January 2011 9th issue

In this issue, we are glad to have the author of New Progress in Junior Mathematics and New Progress in Senior Mathematics (Extended Part), Mr. H. M. Chan, to explore the question trend in the Hong Kong Diploma of Secondary Education (HKDSE) Examination — Mathematics (Compulsory Part) Paper I.

IntroductionCompared with the Hong Kong Certificate Education (HKCE) curriculum, there has been a radical change in the Mathematics (Compulsory Part) curriculum in HKDSE. Let us explore the changes in topics first.

Changes in TopicsSince the senior secondary curriculum changed from a two-year to a three-year scheme, some new topics were added to the HKDSE curriculum. These topics can be divided into two categories.1. Brand new topics, for instance, ‘Permutation and combination’.2. Topics that have been removed from the HKCE curriculum since 2006.In addition, the distribution of Foundation Topics and Non-Foundation Topics has been modified. There are some Non-Foundation Topics (such as ‘Solving trigonometric Equations’ and ‘Equations of circles’) which have become Foundation Topics in the HKDSE curriculum. However, only the Foundation Topic ‘Solving simultaneous equations in two unknowns, one quadratic and one linear, by the graphical method’ has changed to a Non-Foundation Topic.

Comparing the HKDSE Examination Sample Paper with the HKCE Examination Past PaperAs the Hong Kong Examinations and Assessment Authority (HKEAA) has published a sample paper for the HKDSE Examination, we can predict the question trend through analyzing the differences in the distribution of exam questions between the HKCE Examination and the HKDSE Examination under three strands (‘Number and Algebra’, ‘Measures, Shape and Space’ and ‘Data Handling’). The following shows the comparison of the marks covered in each topic in Paper I from the sample paper and the HKCE Examination (average marks taken from 1998 to 2010).

Mr. H. M. Chan,TWGHs Lee Ching Dea

Memorial College

Exploring the Question Trend in the HKDSE Examination — Mathematics (Compulsory Part) Paper I

Coming issue: The Giant of Maths, Prof. Shing-

Tung Yau — Exploring The Geometry of

The Universe

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Mathematics Express January 2011

Number and Algebra1. The three high-weighted topics ‘More about

polynomials’, ‘Variations’ and ‘Arithmetic and geometric sequences and their summations’, retain their high weightings in the sample paper.

2. Each question in Paper I Section B carries 11 marks in the HKCE Examination while that in the HKDSE Examination may vary (from 4 to 13 marks in the sample paper). This change increases the flexibility to construct questions of Non-Foundation Topics. Therefore, it is more likely to cover questions related to such topics in the HKDSE Examination. For example, there is one simple question about ‘Exponential and logarithmic functions’ appearing in the sample paper.

Noticeable topic: Exponential and logarithmic functions

Table 1

HKDSE Examination

HKCE Examination (98 – 10)*

Quadratic equations in one unknown 1

Functions and graphs 2Exponential and logarithmic functions 5 0

More about polynomials 6 4More about equations 0Variations 6 5Arithmetic and geometric sequences and their summations

4 7

Inequalities and linear programming 4

More about graphs of functions 0

Number and Algebra 21 23

Table 2

Table 1

HKDSE Examination

HKCE Examination (98 – 10)*

Basic properties of circles 10 6Locus 2 0Equations of straight lines and circles 14 12

More about trigonometry 8 12Measures, Shape and Space 34 30

HKDSE Examination

HKCE Examination (98 – 10)*

Permutation and combination 0

More about probability 5 7Measures of dispersion 4Uses and abuses of statistics 9 0Data Handling 14 11

Table 3* Correct to the nearest integer

* Correct to the nearest integer

* Correct to the nearest integer

Measures, Shape and Space1. Since parts of the ‘Basic properties of circles’

have become Foundation Topics, there may be some simple questions about the properties of circles in Section A.

2. The concept of the relationship between a circle and a straight line was frequently asked in the HKCE Examination and it is reintroduced in ‘Equations of straight lines and circles’. The concept may also be tested in the HKDSE Examination.

3. In the sample paper, there is one question requiring students to describe the locus. Students are weak in handling such a question and teachers may need to strengthen the students’ ability to give written explanations for their answers.

Noticeable topic: Locus

Data HandlingQuestions about ‘Uses and abuses of statistics’ are weighted about 10% in Paper I, so the importance of this topic cannot be neglected.

Noticeable topic: Uses and abuses of statistics

Other than the three noticeable topics mentioned above, topics that are not covered in the sample paper could be tested in the 2012 HKDSE Examination Paper I, such as ‘Inequalities and linear programming’, ‘Permutation and combination’, ‘Operations of complex numbers’ and ‘Further applications’.

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Mathematics ExpressJanuary 2011

Fig. 1

Question TrendWe are going to predict the question trend for some topics that are not covered in the sample paper. The following shows some examples of those topics.

Permutation and CombinationExample:

In the game of Tic Tac Toe, there are 5 ‘ ’s and 4 ‘✗’s. If the 5 ‘ ’s cannot form a diagonal, find the probability of the condition. (Fig. 1 is an example.)

Analysis:

Questions about probability are usually framed with daily-life activities, such as TV game shows. Teachers can frame the problems about ‘Permutation and combination’ and ‘Probability’ with examples of game shows, asking students to calculate different probabilities. It raises students’ interest in learning and trains them in solving daily-life problems in a mathematical way, which will also help them with the HKDSE Examination in the future.

Operations of Complex NumbersExample:

If ( )A Bii

+ =−

2 1695 12

, find the values of A and B.

Analysis:

In the HKDSE curriculum, ‘Operations of complex numbers’ is a Non-Foundation Topic, so questions related to this topic may appear in Paper I (Section B). The above example includes addition, subtraction, multiplication and division of complex numbers and also the concepts of identity in complex numbers. The question trend of ‘Complex numbers’ would be to include multiple concepts in one question.

Further ApplicationsExample:

Table 4 shows a bank statement of Mary.

A Open a bank account

Ｂ Buy a desktop computer by EPS

C Pay salaries tax through PPS

D Sell a stock and deposit the earnings into the account

E Interest

F Sell the desktop computer bought on 2004/3/27 and deposit the income into the account

(a) If the depreciation rate of the desktop computer is x% per annum, find the value of x.

(b) The deductions and allowances of Mary are $200 000 and $20 000 respectively. Refer to table 5, and find her yearly salary.

(c) (i) Mary bought a stock at a price of $S . The shareholders receive 1% dividends each year. Mary uses the dividends to buy the same stock each year. Suppose the stock price increases by 5% yearly. Show that the total stock price after n

years is105 1 05 1 014

( . . )n n− S.

Net Chargeable Income Tax Rate

Net Chargeable Income Rate

First $30 000 2%Next $30 000 8%Next $30 000 14%

Remainder 20%

Table 5

✗ ✗

✗

✗

Tax

Subtraction and division of complex numbers

Addition and multiplication of complex numbers

Date Deposit Withdrawl Balance Reference

2004/1/1 $50 000.00 A

2004/3/27 $4000.00 $46 000.00 Ｂ

2004/4/20 $41 200.00 $4 800.00 C

2004/6/30 $137 621.53 $142 421.53 D

2004/12/31 $1439.18 $143 860.71 E

2005/3/27 $2624.40 $146 485.11 F

Table 4

Depreciation

Geometric series

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Mathematics Express January 2011

Fig. 2

(ii) Fig. 2 shows the graph of y = 1.05n – 1.01n. Mary bought the stock on 1994/6/30 and sold it on

2004/6/30. Using the graph, find the value of S.

(b) Mary received interest of $1 439.18 from 2004/6/30 to 2004/12/31. If the interest rate is y%, compounded daily, find the value of y.

Analysis:

The above example is based on the financial record of a person. It includes ‘Applications of percentage’ such as depreciation, compound interest and tax, together with ‘Geometric sequences’ and ‘Using graphical method’. The question trend of ‘Further applications’ would be using scenarios from daily-life to combine several topics and test students’ different mathematical skills. It could probably be a high-weighted conventional question.

General RecommendationsIn order to perform well in the HKDSE Examination, students are required to understand the question trend and answering skills. Moreover, they need to have a good mathematical foundation. The following are some recommendations based on reviewing the performance of students in past public examinations.

1. Develop a better spatial sense, for example, distinguishing right-angled triangles from non right-angled triangles in 3-dimensional diagrams

2. Identify different situations in probability problems

3. Thoroughly study the relation between the lengths and the areas of similar figures

4. Practise more on problems involving geometric proofs

ConclusionWith the above analysis, we hope that everyone gets to know more about the question trend of the HKDSE Examination and hence develop a tailor-made teaching strategy for students to attend the HKDSE Examination. (The above analysis and opinions are for reference only.)

Graphical method

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Compound interest