2msu-ar-2007

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Mathematical Studies

2 0 0 7 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

2007 ASSESSMENT REPORT

GENERAL COMMENTS

The 2007 examination provided students with the opportunity to demonstrate their skills and

understanding in relation to content drawn from across the Mathematical Studies curriculum.

To achieve to a high level, students needed to be able to express an understanding of the key

concepts of this course, to be proficient algebraic manipulators, to be able to work with

rational and irrational numbers, and to be discerning users of electronic technology. The

examination provided many with a successful final experience of secondary mathematics,

with a mean of 60% and a significant proportion of students scoring better than 70%.

Importantly, the examination provided sufficient opportunities for the very best students to

distinguish themselves, with a good degree of differentiation at the top end of the cohort.

These more challenging elements were found in the last parts of a number of questions andwere not concentrated in the final questions, meaning that most students were able to access

all parts of the paper in a satisfactory manner.

At a number of points throughout the paper, where students were required to display algebraic

skills, it was commonplace to see marks lost due to poor use of brackets and a lack of

confidence with negative signs.

There were parts of the paper where students were asked to work in terms of exact values.

Given the availability of technology in this examination, the requirement of a solution in

terms of exact values is an important way of assessing that students have the algebraic skills

to complete solutions that could otherwise be undertaken with the use of technology, leading

to a non-exact solution. Some otherwise strong students seemed uncomfortable with the useof exact values within mathematical processes. In relation to exactness, it is important that

students are clear that an answer obtained using electronic technology or a decimal value will

not satisfy this requirement and so will not receive full credit in these questions. In a similar

vein, the questions involving unknown constants saw some students attempt the assignation of

values rather than working confidently in terms of this unknown.

One issue of concern was the number of students who presented several solutions to the same

question and who did not cross out the solution(s) that they considered incorrect, despite the

instruction on the front cover to use a single line to do so. The issue of giving full credit to a

student who has provided multiple solutions and is unable or unwilling to determine that a

solution is, in fact, correct is one that will be reviewed as part of the marking panel’s ongoing

refinement of marking practices.

ASSESSMENT COMPONENT 3: EXAMINATION

Question 1

This question was answered very well, with the majority of students obtaining full marks. In

Part (b), a number of students proceeded to solve the determinant equal to zero—an

unnecessary procedure.

Question 2

A good proportion of students scored highly for this question. The students who scored less

highly tended to be those who misinterpreted the requirements of Part (a) and either failed to

provide an equation in Part (i) or differentiated rather than integrated in Part (ii).

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Question 3

Part (a) of this question was completed well. Some students made errors when working with

the language of probability like ‘no more than …’ and ‘at least one …’. Given the strong

progress through Part (a), relatively few students were successful in tackling Part (b).

Of those that saw the link to Part (a), many successfully pursued a trial and error solution.

Some lost marks by giving an answer that generated a proportion that rounded to one-quarter,

rather than giving an answer that generated a proportion ‘less than one-quarter …’ as

required. A notable proportion of students who successfully answered Part (b) used an

analytic method.

Question 4

Students who read the question carefully completed this well, with over 25% receiving full

marks. Marks were lost in Part (a) for failing to provide an overestimate or for not drawing

the correct number or width of rectangles. In Part (b), some students did not appreciate that

attempting to read function values from a graph is generally an inaccurate method and not one

that will provide answers to two decimal places, as was required. In Part (c), the commonest

way to lose marks was to ignore the requirement of an ‘exact’ value.

Question 5

This question was responded to well, with strong results in Parts (a) to (c). The loss of marks,

where it occurred, was most often in Part (d). In Part (c), some students did not adequately

define the row operations required to obtain the reduced system that was given. In Part (d),

the errors arose from the execution of a single solution in terms of an existing parameter, with

some attempting a parametric solution of the form ‘let z = t …’.

Question 6

This question examined both the concept of a rate of change (subtopic 2.4) and use of first

principles (subtopic 2.6). The first principles algebraic structure in Part (b) was generally

done well, with student errors occurring due to a failure to use brackets wisely or an inabilityto successfully manipulate algebraic fractions to obtain a common denominator. The concepts

asked for in Parts (a) and (c) were, at times, poorly expressed. In Part (a), neglect of the

concept of ‘average’, in terms of rates of change, weakened many students’ responses. In

Part (c), phrases referring to slopes of functions instead of slopes of tangents (to functions)

lacked a key element of the interpretation that was required.

Question 7

While Part (a) of this question was generally answered well, many students struggled with the

conceptual relationship between a derivative and its function of origin. The link between the

maxima/minima of a derivative and the inflection point of a function, as seen with logistic

functions, helped many students obtain partial credit in Part (b). Too few students utilised the

other information that was provided by the derivative, like the fact that it was negative,implying that the function was decreasing.

Question 8

This question was answered well. In Part (a), some students lost marks by failing to note the

importance of the confidence interval lying below the critical value. In Part (b), a surprising

number of students struggled to select or use appropriately the correct formula, including the

necessary rounding up.

Question 9

Many students scored poorly in this question, with some making no attempt to tackle even the

relative simple calculations required in Part (a). This suggests that some of these students

have had limited exposure to a dynamic representation such as this, where a parameter like k

needs to be considered as taking on a number of values. Those who attempted them scored

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well in Parts (a) and (b) (i). The proof in Part (b) (ii) caused difficulty for many students. One

concerning error was to assign values to k. This received little credit as it was essentially

repeating Part (a). This type of error suggests that some students have seen little by way of

proof within their course of study. It was pleasing to see that, of the students who tackled the

proof in Part (b) (ii), a good proportion received full marks, showing that the required algebra

was within the grasp of those prepared to give it a go.

Question 10

Given the relative simplicity of this question as an example of a hypothesis testing question,

and given its similarity to the available exemplar materials, it was of some concern that over

20% of students received 0 or 1 mark for this question. Many were unfamiliar with the

structure of a z-test and some attempted to answer this question through the use of confidence

intervals, an inappropriate method for this question. Of those who were familiar with this

component of the course and scored well in Parts (a) to (c), the commonest errors were in

showing an understanding of the significance of obtaining a p-value of greater magnitude than

the significance level. Too many ‘accepted the null hypothesis’, which in this area of

mathematics is vastly different to correctly ‘not rejecting the null hypothesis’. The focus on

whether or not to reject was emphasised in the wording of Part (d) and should be at theforefront of students’ work in this area. A similar line of reasoning caused a loss of marks in

Part (e). Too few students seemed to understand that failing to reject the null hypothesis

meant that there was no evidence (at the 95% level of confidence) that the proportion under

consideration was not three-quarters. Rather than choosing appropriate language like ‘the

proportion (under consideration) could be three-quarters’, students frequently concluded that

the proportion ‘was three-quarters’, which is akin to accepting the null hypothesis. Relatively

few students received full marks for this question.

Question 11

This question was done well, with almost all students making some progress and quite a

number obtaining full credit. Marks were lost in Part (a) by the presentation of sketches that

did not adequately capture the features of the graph, due to student’s lack of attention to thescale that had been provided. In Part (c) (ii), the requirement of exact values was overlooked

to the detriment of some. The requested simplification required the evaluation of ln e and ln 1;

this was not always carried out.

Question 12

Parts (a) and (b) were done well. Some students had problems in dealing with the surds in

Part (b) (i), but most achieved high marks in these sections. Many students struggled to

simplify the expression provided in Part (c), despite the fact that all that was required was

knowledge of the matrix fundamentals I P P =−1

, D DI = and that repeated matrix

multiplication could be represented by . Part (d) caused many students difficulty. Some

provided the correct expression for without any evidence of use of Parts (a) and (c) and socould not be awarded full credit. Students should be aware that, in a situation where a method

is specified and 3 marks are allocated, a one-line answer is very unlikely to be sufficient.

n Dn

A

Question 13

Students generally progressed well through the first parts of this question, although algebraic

errors cost many of them marks in various parts. Part (c) provided a point where good

students could demonstrate their skills by their successful handling of a quadratic with

unknown coefficients.

Question 14

Virtually all students made some inroads into this question but, in general, the responses

indicated that a significant proportion of students did not have the conceptual frameworknecessary to maximise their results in this sort of question. Too many students answered

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Part (a) carelessly and failed to gain this mark. In Part (b) (ii), many students did not have a

picture of a distribution of sample means as both tighter and taller than the population of

individuals. Part (c) was unattempted by some, or not marked in an illustrative manner by

others. Part (d) was done well but errors marred the work of many in Part (e). Part (e) (ii)

required a probability-based reason, which is to be expected given its link with Part (i).

Question 15

Most students made good progress with this question, with over 20% receiving 9 or 10 out

of 10. Many of the remaining students lost marks due to their discomfort in dealing with exact

values in Part (b) (ii). In Part (c), it was disappointing to see that a significant number of

students opted for a difficult algebraic method, and struggled to gain the marks, when

electronic technology provided an efficient alternative. The viability of this option was

re-enforced by the specification of a three decimal place answer and the allocation of only

2 marks to what was, when attempted algebraically, a challenging piece of mathematics.

Question 16

Parts (a) to (c) were done well by most students. The need to provide integer results, when

working with a population model such as this, was overlooked by some, but this was not penalised heavily. Some students missed the ‘two year per transition’ nature of this model, but

were able to complete all parts of the question by following this error through. Part (d) tested

the students’ understanding of the mechanics of the type of model that they had been working

with. Whilst some were very successful, a number of students made errors in their formation

of the matrix L. In most cases, Part (e) was able to be completed regardless of errors made in

Part (d), but a number of students did not provide the information that markers required to

award follow through marks in this part. It is important to make clear to students that, when

calculations are taking place on a calculator, it is in students’ best interests to provide

evidence of what they are calculating, and any intermediate results, so that markers can give

credit where it is due.

Chief Assessor

Mathematical Studies

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2msu-ar-2007