STAGE 2 MATHEMATICS PATHWAYS
FOLIO TASK
Transition Matrices - Limitations of Matrix Models
Topic:
Matrices
Subtopics from the Stage 2 Mathematical Applications Subject Outline:
4.3 – Transition Matrices
A completed investigation should include:
 an introduction that outlines the problem to be explored, including it significance, its features,
and the context
 the method required to find a solution, in terms of the mathematical model or strategy to be used
 the appropriate application of the mathematical model or strategy, including
- the generation or collection of relevant data and/or information, with details of the process of
collection
- mathematical calculations and results, and appropriate representations
- the analysis and interpretation of results
- reference to the limitations of the original problem
 a statement of the results and conclusions in the context of the original problem
 appendices and a bibliography, as appropriate.
Learning Requirements
1.
2.
3.
4.
5.
6.
Assessment Design Criteria
Capabilities
Demonstrate an understanding
of mathematical concepts and
relationships.
Mathematical Knowledge and Skills and Their
Application
Communication
Identify, collect, and organise
mathematical information
relevant to investigating and
finding solutions to
questions/problems.

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.
Recognise and apply the
mathematical techniques needed
when analysing and finding a
solution to a question/problem in
context.
Make informed use of electronic
technology to aid and enhance
understanding.
Interpret results, draw
conclusions, and reflect on the
reasonableness of these in the
context of the question/problem.
Communicate mathematical
ideas and reasoning using
appropriate language and
representations.
The specific features are as follows:

Citizenship
Personal
Development
Work
Learning
MKSA3 Application of knowledge and skills to
answer questions 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:
Page 1 of 3

CMI1 Communication of mathematical ideas and
reasoning to develop logical arguments.

CMI2 Use of appropriate mathematical notation,
representations, and terminology.
Stage 2 Mathematics Pathways Matrices task
Ref: A203780 (revised February 2016)
© SACE Board of South Australia 2010
PERFORMANCE STANDARDS FOR STAGE 2 MATHEMATICS PATHWAYS
Mathematical Knowledge and Skills
and Their Application
A
Comprehensive knowledge of content and
understanding of concepts and relationships.
Development and effective application of
mathematical models.
Appropriate selection and use of
mathematical algorithms and techniques
(implemented electronically where
appropriate) to find efficient solutions to
complex questions.
Complete, concise, and accurate solutions to
mathematical problems set in applied contexts.
Highly effective and accurate application of
knowledge and skills to answer questions set
in applied contexts.
B
Use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find some correct
solutions to complex questions.
Mostly accurate and complete solutions to
mathematical problems set in applied contexts.
Generally competent knowledge of content
and understanding of concepts and
relationships.
Generally accurate application of knowledge
and skills to answer questions 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.
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.
Basic knowledge of content and some
understanding of concepts and relationships.
Application of a mathematical model, with partial
effectiveness.
Some use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find some correct
solutions to routine questions.
Partly accurate and generally incomplete solutions
to mathematical problems set in applied contexts.
Sometimes accurate application of
knowledge and skills to answer questions set
in applied contexts.
E
In-depth understanding of the reasonableness and
possible limitations of the interpreted results, and
recognition of assumptions made.
Attempted development and appropriate
application of mathematical models.
Use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find mostly correct
solutions to routine questions.
D
Concise interpretation of the mathematical results
in the context of the problem.
Some depth of knowledge of content and
understanding of concepts and relationships.
Accurate application of knowledge and skills
to answer questions set in applied contexts.
C
Mathematical Modelling and Problemsolving
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.
Page 2 of 3
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.
Communication of
Mathematical
Information
Highly effective
communication of
mathematical ideas and
reasoning to develop
logical arguments.
Proficient and accurate use
of appropriate notation,
representations, and
terminology.
Effective communication of
mathematical ideas and
reasoning to develop
mostly logical arguments.
Mostly accurate use of
appropriate notation,
representations, and
terminology.
Appropriate communication
of mathematical ideas and
reasoning to develop some
logical arguments.
Use of generally
appropriate notation,
representations, and
terminology, with some
inaccuracies.
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.
Stage 2 Mathematics Pathways Matrices task
Ref: A203780 (revised February 2016)
© SACE Board of South Australia 2010
STAGE 2 MATHEMATICS PATHWAYS
FOLIO TASK
Transition Matrices - Limitations of Matrix Models
Introduction
There are currently 2 similar fruit and vegetable shops in one area – shop A and shop B. The
current local area market share of shop A is X%.
A survey of customers carrying out their weekly grocery shopping at each shop revealed that 76%
of customers buying their weekly groceries at shop A intended to do their grocery shopping at shop
A the following week, and 61% of customers buying their weekly groceries at shop B intended to
do their weekly grocery shopping at shop B the following week.
Mathematical Investigations
Using matrix methods investigate:



the market share trends in the long run, if conditions remain the same
the impact on the long term trends of varying the initial market share for the two shops
the effect of one of the shops mounting a strong advertising campaign.
A third cheaper shop (shop C) enters the market a few months later, and achieves an initial market
share of no more than 15%. A new survey reveals the following trends:
-
12% of shop A customers will shop at shop B the following week
36% of shop A customers will shop at shop C the following week
40% of shop B customers will shop at shop B the following week
44% of shop B customers will shop at shop C the following week
14% of shop C customers will shop at shop A the following week
7% of shop C customers will shop at shop B the following week.
Using matrix methods consider:




the initial market share of the three shops, providing reasons for the values in the initial matrix
the market trends after different periods of time (weeks)
the impact on the long-term market share of each shop if the survey results were significantly
different
the effect of changes to the scenario, e.g. one shop extending trading hours to encourage
customers to change shopping habits. You should investigate at least two changes to the
scenario, and therefore variations to the transition matrix.
Analysis/Discussion
Critically analyse your results, considering:


the information your calculations have provided
possible implications of the investigation.
Conclusion:
The conclusions should include a summary of results, comments on the appropriateness of the
model used and any assumptions and limitations of the investigation.
Notes to teacher:
The subject outline requires matrices that are 3×3 systems or higher, and therefore it is important
that the majority of the calculations use matrices that fit this requirement. The 2×2 transition matrix
used in the initial investigations is only considered routine; however for the purpose of the initial
investigations it is appropriate.
Teachers may consider adding visual aids to the task to assist students who need support in
accessing the requirements of the task.
Page 3 of 3
Stage 2 Mathematics Pathways Matrices task
Ref: A203780 (revised February 2016)
© SACE Board of South Australia 2010