population of your regional city. All solutions are to be supported with calculations and appropriate representation and notation.
Assessment conditions
You have three weeks to complete this assessment task. Use of technology is required. Your investigation should include:
(a) an introduction that demonstrates your understanding of the problem to be explored
(b) evidence of the developed solution and the solution reached
(c) analysis and interpretation of the results
(d) a conclusion
(e) acknowledgements, appendixes, and bibliography (as required).
Learning Requirements Assessment Design Criteria Capabilities
1. Understand fundamental mathematical concepts and relationships.
2. Identify, collect, and organise mathematical information relevant to investigating and finding solutions to questions/problems taken from social, scientific, economic, or historical contexts.
3. Recognise and apply the mathematical techniques needed when analysing and finding a solution to a question/problem in context.
4. Make informed use of electronic technology to provide numerical results and graphical representations.
5. Interpret results, draw conclusions, and reflect on the reasonableness of these in the context of the question/problem.
6. Communicate mathematical ideas and reasoning using appropriate language and representations.
7. Work both independently and cooperatively in planning, organising, and carrying out mathematical activities.
Mathematical Knowledge and Skills and Their Application
The specific features are as follows:
 MKSA1 Knowledge of content and understanding of mathematical concepts and relationships.
 MKSA2 Use of mathematical algorithms and techniques
(implemented electronically where appropriate) to find solutions to routine and complex questions.
 MKSA3 Application of knowledge and skills to answer questions set in applied contexts.
Mathematical Modelling and Problem-solving
The specific features are as follows:
 MMP1 Application of mathematical models.
 MMP2 Development of mathematical results for problems set in applied contexts.
 MMP3 Interpretation of the mathematical results in the context of the problem.
 MMP4 Understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made.
Communication of Mathematical Information
The specific features are as follows:
 CMI1 Communication of mathematical ideas and reasoning to develop logical arguments.
 CMI2 Use of appropriate mathematical notation, representations, and terminology.
Communication
Citizenship
Personal
Development
Work
Learning
Page 1 of 4 Stage 2 Mathematical Applications task
Ref: A185534 (revised January 2013)
© SACE Board of South Australia 2013

PERFORMANCE STANDARDS FOR STAGE 2 MATHEMATICAL APPLICATIONS
Mathematical Knowledge and
Skills and Their Application
Mathematical Modelling and Problemsolving
Communication of
Mathematical Information
A Comprehensive knowledge of content and understanding of concepts and relationships.
Appropriate selection and use of mathematical algorithms and techniques (implemented electronically where appropriate) to find efficient solutions to complex questions.
Highly effective and accurate application of knowledge and skills to answer questions set in applied contexts.
B Some depth of knowledge of content and understanding of concepts and relationships.
Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find some correct solutions to complex questions.
Accurate application of knowledge and skills to answer questions set in applied contexts.
Attempted development and appropriate application of mathematical models.
Mostly accurate and complete solutions to mathematical problems set in applied contexts.
Complete interpretation of the mathematical results in the context of the problem.
Some depth of understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made.
C Generally competent knowledge of content and understanding of concepts and relationships.
Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find mostly correct solutions to routine questions.
Generally accurate application of knowledge and skills to answer questions set in applied contexts.
Effective communication of mathematical ideas and reasoning to develop mostly logical arguments.
Mostly accurate use of appropriate notation, representations, and terminology.
Appropriate application of mathematical models.
Some accurate and generally complete solutions to mathematical problems set in applied contexts.
Generally appropriate interpretation of the mathematical results in the context of the problem.
Some understanding of the reasonableness and possible limitations of the interpreted results and some recognition of assumptions made.
Appropriate communication of mathematical ideas and reasoning to develop some logical arguments.
Use of generally appropriate notation, representations, and terminology, with some inaccuracies.
D Basic knowledge of content and some understanding of concepts and relationships.
Some use of mathematical algorithms and techniques (implemented electronically where appropriate) to find some correct solutions to routine questions.
Sometimes accurate application of knowledge and skills to answer questions set in applied contexts.
E Limited knowledge of content.
Attempted use of mathematical algorithms and techniques
(implemented electronically where appropriate) to find limited correct solutions to routine questions.
Attempted application of knowledge and skills to answer questions set in applied contexts, with limited effectiveness.
Development and effective application of mathematical models.
Complete, concise, and accurate solutions to mathematical problems set in applied contexts.
Concise interpretation of the mathematical results in the context of the problem.
In-depth understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made.
Application of a mathematical model, with partial effectiveness.
Partly accurate and generally incomplete solutions to mathematical problems set in applied contexts.
Attempted interpretation of the mathematical results in the context of the problem.
Some awareness of the reasonableness and possible limitations of the interpreted results.
Attempted application of a basic mathematical model.
Limited accuracy in solutions to one or more mathematical problems set in applied contexts.
Limited attempt at interpretation of the mathematical results in the context of the problem.
Limited awareness of the reasonableness and possible limitations of the results.
Highly effective communication of mathematical ideas and reasoning to develop logical arguments.
Proficient and accurate use of appropriate notation, representations, and terminology.
Some appropriate communication of mathematical ideas and reasoning.
Some attempt to use appropriate notation, representations, and terminology, with occasional accuracy.
Attempted communication of emerging mathematical ideas and reasoning.
Limited attempt to use appropriate notation, representations, or terminology, and with limited accuracy.
Page 2 of 4 Stage 2 Mathematical Applications task
Ref: A185534 (revised January 2013)
© SACE Board of South Australia 2013

FOLIO - INVESTIGATION 1
MATRICES
Introduction
In a discussion with a group of your friends someone in the group said that
___________ has too many fast-food outlets for the current population. They went on to say that they could not see ___________ being able to support all of these outlets into the future. A heated discussion went on for several minutes but neither side had any real data on which to base their argument. You decided that you could use your knowledge of Transition Matrices to investigate the situation and have a look at the long term prospects for fast food outlets in ___________ .
Read the entire task and write an informative introduction that demonstrates your understanding of the situation to be investigated.
Mathematical Investigation
Use matrix methods to investigate the following questions.
1. Categorise the fast-food meals available in ___________.
2. Conduct a survey to find out what the last fast-food meal was and what their next fast-food meal is likely to be.
(a) Decide on a sampling method and justify your choice.
(b) How well does your sample reflect the population of ___________?
(c) Discuss any possible errors that could occur when using this sampling method.
3. Construct an initial state matrix showing their last fast-food meal.
4. Construct a transition matrix showing the probabilities associated with the choice of their next fast-food meal.
5. Use your matrices to predict how many of your sample will purchase each fastfood meal at their next choice and discuss your findings.
6. Find the steady state matrix. Predict how many of your sample would buy each meal in the long run.
7. How well do you think your data could be extrapolated to the ___________ population? Explain.
8. What do you think are the current long term prospects for fast food outlets in
___________? Justify your answer.
9. Comment on the reliability of your answers and on any assumptions you have made. Discuss the limitations of the model used.
10. There are many things that can influence market share. Consider the impact on the long term prospects for fast-food outlets in ___________ if there are changes made to:
(a) The initial state matrix
(b) The transition matrix.
Provision of clear directions and scaffolding is recommended in the first investigation to support students to develop appropriate techniques for undertaking and presenting folio tasks. Subsequent investigations should be less directed.
Students have opportunities to appropriately select and use mathematical algorithms and techniques as they apply their knowledge and skills to find solutions in an applied context.
Page 3 of 4 Stage 2 Mathematical Applications task
Ref: A185534 (revised January 2013)
© SACE Board of South Australia 2013

Analysis /Discussion
11. Critically analyse your results, considering the information your investigation has provided and the possible implications of the investigation.
Conclusion
12. Your conclusion should include a summary of results, comments on the appropriateness of the model used and any assumptions and limitations of the investigation.
Question 11 and
12 provide the opportunity to develop logical arguments through highly effective communication of mathematical ideas and reasoning.
Mathematical Modelling and Problem-solving
Students have the opportunity to develop a mathematical model that leads to mathematical results (i.e. the long-term prospects). Students are asked to interpret their mathematical results in the context of the problem. The recognition and discussion of error sources/assumptions provides further opportunity to demonstrate their understanding of the reasonableness and possible limitations of the results.
Communication of Mathematical Information
Students have many opportunities to provide evidence of proficient and accurate use of appropriate notation (e.g. labelling of matrices), representations (e.g. tabulated or matrix form of presenting data), and terminology (e.g. appropriateness of sampling language).
Page 4 of 4 Stage 2 Mathematical Applications task
Ref: A185534 (revised January 2013)
© SACE Board of South Australia 2013