STAGE 2 MATHEMATICS PATHWAYS
EXTERNAL ASSESSMENT – INVESTIGATION
This is a written investigation to be completed under supervision and consisting of a series of connected questions.
Students are required to draw their own conclusions and complete a report about the investigation.
Students are required to demonstrate their use of problem-solving strategies, as well as their knowledge, skills, and
understanding. The exploration of any patterns and structures, using changing parameters, is included.
Students are expected to make use of electronic technology where appropriate.
The total time spent by students in completing the series of connected questions and the report should be 1½
hours for a 10-credit subject.
The report for an investigation should include:
 an introduction that demonstrates an understanding of the features of the problem or the situation investigated
 evidence that the student has followed instructions
 mathematical calculations and results, and appropriate representations
 a summary of results or findings and conclusions drawn.
Learning Requirements
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.
Page 1 of 8
Assessment Design Criteria
Capabilities
Mathematical Knowledge and Skills and Their
Application
Communication
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.
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 Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011
Performance Standards for Stage 2 Mathematics Pathways
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.
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 and
theoretical contexts.
Highly effective communication of mathematical
ideas and reasoning to develop logical
arguments.
Highly effective and accurate application of
knowledge and skills to answer questions set
in applied and theoretical contexts.
B
C
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 and
theoretical 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
and theoretical contexts.
Effective communication of mathematical ideas
and reasoning to develop mostly logical
arguments.
Mostly accurate use of appropriate notation,
representations, and terminology.
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 and theoretical contexts.
Generally appropriate interpretation of the
mathematical results in the context of the
problem.
Appropriate communication of mathematical
ideas and reasoning to develop some logical
arguments.
Use of generally appropriate notation,
representations, and terminology, with some
inaccuracies.
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 appropriate communication of
mathematical ideas and reasoning.
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 or theoretical contexts.
Some attempt to use appropriate notation,
representations, and terminology, with
occasional accuracy.
Sometimes accurate application of knowledge
and skills to answer questions set in applied or
theoretical 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 and theoretical
contexts.
Proficient and accurate use of appropriate
notation, representations, and terminology.
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 or
theoretical contexts, with limited effectiveness.
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.
Attempted communication of emerging
mathematical ideas and reasoning.
Limited accuracy in solutions to one or more
mathematical problems set in applied or
theoretical contexts.
Limited attempt to use appropriate notation,
representations, or terminology, and with
limited accuracy.
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.
Page 2 of 8
Stage 2 Mathematics Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011
Introduction
You are a tradesperson who must decide whether to buy a 4-wheel drive or a 2-wheel drive vehicle for your work.
You have collected the following data from a selection of 220 4-wheel drive and 220
2-wheel drive vehicles that a recent “Vehicle Trading” magazine had listed for sale.
4-wheel drives
WEIGHT (kg)
1855
1570
1600
2178
2234
2245
2303
1800
1833
1863
1868
1996
1996
1595
1595
1490
1525
1570
1480
1540
1855
1745
FUEL CONSUMPTION(L/100km)
11.3
10.0
10.0
13.8
14.5
13.8
14.5
11.0
10.4
10.6
9.5
9.9
9.5
10.5
10.3
8.9
9.7
9.1
9.3
9.6
11.9
10.3
2-wheel drives
WEIGHT (kg) FUEL CONSUMPTION(L/100km)
1540
6.1
1650
8.0
1205
7.3
1335
7.6
1455
7.6
1029
6.6
1050
7.5
1070
6.8
1268
7.1
1277
8.0
1454
7.5
1465
8.3
1562
9.5
1604
7.3
1735
11.7
1790
16.2
1790
12.5
1704
10.5
1480
7.8
1510
8.9
1615
9.7
865
5.0
You will need to:
 state the process you could have used to select your sample.
 compare the weights and fuel consumption of each vehicle type.
 check to see how much the weight of the vehicle affects its fuel consumption (is there a significant correlation
between vehicle weight and fuel consumption?).
 use the statistical evidence and any other factors to make a decision on which type of vehicle is most
appropriate for your work purposes.
Page 3 of 8
Stage 2 Mathematics Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011
Set your work out in the spaces provided on the following pages:
Step 1. Introduction
After reading through the entire paper, write a brief introduction to this investigation in your own words.
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Step 2. Sampling method
Explain the sampling method that could have been used to select these 44 cars from a full list of 440 cars (220 2wheel drives and 220 4-wheel drives) in the magazine.
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Step 3. Comparing mean weights.
a) Calculate the mean (average) weight of the 4-wheel drive vehicles.
b) Calculate the mean (average) weight of the 2-wheel drive vehicles.
c) What conclusion can you draw? Discuss the reasonableness of your answer.
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Page 4 of 8
Stage 2 Mathematics Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011
Step 4. Comparing median fuel consumption figures
a) Calculate the median fuel consumption for the 4-wheel drive vehicles selected in your sample.
b) Calculate the median fuel consumption for the 2-wheel drive vehicles selected in your answer.
c) (i) Using the medians, calculate the cost of travelling 100km trip costs $1.25 per litre.
Cost for 4-wheel drive:
Cost for 2-wheel drive:
(ii) What is the cost difference for the 100 km trip?
d) Discuss the assumptions made in determining the difference spread of each set of data.
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Step 5. Consistency of data
Calculate the standard deviation for each of the 4 sets of data and comment on the data.
WEIGHT
Standard Deviation
FUEL CONSUMPTION
Standard Deviation
4 WD
Sx =
4WD
Sx =
2WD
Sx=
2WD
SX=
e) Comment on the data:
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Page 5 of 8
Stage 2 Mathematics Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011
Step 6. Fuel consumption of 4-wheel drive vs 2-wheel drive - box and whisker plots
a) Complete the 5 number summaries for fuel consumption for both 4-wheel drive and 2-wheel drive vehicles
in the table below:
4 Wheel Drive
2 Wheel Drive
The minimum value (Min):
The minimum value (Min):
The lower quartile:
The lower quartile:
The median:
The median:
The upper quartile:
The upper quartile:
The maximum value (Max):
The maximum value (Max):
b) Draw box and whisker plots on the scale below to compare the 2 data sets.
--------------------------------------------------------
5
10
15
L/100km
Scale
Comment on the differences.
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Step 7. Correlation - Weight vs fuel consumption for 4–wheel drive vehicles
a) Consider the 4-wheel drive vehicles only. Plot fuel consumption against weight and draw in the line of best
fit. (You may use a graphics calculator and draw the graph by hand on the graph paper supplied, or
produce and print the graph from a computer.)
Attach your graph in the space below.
Page 6 of 8
Stage 2 Mathematics Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011
b) (i) State the r and r-squared values
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(ii) Comment on the strength of the correlation.
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c) Find the equation of the line of best fit (Trend line)
d) The gradient of the line of best fit tells us that for every _____ kg of extra weight, fuel consumption
increases/decreases by ______L/100km.
e) If you wanted a car that used no more than 7 litres per 100 km of fuel, use the equation of the line of best fit to
predict the maximum weight for your car.
Step 8. Discussion and conclusion
Discuss the statistics above, including their limitations, as well as discussing any other relevant work-related factors
that would affect your choice of vehicle.
Draw a conclusion about which type of vehicle you will choose.
(Discussion should be about ½ to 1 page in length. Refer to figures to back up your argument.)
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Page 7 of 8
Stage 2 Mathematics Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011
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Page 8 of 8
Stage 2 Mathematics Pathways annotated task for use from 2011
106742466 (revised March 2011)
© SACE Board of South Australia 2011