STAGE 2 MATHEMATICAL STUDIES
FOLIO TASK – STATISTICAL INVESTIGATION OF INTERNET USAGE PATTERNS
Topic:
Subtopics:
Working with Functions and Graphs Using Calculus
Subtopic 1.1: Normal Distributions
Subtopic 1.5: Binomial Distributions
Subtopic 1.6: Testing Claims about a Population Proportion
Subtopic 1.7 Confidence Intervals for a Population Proportion
A completed investigation should include:
 an introduction that outlines the problem to be explored, including its 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
 a statement of the results and conclusions in the context of the original problem
 appendices and a bibliography, as appropriate.
Learning Requirements
Assessment Design Criteria
Capabilities
1. understand fundamental
mathematical concepts,
demonstrate mathematical
skills, and apply routine
mathematical procedures
2. use mathematics as a tool
to analyse data and other
information elicited from the
study of situations taken
from social, scientific,
economic, or historical
contexts
3. think mathematically by
posing questions/problems,
making and testing
conjectures, and looking for
reasons that explain the
results
4. make informed and critical
use of electronic technology
to provide numerical results
and graphical
representations
5. communicate
mathematically and present
mathematical information in
a variety of ways
6. work both individually and
cooperatively in planning,
organising, and carrying out
mathematical activities
Mathematical Knowledge and Skills and
Their Application
Communication
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Learning
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 in applied and theoretical contexts.
Mathematical Modelling and Problem-solving
The specific features are as follows:
MMP1 Application of mathematical models.
MMP2 Development of solutions to mathematical
problems set in applied and theoretical 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.
MMP5 Development and testing of conjectures, with
some attempt at proof.
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.
Stage 2 Mathematics studies annotated task
Ref: A153516 (January 2012)
© SACE Board of South Australia 2012
Performance Standards for Stage 2 Mathematical Studies
A
Mathematical Knowledge and
Skills and Their Application
Mathematical Modelling and
Problem-solving
Communication of
Mathematical
Information
Comprehensive knowledge of content and
understanding of concepts and
relationships.
Development and effective application of mathematical models.
Highly effective communication of
mathematical ideas and
reasoning to develop logical
arguments.
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 and theoretical 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 and
theoretical contexts.
Complete, concise, and accurate solutions to mathematical
problems set in applied and theoretical 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.
Proficient and accurate use of
appropriate notation,
representations, and terminology.
Development and testing of valid conjectures, with proof.
Attempted development and appropriate application of
mathematical models.
Mostly accurate and complete solutions to mathematical
problems set in applied and theoretical contexts.
Complete interpretation of the mathematical results in the
context of the problem.
Effective communication of
mathematical ideas and
reasoning to develop mostly
logical arguments.
Mostly accurate use of
appropriate notation,
representations, and terminology.
Some depth of understanding of the reasonableness and
possible limitations of the interpreted results, and recognition of
assumptions made.
Development and testing of reasonable conjectures, with
substantial attempt at proof.
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 and theoretical contexts.
D
E
Basic knowledge of content and some
understanding of concepts and
relationships.
Appropriate application of mathematical models.
Some accurate and generally complete solutions to
mathematical problems set in applied and theoretical 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.
Application of a mathematical model, with partial effectiveness.
Partly accurate and generally incomplete solutions to
mathematical problems set in applied or theoretical contexts.
Attempted interpretation of the mathematical results in the
context of the problem.
Sometimes accurate application of
knowledge and skills to answer questions
set in applied or theoretical contexts.
Attempted development or testing of a reasonable conjecture.
Some awareness of the reasonableness and possible
limitations of the interpreted results.
Limited knowledge of content.
Attempted application of a basic mathematical model.
Attempted use of mathematical algorithms
and techniques (implemented electronically
where appropriate) to find limited correct
solutions to routine questions.
Limited accuracy in solutions to one or more mathematical
problems set in applied or theoretical contexts.
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Use of generally appropriate
notation, representations, and
terminology, with some
inaccuracies.
Development and testing of reasonable conjectures, with some
attempt at proof.
Some use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find some correct
solutions to routine questions.
Attempted application of knowledge and
skills to answer questions set in applied or
theoretical contexts with limited
effectiveness.
Appropriate communication of
mathematical ideas and
reasoning to develop some
logical arguments.
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.
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.
Limited attempt to develop or test a conjecture.
Stage 2 Mathematics studies annotated task
Ref: A153516 (January 2012)
© SACE Board of South Australia 2012
STAGE 2 MATHEMATICAL STUDIES
FOLIO TASK – STATISTICAL INVESTIGATION OF INTERNET USAGE PATTERNS
The data in the table below was taken from the 2006 CensusAtSchool. The figures given can be considered to be
representative of the 2006 population of students who took part in the project. That is, the data represented males and
females from all year levels, from all across Australia.
The data are from the question:
How do you usually spend your time on the internet? 2006 (Q.35)
Internet Activity
Buying and/or selling things
Emailing friends and family
Researching for school work
Playing games and simulations
Never
6.7%
17.8%
15.1%
79.1%
Rarely
20.1%
17.3%
24.5%
14.1%
Sometimes
45.9%
35.6%
30.7%
4.2%
Often
27.3%
29.3%
29.7%
2.6%
1) Go to the internet and log on to the Census at Schools home page at
http://www.abs.gov.au/websitedbs/CaSHome.nsf/Home/Home
2) Use the Random Sampler to select a sample of students from which you can extract appropriate data to
determine if internet usage has changed between 2006 and the data currently available.
3) Do appropriate statistical analysis of the relevant data. You may seek help with the organisation of the data you
download from the internet but you need to make decisions about the best way to analyse the data.
4) Prepare a report that includes:
 an introduction explaining what you are investigating
 an explanation of your method
 mathematical calculations and results, and appropriate representations
 an analysis of your results
 a statement of the results and conclusions in the context of the original problem, and considering the
reasonableness and possible limitations of the interpreted results and recognition of any assumptions made.
You do not have to attach a printout of the raw data but should save the data as an Excel file and provide in an
appendix details of your username and the location of the file on the school’s intranet.
You must also submit a printed or hand-written version of your report.
Page 3 of 4
Stage 2 Mathematics Studies annotated task
Ref: A153516 (January 2013)
© SACE Board of South Australia 2012
Mathematical Knowledge and Skills and Their Application
In this task, there are ample opportunities to demonstrate a comprehensive knowledge of content and understanding of concepts
and relationships pertaining to subtopics 1.1, 1.6 and 1.7 (normal distributions, testing claims, and estimating confidence intervals
for a population proportion). There is a wide range of possible approaches to producing a solution to the problem, providing
opportunities to make appropriate selection and to demonstrate the use of appropriate mathematical algorithms and techniques to
find efficient solutions to this complex problem. Efficient use of technology will make any repetitive tasks much less time
consuming.
Depending on their background and prior opportunities to use spreadsheets, students will have varying skills in using this kind of
software. Some students will do the calculations on a graphics calculator and use the spreadsheet as a display tool rather than
fully use the power of a spreadsheet. This will not, on its own, prevent them from being assessed at the highest level in
Mathematical Knowledge and Skills and Their Application.
Mathematical Modelling and Problem-solving
Some students may choose to focus on one aspect of internet activity (e.g. Research for School) and in demonstrating a change
in the pattern of use there, consider this sufficient evidence that patterns of internet use have changed. A more complete solution
would examine all aspects of internet use and explain in what way the use of the internet has changed between 2006 and the
present.
The way students compare the data and calculated values from the 2006 and current cohorts provides them with opportunities to
demonstrate their ability to interpret their results in this context.
The use of the “Random Sampler” provides opportunities to comment on the reasonableness and possible limitations of their
interpreted results. Where the random sampler provides anomalous results, students have the opportunity to explain how they
dealt with those results and to comment on any assumptions they made. Classroom discussion about the nature of the data
collected in the CensusatSchool web site, prior to doing this task assists in making decisions about the reliability and
reasonableness of results from their data.
Examination of the data for 2006 and the current year and the results of the hypothesis testing and confidence intervals calculated
lead students to develop a conjecture as to whether or not internet use has changed. The way they go about explaining and
justifying their point of view by referring to their data and calculated values provides opportunities for assessment of how well they
have substantiated this conjecture.
As it stands, this task does not provide an opportunity for students to work in a group; however, this specific feature could be
addressed if the task was altered. For example, students could be given the opportunity to work together on a joint approach to
the task, explaining their intended method but each finding their own sample and drawing their own conclusions. Sharing and
comparison of results might also provide further opportunity to discuss reasonableness and possible limitations of their interpreted
results.
Communication of Mathematical Information:
The collation of values calculated from the sample data and comparison of these values with the data representative of the 2006
cohort provides the opportunity to demonstrate reasoning to develop a logical argument in support of their hypothesis. There is
clearly a requirement to communicate this argument and support it with the values they calculate. Graphical representations (for
example, of the position of a 2006 value compared to the confidence interval for the current value) could enhance communication
of written arguments.
Though much of the mathematical calculations for p-values, Z-scores and confidence intervals are done with technology, some
students may choose to provide hand written mathematical working of these values as an example of what is being done with the
technology. Hence the opportunity exists for the students to demonstrate proficient and accurate use of appropriate notation.
Written comments making interpretations, discussing reasonableness and limitations of results and supporting conjectures provide
opportunities to use appropriate terminology.
Page 4 of 4
Stage 2 Mathematics Studies annotated task
Ref: A153516 (January 2013)
© SACE Board of South Australia 2012