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Mathematical Studies

2 0 0 6 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

2006 ASSESSMENT REPORT

GENERAL COMMENTS

The examination is designed to assess primarily Learning Outcomes 1 to 5 (SSABSA,

curriculum statement, 2006) the five learning outcomes being that students should be able to:

1. 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;

2. understand fundamental concepts, demonstrate mathematical skills, and apply routine

mathematical procedures;

3. think mathematically by posing questions, making and testing conjectures, and looking for

reasons that explain the results of the mathematics;

4. make informed and critical use of electronic technology to provide numerical results andgraphical representations, and to refine and extend mathematical knowledge;

5. communicate mathematically, and present mathematical information in a variety of ways.

The 2006 results suggest that the examination did not hinder students in demonstrating their

knowledge and skills as they pertain to the above outcomes. The mark distribution had a mean

of 60% with a skewed-left bell-like shape. The examination succeeded in differentiating

between individual students within the group who scored highly.

A number of questions required the students to make links between parts of the question if

they were to score highly. It is pleasing to see that many of the students unable to make the

links were able to complete the parts of the questions (including parts in the last few

questions) that might be described as ‘meat and potatoes’, hence showing that they can applyroutine mathematical procedures. It is equally pleasing to see a good number of students

displaying strong skills, as described by Learning Outcomes 1, 3, 4, and 5. To improve

student performance of the skills in these outcomes, their learning should include an

immersion into larger problems well before the ‘exam swatting’ period.

Numerous questions supplied the students with graphical displays of functions. One potential

reason for this is that the student can check that their entry of data into the electronic

technology they are using. It is evident that some students do not use the supplied graphs in

this manner.

Success in this examination required students to be capable symbolic manipulators who show

all of their working. Students are instructed on the front cover of the exam that, ‘Appropriatesteps of logic and correct answers are required for full marks’. Many students could not be

awarded partial credit because they chose not to show all working. This was particularly true

in questions where the student was asked to ‘show’ a result to be true. Many students also

found difficulty when the symbolic manipulation required them to deal with brackets and

negative signs.

It is pleasing to report that students generally paid attention to the sensible rounding of

numeric results provided by the electronic technology they used.

The curriculum statement calls for students to be ‘discerning in their use of electronic

technology to solve questions in examinations’. It is evident that many students are not

discerning. There was a desire by one group of students to first attempt a question using a ‘by

head’ method and then, if stuck, revert to using electronic technology. This caused students to

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waste a great deal of time in some cases. See question specific comments for more detail.

ASSESSMENT COMPONENT: EXAMINATION

Question 1

The majority of students scored very well in this question. However, some students were

unaware of the notation used for the determinant and some failed to integrate x .

Question 2

The majority of students also scored highly in this question. It did, however, distinguish

between those students who understood the process of matrix multiplication and those who

could simply compute using electronic technology.

In part a (i), most students chose to use electronic technology to compute the determinant.

Many students did not see, or chose to ignore, the conceptual link between part (b) and part

(c).

In answering part (d) some students did not demonstrate an understanding that C needed to be

pre-multiplied by A−1 , attempting to compute CA . Some students, who did write down−1

A−1C , ended by writing down matrices of the wrong shape.

Question 3

The majority of students displayed they were able to carry out the basic computations

associated with binomial probabilities. The notion of ‘at least one’ did cause some studentsdifficulty.

Quite a few students used their electronic technology sensibly in part (b) using a ‘trial and

error’ approach. A few employed an analytical technique in this part while the majority of

students failed to make much progress.

Very few students found success in part (c), being unable to provide a binomial assumption

that was not satisfied.

Question 4

Success in this question required that a student understood how a chord is able to approximatethe slope of a tangent at a point and the concept of the derivative function. More than a few

students chose to draw a tangent rather than a chord (as clearly requested) for part (a).

It was pleasing to see many good attempts to part (b). Many of the good attempts scored two

marks out of three as students failed to include a point of inflection on y = h '( x) at some point

where( x,h'( x)) a < x < f . There were, however, slightly more poor or null attempts than

good attempts.

Question 5

Part (b) of this question could be approached either by standardising or by using a trial anderror approach aided by the use of electronic technology. Many students did neither. While

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most students were able to successfully compute the answer to part (c) (ii), relatively few

were able to use the number gained to provide a sensible answer to the question posed in part

(c) (ii).

Question 6

Many students were able to score three or four marks for this question. Some students found

difficulty in applying the necessary logarithm law to isolate 2a. Some students did not pay

attention to the requirement of an ‘exact value’ for a and provided a decimal result.

Question 7

It was very pleasing that most students were able to compute the confidence interval required

in part (a) and also use it sensibly to provide a good answer to part (b).

Some students were unsure of the meaning of the ‘width’ of a confidence interval given they

doubled 0.02 prior to substituting it into the required formula. Some students either did not

round, or rounded incorrectly, the value for n given by the formula.

Question 8

Parts (a) and (b) of this question required students to use electronic technology and to be able

to display their results neatly and accurately on paper. While the vast majority of students

were able to do this, too many failed to meet the requirements. Poorly drawn graphs with lack

of accuracy when plotting intercepts and the like were common. More than a few students

attempted part (b) using analytical methods, spending more time on that part than was

required. An exact result was not asked for in this part and so a decimal approximation as

given by the electronic technology was acceptable.

Part (c) was generally well done, the main error being incorrect use of brackets.

It was pleasing that most students started well in part (d). Many students arrived at the

appropriate quadratic, but then some struggled to solve the quadratic, not knowing how to

deal with the ‘a’. Few students added the fact that a>0 was the condition that proved there

were exactly two solutions.

Question 9

Most students answered part (a) well. However, some made computational errors.

It was clear that many students were able to see the pattern evident as a result of calculationsin part (a). However, most were unable to reason well enough to justify the answer they

provided to part (b). Students needed to say that tends to zero as n tends to positive

infinity.

n6.0

Question 10

In part (c) (i) more than a few students failed to include the integral limits. In part (c) (ii) it

was disappointing to see both computational and rounding errors made. Students need to

know and be well practised in the most efficient way to compute definite integrals using their

electronic technology.

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Part (c) (iii) required students to provide an interpretation of the value computed in part (c)

(ii). Many students understood that a difference was involved but most did not realise the

quantity involved was the total flow (amount) and not the flow rate.

Question 11

This question separated the cohort into three groups: those who did not put pen to paper, those

who could write down some equations but not use them in any way, and those who did all or

most of the question successfully.

In part (a), it was quite common for students to write down something like

A + B + C = A + 5.5 +3 = 4 + B + 3.5 and then not be able to proceed to a format that was of

use to them.

In part (b) (ii) it was surprising how many students could write down D + E = D + F but then

not conclude that E = F . Some students attempted this part using inverse logic; such an

approach was not awarded any marks.

Question 12

Many students performed very well in this question, ultimately being able to work with the

specific case of the tangent at x = 3, then being able to work with other cases, and ultimately

the general case.

Parts (b) and (c) required the student to work in an ‘exact’ manner, giving the values of A, B,

and C as integers. This was ignored by some students who converted fractions to decimal

approximations at the first opportunity. Some students unfortunately chose to read the values

of the x and y intercepts of the tangent at x = 3 from the graph.

Responses to part (d) suggested that many students had not encountered a situation wherethey were to make a conjecture of the type expected (that the area of triangle OEF was 9/k ).

Some students made alternative (but correct) conjectures and were rewarded, but then

struggled to prove their conjecture as asked in part (d) (ii).

It was pleasing to see that about 20% of the students made excellent attempts at proving their

conjecture (with most being totally successful).

Question 13

This was the first year that a question involving a Z-test had been included. From the response

to part (a), it was clear that many students had been well taught and that the support materials

made available by SSABSA had been well utilised.

Some students performed a one-tailed test. Students should be made aware that this course

only includes two-tailed procedures. Some students were unaware that the null distribution for

a Z-test is the Z distribution N(0,1).

In part (b), most students calculated the required confidence interval correctly, but many were

unable to fully explain that the breeder’s claim was justified. Students needed to comment

that the 25 microns was greater than the upper limit of the confidence interval’s upper limit.

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Question 14

In part (a) there were numerous students who attempted to carry out the integration of g( x)

using an analytical method. In general, it seems that students need to become better at

deciding when, and when not to, use electronic technology to compute things for which they

think may also be approached analytically.

Part (c) was generally very well done, with poor use of brackets and negative signs causing

some students not to show what was required.

Part (d) was not done well, with many students not seeing the link between what was given in

part (c) and what was required in part (d). Of those who did see and use the link, some found

only f (k ) instead of f (k ) - f (0).

The requirement to interpret in part (e) was handled poorly by most students. Students who

reached a wrong answer in part (d) often gave an answer to part (e) that contradicted the

graph supplied. In such cases, students should check for consistency and comment on the lack

of consistency if present. The students who correctly reported that as k tended to positiveinfinity, A(k ) approached 2, rarely related this fact back to the graph as required.

Question 15

Most students scored well in this question, demonstrating good skills in relating matrix

calculations to a real-life context.

The calculation of the determinant in part (c) (ii) caused some students problems. Incorrect

use of brackets and the poor handling of fractions and symbols were common problems.

In part (c) (v) working with fractions again caused problems for some students. Many

students could not be awarded partial credit as they did not show the result of each rowoperation clearly.

A pleasing number of students were able to successfully answer part (c) (vi).

Question 16

An efficient way to complete part (a) of this question was to use electronic technology. Many

students chose not to do this. The point of part (a) was for students to compute the derivatives

and, if they considered the values, realise that the rate of change of V (t ) was decreasing post

t = 40.

Part (b) was not handled well by most students. It required students to take great care

algebraically. Given the students were required to ‘show that’, clearly showing the removal of

the common factor(s) was a critical step in the problem.

It was pleasing that many students indicated that the point of inflection was important in

answering part (c). However, despite the second derivative being given in part (b), relatively

few students determined that the inflection occurred 30 years after planting. Even fewer

identified the correct time period as beyond the 30-year mark.

Relatively few students attempted parts (d) and (e). However, it was very pleasing to see

numerous excellent solutions with a good number of students gaining full marks for this

question.

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In part (e) some students indicated the solution required them to find the point of intersection

of two functions, but were unable to do this. This was a case where the use of electronic

technology offered an efficient way to proceed.

Chief Assessor

Mathematical Studies

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