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Mathematical Studies
2 0 0 4 A S S E S S M E N T R E P O R T
Mathematics Learning Area
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MATHEMATICAL STUDIES
2004 ASSESSMENT REPORT
GENERAL COMMENTS
This was the second examination that assessed student learning
based on the MathematicalStudies Curriculum Statement. The overall
results differed little from those of 2003.
Students provided fully correct solutions for each of the
fourteen questions.
Again, the examining team wrote an examination that provided an
honest assessment of thelearning outcomes and content of this
subject as directed by the curriculum statement.
The examination included questions that required students to:•
‘use mathematics as a tool to analyse data and other information
elicited from the study of
situations’;• ‘understand fundamental concepts, demonstrate
mathematical skills, and apply routine
mathematical procedures’;• look for reasons that explain
results;• communicate mathematically.
It was evident many students were well rehearsed in the routine
procedures included in thiscourse. It was also evident that many
students found great difficulty in being able to do more thanthey
had rehearsed. Applying routine mathematical procedures is one
(important) part of thiscourse, but not the only part – see the
learning outcomes in the curriculum statement.
As in 2003, where students were required to demonstrate a simple
mathematical skill, their level
of communication was good. However, when the demonstration of a
deeper understanding wasrequired, the quality of the responses fell
dramatically.
Some students seem to have found contextual questions that are
composed of connected partschallenging. Such questions could be
defined simply as problem-solving type questions.
It seems unlikely that the traditional teaching approach of
‘type example followed by completionof an exercise set’, if the
only approach used, will prepare students adequately to fulfil
thelearning outcomes and address the key questions and key ideas of
this course.
Too many students were unsure of when and where to use
electronic technology. See specificquestion comments for more
detail.
While many excellent solutions were written, in general, the
students’ ability to communicate thelogic used to arrive at an
answer was not strong. To be awarded part marks for a question
theymust write down the steps of mathematical logic they have
employed to reach the answer.
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ASSESSMENT COMPONENT 3: EXAMINATION
Question 1
Parts (a) and (b) were generally done well. Part (c), a very
similar question to that from the 2003examination, was handled less
well. Too many students employed a variety of incorrect
methods.Some students attempted to integrate each part of the
product.
Question 2
This question was done quite well. When indicating something on
a graph (part (a)), studentsmust be sure to do so very clearly. The
main error made by students was to not use values
provided in the table and take estimates from the graph.
It was disappointing that many students, despite having
calculated estimates for v( t ) dt 20
40
! , wereunable to provide a correct meaning of this integral in
terms of the motion of the car.
Question 3
Most students were able to compute the answer to part (a)
correctly but quite a few students didnot then use this value in
answering part (b). Clearly these students were looking at the
parts ofthis questions as unrelated.
In part (b) (ii), the phrase ‘at least two’ caused problems for
some students. Students should beencouraged to write down their
logic in such questions. If an incorrect answer is all that is
writtendown, then the student can be awarded no marks from the
three marks allocated. If the studentwrites down P (Y ≥ 2) = 1 − P
(Y ≤ 2) , then the marker can determine where the error in logic
isand award part marks accordingly.
If students have a form of electronic technology that allows a
calculation, like that in part (b) (ii),to be computed directly,
then great care must be taken to ensure they compute it
correctly.
Question 4
The large majority of students had rehearsed the routine process
of first principles differentiationvery well. Those students who
substituted 4 for x at the beginning of the process had a far
easierroute to success. Common errors included:
• starting with3 x + h
x+
h−
2
instead of3( x + h )
x+
h−
2
;
• placing the limit statement on the wrong side of the equal to
sign;• removing the limit statement incorrectly;• algebraic
manipulation of the fraction.
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Question 5
Students wrote generally good solutions to this question. The
algebraic nature of the determinantcaused problems for some
students.
When applying one of the determinant algorithms, some students
confused the symbol for thedeterminant of a two by two matrix for
brackets and then proceeded to find a two by two matrixas an
answer.
Most students saw the need to equate the determinant to zero in
part (c), but struggled to makesense of the result of this if they
had arrived at unusual results in part (b). If students made
errorsin part (b) they could still achieve full marks in part (c),
but had to deal with the solutions to theequation sensibly should
they have not been real numbers. Most students seemed
comfortableusing electronic technology to find the solutions to the
equation they derived in part (b).
Question 6
Many students wrote very good solutions to this question.
Many students handled the implicit differentiation process well.
The most common error here wasnot using brackets when
differentiating –2 xy.
Some students wasted time by finding both tangents in part (b).
Others were able to make theappropriate substitution for y, but
then could not proceed to simplify the equation to a
simplequadratic and find its solutions.
Many students were able to realise they had to equatedydx
to 1 to begin part (c). Some students
did not know how to proceed after arriving at 3( y − 1) 2 = − 2
y . Some students arrived at
3 y2
− 4 y + 3 = 0 but failed to clearly state this equation had no
real solutions. Using electronictechnology for this was
appropriate, as was the use of the discriminant.
It should be noted that using a numerical method (‘solver’ on
electronic technology) on theequation 3( y − 1) 2 = − 2 y was not
helpful.
Question 7
A pleasing number of excellent solutions were written to this
question. However, too many poorsolutions were also offered.
This question required the students to apply some fundamental
concepts and skills to a problemthey had not seen before – an
unrehearsed situation.
In part (a) the students had to clearly show the logic that
explained how the traffic flow at eachintersection gave rise to the
augmented matrix. Often, students simply reconstructed the
equationsfrom the augmented matrix.
Most students coped well with the row operations expected in
part (b). Some students made thequestion more difficult by
operating before substituting the values for a , b, and so on.
Some
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students lost marks because they used electronic technology to
reduce the matrix to row echelonform. The question clearly asked
the student to use row operations and to show each step of
theworking.
In part (c) (i) some students did not use the work they had done
in part (b). Time could have beensaved if they had. Some students
clearly had difficulties using parametric equations to representthe
infinite set of solutions to this system.
While many students answered part (c) (ii) correctly it was
disappointing that many more studentsdid not. It was disappointing
that more students did not simply look at the diagram and see that
if x1 = 0 then x 2 = a . Some students, who had answered part (c)
(i) correctly, did not make thesimple connection that x1 = 0 and
substitute accordingly or use the diagram as explained above.
Students should be encouraged to attempt the last parts of
questions whether or not they had feltthey had been successful in
the previous part.
Question 8
Again, many excellent solutions were written for this
question.
In this sort of question students must be encouraged to write
down each line of their working, so points of error can be clearly
found and the marker is able to award the marks that have
beenearned.
The inclusion of a parameter ( a in this case) did cause some
students some concern.
Question 9
This question assessed students’ understanding of the
relationship between a function and itsderivative, from a graphical
perspective – a fundamental concept.
In helping students to understand such concepts, the use of
electronic technology is invaluable,even though it is of no direct
use in answering this question.
In general, student performance in this question highlighted the
fact that even though moststudents are competent at the application
of routine processes, their understanding of thefundamental concept
examined here was not strong.
Question 10
This question was not handled well by many students.
The key questions and ideas from Subtopic 2.1: The Utility of
Functional Models are thefoundation of this question.
One model listed is the quadratic model. The following comments
are given in the curriculumstatement as supporting ideas of how to
address this section of the curriculum statement.
• For example, an algebraic model could be derived from
assumptions or given properties,….
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• It is envisaged that this approach would work effectively for:
quadratic models for the area of(say) rectangles with fixed
perimeters; ….
Most students were able to answer part (a) correctly. While many
good solutions (and someexquisite ones) were written for part (b),
many students lost their way after starting. The simplestway to
proceed was to use the fact that the triangle FDC had two lengths
that formed two thirds ofa Pythagorean triad (3, 4, 5). A number of
other approaches were possible and were used.
Even if unsuccessful in part (b), students were able to continue
and still gain all marks thatfollowed. It was disappointing to see
that many students could not apply the area of a triangleformula to
find the area of triangle DEF. The fact that AD was the height of
this triangle was notobvious to many.
Question 11
The routine parts of this question were generally done well.
Many students had difficulty in recognising the importance of
the range 1015 grams to 1025grams in answering part (b). Many did
not recognise they had already done the necessarycalculations to
answer Part (b) in Part ( !!! ).
Part (c) (iii) was not handled well. Few students were able to
see how their answers to part (c) (i)and (ii) could be used in part
(c).
Clearly students would benefit from more opportunities to solve
problems that require theapplication of routine process (like the
calculation of a confidence interval) and some simplelogic.
Question 12
This question assessed student achievement in the following key
questions/ideas from Subtopic2.3: Introduction to the Definite
Integral:
• The definite integral as the unique number between the upper
and lower sums, interpreted both as an area and as distance
travelled.
• Areas that can be determined by geometric means, such as:-
linear functions or the circle areas under the graph of simple
polynomial, rational,
exponential, or root functions ……..
This was the most poorly done question in the examination. It
was evident that students did notunderstand that the definite
integral could be interpreted as an area.
Many students attempted to integrate 9 − x 2 using some
‘invented’ algorithm.
It was also evident that many students did not understand the
phrase ‘exact value’. Many used
electronic technology to find a decimal approximation for 9 − x
2−3
3
! dx
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In part (c), many students had difficulty interpreting the
meaning of the integral presented.
Few students saw how they were able to use the result presented
in part (c) to complete part(d) (i).
Of the students who made a good attempt at the final part, too
many wrote incomplete solutions,failing to explicitly state that
the integral provided the value of an area. It was sufficient for
astudent to draw a picture of a ‘flattened’ circle with appropriate
dimensions and state the integralcalculated the area of this
figure. Knowledge of an ellipse was not assumed.
Question 13
Most students did very well in parts (a), (b) and (d) of this
question.
In part (c), many students did not ‘explain’. For full marks
students had to make clear that thematrix C , when pre-multiplied
by A gave the predicted numbers in each age group on day 1 andthen
the matrix AC being pre-multiplied by A (resulting in A 2C ) gave
the same for day 2.
In part (d) many students saw a need to include k in their
answer. Clearly the cyclic naturehighlighted by the answers to
parts (d) (i) and (ii) was missed by many students.
Many students were able to perform the algebraic manipulation
required in part (e) (i) and (ii), butrelatively few were able to
use the results in answering part (f). In part (e) (i), of those
who found MN , some did not then use the given result that MN = N
and consequently stopped. Of those whodid use the given result,
some then struggled to use the resulting equations to show the
requiredresult.
Question 14
More than a few students did not attempt this question at all.
This is a disturbing result givenmany of the marks in this question
were awarded to routine calculations.
Parts (a) and (b) were answered well by many students. However,
some students’ use of notationin part (b) was poor. Many stated
that X = µ rather than µ ( X20 ) = µ .
Many students provided an incorrect answer to part (c). They
seem to believe that, provided n is30 or greater, the distribution
of sample means will be approximately normal. This is notgenerally
true.
Most students coped well with part (d).
Many students who chose the correct option for part (e) were
unable to explain why.
Part (f) of this question required students to have understood
the question as a whole. This wasnot done well by many students. Of
those students who made a good attempt, many did notrealise that
their answer to part (d) would have been of help in part (f) (ii)
and re-calculated it.Many also failed to show that 19 batteries
were not enough.
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Question 15
Too many students drew very poor graphs in part (a). No
indication of scale, carelessly drawncurves and a general lack of
accuracy were among the problems.
It was common for students to take an algebraic approach to part
(b) (i). This was unfortunate.Electronic technology provided an
efficient method to approach this task.
It was disappointing that many students were unable to recognise
C = ab t , t ≥ 0 as anexponential function. Many students were
unable to make the link that an exponential functionincreases (at
an increasing rate) as t increases and that such a model would be
inappropriate tomodel the cumulative number of sales of digital
television.
Relatively few students were able to answer part (d) well. Few
students used the fact that theinflection point of a logistic
function occurs at half the limiting value of the function and that
thedata presented seemed to have the point of maximum rate of
change at a percentage lower thanthe limiting percentage (even if
the limiting value was taken to be 90%).
Providing an interpretation of the point of inflection provided
in part (d) (ii) was done poorly.
Chief AssessorMathematical Studies
