secondary 3 assessment design - hkeaa

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Results of Secondary 3 Mathematics in TSA 2015

The territory-wide percentage of S.3 students achieving Mathematics Basic Competency

in TSA 2015 was 79.9%. The proportion achieving basic competency in 2015 was

almost the same as that of last year.

Secondary 3 Assessment Design

The design of assessment tasks for S.3 was based on the documents Mathematics

Curriculum: Basic Competency for Key Stage 3 (Tryout Version) and Syllabuses for

Secondary Schools – Mathematics (Secondary 1 – 5), 1999. The tasks covered the

three dimensions of the mathematics curriculum, namely Number and Algebra,

Measures, Shape and Space, and Data Handling. They focused on the Foundation

Part of the S1 – 3 syllabuses in testing the relevant concepts, knowledge, skills and

applications.

The Assessment consisted of various item types including multiple-choice questions, fill

in the blanks, answers-only questions and questions involving working steps. The item

types varied according to the contexts of the questions. Some test items consisted of

sub-items. Besides finding the correct answers, students were also tested in their ability

to present solutions to problems. This included writing out the necessary statements,

mathematical expressions and explanations.

The Assessment consisted of 160 test items (217 score points), covering all of the 129

Basic Competency Descriptors. These items were organized into four sub-papers, each

65 minutes in duration and covering all three Dimensions. Some items appeared in

more than one sub-paper to act as inter-paper links. Each student was required to

attempt one sub-paper only. The number of items on the various sub-papers is

summarized in Table 8.7. These numbers include several overlapping items that

appear in more than one sub-paper to enable the equating of test scores.

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Table 8.7 Number of Items and Score Points for S.3

Subject No. of Items (Score Points)

Paper 1 Paper 2 Paper 3 Paper 4 Total* Mathematics Written Paper

Number and Algebra 23 (31) 23 (30) 23 (30) 22 (28) 71 (91) Measures, Shape and Space 21 (29) 21 (28) 21 (30) 22 (31) 71 (97)

Data Handling 6 (9) 6 (11) 6 (9) 6 (10) 18 (29) Total 50 (69) 50 (69) 50 (69) 50 (69) 160 (217)

* Items that appear in different sub-papers are counted once only.

The item types of the sub-papers were as follows:

Table 8.8 Item Types of the Sub-papers

Section Percentage of Score Points Item Types

A ~ 30% • Multiple-choice questions: choose the best answer from among four options

B ~ 30% • Calculate numerical values • Give brief answers

C ~ 40%

• Solve application problems showing working steps

• Draw diagrams or graphs • Open-ended questions requiring reasons or

explanations

Performance of S.3 Students with Minimally Acceptable Levels of Basic Competence in TSA 2015

S.3 Number and Algebra Dimension

S.3 students performed more than satisfactorily in this Dimension. The majority of

students demonstrated recognition of the basic concepts of directed numbers,

approximation and errors, formulating problems with algebraic language and linear

equations in one unknown. Performance was only fair in items related to using

percentages, manipulations of simple polynomials and formulas. Comments on students’

performances are provided with examples cited where appropriate (question number x /

sub-paper y quoted as Qx/My). More examples may also be found in the section

General Comments.

Number and Number Systems

Directed Numbers and the Number Line: The performance of students was good.

They could demonstrate recognition of the ordering of integers on the number line.

They were able to handle the simple operation of directed numbers. However, some

students could not use directed numbers to correctly describe real life situations.

Q21/M2

Example of Student Work (Could not use a directed number to represent each of the

following situations)

Numerical Estimation: The majority of students were able to determine whether the

value mentioned in a simple context is obtained by estimation or by computation of

the exact value. They were able to judge the reasonability of answers by the given

information of the question. More than half of the students could estimate values

with reasonable justifications. Nevertheless, some students were not able to use

correct methods to estimate the approximate values.

Q48/M3

Example of Student Work (Estimate the maximum number of machines that can be

carried by the lorry each time – without giving an approximation for the weight of

each machine)

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Table 8.7 Number of Items and Score Points for S.3

Subject No. of Items (Score Points)

Paper 1 Paper 2 Paper 3 Paper 4 Total* Mathematics Written Paper

Number and Algebra 23 (31) 23 (30) 23 (30) 22 (28) 71 (91) Measures, Shape and Space 21 (29) 21 (28) 21 (30) 22 (31) 71 (97)

Data Handling 6 (9) 6 (11) 6 (9) 6 (10) 18 (29) Total 50 (69) 50 (69) 50 (69) 50 (69) 160 (217)

* Items that appear in different sub-papers are counted once only.

The item types of the sub-papers were as follows:

Table 8.8 Item Types of the Sub-papers

Section Percentage of Score Points Item Types

A ~ 30% • Multiple-choice questions: choose the best answer from among four options

B ~ 30% • Calculate numerical values • Give brief answers

C ~ 40%

• Solve application problems showing working steps

• Draw diagrams or graphs • Open-ended questions requiring reasons or

explanations

Performance of S.3 Students with Minimally Acceptable Levels of Basic Competence in TSA 2015

S.3 Number and Algebra Dimension

S.3 students performed more than satisfactorily in this Dimension. The majority of

students demonstrated recognition of the basic concepts of directed numbers,

approximation and errors, formulating problems with algebraic language and linear

equations in one unknown. Performance was only fair in items related to using

percentages, manipulations of simple polynomials and formulas. Comments on students’

performances are provided with examples cited where appropriate (question number x /

sub-paper y quoted as Qx/My). More examples may also be found in the section

General Comments.

Number and Number Systems

Directed Numbers and the Number Line: The performance of students was good.

They could demonstrate recognition of the ordering of integers on the number line.

They were able to handle the simple operation of directed numbers. However, some

students could not use directed numbers to correctly describe real life situations.

Q21/M2

Example of Student Work (Could not use a directed number to represent each of the

following situations)

Numerical Estimation: The majority of students were able to determine whether the

value mentioned in a simple context is obtained by estimation or by computation of

the exact value. They were able to judge the reasonability of answers by the given

information of the question. More than half of the students could estimate values

with reasonable justifications. Nevertheless, some students were not able to use

correct methods to estimate the approximate values.

Q48/M3


carried by the lorry each time – without giving an approximation for the weight of

each machine)

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Q48/M3


carried by the lorry each time – mistakenly rounding off the weight of each machine)

Example of Student Work (Good performance)

• Approximation and Errors: When a number is greater than 1, students in general

were able to correctly round the number to 3 significant figures. In addition, most

students were able to convert numbers in scientific notation to integers.

• Rational and Irrational Numbers: Most students were able to demonstrate recognition of

the integral part of a . Many students could represent fractions on the number line.

Comparing Quantities

• Using Percentages: Students did well in solving problems regarding selling prices

and depreciations. However, finding the cost price and the principal were still their

weak spots. Some students confused the formulae of finding simple interest with that

of compound interest.

Q42/M4

Exemplar Item (Find the cost price)

Peter sells a jacket at a profit of 55%. If the profit is $330, find the cost price of the

jacket.

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Q48/M3


carried by the lorry each time – mistakenly rounding off the weight of each machine)


• Approximation and Errors: When a number is greater than 1, students in general

were able to correctly round the number to 3 significant figures. In addition, most

students were able to convert numbers in scientific notation to integers.

• Rational and Irrational Numbers: Most students were able to demonstrate recognition of

the integral part of a . Many students could represent fractions on the number line.

Comparing Quantities

• Using Percentages: Students did well in solving problems regarding selling prices

and depreciations. However, finding the cost price and the principal were still their

weak spots. Some students confused the formulae of finding simple interest with that

of compound interest.

Q42/M4

Exemplar Item (Find the cost price)

Peter sells a jacket at a profit of 55%. If the profit is $330, find the cost price of the

jacket.

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Q42/M2

Example of Student Work (Mistakenly used the formula for finding the amount by

taking compound interest as the formula for finding the simple interest)

Q43/M1

Exemplar Item (Find the compound interest)

David deposits $7500 in a bank at an interest rate of 6% p.a. compounded yearly.

Find the interest he will receive after 2 years.

Example of Student Work (Confused compound interest with simple interest, as well

as interest with amount)

Example of Student Work (Confused compound interest with amount)

• Rate and Ratio: The performance of students was quite good, though there was room

for improvement for some students when using ratio to solve real-life problems.

Observing Patterns and Expressing Generality

• Formulating Problems with Algebraic Language: Students did quite well. They could

distinguish the difference between (–3)2 and –32 ; x2 and 2x. They were able to

translate word phrases/contexts into algebraic languages, substitute values into

formulas and find the value of a variable and formulate simple equations from simple

contexts. They could also write down the next few terms in sequences from several

consecutive terms that were given. Nevertheless, some students could not intuitively

find the nth term of a simple number sequence.

Q24/M1

Exemplar Item (Find the nth term of a number sequence)

Figure 1 to Figure 4 consist of 1, 4, 9 and 16 dots respectively.

Figure 1

Figure 2

Figure 3

Figure 4

According to the above pattern, how many dots does Figure n consist of ? (Express

the answer in terms of n)

Example of Student Work (Confused nn with n2)

• Manipulations of Simple Polynomials: The majority of students were able to

distinguish like terms and unlike terms in polynomials. On the other hand, many

students could not distinguish polynomials from algebraic expressions. Their

performance was still fair in dealing with the manipulations of simple polynomials.

Q24/M3

Exemplar Item (Simplify a polynomial)

Simplify )67()35( 22 xxxx −−+ .

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Q42/M2

Example of Student Work (Mistakenly used the formula for finding the amount by

taking compound interest as the formula for finding the simple interest)

Q43/M1

Exemplar Item (Find the compound interest)

David deposits $7500 in a bank at an interest rate of 6% p.a. compounded yearly.

Find the interest he will receive after 2 years.

Example of Student Work (Confused compound interest with simple interest, as well

as interest with amount)

Example of Student Work (Confused compound interest with amount)

• Rate and Ratio: The performance of students was quite good, though there was room

for improvement for some students when using ratio to solve real-life problems.

Observing Patterns and Expressing Generality

• Formulating Problems with Algebraic Language: Students did quite well. They could

distinguish the difference between (–3)2 and –32 ; x2 and 2x. They were able to

translate word phrases/contexts into algebraic languages, substitute values into

formulas and find the value of a variable and formulate simple equations from simple

contexts. They could also write down the next few terms in sequences from several

consecutive terms that were given. Nevertheless, some students could not intuitively

find the nth term of a simple number sequence.

Q24/M1

Exemplar Item (Find the nth term of a number sequence)

Figure 1 to Figure 4 consist of 1, 4, 9 and 16 dots respectively.

Figure 1

Figure 2

Figure 3

Figure 4

According to the above pattern, how many dots does Figure n consist of ? (Express

the answer in terms of n)

Example of Student Work (Confused nn with n2)

• Manipulations of Simple Polynomials: The majority of students were able to

distinguish like terms and unlike terms in polynomials. On the other hand, many

students could not distinguish polynomials from algebraic expressions. Their

performance was still fair in dealing with the manipulations of simple polynomials.

Q24/M3

Exemplar Item (Simplify a polynomial)

Simplify )67()35( 22 xxxx −−+ .

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Q24/M3

Example of Student Work (Mistakenly performed addition and subtraction of unlike

terms)

Example of Student Work (Could not simplify )6(3 xx −− correctly)

• Laws of Integral Indices: The performance of students was fair in simplifying

algebraic expressions by laws of integral indices.

Q44/M4

Example of Student Work (Has mistakenly taken nmnm xx +=)( )

Example of Student Work (Has mistakenly taken nmnm xx -)( = and mnn

m

xxx ÷= )

Factorization of Simple Polynomials: Student performed quite well in factorizing

simple polynomials by using the perfect square expressions. Their performance was

fair in using the cross method and grouping terms.

Q26/M1

Exemplar Item (Factorize the expression by grouping terms)

Factorize ykyxkx .

Example of Student Work (Student knew to group some of the terms, but the

expression could not be factorized completely)

Q27/M3

Exemplar Item ( Factorize the expression by using the cross method)

Factorize 652 xx .

Example of Student Work


Algebraic Relations and Functions

Linear Equations in One Unknown: Students were able to solve simple equations and

formulate equations from simple contexts.

Linear Equations in Two Unknowns: By providing a table with given x values and

one corresponding y value, students were more able to plot graphs of linear equations

in 2 unknowns. They could use graphical methods and algebraic methods to solve

linear simultaneous equations. Besides, they could demonstrate recognition that

graphs of equations of the form ax + by + c = 0 are straight lines. However, there is

still room for improvement in plotting straight lines.

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Q24/M3

Example of Student Work (Mistakenly performed addition and subtraction of unlike

terms)

Example of Student Work (Could not simplify )6(3 xx −− correctly)

• Laws of Integral Indices: The performance of students was fair in simplifying

algebraic expressions by laws of integral indices.

Q44/M4

Example of Student Work (Has mistakenly taken nmnm xx +=)( )

Example of Student Work (Has mistakenly taken nmnm xx -)( = and mnn

m

xxx ÷= )

Factorization of Simple Polynomials: Student performed quite well in factorizing

simple polynomials by using the perfect square expressions. Their performance was

fair in using the cross method and grouping terms.

Q26/M1

Exemplar Item (Factorize the expression by grouping terms)

Factorize ykyxkx .

Example of Student Work (Student knew to group some of the terms, but the

expression could not be factorized completely)

Q27/M3

Exemplar Item ( Factorize the expression by using the cross method)

Factorize 652 xx .



Algebraic Relations and Functions

Linear Equations in One Unknown: Students were able to solve simple equations and

formulate equations from simple contexts.

Linear Equations in Two Unknowns: By providing a table with given x values and

one corresponding y value, students were more able to plot graphs of linear equations

in 2 unknowns. They could use graphical methods and algebraic methods to solve

linear simultaneous equations. Besides, they could demonstrate recognition that

graphs of equations of the form ax + by + c = 0 are straight lines. However, there is

still room for improvement in plotting straight lines.

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Q44/M3

Example of Student Work (Has mistakenly marked the point (–2, 4) on the position of

(4, –2))

Example of Student Work (Did not extend at two ends)


Q44/M2

Example of Student Work (Solving simultaneous equations – the student tried to use

the method of elimination, but mistakes occurred in the steps)

Example of Student Work (Solving simultaneous equations – although the student

knew how to use the method of substitution, mistakes occurred in the computation )


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Q44/M3

Example of Student Work (Has mistakenly marked the point (–2, 4) on the position of

(4, –2))

Example of Student Work (Did not extend at two ends)


Q44/M2

Example of Student Work (Solving simultaneous equations – the student tried to use

the method of elimination, but mistakes occurred in the steps)

Example of Student Work (Solving simultaneous equations – although the student

knew how to use the method of substitution, mistakes occurred in the computation )


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• Identities: Many students were able to distinguish equations from identities. However,

their performance was only fair in using the perfect square expressions to expand

simple algebraic expressions.

Q29/M3

Exemplar Item (Expand simple algebraic expressions by using the perfect square

expressions)

Expand 2)2( x− .

Example of Student Work (Although the student knew how to expand the expression

by perfect square, mistakes occurred in computation)

Example of Student Work (Has mistakenly taken 222)( baba −=− as identity)

• Formulas: Many students could find the value of a specified variable in the formula.

The performance was fair in simplifying algebraic fractions and performing change

of subject in simple formulas.

Q29/M1

Exemplar Item (Simplifying algebraic fractions)

Simplify yx

yx

3− .

Example of Student Work (Although the student was able to find the common

denominator, the algebraic fractions was not simplified)

Example of Student Work (Could not simplify the algebraic fraction to its simplest

form)

Q29/M1

Example of Student Work (Could not demonstrate recognition of the manipulation of

algebraic fractions)

• Linear Inequalities in One Unknown: Students could formulate inequalities from

simple contexts. They demonstrated good recognition of the properties of inequalities.

Their performance was fair in solving simple linear inequalities.

Q31/M4

Exemplar Item (Solving inequalities)

Solve the inequality 712 −>+x .

Example of Student Work (Mistakes occurred in computation)

Example of Student Work (Mistakes occurred in computation and could not express

the answer with inequality sign)


S.3 Measures, Shape and Space Dimension

S.3 students performed steadily in this Dimension. They could find areas and volumes in

2-D and 3-D figures, angles related with lines and rectilinear figures, solve problems

regarding transformation and symmetry, quadrilaterals and trigonometry. However,

more improvement could be shown in items related to definitions of common terms in

geometry as well as deductive geometry. Comments on students’ performances are

provided with examples cited where appropriate (question number x /sub-paper y quoted as

Qx/My). More items may also be found in the section General Comments.

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• Identities: Many students were able to distinguish equations from identities. However,

their performance was only fair in using the perfect square expressions to expand

simple algebraic expressions.

Q29/M3

Exemplar Item (Expand simple algebraic expressions by using the perfect square

expressions)

Expand 2)2( x− .

Example of Student Work (Although the student knew how to expand the expression

by perfect square, mistakes occurred in computation)

Example of Student Work (Has mistakenly taken 222)( baba −=− as identity)

• Formulas: Many students could find the value of a specified variable in the formula.

The performance was fair in simplifying algebraic fractions and performing change

of subject in simple formulas.

Q29/M1

Exemplar Item (Simplifying algebraic fractions)

Simplify yx

yx

3− .

Example of Student Work (Although the student was able to find the common

denominator, the algebraic fractions was not simplified)

Example of Student Work (Could not simplify the algebraic fraction to its simplest

form)

Q29/M1

Example of Student Work (Could not demonstrate recognition of the manipulation of

algebraic fractions)

• Linear Inequalities in One Unknown: Students could formulate inequalities from

simple contexts. They demonstrated good recognition of the properties of inequalities.

Their performance was fair in solving simple linear inequalities.

Q31/M4

Exemplar Item (Solving inequalities)

Solve the inequality 712 −>+x .


Example of Student Work (Mistakes occurred in computation and could not express

the answer with inequality sign)


S.3 Measures, Shape and Space Dimension

S.3 students performed steadily in this Dimension. They could find areas and volumes in

2-D and 3-D figures, angles related with lines and rectilinear figures, solve problems

regarding transformation and symmetry, quadrilaterals and trigonometry. However,

more improvement could be shown in items related to definitions of common terms in

geometry as well as deductive geometry. Comments on students’ performances are

provided with examples cited where appropriate (question number x /sub-paper y quoted as

Qx/My). More items may also be found in the section General Comments.

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Measures in 2-D and 3-D Figures

• Estimation in Measurement: Students were able to find the range of measures from a

measurement of a given degree of accuracy, choose an appropriate unit and the

degree of accuracy for real-life measurements. Quite a number of students could

estimate measures with justification.

Q48/M2

Example of Student Work (Estimate the length of the corridor – using a ruler to

measure instead of using estimation strategies)


• Simple Idea of Areas and Volumes: Many students could use the formulas for

circumferences and areas of circles to solve problems. Their performance was quite

good in using formulas for the surface areas and volumes of solids.

• More about Areas and Volumes: Many students could use formulas to calculate arc

lengths, areas of sectors, volumes of spheres and surface areas of cones. Almost half

of the students were able to use relationships between the sides and volumes of

similar figures to solve problems and distinguish among formulas for surface areas

by considering dimensions.

Q47/M1

Exemplar Item (Calculate the curved surface area)

The figure shows a right circular cone of

height 21 cm and base radius 20 cm.

Its slant height is 29 cm. Find the curved

surface area of the cone in terms of π.

Example of Student Work (Used the formula incorrectly)

Example of Student Work (Wrong unit)

Learning Geometry through an Intuitive Approach

• Introduction to Geometry: Students could generally identify types of angles and 3-D

solids from given nets. Quite a number of students were able to sketch a diagram of a

triangular prism. However, their performance was weak in the recognition of regular

polygons and equiangular polygons.

29 cm

20 cm

21 cm

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Measures in 2-D and 3-D Figures

• Estimation in Measurement: Students were able to find the range of measures from a

measurement of a given degree of accuracy, choose an appropriate unit and the

degree of accuracy for real-life measurements. Quite a number of students could

estimate measures with justification.

Q48/M2

Example of Student Work (Estimate the length of the corridor – using a ruler to

measure instead of using estimation strategies)


• Simple Idea of Areas and Volumes: Many students could use the formulas for

circumferences and areas of circles to solve problems. Their performance was quite

good in using formulas for the surface areas and volumes of solids.

• More about Areas and Volumes: Many students could use formulas to calculate arc

lengths, areas of sectors, volumes of spheres and surface areas of cones. Almost half

of the students were able to use relationships between the sides and volumes of

similar figures to solve problems and distinguish among formulas for surface areas

by considering dimensions.

Q47/M1

Exemplar Item (Calculate the curved surface area)

The figure shows a right circular cone of

height 21 cm and base radius 20 cm.

Its slant height is 29 cm. Find the curved

surface area of the cone in terms of π.

Example of Student Work (Used the formula incorrectly)

Example of Student Work (Wrong unit)

Learning Geometry through an Intuitive Approach

• Introduction to Geometry: Students could generally identify types of angles and 3-D

solids from given nets. Quite a number of students were able to sketch a diagram of a

triangular prism. However, their performance was weak in the recognition of regular

polygons and equiangular polygons.

29 cm

20 cm

21 cm

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Q32/M4

Example of Student Work (Sketch a diagram of a triangular prism – the dotted line

was not drawn in a right position)

Example of Student Work (Sketch a diagram of a triangular prism – the student didn’t

use solid lines and dotted lines appropriately)

• Transformation and Symmetry: Students’ performance was good. They were able to

grasp the concepts of axes of symmetry and the order of rotational symmetry, and

they could also identify the effect on the size and shape of a figure under a single

transformation.

• Congruence and Similarity: Students demonstrated recognition of the conditions for

similar triangles. Nonetheless, some of them confused the reasons for congruent

triangles with that for similar triangles. Quite a number of students were not able to

identify the corresponding angles of two similar triangles.

• Angles related with Lines and Rectilinear Figures: Students did well in solving

simple geometric problems. Nevertheless, some students were weak in recognition of

adjacent angles and applying the formula for exterior angles of convex polygons.

• More about 3-D figures: Students were able to identify the nets of cubes and match

3-D objects with various views. Nearly half of the students could name the angle

between 2 planes. Nevertheless, some of them could not identify the angle between a

line and a horizontal plane.

Q36/M4

Exemplar Item (Identify the angle between a line and a horizontal plane)

In the figure, VABCD is a pyramid with a rectangular base. Its base ABCD is a horizontal plane and its height is VC. Name the angle between VB and the plane ABCD.

Example of Student Work (Could not identify the correct angle)

(1)

(2)

(3)

(4)

Learning Geometry through a Deductive Approach

• Simple Introduction to Deductive Geometry: Students performed quite well in using

the conditions for congruent and similar triangles to perform simple proofs. Their

performance was quite poor when dealing with problems about angles and lines.

Many of students could not provide reasons or complete the proof correctly.

Moreover, quite a number of students were not able to identify perpendicular

bisectors of a triangle.

V

A

B C

D

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Q32/M4

Example of Student Work (Sketch a diagram of a triangular prism – the dotted line

was not drawn in a right position)

Example of Student Work (Sketch a diagram of a triangular prism – the student didn’t

use solid lines and dotted lines appropriately)

• Transformation and Symmetry: Students’ performance was good. They were able to

grasp the concepts of axes of symmetry and the order of rotational symmetry, and

they could also identify the effect on the size and shape of a figure under a single

transformation.

• Congruence and Similarity: Students demonstrated recognition of the conditions for

similar triangles. Nonetheless, some of them confused the reasons for congruent

triangles with that for similar triangles. Quite a number of students were not able to

identify the corresponding angles of two similar triangles.

• Angles related with Lines and Rectilinear Figures: Students did well in solving

simple geometric problems. Nevertheless, some students were weak in recognition of

adjacent angles and applying the formula for exterior angles of convex polygons.

• More about 3-D figures: Students were able to identify the nets of cubes and match

3-D objects with various views. Nearly half of the students could name the angle

between 2 planes. Nevertheless, some of them could not identify the angle between a

line and a horizontal plane.

Q36/M4

Exemplar Item (Identify the angle between a line and a horizontal plane)

In the figure, VABCD is a pyramid with a rectangular base. Its base ABCD is a horizontal plane and its height is VC. Name the angle between VB and the plane ABCD.

Example of Student Work (Could not identify the correct angle)

(1)

(2)

(3)

(4)

Learning Geometry through a Deductive Approach

• Simple Introduction to Deductive Geometry: Students performed quite well in using

the conditions for congruent and similar triangles to perform simple proofs. Their

performance was quite poor when dealing with problems about angles and lines.

Many of students could not provide reasons or complete the proof correctly.

Moreover, quite a number of students were not able to identify perpendicular

bisectors of a triangle.

V

A

B C

D

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Q45/M1

Exemplar Item (Geometric proof)

In the figure, ABCD is a straight line.

∠ABE = 67° and ∠FCD = 113°.

Prove that CFBE // .

Example of Student Work (Could not provide reasons, and some necessary steps were

missing)

Example of Student Work (Wrongly used ‘EBC’ and ‘FCD’ to represent the angles in

the diagram, and correct reasons were not stated)


• Pythagoras’ Theorem: Many students could use Pythagoras’ Theorem and the

converse of Pythagoras’ Theorem to solve simple problems.

• Quadrilaterals: Students performed well. They could use the properties of rhombuses

in numerical calculations.

F C

D

B E 67°

113°

A

Learning Geometry through an Analytic Approach

• Introduction to Coordinates: Students could grasp the basic concepts of the

rectangular coordinate system, but they only performed fair in problems regarding

polar coordinates. They did not do well in items related to calculating areas of simple

figures and a single transformation including rotation.

Q49/M1

Exemplar Item (Find the area of a figure)

Find the area of the pentagon ABCDE in the figure.

Example of Student Work (Wrongly calculated the height of CDE and its area)

• Coordinate Geometry of Straight Lines: Students could use the distance formula and

the mid-point formula. However, their performance was only fair in finding slopes of

straight lines and applying the conditions for parallel lines.

Trigonometry

• Trigonometric Ratios and Using Trigonometry: Students could grasp the basic

concepts of sine, cosine and tangent ratios. Their performance was satisfactory in

recognition of the idea of bearing. Quite a number of students were able to solve

simple 2-D problems involving one right-angled triangle.

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Q45/M1

Exemplar Item (Geometric proof)

In the figure, ABCD is a straight line.

∠ABE = 67° and ∠FCD = 113°.

Prove that CFBE // .

Example of Student Work (Could not provide reasons, and some necessary steps were

missing)

Example of Student Work (Wrongly used ‘EBC’ and ‘FCD’ to represent the angles in

the diagram, and correct reasons were not stated)


• Pythagoras’ Theorem: Many students could use Pythagoras’ Theorem and the

converse of Pythagoras’ Theorem to solve simple problems.

• Quadrilaterals: Students performed well. They could use the properties of rhombuses

in numerical calculations.

F C

D

B E 67°

113°

A

Learning Geometry through an Analytic Approach

• Introduction to Coordinates: Students could grasp the basic concepts of the

rectangular coordinate system, but they only performed fair in problems regarding

polar coordinates. They did not do well in items related to calculating areas of simple

figures and a single transformation including rotation.

Q49/M1

Exemplar Item (Find the area of a figure)

Find the area of the pentagon ABCDE in the figure.

Example of Student Work (Wrongly calculated the height of CDE and its area)

• Coordinate Geometry of Straight Lines: Students could use the distance formula and

the mid-point formula. However, their performance was only fair in finding slopes of

straight lines and applying the conditions for parallel lines.

Trigonometry

• Trigonometric Ratios and Using Trigonometry: Students could grasp the basic

concepts of sine, cosine and tangent ratios. Their performance was satisfactory in

recognition of the idea of bearing. Quite a number of students were able to solve

simple 2-D problems involving one right-angled triangle.

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S.3 Data Handling Dimension

The performances of S.3 students were satisfactory in this Dimension. They did well in

items related to using simple methods to collect data, interpret simple statistical charts

and choose appropriate diagrams to present a set of data. However, performance was

weak when students were asked to construct stem-and-leaf diagrams, compare the

presentations of the same set of data by using statistical charts and find the mean from a

set of grouped data. Comments on students’ performance are provided below with

examples cited where appropriate (question number x / sub-paper y quoted as Qx/My).

More examples may also be found in the section General Comments.

Organization and Representation of Data

• Introduction to Various Stages of Statistics: Students were able to use simple

methods to collect data and organize data, organize the same set of data by different

grouping methods. About half of the students could distinguish discrete and

continuous data.

• Construction and Interpretation of Simple Diagrams and Graphs: Students could

interpret simple statistical charts and choose appropriate diagrams/graphs to present a

set of data. Nonetheless, many students were not able to construct stem-and-leaf

diagrams correctly.

Q49/M2

Example of Student Work (Construction of stem-and-leaf diagrams – the student

mistakenly added the commas between the data)

Q49/M2


mistakenly arranged the data on the stem from largest to smallest)

Analysis and Interpretation of data

Measures of Central Tendency: In finding averages from a set of ungrouped data,

students performed quite well. On the other hand, many students could not find the

mean from a set of grouped data. More than half of the students could not identify

sources of deception in cases of misuse of averages.

Q50/M4

Exemplar Item (Find the mean from a set of grouped data)

The table below shows the weights of 60 suitcases.

Weight (kg) 16 – 20 21 – 25 26 – 30 31 – 35Frequency 8 32 14 6

Find the mean weight of the 60 suitcases.

Example of Student Work (Mistakenly considered the mean of frequencies)

Example of Student Work (Correct formula but errors occurred in steps)

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355

S.3 Data Handling Dimension

The performances of S.3 students were satisfactory in this Dimension. They did well in

items related to using simple methods to collect data, interpret simple statistical charts

and choose appropriate diagrams to present a set of data. However, performance was

weak when students were asked to construct stem-and-leaf diagrams, compare the

presentations of the same set of data by using statistical charts and find the mean from a

set of grouped data. Comments on students’ performance are provided below with

examples cited where appropriate (question number x / sub-paper y quoted as Qx/My).

More examples may also be found in the section General Comments.

Organization and Representation of Data

• Introduction to Various Stages of Statistics: Students were able to use simple

methods to collect data and organize data, organize the same set of data by different

grouping methods. About half of the students could distinguish discrete and

continuous data.

• Construction and Interpretation of Simple Diagrams and Graphs: Students could

interpret simple statistical charts and choose appropriate diagrams/graphs to present a

set of data. Nonetheless, many students were not able to construct stem-and-leaf

diagrams correctly.

Q49/M2


mistakenly added the commas between the data)

Q49/M2


mistakenly arranged the data on the stem from largest to smallest)

Analysis and Interpretation of data

Measures of Central Tendency: In finding averages from a set of ungrouped data,

students performed quite well. On the other hand, many students could not find the

mean from a set of grouped data. More than half of the students could not identify

sources of deception in cases of misuse of averages.

Q50/M4

Exemplar Item (Find the mean from a set of grouped data)

The table below shows the weights of 60 suitcases.

Weight (kg) 16 – 20 21 – 25 26 – 30 31 – 35Frequency 8 32 14 6

Find the mean weight of the 60 suitcases.

Example of Student Work (Mistakenly considered the mean of frequencies)

Example of Student Work (Correct formula but errors occurred in steps)

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Probability

• Simple Idea of Probability: Students were able to find the theoretical probability by

listing. Their performance was only fair in calculating the empirical probability.

Q50/M3

Exemplar Item (Calculate the theoretical probability)

Consider a family with 3 children.

(a) Let B represent a boy and G represent a girl. Complete the tree diagram provided

in the ANSWER BOOKLET to list out all possible outcomes.

(b) Find the probability that only one child is a girl in the family.

Example of Student Work (Could not find the three possible outcomes which satisfy

the condition from the tree diagram)


General Comments on S.3 Student Performances

The overall performance of S.3 students was steady. They did quite well in the Number

and Algebra Dimension. Performance was satisfactory in the Measures, Shape and

Space Dimension and in the Data Handling Dimension.

The areas in which students demonstrated adequate skills are listed below:

Directed Numbers and the Number Line

• Use positive numbers, negative numbers and zero to describe situations like profit

and loss, floor levels relative to the ground level (e.g. Q21/M1).

• Demonstrate recognition of the ordering of integers on the number line

(e.g. Q21/M3).

• Add, subtract, multiply and divide directed numbers (e.g. Q21/M4).

Numerical Estimation

• Determine whether to estimate or to compute the exact value in a simple context (e.g.

Q1/M3).

• Judge, without actual calculations, the reasonableness of answers from computations

(e.g. Q1/M2).

Approximation and Errors

• Round off a number to 3 significant figures (e.g. Q26/M3).

• Represent a large or a small number in scientific notation (e.g. Q2/M1).

• Convert numbers in scientific notation to integers or decimals (e.g. Q3/M2).

Rational and Irrational Numbers

• Demonstrate, without using calculators, recognition of the integral part of a ,

where a is a positive integer not greater than 200 (e.g. Q3/M1).

Using Percentages

• Solve simple selling problems (e.g. Q42/M3).

Rate and Ratio

• Represent a ratio in the form a : b (or ba

), a : b : c (e.g. Q22/M4).

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357

Probability

• Simple Idea of Probability: Students were able to find the theoretical probability by

listing. Their performance was only fair in calculating the empirical probability.

Q50/M3

Exemplar Item (Calculate the theoretical probability)

Consider a family with 3 children.

(a) Let B represent a boy and G represent a girl. Complete the tree diagram provided

in the ANSWER BOOKLET to list out all possible outcomes.

(b) Find the probability that only one child is a girl in the family.

Example of Student Work (Could not find the three possible outcomes which satisfy

the condition from the tree diagram)


General Comments on S.3 Student Performances

The overall performance of S.3 students was steady. They did quite well in the Number

and Algebra Dimension. Performance was satisfactory in the Measures, Shape and

Space Dimension and in the Data Handling Dimension.

The areas in which students demonstrated adequate skills are listed below:

Directed Numbers and the Number Line

• Use positive numbers, negative numbers and zero to describe situations like profit

and loss, floor levels relative to the ground level (e.g. Q21/M1).

• Demonstrate recognition of the ordering of integers on the number line

(e.g. Q21/M3).

• Add, subtract, multiply and divide directed numbers (e.g. Q21/M4).

Numerical Estimation


Q1/M3).

• Judge, without actual calculations, the reasonableness of answers from computations

(e.g. Q1/M2).


• Round off a number to 3 significant figures (e.g. Q26/M3).

• Represent a large or a small number in scientific notation (e.g. Q2/M1).


Rational and Irrational Numbers



Using Percentages


Rate and Ratio

• Represent a ratio in the form a : b (or ba

), a : b : c (e.g. Q22/M4).

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358

• Find the other quantity from a given ratio a : b and the value of either a or b

(e.g. Q22/M3).

• Use rate and ratio to solve simple real-life problems (e.g. Q23/M2).

Formulating Problems with Algebraic Language

• Distinguish the difference between 2x and 2 + x; (–2)n and –2n; x2 and 2x, etc.

(e.g. Q3/M4).

• Translate word phrases/contexts into algebraic languages (e.g. Q4/M1).

• Substitute values into some common and simple formulas and find the value of a

specified variable (e.g. Q23/M1).

• Formulate simple equations/inequalities from simple contexts (e.g. Q4/M3).

• Describe patterns by writing the next few terms in arithmetic sequences, geometric

sequences, Fibonacci sequence or sequences of polygonal numbers from several

consecutive terms of integral values (e.g. Q23/M4).

Linear Equations in One Unknown

• Solve simple equations (e.g. Q6/M1).

Linear Equations in Two Unknowns

• Demonstrate recognition that graphs of equations of the form ax + by + c = 0 are

straight lines (e.g. Q7/M3).

Linear Inequalities in One Unknown

• Represent inequalities, such as x < –2, x ≥ 3, etc., on the number line and vice versa

(e.g. Q9/M1).

Estimation in Measurement

• Find the range of measures from a measurement of a given degree of accuracy (e.g.

Q9/M3).

• Choose an appropriate unit and the degree of accuracy for real-life measurements

(e.g. Q10/M2).

More about Areas and Volumes

• Calculate volumes of pyramids, circular cones and spheres (e.g. Q11/M2).

Introduction to Geometry

• Identify types of angles with respect to their sizes (e.g. Q13/M2).

• Make 3-D solids from given nets (e.g. Q13/M4).

Transformation and Symmetry

• Determine the order of rotational symmetry from a figure and locate the centre of

rotation (e.g. Q32/M2).

• Name the single transformation involved in comparing the object and its image

(e.g. Q14/M3).

Congruence and Similarity

• Demonstrate recognition of the conditions for congruent and similar triangles

(e.g. Q15/M4).

• Demonstrate recognition of the properties of congruent and similar triangles

(e.g. Q33/M1).

Angles related with Lines and Rectilinear Figures

• Use the angle properties associated with intersecting lines/parallel lines to solve

simple geometric problems (e.g. Q34/M3).

• Use the properties of angles of triangles to solve simple geometric problems

(e.g. Q33/M2).

Quadrilaterals

• Use the properties of rhombuses in numerical calculations (e.g. Q36/M2).

Introduction to Coordinates

• Use an ordered pair to describe the position of a point in the rectangular coordinate

plane and locate a point of given rectangular coordinates (e.g. Q17/M3 and Q37/M3).

• Match a point under a single transformation with its image in the rectangular

coordinate plane (e.g. Q38/M4).

Coordinate Geometry of Straight Lines

• Use the distance formula (e.g. Q17/M2).

S3MATHEMATICS

359

• Find the other quantity from a given ratio a : b and the value of either a or b

(e.g. Q22/M3).

• Use rate and ratio to solve simple real-life problems (e.g. Q23/M2).

Formulating Problems with Algebraic Language

• Distinguish the difference between 2x and 2 + x; (–2)n and –2n; x2 and 2x, etc.

(e.g. Q3/M4).





• Describe patterns by writing the next few terms in arithmetic sequences, geometric

sequences, Fibonacci sequence or sequences of polygonal numbers from several

consecutive terms of integral values (e.g. Q23/M4).

Linear Equations in One Unknown



• Demonstrate recognition that graphs of equations of the form ax + by + c = 0 are

straight lines (e.g. Q7/M3).

Linear Inequalities in One Unknown

• Represent inequalities, such as x < –2, x ≥ 3, etc., on the number line and vice versa

(e.g. Q9/M1).

Estimation in Measurement


Q9/M3).

• Choose an appropriate unit and the degree of accuracy for real-life measurements

(e.g. Q10/M2).




• Identify types of angles with respect to their sizes (e.g. Q13/M2).

• Make 3-D solids from given nets (e.g. Q13/M4).

Transformation and Symmetry

• Determine the order of rotational symmetry from a figure and locate the centre of

rotation (e.g. Q32/M2).

• Name the single transformation involved in comparing the object and its image

(e.g. Q14/M3).

Congruence and Similarity

• Demonstrate recognition of the conditions for congruent and similar triangles

(e.g. Q15/M4).

• Demonstrate recognition of the properties of congruent and similar triangles

(e.g. Q33/M1).




• Use the properties of angles of triangles to solve simple geometric problems

(e.g. Q33/M2).

Quadrilaterals


Introduction to Coordinates

• Use an ordered pair to describe the position of a point in the rectangular coordinate

plane and locate a point of given rectangular coordinates (e.g. Q17/M3 and Q37/M3).

• Match a point under a single transformation with its image in the rectangular

coordinate plane (e.g. Q38/M4).

Coordinate Geometry of Straight Lines

• Use the distance formula (e.g. Q17/M2).

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360

Introduction to Various Stages of Statistics

• Use simple methods to collect data (e.g. Q20/M3).

• Organize the same set of data by different grouping methods (e.g. Q50/M1).

Construction and Interpretation of Simple Diagrams and Graphs

• Choose appropriate diagrams/graphs to present a set of data (e.g. Q19/M1).

Other than items in which students performed well, the Assessment data also provided

some entry points to strengthen teaching and learning. Items worthy of attention are

discussed below:


• Round off a number to a certain number of decimal places (e.g. Q2/M2): Quite a

number of students chose the correct answer, option C. However, options B and D

were chosen by about 20% of students respectively. They could not judge whether

zeros should be added after the digit ‘8’ or how many zeros should be added.

Q2/M2

Round off 0.079 95 to 3 decimal places.

A. 0.079 9 B. 0.08 C. 0.080 D. 0.080 0

Manipulations of Simple Polynomials

• Distinguish polynomials from algebraic expressions (e.g. Q4/M4): Only some

students chose the correct answer, option B. Each of the remaining options was

chosen by about 20% of students respectively. They did not grasp the concept of

polynomials.

Q4/M4

Which of the following is a polynomial?

A. xx 32 2 +

B. xx 32

2

+

C. xx 32 +

D. x

x312 2 +


• Plot graphs of linear equations in 2 unknowns (e.g. Q27/M2 and Q44/M3): Two

different items about plotting graphs of linear equations in 2 unknowns were set in

the assessment in different sub-papers. One of the items asked students to plot the

graph directly. The other item provided a table preset with some coordinates to assist

plotting.

Q27/M2

Draw the graph of the equation 02 =−+ yx on the rectangular coordinate plane

given in the ANSWER BOOKLET. The range of x must include the values from – 2

to 2.

Q44/M3

Complete the table for the equation 02 =−+ yx in the ANSWER BOOKLET.

x – 2 1 2 y 1

According to the table, draw the graph of this equation on the rectangular coordinate

plane given in the ANSWER BOOKLET.

• According to the facilities, students did well in Q44/M3. However, the facility of

Q27/M2 was only just over half of that of Q44/M3.

• Determine whether a point lies on a straight line with a given equation (e.g. Q7/M1):

More than half of the students chose the correct answer, option D. However, many

students still chose options B and C. They might only have considered 3 + 3 = 6 and

0 + 3 = 3 instead of considering the x-coordinate is –3 or not.

Q7/M1

Which of the following points lie on the straight line 03 =+x ?

P(3 , 6) , Q(–3 , 0) , R(0 , 3) , S(–3 , 6) A. P and R B. P and S C. Q and R D. Q and S

S3MATHEMATICS

361

Introduction to Various Stages of Statistics


• Organize the same set of data by different grouping methods (e.g. Q50/M1).


• Choose appropriate diagrams/graphs to present a set of data (e.g. Q19/M1).

Other than items in which students performed well, the Assessment data also provided

some entry points to strengthen teaching and learning. Items worthy of attention are

discussed below:


• Round off a number to a certain number of decimal places (e.g. Q2/M2): Quite a

number of students chose the correct answer, option C. However, options B and D

were chosen by about 20% of students respectively. They could not judge whether

zeros should be added after the digit ‘8’ or how many zeros should be added.

Q2/M2

Round off 0.079 95 to 3 decimal places.

A. 0.079 9 B. 0.08 C. 0.080 D. 0.080 0

Manipulations of Simple Polynomials

• Distinguish polynomials from algebraic expressions (e.g. Q4/M4): Only some

students chose the correct answer, option B. Each of the remaining options was

chosen by about 20% of students respectively. They did not grasp the concept of

polynomials.

Q4/M4

Which of the following is a polynomial?

A. xx 32 2 +

B. xx 32

2

+

C. xx 32 +

D. x

x312 2 +


• Plot graphs of linear equations in 2 unknowns (e.g. Q27/M2 and Q44/M3): Two

different items about plotting graphs of linear equations in 2 unknowns were set in

the assessment in different sub-papers. One of the items asked students to plot the

graph directly. The other item provided a table preset with some coordinates to assist

plotting.

Q27/M2

Draw the graph of the equation 02 =−+ yx on the rectangular coordinate plane

given in the ANSWER BOOKLET. The range of x must include the values from – 2

to 2.

Q44/M3

Complete the table for the equation 02 =−+ yx in the ANSWER BOOKLET.

x – 2 1 2 y 1

According to the table, draw the graph of this equation on the rectangular coordinate

plane given in the ANSWER BOOKLET.

• According to the facilities, students did well in Q44/M3. However, the facility of

Q27/M2 was only just over half of that of Q44/M3.

• Determine whether a point lies on a straight line with a given equation (e.g. Q7/M1):

More than half of the students chose the correct answer, option D. However, many

students still chose options B and C. They might only have considered 3 + 3 = 6 and

0 + 3 = 3 instead of considering the x-coordinate is –3 or not.

Q7/M1

Which of the following points lie on the straight line 03 =+x ?

P(3 , 6) , Q(–3 , 0) , R(0 , 3) , S(–3 , 6) A. P and R B. P and S C. Q and R D. Q and S

S3 MATHEMATICS

362

Formulas

• Perform change of subject in simple formulas but not including radical sign (e.g.

Q28/M2 and Q30/M4): Two different items about change of subject in simple

formulas were set in the assessment in different sub-papers. The two items are

basically the same; the only difference is the two terms on the right hand side of the

formulas are interchanged.

Q28/M2

Make P the subject of the formula 14+=

PN .

Q30/M4

Make P the subject of the formula 4

1 PN += .

• According to the result, the facility of Q30/M4 was higher than that of Q28/M2. In

Q28/M2, more students rewrote the formula as P = 4N – 1.


• Distinguish among formulas for lengths, areas and volumes by considering

dimensions (e.g. Q11/M3): Almost half of the students chose the correct answer,

option A. However, about 40% of students chose options C and 10% of students

chose D.

Q11/M3

The ratio of the volumes of two similar cones is 1 : 27. Which of the following is

the ratio of their corresponding heights?

A. 1 : 3

B. 12 : 32

C. 13 : 33

D. 13 : 273

• Distinguish among formulas for lengths, areas, volumes by considering dimensions

(e.g. Q12/M1): More than half of the students chose the correct answer, option B.

However, options A and C were chosen by about 20% of students respectively.

Q12/M1

In the figure, the roly-poly toy can be divided into a head and a body. The head is formed by 3 spheres. The diameters of the spheres are 2a, 2a and 2b respectively. The body is formed by a cylinder and a hemisphere. The base radius and the height of the cylinder are b and h respectively. The radius of the hemisphere is also b. By considering the dimensions, determine which of the following could express the total surface area of the roly-poly toy.

A. )36(83

233 hbba ++π

B. )27(8 2 hbba ++ππ

C. hba 254 ++ ππ

D. 2a + 3b + h

h

2b

2a

b

2a

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363

Formulas

• Perform change of subject in simple formulas but not including radical sign (e.g.

Q28/M2 and Q30/M4): Two different items about change of subject in simple

formulas were set in the assessment in different sub-papers. The two items are

basically the same; the only difference is the two terms on the right hand side of the

formulas are interchanged.

Q28/M2

Make P the subject of the formula 14+=

PN .

Q30/M4

Make P the subject of the formula 4

1 PN += .

• According to the result, the facility of Q30/M4 was higher than that of Q28/M2. In

Q28/M2, more students rewrote the formula as P = 4N – 1.


• Distinguish among formulas for lengths, areas and volumes by considering

dimensions (e.g. Q11/M3): Almost half of the students chose the correct answer,

option A. However, about 40% of students chose options C and 10% of students

chose D.

Q11/M3

The ratio of the volumes of two similar cones is 1 : 27. Which of the following is

the ratio of their corresponding heights?

A. 1 : 3

B. 12 : 32

C. 13 : 33

D. 13 : 273

• Distinguish among formulas for lengths, areas, volumes by considering dimensions

(e.g. Q12/M1): More than half of the students chose the correct answer, option B.

However, options A and C were chosen by about 20% of students respectively.

Q12/M1

In the figure, the roly-poly toy can be divided into a head and a body. The head is formed by 3 spheres. The diameters of the spheres are 2a, 2a and 2b respectively. The body is formed by a cylinder and a hemisphere. The base radius and the height of the cylinder are b and h respectively. The radius of the hemisphere is also b. By considering the dimensions, determine which of the following could express the total surface area of the roly-poly toy.

A. )36(83

233 hbba ++π

B. )27(8 2 hbba ++ππ

C. hba 254 ++ ππ

D. 2a + 3b + h

h

2b

2a

b

2a

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364


Determine whether a polygon is regular, convex, concave, equilateral or equiangular

(e.g. Q32/M3): Only some of the students chose the correct answers (Figure P and

Figure S). Quite a number of students thought that Figure Q and Figure R were also

equiangular.

Q32/M3

Which of the following polygons MUST be equiangular? (May be more than one answer)

P. Q. R. S.


Demonstrate recognition of the angles with respect to their positions relative to lines

and polygons (e.g. Q15/M3): Only half of the students chose the correct answer,

option D. Quite a number of students mistakenly thought that a and b are angles at a

point or interior angles on the same side.

Q15/M3

In the figure, a and b are

A. vertically opposite angles.

B. angles at a point.

C. interior angles on the same side.

D. adjacent angles.

a b


• Compare the presentations of the same set of data by using statistical charts (e.g.

Q19/M4): About half of the students chose option A. They were not aware that the

vertical axis of the histogram in each option is labelled as ‘frequency’ instead of

‘cumulative frequency’.

Q19/M4

The cumulative frequency polygon below shows the results of the 50 m freestyle event of 14 boys in a swimming gala.

Which of the following histograms can be used to construct the above diagram? A. B.

C. D.

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365


Determine whether a polygon is regular, convex, concave, equilateral or equiangular

(e.g. Q32/M3): Only some of the students chose the correct answers (Figure P and

Figure S). Quite a number of students thought that Figure Q and Figure R were also

equiangular.

Q32/M3

Which of the following polygons MUST be equiangular? (May be more than one answer)

P. Q. R. S.


Demonstrate recognition of the angles with respect to their positions relative to lines

and polygons (e.g. Q15/M3): Only half of the students chose the correct answer,

option D. Quite a number of students mistakenly thought that a and b are angles at a

point or interior angles on the same side.

Q15/M3

In the figure, a and b are

A. vertically opposite angles.

B. angles at a point.

C. interior angles on the same side.

D. adjacent angles.

a b


• Compare the presentations of the same set of data by using statistical charts (e.g.

Q19/M4): About half of the students chose option A. They were not aware that the

vertical axis of the histogram in each option is labelled as ‘frequency’ instead of

‘cumulative frequency’.

Q19/M4

The cumulative frequency polygon below shows the results of the 50 m freestyle event of 14 boys in a swimming gala.

Which of the following histograms can be used to construct the above diagram? A. B.

C. D.

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366

Best Performance of S.3 Students in TSA 2015

Students sitting for each sub-paper were ranked according to their scores and the

performances of approximately the top 10% were singled out for further analysis. The

performances of these students are described below.

Among these students, the majority of them achieved a full score or lost at most five

score points in the whole assessment. They demonstrated good understanding and

mastery of the concepts and skills assessed by the sub-papers they attempted.

These students were able to


Q1/M4).





• Solve simple problems on growths and depreciations (e.g. Q43/M3).

• Find the other quantity from a given ratio a : b and the value of either a or b (e.g.

Q22/M3).

• Use rate and ratio to solve simple real-life problems including mensuration problems

(e.g. Q23/M2).






• Plot graphs of linear equations in 2 unknowns (e.g. Q44/M3).

• Solve a system of simple linear simultaneous equations by algebraic methods (e.g.

Q44/M2).


Q9/M3).

• Use the formulas for circumferences and areas of circles (e.g. Q46/M3).

• Calculate arc lengths (e.g. Q46/M2).

• Calculate areas of sectors (e.g. Q46/M1).


• Name the single transformation involved in comparing the object and its image (e.g.

Q14/M3).

• Demonstrate recognition of the conditions for congruent and similar triangles (e.g.

Q33/M4).



• Use the relations between sides and angles associated with isosceles/equilateral

triangles to solve simple geometric problems (e.g. Q47/M3).


• Use the mid-point formula (e.g. Q18/M4).

• Solve simple 2-D problems involving one right-angled triangle (e.g. Q49/M4).


• Interpret simple statistical charts (e.g. Q39/M4).

• Calculate the theoretical probability by listing (e.g. Q50/M3).

The examples of work by these students are illustrated as follows:

Students with the best performance were able to construct simple statistical charts by

using the given data.

Q49/M3

Example of Student Work (Construct simple statistical charts)

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367

Best Performance of S.3 Students in TSA 2015

Students sitting for each sub-paper were ranked according to their scores and the

performances of approximately the top 10% were singled out for further analysis. The

performances of these students are described below.

Among these students, the majority of them achieved a full score or lost at most five

score points in the whole assessment. They demonstrated good understanding and

mastery of the concepts and skills assessed by the sub-papers they attempted.

These students were able to


Q1/M4).





• Solve simple problems on growths and depreciations (e.g. Q43/M3).

• Find the other quantity from a given ratio a : b and the value of either a or b (e.g.

Q22/M3).

• Use rate and ratio to solve simple real-life problems including mensuration problems

(e.g. Q23/M2).






• Plot graphs of linear equations in 2 unknowns (e.g. Q44/M3).

• Solve a system of simple linear simultaneous equations by algebraic methods (e.g.

Q44/M2).


Q9/M3).

• Use the formulas for circumferences and areas of circles (e.g. Q46/M3).

• Calculate arc lengths (e.g. Q46/M2).

• Calculate areas of sectors (e.g. Q46/M1).


• Name the single transformation involved in comparing the object and its image (e.g.

Q14/M3).

• Demonstrate recognition of the conditions for congruent and similar triangles (e.g.

Q33/M4).



• Use the relations between sides and angles associated with isosceles/equilateral

triangles to solve simple geometric problems (e.g. Q47/M3).


• Use the mid-point formula (e.g. Q18/M4).

• Solve simple 2-D problems involving one right-angled triangle (e.g. Q49/M4).


• Interpret simple statistical charts (e.g. Q39/M4).

• Calculate the theoretical probability by listing (e.g. Q50/M3).

The examples of work by these students are illustrated as follows:

Students with the best performance were able to construct simple statistical charts by

using the given data.

Q49/M3

Example of Student Work (Construct simple statistical charts)

S3 MATHEMATICS

368

Students with the best performance could solve the problem correctly with complete and

clear presentation.

Q44/M1

Example of Student Work (Solve simultaneous equations)

Students with the best performance were able to make good use of the given conditions

and solve the problem systematically.

Q48/M4

Example of Student Work (Estimate the length of the corridor)

Q45/M2

Example of Student Work (Geometric proof)

Q45/M3

Example of Student Work (Geometric Proof)

Some common weaknesses of high-achieving students are as follows:

• Some students could not compare the presentations of the same set of data by using

statistical charts.

• Some students could not distinguish polynomials from algebraic expressions.

• Some students could not determine whether a polygon is regular, convex, concave,

equilateral or equiangular.

S3MATHEMATICS

369

Students with the best performance could solve the problem correctly with complete and

clear presentation.

Q44/M1

Example of Student Work (Solve simultaneous equations)

Students with the best performance were able to make good use of the given conditions

and solve the problem systematically.

Q48/M4

Example of Student Work (Estimate the length of the corridor)

Q45/M2

Example of Student Work (Geometric proof)

Q45/M3

Example of Student Work (Geometric Proof)

Some common weaknesses of high-achieving students are as follows:

• Some students could not compare the presentations of the same set of data by using

statistical charts.

• Some students could not distinguish polynomials from algebraic expressions.

• Some students could not determine whether a polygon is regular, convex, concave,

equilateral or equiangular.

S3 MATHEMATICS

370

Overview of Student Performances in Mathematics at Secondary 3 TSA 2013-2015

This was the tenth year that Secondary 3 students participated in the Territory-wide System

Assessment. The percentage of students achieving Basic Competency this year was 79.9%

which was about the same as last year.

The percentages of students achieving Basic Competency from 2013 to 2015 are listed

below:

Table 8.9 Percentages of S.3 Students Achieving Mathematics Basic Competency from 2013 to 2015

Year % of Students Achieving Mathematics Basic Competency

2013 79.7

2014 79.9

2015 79.9

The performances of S.3 students over the past three years in each Dimension of

Mathematics are summarized as follows:

Tabl

e 8.

10

Ove

rvie

w o

f Stu

dent

Per

form

ance

s in

Mat

hem

atic

s at S

.3 T

SA 2

013-

2015

Y

ear

Num

ber

a nd

Alg

ebra

20

13

2014

20

15

Rem

arks

Stre

ngth

s

• S t

uden

ts di

d w

ell i

n th

e op

erat

ions

o f

dire

cted

num

bers

. Th

ey c

ould

al

so

use

dire

cted

nu

mbe

rs

to

desc

ribe

real

life

situ

atio

ns.

• St

uden

ts co

uld

dete

rmin

e w

heth

er

to e

stim

ate

or to

com

pute

the

exac

t va

lue

in a

sim

ple

cont

ext.

• St

uden

ts co

uld

conv

ert n

umbe

rs in

sc

ient

ific

nota

tion

to in

tege

rs.

• S t

uden

ts co

uld

solv

e sim

ple

prob

lem

s by

usin

g ra

te.

• St

uden

ts co

uld

form

ulat

e eq

uatio

ns fr

om si

mpl

e co

ntex

ts.

• St

uden

ts co

uld

obse

rve

the

patte

rn

o f

the

num

ber

sequ

ence

s an

d w

rote

dow

n th

e ne

xt fe

w te

rms.

• S t

uden

ts di

d w

ell i

n th

e op

erat

ions

of

di

rect

ed

num

bers

. Th

ey

dem

onstr

ated

re

cogn

ition

of

th

e nu

mbe

r lin

e.

• St

uden

ts co

uld

dete

rmin

e w

heth

er

to e

stim

ate

or to

com

pute

the

exac

t va

lue

in a

sim

ple

cont

ext.

• St

uden

ts co

uld

conv

ert n

umbe

rs in

sc

ient

ific

nota

tion

to d

ecim

als.

• S t

uden

ts co

uld

solv

e sim

ple

prob

lem

s by

usin

g ra

te.

• St

uden

ts co

uld

trans

late

w

ord

phra

ses/c

onte

xts

into

al

gebr

aic

lang

uage

s. •

Mo s

t stu

dent

s w

ere

capa

ble

of

s olv

ing

simpl

e eq

uatio

ns.

They

c o

uld

also

sub

stitu

te v

alue

s in

to

form

ulas

to

fin

d th

e un

know

n va

lue.

•

S tud

ents

coul

d fo

rmul

ate e

quati

ons

from

sim

ple

cont

exts.

•

Stud

ents

dem

onstr

ated

reco

gniti

on

of th

e pr

oper

ties o

f ine

qual

ities

.

• S t

uden

ts de

mon

strat

ed re

cogn

ition

o f

the

num

ber

line.

The

y co

uld

also

us

e di

rect

ed

num

bers

to

de

scrib

e re

al li

fe si

tuat

ions

. •

Stud

ents

wer

e ab

le t

o de

term

ine

whe

ther

to e

stim

ate

or to

com

pute

th

e ex

act v

alue

in a

sim

ple

cont

ext.

• St

uden

ts w

ere

able

to ro

und

off a

nu

mbe

r to

a c

erta

in n

umbe

r of

s ig

nific

ant f

igur

es. T

hey d

emon

strate

d re

cogn

ition

of s

cient

ific n

otat

ion.

•

Stud

ents

wer

e ab

le to

solv

e sim

ple

selli

ng p

robl

ems

and

prob

lem

s on

de

prec

iatio

ns.

• S t

uden

ts w

ere

able

to

so

lve

prob

lem

s by

usin

g ra

tio.

• S t

uden

ts w

ere

able

to

tra

nsla

te

wor

d ph

rase

s/con

texts

into

alg

ebra

ic la

ngua

ges.

• S t

uden

ts w

ere

able

to

subs

titut

e v a

lues

int

o fo

rmul

as t

o fin

d th

e un

know

n va

lue.

•

Stud

ents

wer

e ab

le t

o fo

rmul

ate

equa

tions

from

sim

ple

cont

exts.

• M

any

stude

nts

wer

e no

t fa

mili

ar

with

th

e co

ncep

ts of

so

me

term

inol

ogie

s (e

.g.

loss

pe

rcen

t, co

mpo

und

inte

rest)

and

so

they

us

ed th

e fo

rmul

as in

corre

ctly

. •

T he

pres

enta

tion

of so

me

stude

nts’

answ

ers

rem

aine

d in

com

plet

e an

d ca

rele

ss

mist

akes

oc

curre

d fre

quen

tly in

solv

ing

prob

lem

s. •

Man

y stu

dent

s di

d no

t use

a r

uler

to

dra

w st

raig

ht li

nes.

• A

nsw

ers

wer

e of

ten

not c

orre

cted

to

the

requ

ired

degr

ee o

f acc

urac

y. •

Uni

ts w

ere

ofte

n om

itted

in

the

answ

er.

• S t

uden

ts co

uld

not m

aste

r abs

tract

co

ncep

ts (s

uch

as th

e nt

h te

rm o

f a

sequ

ence

).

S3MATHEMATICS

371

Overview of Student Performances in Mathematics at Secondary 3 TSA 2013-2015

This was the tenth year that Secondary 3 students participated in the Territory-wide System

Assessment. The percentage of students achieving Basic Competency this year was 79.9%

which was about the same as last year.

The percentages of students achieving Basic Competency from 2013 to 2015 are listed

below:

Table 8.9 Percentages of S.3 Students Achieving Mathematics Basic Competency from 2013 to 2015

Year % of Students Achieving Mathematics Basic Competency

2013 79.7

2014 79.9

2015 79.9

The performances of S.3 students over the past three years in each Dimension of

Mathematics are summarized as follows:

Tabl

e 8.

10

Ove

rvie

w o

f Stu

dent

Per

form

ance

s in

Mat

hem

atic

s at S

.3 T

SA 2

013-

2015

Y

ear

Num

ber

a nd

Alg

ebra

20

13

2014

20

15

Rem

arks

Stre

ngth

s

• S t

uden

ts di

d w

ell i

n th

e op

erat

ions

of

dire

cted

num

bers

. Th

ey c

ould

al

so

use

dire

cted

nu

mbe

rs

to

desc

ribe

real

life

situ

atio

ns.

• S t

uden

ts co

uld

dete

rmin

e w

heth

er

to e

stim

ate

or to

com

pute

the

exac

t va

lue

in a

sim

ple

cont

ext.

• St

uden

ts co

uld

conv

ert n

umbe

rs in

sc

ient

ific

nota

tion

to in

tege

rs.

• S t

uden

ts co

uld

solv

e sim

ple

prob

lem

s by

usin

g ra

te.

• St

uden

ts co

uld

form

ulat

e eq

uatio

ns fr

om si

mpl

e co

ntex

ts.

• S t

uden

ts co

uld

obse

rve

the

patte

rn

of

the

num

ber

sequ

ence

s an

d w

rote

dow

n th

e ne

xt fe

w te

rms.

• S t

uden

ts di

d w

ell i

n th

e op

erat

ions

of

di

rect

ed

num

bers

. Th

ey

dem

onstr

ated

re

cogn

ition

of

th

e nu

mbe

r lin

e.

• S t

uden

ts co

uld

dete

rmin

e w

heth

er

to e

stim

ate

or to

com

pute

the

exac

t va

lue

in a

sim

ple

cont

ext.

• St

uden

ts co

uld

conv

ert n

umbe

rs in

sc

ient

ific

nota

tion

to d

ecim

als.

• S t

uden

ts co

uld

solv

e sim

ple

prob

lem

s by

usin

g ra

te.

• St

uden

ts co

uld

trans

late

w

ord

phra

ses/c

onte

xts

into

al

gebr

aic

lang

uage

s. •

Mo s

t stu

dent

s w

ere

capa

ble

of

solv

ing

simpl

e eq

uatio

ns.

They

co

uld

also

sub

stitu

te v

alue

s in

to

form

ulas

to

fin

d th

e un

know

n va

lue.

•

Stud

ents

coul

d fo

rmul

ate e

quati

ons

from

sim

ple

cont

exts.

•

S tud

ents

dem

onstr

ated

reco

gniti

on

of th

e pr

oper

ties o

f ine

qual

ities

.

• S t

uden

ts de

mon

strat

ed re

cogn

ition

of

the

num

ber

line.

The

y co

uld

also

us

e di

rect

ed

num

bers

to

de

scrib

e re

al li

fe si

tuat

ions

. •

S tud

ents

wer

e ab

le t

o de

term

ine

whe

ther

to e

stim

ate

or to

com

pute

th

e ex

act v

alue

in a

sim

ple

cont

ext.

• St

uden

ts w

ere

able

to ro

und

off a

nu

mbe

r to

a c

erta

in n

umbe

r of

sig

nific

ant f

igur

es. T

hey d

emon

strate

d re

cogn

ition

of s

cient

ific n

otat

ion.

•

Stud

ents

wer

e ab

le to

solv

e sim

ple

selli

ng p

robl

ems

and

prob

lem

s on

de

prec

iatio

ns.

• S t

uden

ts w

ere

able

to

so

lve

prob

lem

s by

usin

g ra

tio.

• St

uden

ts w

ere

able

to

tra

nsla

te

wor

d ph

rase

s/con

texts

into

alg

ebra

ic la

ngua

ges.

• St

uden

ts w

ere

able

to

subs

titut

e va

lues

int

o fo

rmul

as t

o fin

d th

e un

know

n va

lue.

•

S tud

ents

wer

e ab

le t

o fo

rmul

ate

equa

tions

from

sim

ple

cont

exts.

• M

any

stude

nts

wer

e no

t fa

mili

ar

with

th

e co

ncep

ts of

so

me

term

inol

ogie

s (e

.g.

loss

pe

rcen

t, co

mpo

und

inte

rest)

and

so

they

us

ed th

e fo

rmul

as in

corre

ctly

. •

T he

pres

enta

tion

of so

me

stude

nts’

answ

ers

rem

aine

d in

com

plet

e an

d ca

rele

ss

mist

akes

oc

curre

d fre

quen

tly in

solv

ing

prob

lem

s. •

Man

y stu

dent

s di

d no

t use

a r

uler

to

dra

w st

raig

ht li

nes.

• A

nsw

ers

wer

e of

ten

not c

orre

cted

to

the

requ

ired

degr

ee o

f acc

urac

y. •

Uni

ts w

ere

ofte

n om

itted

in

the

answ

er.

• St

uden

ts co

uld

not m

aste

r abs

tract

co

ncep

ts (s

uch

as th

e nt

h te

rm o

f a

sequ

ence

).

S3 MATHEMATICS

372

Yea

r

Num

ber

a nd

Alg

ebra

2013

2014

2015

Re m

arks

We a

knes

ses

W

hen

stude

nts

wer

e as

ked

to

roun

d of

f a

num

ber

to a

cer

tain

nu

mbe

r of

dec

imal

pla

ces,

they

m

istak

enly

ro

unde

d of

f th

e nu

mbe

r by

con

sider

ing

signi

fican

t fig

ures

.

Stud

ents’

per

form

ance

was

wea

k in

usin

g pe

rcen

tage

s to

fin

d th

e or

igin

al v

alue

of a

n ob

ject

.

S tud

ents

mix

ed u

p th

e fo

rmul

as

for

findi

ng

simpl

e in

tere

st an

d co

mpo

und

inte

rest.

Man

y stu

dent

s co

uld

not

intu

itive

ly f

ind

the

nth

term

of

a nu

mbe

r seq

uenc

e.

Perfo

rman

ce

was

w

eak

whe

n stu

dent

s w

ere

aske

d to

sim

plify

al

gebr

aic

expr

essio

ns

with

ne

gativ

e in

dice

s.

Stud

ents’

per

form

ance

was

onl

y fa

ir in

fa

ctor

izat

ion

of

simpl

e po

lyno

mia

ls.

S tud

ents

wer

e qu

ite

wea

k in

re

cogn

izin

g th

e m

eani

ng o

f ro

ots

of e

quat

ions

.

S tud

ents

coul

d no

t di

sting

uish

w

heth

er a

n eq

ualit

y is

an e

quat

ion

or a

n id

entit

y.

Stud

ents

wer

e qu

ite

wea

kin

re

cogn

izin

g th

e pr

oper

ties

of

ineq

ualit

ies.

W

hen

stude

nts

wer

e as

ked

to

roun

d of

f a

num

ber

to a

cer

tain

nu

mbe

r of

dec

imal

pla

ces,

they

m

istak

enly

ro

unde

d of

f th

e nu

mbe

r by

con

sider

ing

signi

fican

t fig

ures

.

Man

y stu

dent

s co

uld

not r

epre

sent

a

num

ber i

n sc

ient

ific

nota

tion.

S t

uden

ts w

ere

quite

w

eak

in

reco

gniz

ing

the

conc

epts

of

perc

enta

ge

chan

ge,

perc

enta

ge

decr

ease

and

loss

per

cent

age.

The

y m

istak

enly

su

bstit

uted

th

e co

st pr

ice

and

selli

ng

pric

e in

th

e fo

rmul

as.

St

uden

ts m

ixed

up

the

form

ulas

fo

r fin

ding

sim

ple

inte

rest

and

com

poun

d in

tere

st.

Stud

ents

coul

d no

t di

sting

uish

po

lyno

mia

ls fro

m

alge

brai

c ex

pres

sions

.

S tud

ents

wer

e w

eak

in re

cogn

izin

g th

e te

rmin

olog

ies o

f pol

ynom

ials.

St

uden

ts’ p

erfo

rman

ce w

as o

nly

fair

in

fact

oriz

atio

n of

sim

ple

poly

nom

ials.

S t

uden

ts co

uld

not

disti

ngui

sh

whe

ther

an

equa

lity

is an

equ

atio

n or

an

iden

tity.

St

uden

ts w

ere

quite

w

eak

in

reco

gniz

ing

the

conc

epts

of p

rofit

, se

lling

pric

e an

d so

man

y of

them

co

uld

not

find

the

cost

pric

e co

rrect

ly.

M

any

stude

nts

conf

used

co

mpo

und

inte

rest

with

sim

ple

inte

rest,

as

wel

l as

am

ount

with

in

tere

st. C

onse

quen

tly,

they

use

d w

rong

met

hods

.

Man

y stu

dent

s co

uld

not

disti

ngui

sh

poly

nom

ials

from

al

gebr

aic

expr

essio

ns.

S t

uden

ts’ p

erfo

rman

ce w

as o

nly

fair

in fa

ctor

izat

ion

and

expa

nsio

n of

sim

ple

poly

nom

ials.

With

out

bein

g gi

ven

a ta

ble

to

assis

t ca

lcul

atio

n of

coo

rdin

ates

, m

any

stude

nts

wer

e no

t ab

le t

o pl

ot th

e gr

aph

of a

line

ar e

quat

ion

corre

ctly

.

S tud

ents’

per

form

ance

was

wea

k w

hen

they

wer

e as

ked

to p

erfo

rm

chan

ge

of

subj

ect

in

simpl

e fo

rmul

as.

Y

ear

Mea

sure

s,

S hap

e an

d Sp

ace

2013

20

14

2015

R

emar

ks

Stre

ngth

s

• St

uden

ts w

ere

able

to

fin

d th

e ar

eas o

f sec

tors

. •

Stud

ents

coul

d id

entif

y th

e r e

latio

nshi

p be

twee

n sim

ple

3-D

so

lids a

nd th

eir c

orre

spon

ding

2-D

fig

ures

. The

y co

uld

also

reco

gniz

e th

e cr

oss-

sect

ions

of s

impl

e so

lids.

• St

uden

ts co

uld

draw

the

axe

s of

sy

mm

etry

. •

Whe

n th

e ob

ject

and

its

imag

e w

ere

give

n, st

uden

ts co

uld

iden

tify

the

singl

e tra

nsfo

rmat

ion

invo

lved

. •

S tud

ents

coul

d us

e th

e an

gle

prop

ertie

s as

soci

ated

w

ith

inte

rsec

ting

lines

/par

alle

l lin

es a

nd

the

prop

ertie

s of

tria

ngle

s to

sol

ve

simpl

e ge

omet

ric p

robl

ems.

• S t

uden

ts co

uld

use

the

prop

ertie

s o f

an

gles

of

tri

angl

es

and

the

rela

tions

bet

wee

n sid

es a

nd a

ngle

s as

soci

ated

with

isos

cele

s tri

angl

es

to

solv

e sim

ple

geom

etric

pr

oble

ms.

• St

uden

ts w

ere

able

to

us

e th

e pr

oper

ties

of

quad

rilat

eral

s in

nu

mer

ical

cal

cula

tions

. •

Stud

ents

had

good

kno

wle

dge

of

the

rect

angu

lar c

oord

inat

e sy

stem

.

• St

uden

ts w

ere

able

to

fin

d th

e ra

nge

of

mea

sure

s fro

m

a m

easu

rem

ent o

f a g

iven

deg

ree

of

accu

racy

an

d ch

oose

an

a p

prop

riate

uni

t and

the

degr

ee o

f ac

cura

cy fo

r rea

l-life

mea

sure

men

ts.

• St

uden

ts co

uld

choo

se th

e m

etho

d fro

m th

e gi

ven

optio

ns th

at g

ave

a m

ore

accu

rate

read

ing.

•

S tud

ents

wer

e ab

le

to

find

the

area

s of

sec

tors

and

the

sur

face

ar

eas o

f sph

eres

. •

S tud

ents

coul

d de

mon

strat

e r e

cogn

ition

of

com

mon

ter

ms

in

geom

etry

. •

Stud

ents

coul

d id

entif

y th

e re

latio

nshi

p be

twee

n sim

ple

3-D

so

lids a

nd th

eir c

orre

spon

ding

2-D

fig

ures

. •

Stud

ents

coul

d de

term

ine

the

num

ber

of a

xes

of s

ymm

etry

and

l o

cate

the

cent

re o

f rot

atio

n fro

m a

f ig

ure.

•

Whe

n th

e ob

ject

and

its

imag

e w

ere

give

n, st

uden

ts co

uld

iden

tify

the

singl

e tra

nsfo

rmat

ion

invo

lved

. •

Stud

ents

coul

d us

e th

e an

gle

prop

ertie

s as

soci

ated

w

ith

inte

rsec

ting

lines

/par

alle

l lin

es a

nd

the

prop

ertie

s of

tria

ngle

s to

sol

ve

simpl

e ge

omet

ric p

robl

ems.

• St

uden

ts ha

d go

od k

now

ledg

e of

th

e re

ctan

gula

r coo

rdin

ate

syste

m.

• St

uden

ts w

ere

able

to

fin

d th

e ra

nge

of

mea

sure

s fro

m

a m

easu

rem

ent o

f a g

iven

deg

ree

of

accu

racy

an

d ch

oose

an

a p

prop

riate

uni

t and

the

degr

ee o

f ac

cura

cy fo

r rea

l-life

mea

sure

men

ts.

• St

uden

ts w

ere

able

to

fin

d th

e ar

eas o

f sec

tors

and

the

volu

mes

of

pyra

mid

s. •

S tud

ents

wer

e ab

le to

iden

tify

the

rela

tions

hip

betw

een

simpl

e 3-

D

solid

s and

thei

r cor

resp

ondi

ng 2

-D

figur

es.

• S t

uden

ts w

ere

able

to

dete

rmin

e th

e or

der

of r

otat

iona

l sy

mm

etry

fro

m a

figu

re.

• W

hen

the

obje

ct a

nd i

ts im

age

wer

e gi

ven,

stu

dent

s w

ere

able

to

iden

tify

the

singl

e tra

nsfo

rmat

ion

invo

lved

. •

Stud

ents

wer

e ab

le to

dem

onstr

ate

r eco

gniti

on o

f th

e co

nditi

ons

for

c ong

ruen

t and

sim

ilar t

riang

les.

• St

uden

ts w

ere

able

to u

se th

e an

gle

prop

ertie

s as

soci

ated

w

ith

inte

rsec

ting

lines

/par

alle

l lin

es a

nd

the

prop

ertie

s of

tria

ngle

s to

sol

ve

simpl

e ge

omet

ric p

robl

ems.

• St

uden

ts ha

d go

od k

now

ledg

e of

th

e re

ctan

gula

r coo

rdin

ate

syste

m.

• St

uden

ts di

d w

ell

in

strai

ghtfo

rwar

d qu

estio

ns (

such

as

num

eric

al

calc

ulat

ion

in

simpl

e g e

omet

ric

prob

lem

s).

How

ever

, t h

eir p

erfo

rman

ce w

as o

nly

fair

in

item

s inv

olvi

ng m

ore

judg

men

ts.

• St

uden

ts w

ere

will

ing

to a

ttem

pt

geom

etric

pro

ofs.

How

ever

, th

ey

usua

lly

coul

d no

t us

e lo

gica

l r e

ason

ing

and

corre

ct r

easo

ns t

o co

mpl

ete

the

proo

fs.

• St

uden

ts co

uld

not m

aste

r abs

tract

co

ncep

ts (s

uch

as p

roje

ctio

n of

an

edge

on

a

plan

e,

p lan

es

of

refle

ctio

nal s

ymm

etrie

s of c

ubes

). •

Inap

prop

riate

or

in

corre

ct

pres

enta

tion

frequ

ently

oc

curre

d ( s

uch

as

conf

used

∠

ABC

with

ABC,

AB

= BC

with

AB

// BC

). •

Ans

wer

s w

ere

ofte

n no

t cor

rect

ed

to th

e re

quire

d de

gree

of a

ccur

acy.

• U

nits

wer

e of

ten

omitt

ed i

n th

e an

swer

. •

Man

y stu

dent

s w

ere

not

fam

iliar

w

ith th

e fo

rmul

as.

S3MATHEMATICS

373

Y

ear

Mea

sure

s,

S hap

e an

d Sp

ace

2013

20

14

2015

R

emar

ks

Stre

ngth

s

• St

uden

ts w

ere

able

to

fin

d th

e ar

eas o

f sec

tors

. •

S tud

ents

coul

d id

entif

y th

e re

latio

nshi

p be

twee

n sim

ple

3-D

so

lids a

nd th

eir c

orre

spon

ding

2-D

fig

ures

. The

y co

uld

also

reco

gniz

e th

e cr

oss-

sect

ions

of s

impl

e so

lids.

• S t

uden

ts co

uld

draw

the

axe

s of

sy

mm

etry

. •

Whe

n th

e ob

ject

and

its

imag

e w

ere

give

n, st

uden

ts co

uld

iden

tify

the

singl

e tra

nsfo

rmat

ion

invo

lved

. •

Stud

ents

coul

d us

e th

e an

gle

prop

ertie

s as

soci

ated

w

ith

inte

rsec

ting

lines

/par

alle

l lin

es a

nd

the

prop

ertie

s of

tria

ngle

s to

sol

ve

simpl

e ge

omet

ric p

robl

ems.

• St

uden

ts co

uld

use

the

prop

ertie

s of

an

gles

of

tri

angl

es

and

the

rela

tions

bet

wee

n sid

es a

nd a

ngle

s as

soci

ated

with

isos

cele

s tri

angl

es

to

solv

e sim

ple

geom

etric

pr

oble

ms.

• St

uden

ts w

ere

able

to

us

e th

e pr

oper

ties

of

quad

rilat

eral

s in

nu

mer

ical

cal

cula

tions

. •

Stud

ents

had

good

kno

wle

dge

of

the

rect

angu

lar c

oord

inat

e sy

stem

.

• St

uden

ts w

ere

able

to

fin

d th

e ra

nge

of

mea

sure

s fro

m

a m

easu

rem

ent o

f a g

iven

deg

ree

of

accu

racy

an

d ch

oose

an

ap

prop

riate

uni

t and

the

degr

ee o

f ac

cura

cy fo

r rea

l-life

mea

sure

men

ts.

• S t

uden

ts co

uld

choo

se th

e m

etho

d fro

m th

e gi

ven

optio

ns th

at g

ave

a m

ore

accu

rate

read

ing.

•

Stud

ents

wer

e ab

le

to

find

the

area

s of

sec

tors

and

the

sur

face

ar

eas o

f sph

eres

. •

Stud

ents

coul

d de

mon

strat

e re

cogn

ition

of

com

mon

ter

ms

in

geom

etry

. •

S tud

ents

coul

d id

entif

y th

e re

latio

nshi

p be

twee

n sim

ple

3-D

so

lids a

nd th

eir c

orre

spon

ding

2-D

fig

ures

. •

S tud

ents

coul

d de

term

ine

the

num

ber

of a

xes

of s

ymm

etry

and

lo

cate

the

cent

re o

f rot

atio

n fro

m a

fig

ure.

•

Whe

n th

e ob

ject

and

its

imag

e w

ere

give

n, st

uden

ts co

uld

iden

tify

the

singl

e tra

nsfo

rmat

ion

invo

lved

. •

Stud

ents

coul

d us

e th

e an

gle

prop

ertie

s as

soci

ated

w

ith

inte

rsec

ting

lines

/par

alle

l lin

es a

nd

the

prop

ertie

s of

tria

ngle

s to

sol

ve

simpl

e ge

omet

ric p

robl

ems.

• St

uden

ts ha

d go

od k

now

ledg

e of

th

e re

ctan

gula

r coo

rdin

ate

syste

m.

• St

uden

ts w

ere

able

to

fin

d th

e ra

nge

of

mea

sure

s fro

m

a m

easu

rem

ent o

f a g

iven

deg

ree

of

accu

racy

an

d ch

oose

an

ap

prop

riate

uni

t and

the

degr

ee o

f ac

cura

cy fo

r rea

l-life

mea

sure

men

ts.

• S t

uden

ts w

ere

able

to

fin

d th

e ar

eas o

f sec

tors

and

the

volu

mes

of

pyra

mid

s. •

Stud

ents

wer

e ab

le to

iden

tify

the

rela

tions

hip

betw

een

simpl

e 3-

D

solid

s and

thei

r cor

resp

ondi

ng 2

-D

figur

es.

• St

uden

ts w

ere

able

to

dete

rmin

e th

e or

der

of r

otat

iona

l sy

mm

etry

fro

m a

figu

re.

• W

hen

the

obje

ct a

nd i

ts im

age

wer

e gi

ven,

stu

dent

s w

ere

able

to

iden

tify

the

singl

e tra

nsfo

rmat

ion

invo

lved

. •

S tud

ents

wer

e ab

le to

dem

onstr

ate

reco

gniti

on o

f th

e co

nditi

ons

for

cong

ruen

t and

sim

ilar t

riang

les.

• St

uden

ts w

ere

able

to u

se th

e an

gle

prop

ertie

s as

soci

ated

w

ith

inte

rsec

ting

lines

/par

alle

l lin

es a

nd

the

prop

ertie

s of

tria

ngle

s to

sol

ve

simpl

e ge

omet

ric p

robl

ems.

• S t

uden

ts ha

d go

od k

now

ledg

e of

th

e re

ctan

gula

r coo

rdin

ate

syste

m.

• St

uden

ts di

d w

ell

in

strai

ghtfo

rwar

d qu

estio

ns (

such

as

num

eric

al

calc

ulat

ion

in

simpl

e ge

omet

ric

prob

lem

s).

How

ever

, th

eir p

erfo

rman

ce w

as o

nly

fair

in

item

s inv

olvi

ng m

ore

judg

men

ts.

• S t

uden

ts w

ere

will

ing

to a

ttem

pt

geom

etric

pro

ofs.

How

ever

, th

ey

usua

lly

coul

d no

t us

e lo

gica

l re

ason

ing

and

corre

ct r

easo

ns t

o co

mpl

ete

the

proo

fs.

• S t

uden

ts co

uld

not m

aste

r abs

tract

co

ncep

ts (s

uch

as p

roje

ctio

n of

an

edge

on

a

plan

e,

plan

es

of

refle

ctio

nal s

ymm

etrie

s of c

ubes

). •

I nap

prop

riate

or

in

corre

ct

pres

enta

tion

frequ

ently

oc

curre

d (s

uch

as

conf

used

∠

ABC

with

ABC,

AB

= BC

with

AB

// BC

). •

Ans

wer

s w

ere

ofte

n no

t cor

rect

ed

to th

e re

quire

d de

gree

of a

ccur

acy.

• U

nits

wer

e of

ten

omitt

ed i

n th

e an

swer

. •

Man

y stu

dent

s w

ere

not

fam

iliar

w

ith th

e fo

rmul

as.

S3 MATHEMATICS

374

Y

ear

Mea

sure

s,

S hap

e an

d Sp

ace

2013

20

14

2015

R

emar

ks

Wea

knes

ses

• St

uden

ts’ p

erfo

rman

ce w

as w

eak

in fi

ndin

g th

e ar

ea o

f a se

mi-c

ircle

.

• S t

uden

ts w

ere

wea

k in

abs

tract

co

ncep

ts (s

uch

as

usin

g th

e re

latio

nshi

p of

sim

ilar

figur

es t

o fin

d m

easu

res)

. •

S tud

ents

coul

d no

t de

mon

strat

e re

cogn

ition

of

com

mon

ter

ms

in

geom

etry

. •

Stud

ents

coul

d no

t de

term

ine

whe

ther

a p

olyg

on is

con

vex.

•

Man

y stu

dent

s w

ere

unab

le

to

sket

ch si

mpl

e so

lids.

• St

uden

ts co

uld

not

dem

onstr

ate

reco

gniti

on o

f th

e co

nditi

ons

for

cong

ruen

t and

sim

ilar t

riang

les.

• S t

uden

ts w

ere

wea

k in

iden

tifyi

ng

the

plan

es

of

refle

ctio

nal

sym

met

ries

and

axes

of

rota

tiona

l sy

mm

etrie

s of c

ubes

.

• St

uden

ts w

ere

unab

le

to

disti

ngui

sh

amon

g fo

rmul

as

for

volu

mes

by

co

nsid

erin

g di

men

sions

. •

Stud

ents

coul

d no

t de

term

ine

whe

ther

a p

olyg

on is

regu

lar.

• Q

uite

a n

umbe

r of

stu

dent

s co

uld

not i

dent

ify th

e im

age

of a

fig

ure

afte

r ref

lect

ion.

•

Stud

ents

coul

d no

t de

mon

strat

e re

cogn

ition

of

the

cond

ition

s fo

r co

ngru

ent a

nd si

mila

r tria

ngle

s. •

Stud

ents

wer

e w

eak

in id

entif

ying

th

e pl

anes

of

re

flect

iona

l sy

mm

etrie

s of c

ubes

. •

Man

y stu

dent

s co

uld

not

iden

tify

the

proj

ectio

n of

an

edge

on

a pl

ane.

•

Stud

ents’

per

form

ance

was

onl

y fa

ir in

app

lyin

g th

e co

nditi

ons

for

perp

endi

cula

r lin

es.

• St

uden

ts w

ere

wea

k in

abs

tract

co

ncep

ts (s

uch

as

usin

g re

latio

nshi

p of

sim

ilar

figur

es t

o fin

d m

easu

res)

. •

Man

y stu

dent

s co

uld

not

dete

rmin

e w

heth

er a

pol

ygon

is

equi

angu

lar.

• S t

uden

ts co

uld

not

dem

onstr

ate

reco

gniti

on o

f adj

acen

t ang

les.

• Q

uite

a n

umbe

r of

stu

dent

s co

uld

not

iden

tify

the

angl

e be

twee

n a

line

and

a ho

rizon

tal p

lane

. •

Stud

ents

in

gene

ral

coul

d no

t co

mpl

ete

the

proo

fs

of

simpl

e ge

omet

ric p

robl

ems

rela

ted

with

an

gles

and

line

s. •

Qui

te a

num

ber

of s

tude

nts

coul

d no

t id

entif

y pe

rpen

dicu

lar

bise

ctor

s of a

tria

ngle

.

Y

ear

Dat

a H

andl

ing

2013

20

14

2015

R

emar

ks

Stre

ngth

s

• S t

uden

ts co

uld

orga

nize

the

sam

e se

t of

dat

a by

diff

eren

t gr

oupi

ng

met

hods

. •

Stud

ents’

per

form

ance

was

qui

te

g ood

in

calc

ulat

ing

the

empi

rical

p r

obab

ility

an

d th

e th

eore

tical

pr

obab

ility

by

listin

g.

• S t

uden

ts co

uld

use

simpl

e m

etho

ds

to c

olle

ct d

ata.

• St

uden

ts co

uld

inte

rpre

t sim

ple

statis

tical

cha

rts.

• S t

uden

ts’ p

erfo

rman

ce w

as q

uite

g o

od i

n ca

lcul

atin

g th

e em

piric

al

prob

abili

ty

and

the

theo

retic

al

prob

abili

ty b

y lis

ting.

• S t

uden

ts co

uld

use

simpl

e m

etho

ds

to c

olle

ct d

ata.

• St

uden

ts co

uld

read

inf

orm

atio

n fro

m d

iagr

ams

and

inte

rpre

t th

e in

form

atio

n.

• S t

uden

ts co

uld

choo

se a

ppro

pria

te

diag

ram

s/gra

phs t

o pr

esen

t a se

t of

data

. •

S tud

ents

wer

e ab

le to

cal

cula

te th

e th

eore

tical

pro

babi

lity

by li

sting

.

• M

any

stude

nts d

id n

ot u

se ru

lers

t o

draw

stat

istic

al c

harts

. •

Stud

ents

wer

e w

eak

in re

cogn

izin

g th

e di

scre

te a

nd c

ontin

uous

dat

a.

Wea

knes

ses

• Q

uite

a n

umbe

r of

stu

dent

s co

uld

not

disti

ngui

sh b

etw

een

disc

rete

an

d co

ntin

uous

dat

a. •

Stud

ents

in

gene

ral

coul

d no

t c h

oose

ap

prop

riate

di

agra

ms

/ gra

phs t

o pr

esen

t a se

t of d

ata

and

com

pare

the

pre

sent

atio

ns o

f th

e sa

me

set o

f dat

a by

usin

g sta

tistic

al

char

ts.

• Q

uite

a n

umbe

r of

stu

dent

s w

ere

not a

ble

to fi

nd a

vera

ges f

rom

a se

t of

gro

uped

dat

a.

• S t

uden

ts’ p

erfo

rman

ce w

as o

nly

fair

in d

istin

guish

ing

disc

rete

and

co

ntin

uous

dat

a. •

Stud

ents

in

gene

ral

coul

d no

t c h

oose

ap

prop

riate

di

agra

ms

/ gra

phs t

o pr

esen

t a se

t of d

ata.

• Q

uite

a n

umbe

r of

stu

dent

s w

ere

not a

ble

to fi

nd a

vera

ges f

rom

a se

t of

gro

uped

dat

a.

• S t

uden

ts’ p

erfo

rman

ce w

as o

nly

fair

in d

istin

guish

ing

disc

rete

and

co

ntin

uous

dat

a. •

Stud

ents

in

gene

ral

coul

d no

t c o

nstru

ct s

tem

-and

-l eaf

dia

gram

s co

rrect

ly.

• M

any

stude

nts

coul

d no

t com

pare

th

e pr

esen

tatio

ns o

f the

sam

e se

t of

data

by

usin

g sta

tistic

al c

harts

. •

Qui

te a

num

ber

of s

tude

nts

wer

e no

t abl

e to

find

ave

rage

s fro

m a

set

of g

roup

ed d

ata.

S3MATHEMATICS

375

Y

ear

Dat

a H

andl

ing

2013

20

14

2015

R

emar

ks

Stre

ngth

s

• St

uden

ts co

uld

orga

nize

the

sam

e se

t of

dat

a by

diff

eren

t gr

oupi

ng

met

hods

. •

S tud

ents’

per

form

ance

was

qui

te

good

in

calc

ulat

ing

the

empi

rical

pr

obab

ility

an

d th

e th

eore

tical

pr

obab

ility

by

listin

g.

• St

uden

ts co

uld

use

simpl

e m

etho

ds

to c

olle

ct d

ata.

• S t

uden

ts co

uld

inte

rpre

t sim

ple

statis

tical

cha

rts.

• St

uden

ts’ p

erfo

rman

ce w

as q

uite

go

od i

n ca

lcul

atin

g th

e em

piric

al

prob

abili

ty

and

the

theo

retic

al

prob

abili

ty b

y lis

ting.

• St

uden

ts co

uld

use

simpl

e m

etho

ds

to c

olle

ct d

ata.

• S t

uden

ts co

uld

read

inf

orm

atio

n fro

m d

iagr

ams

and

inte

rpre

t th

e in

form

atio

n.

• St

uden

ts co

uld

choo

se a

ppro

pria

te

diag

ram

s/gra

phs t

o pr

esen

t a se

t of

data

. •

S tud

ents

wer

e ab

le to

cal

cula

te th

e th

eore

tical

pro

babi

lity

by li

sting

.

• M

any

stude

nts d

id n

ot u

se ru

lers

to

draw

stat

istic

al c

harts

. •

S tud

ents

wer

e w

eak

in re

cogn

izin

g th

e di

scre

te a

nd c

ontin

uous

dat

a.

Wea

knes

ses

• Q

uite

a n

umbe

r of

stu

dent

s co

uld

not

disti

ngui

sh b

etw

een

disc

rete

an

d co

ntin

uous

dat

a. •

S tud

ents

in

gene

ral

coul

d no

t ch

oose

ap

prop

riate

di

agra

ms

/gra

phs t

o pr

esen

t a se

t of d

ata

and

com

pare

the

pre

sent

atio

ns o

f th

e sa

me

set o

f dat

a by

usin

g sta

tistic

al

char

ts.

• Q

uite

a n

umbe

r of

stu

dent

s w

ere

not a

ble

to fi

nd a

vera

ges f

rom

a se

t of

gro

uped

dat

a.

• St

uden

ts’ p

erfo

rman

ce w

as o

nly

fair

in d

istin

guish

ing

disc

rete

and

co

ntin

uous

dat

a. •

S tud

ents

in

gene

ral

coul

d no

t ch

oose

ap

prop

riate

di

agra

ms

/gra

phs t

o pr

esen

t a se

t of d

ata.

• Q

uite

a n

umbe

r of

stu

dent

s w

ere

not a

ble

to fi

nd a

vera

ges f

rom

a se

t of

gro

uped

dat

a.

• St

uden

ts’ p

erfo

rman

ce w

as o

nly

fair

in d

istin

guish

ing

disc

rete

and

co

ntin

uous

dat

a. •

S tud

ents

in

gene

ral

coul

d no

t co

nstru

ct s

tem

-and

-leaf

dia

gram

s co

rrect

ly.

• M

any

stude

nts

coul

d no

t com

pare

th

e pr

esen

tatio

ns o

f the

sam

e se

t of

data

by

usin

g sta

tistic

al c

harts

. •

Qui

te a

num

ber

of s

tude

nts

wer

e no

t abl

e to

find

ave

rage

s fro

m a

set

of g

roup

ed d

ata.

S3 MATHEMATICS

376

Comparison of Student Performances in Mathematics at Primary 3, Primary 6 and Secondary 3 TSA 2015

The percentages of P.3, P.6 and S.3 students achieving Basic Competency from 2004 to

2015 are as follows:

Table 8.11 Percentages of Students Achieving Mathematics Basic Competency Year

Level

% of Students Achieving Mathematics BC

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

P.3 84.9 86.8 86.9 86.9 86.9 # 87.0 87.0 87.3 87.5 87.4 87.6

P.6 -- 83.0 83.8 83.8 84.1 # 84.2 84.1 ^ 84.2 ^ 84.0

S.3 -- -- 78.4 79.9 79.8 80.0 80.1 80.1 79.8 79.7 79.9 79.9 # Due to Human Swine Influenza causing the suspension of primary schools, the TSA was cancelled and no data has been provided.

^ The 2012 and 2014 P.6 TSA were suspended. As participation in the 2012 and 2014 P.6 TSA was on a voluntary basis, not all P.6 students were involved and hence no territory-wide data is provided in this report.

A comparison of strengths and weaknesses of P.3, P.6, and S.3 students in TSA enables

teachers to devise teaching strategies and tailor curriculum planning at different key stages

to adapt to students’ needs. The dimensions of Mathematics Curriculum at each key stage

belong to different dimensions as shown below:

Table 8.12 Dimensions of Mathematics Curriculum for Primary 3, Primary 6 and Secondary 3

Primary 3 Primary 6 Secondary 3

Dimension

Number Number

Number and Algebra Algebra

Measures Measures Measures, Shape and Space Shape and Space Shape and Space

Data Handling Data Handling Data Handling

The following table compares students’ performances at P.3, P.6 and S.3 in Mathematics

TSA 2015:

Tabl

e 8.

13

Com

pari

son

of S

tude

nt P

erfo

rman

ces i

n M

athe

mat

ics a

t Pri

mar

y 3,

Pri

mar

y 6

and

Seco

ndar

y 3

TSA

2015

L

evel

D

i men

sion

P.

3 P.

6 S.

3

Num

ber

• M

ost P

.3 st

uden

ts w

ere

capa

ble

of re

cogn

izin

g th

e pl

ace

valu

es in

who

le n

umbe

rs a

nd th

e va

lue

of

the

digi

t. •

T he

maj

ority

of

the

P.3

stude

nts

c oul

d pe

rform

a r

ithm

etic

cal

cula

tions

with

num

bers

up

to 3

di

gits.

•

P .3

stude

nts

wer

e ca

pabl

e of

com

parin

g fra

ctio

ns

a nd

reco

gniz

ing

t he

rela

tions

hip

betw

een

fract

ions

a n

d th

e w

hole

. So

me

stude

nts

coul

d no

t fu

lly

unde

rsta

nd th

e ba

sic c

once

pt o

f a fr

actio

n as

par

ts of

one

who

le.

• P .

3 stu

dent

s w

ere

capa

ble

of

s olv

ing

simpl

e a p

plic

atio

n pr

oble

ms

invo

lvin

g m

ixed

ope

ratio

ns

by

pres

entin

g cl

ear

wor

king

ste

ps

and

expl

anat

ions

, bu

t th

ey s

omet

imes

mis i

nter

pret

ed

ques

tions

due

to c

arel

essn

ess.

•

Man

y stu

dent

s w

ere

able

to

solv

e a p

plic

atio

n pr

oble

ms i

nvol

ving

the

calc

ulat

ion

of m

oney

.

• P.

6 stu

dent

s wer

e ca

pabl

e of

reco

gniz

ing

the

plac

e va

lues

in w

hole

num

bers

and

dec

imal

s.

• P .

6 stu

dent

s co

uld

perfo

rm a

rithm

etic

ope

ratio

ns

o n

who

le

num

bers

, fra

ctio

ns

and

deci

mal

s, i n

clud

ing

smal

l bra

cket

s. H

owev

er, s

ome

stude

nts

forg

ot

the

com

puta

tiona

l ru

le

of

‘per

form

ing

mul

tiplic

atio

n/di

visio

n be

fore

addi

tion/

subt

ract

ion’

. •

P .6

stude

nts u

nder

stood

the

conc

ept o

f fra

ctio

ns a

s pa

rts o

f one

who

le a

nd c

ould

com

pare

frac

tions

. •

P.6

stude

nts

wer

e ca

pabl

e of

cho

osin

g a

suita

ble

mat

hem

atic

al e

xpre

ssio

n fo

r esti

mat

ing

the

valu

e.

• P .

6 stu

dent

s co

uld

solv

e ap

plic

atio

n pr

oble

ms

with

cl

ear

wor

king

ste

ps

a nd

expl

anat

ions

. H

owev

er,

som

e stu

dent

s w

ere

wea

k in

sol

ving

ap

plic

atio

n pr

oble

ms

invo

lvin

g fra

ctio

ns

or

unfa

mili

ar c

onte

xts.

• St

uden

ts co

uld

reco

gniz

e th

e co

ncep

t of d

irect

ed

num

bers

and

thei

r ope

ratio

ns.

• St

uden

ts co

uld

use

rate

an

d r a

tio

to

solv

e pr

oble

ms.

• S t

uden

ts co

uld

jud g

e th

e re

ason

able

ness

of

an

swer

s fro

m c

ompu

tatio

ns.

How

ever

, so

me

of

t hem

us

ed

the

inco

rrect

m

etho

ds

for

the

e stim

atio

n.

Thei

r ex

plan

atio

ns

wer

e al

so

inco

mpl

ete

whe

n th

ey g

ave

justi

ficat

ions

. •

Stud

ents

did

wel

l in

usin

g pe

rcen

tage

s to

sol

ve

simpl

e ap

plic

atio

n pr

oble

ms.

Som

e stu

dent

s co

nfus

ed t

he f

orm

ula

of f

indi

ng s

impl

e in

tere

st w

ith th

at o

f com

poun

d in

tere

st.

Alg

ebra

N

. A.

• P .

6 stu

dent

s w

ere

capa

ble

of u

sing

sym

bols

to

r epr

esen

t num

bers

. •

P.6

stude

nts

wer

e ca

pabl

e of

sol

ving

equ

atio

ns

invo

lvin

g at

mos

t tw

o ste

ps in

the

solu

tions

. •

P .6

stude

nts

wer

e ca

pabl

e of

sol

ving

pro

blem

s by

us in

g sim

ple

equa

tions

.

• S t

uden

ts co

uld

form

ulat

e eq

uatio

ns fr

om s

impl

e c o

ntex

ts an

d so

lve

equa

tions

. •

Stud

ents

coul

d ob

serv

e th

e pa

ttern

of t

he n

umbe

r se

quen

ces

and

wro

te d

own

the

next

few

term

s, b u

t the

y w

ere

not a

ble

to in

tuiti

vely

find

the

nth

term

of a

num

ber s

eque

nce.

• St

uden

ts’

perfo

rman

ce

was

fa

ir in

ad

ditio

n,

subt

ract

ion,

ex

pans

ion

and

fact

oriz

atio

n of

po

lyno

mia

ls.

• St

uden

ts pe

rform

ed

satis

fact

orily

w

hen

they

a p

plie

d th

e la

ws

of

inte

gral

in

dice

s an

d pr

oper

ties o

f ine

qual

ities

to so

lve

prob

lem

s.

S3MATHEMATICS

377

Comparison of Student Performances in Mathematics at Primary 3, Primary 6 and Secondary 3 TSA 2015

The percentages of P.3, P.6 and S.3 students achieving Basic Competency from 2004 to

2015 are as follows:

Table 8.11 Percentages of Students Achieving Mathematics Basic Competency Year

Level

% of Students Achieving Mathematics BC

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

P.3 84.9 86.8 86.9 86.9 86.9 # 87.0 87.0 87.3 87.5 87.4 87.6

P.6 -- 83.0 83.8 83.8 84.1 # 84.2 84.1 ^ 84.2 ^ 84.0

S.3 -- -- 78.4 79.9 79.8 80.0 80.1 80.1 79.8 79.7 79.9 79.9 # Due to Human Swine Influenza causing the suspension of primary schools, the TSA was cancelled and no data has been provided.

^ The 2012 and 2014 P.6 TSA were suspended. As participation in the 2012 and 2014 P.6 TSA was on a voluntary basis, not all P.6 students were involved and hence no territory-wide data is provided in this report.

A comparison of strengths and weaknesses of P.3, P.6, and S.3 students in TSA enables

teachers to devise teaching strategies and tailor curriculum planning at different key stages

to adapt to students’ needs. The dimensions of Mathematics Curriculum at each key stage

belong to different dimensions as shown below:

Table 8.12 Dimensions of Mathematics Curriculum for Primary 3, Primary 6 and Secondary 3

Primary 3 Primary 6 Secondary 3

Dimension

Number Number

Number and Algebra Algebra

Measures Measures Measures, Shape and Space Shape and Space Shape and Space

Data Handling Data Handling Data Handling

The following table compares students’ performances at P.3, P.6 and S.3 in Mathematics

TSA 2015:

Tabl

e 8.

13

Com

pari

son

of S

tude

nt P

erfo

rman

ces i

n M

athe

mat

ics a

t Pri

mar

y 3,

Pri

mar

y 6

and

Seco

ndar

y 3

TSA

2015

L

evel

D

imen

sion

P.

3 P.

6 S.

3

Num

ber

• M

o st P

.3 st

uden

ts w

ere

capa

ble

of re

cogn

izin

g th

e pl

ace

valu

es in

who

le n

umbe

rs a

nd th

e va

lue

of

the

digi

t. •

The

maj

ority

of

the

P.3

stude

nts

coul

d pe

rform

ar

ithm

etic

cal

cula

tions

with

num

bers

up

to 3

d i

gits.

•

P .3

stude

nts

wer

e ca

pabl

e of

com

parin

g fra

ctio

ns

and

reco

gniz

ing

the

rela

tions

hip

betw

een

fract

ions

an

d th

e w

hole

. So

me

stude

nts

coul

d no

t fu

lly

unde

rsta

nd th

e ba

sic c

once

pt o

f a fr

actio

n as

par

ts of

one

who

le.

• P .

3 stu

dent

s w

ere

capa

ble

of

solv

ing

simpl

e ap

plic

atio

n pr

oble

ms

invo

lvin

g m

ixed

ope

ratio

ns

by

pres

entin

g cl

ear

wor

king

ste

ps

and

expl

anat

ions

, bu

t th

ey s

omet

imes

misi

nter

pret

ed

ques

tions

due

to c

arel

essn

ess.

•

Man

y stu

dent

s w

ere

able

to

solv

e ap

plic

atio

n pr

oble

ms i

nvol

ving

the

calc

ulat

ion

of m

oney

.

• P .

6 stu

dent

s wer

e ca

pabl

e of

reco

gniz

ing

the

plac

e va

lues

in w

hole

num

bers

and

dec

imal

s.

• P .

6 stu

dent

s co

uld

perfo

rm a

rithm

etic

ope

ratio

ns

on

who

le

num

bers

, fra

ctio

ns

and

deci

mal

s, in

clud

ing

smal

l bra

cket

s. H

owev

er, s

ome

stude

nts

forg

ot

the

com

puta

tiona

l ru

le

of

‘per

form

ing

mul

tiplic

atio

n/di

visio

n be

fore

addi

tion/

subt

ract

ion’

. •

P .6

stude

nts u

nder

stood

the

conc

ept o

f fra

ctio

ns a

s pa

rts o

f one

who

le a

nd c

ould

com

pare

frac

tions

. •

P.6

stude

nts

wer

e ca

pabl

e of

cho

osin

g a

suita

ble

mat

hem

atic

al e

xpre

ssio

n fo

r esti

mat

ing

the

valu

e.

• P .

6 stu

dent

s co

uld

solv

e ap

plic

atio

n pr

oble

ms

with

cl

ear

wor

king

ste

ps

and

expl

anat

ions

. H

owev

er,

som

e stu

dent

s w

ere

wea

k in

sol

ving

ap

plic

atio

n pr

oble

ms

invo

lvin

g fra

ctio

ns

or

unfa

mili

ar c

onte

xts.

• St

uden

ts co

uld

reco

gniz

e th

e co

ncep

t of d

irect

ed

num

bers

and

thei

r ope

ratio

ns.

• St

uden

ts co

uld

use

rate

an

d ra

tio

to

solv

e pr

oble

ms.

• St

uden

ts co

uld

j udg

e th

e re

ason

able

ness

of

an

swer

s fro

m c

ompu

tatio

ns.

How

ever

, s o

me

of

t hem

us

ed

the

inco

rrect

m

etho

ds

for

the

estim

atio

n.

Thei

r ex

plan

atio

ns

wer

e al

so

inco

mpl

ete

whe

n th

ey g

ave

justi

ficat

ions

. •

Stud

ents

did

wel

l in

usin

g pe

rcen

tage

s to

sol

ve

simpl

e ap

plic

atio

n pr

oble

ms.

Som

e stu

dent

s co

nfus

ed t

he f

orm

ula

of f

indi

ng s

impl

e in

tere

st w

ith th

at o

f com

poun

d in

tere

st.

Alg

ebra

N

.A.

• P.

6 stu

dent

s w

ere

capa

ble

of u

sing

sym

bols

to

repr

esen

t num

bers

. •

P.6

stude

nts

wer

e ca

pabl

e of

sol

ving

equ

atio

ns

invo

lvin

g at

mos

t tw

o ste

ps in

the

solu

tions

. •

P .6

stude

nts

wer

e ca

pabl

e of

sol

ving

pro

blem

s by

usin

g sim

ple

equa

tions

.

• St

uden

ts co

uld

form

ulat

e eq

uatio

ns fr

om s

impl

e co

ntex

ts an

d so

lve

equa

tions

. •

Stud

ents

coul

d ob

serv

e th

e pa

ttern

of t

he n

umbe

r se

quen

ces

and

wro

te d

own

the

next

few

term

s, bu

t the

y w

ere

not a

ble

to in

tuiti

vely

find

the

nth

term

of a

num

ber s

eque

nce.

• St

uden

ts’

perfo

rman

ce

was

fa

ir in

ad

ditio

n,

subt

ract

ion,

ex

pans

ion

and

fact

oriz

atio

n of

po

lyno

mia

ls.

• St

uden

ts pe

rform

ed

satis

fact

orily

w

hen

they

ap

plie

d th

e la

ws

of

inte

gral

in

dice

s an

d pr

oper

ties o

f ine

qual

ities

to so

lve

prob

lem

s.

S3 MATHEMATICS

378

L

evel

D

imen

sion

P.

3 P.

6 S.

3

Mea

sure

s •

T he

maj

ority

of

P.3

stude

nts

coul

d fin

d th

e co

rrect

dat

es a

nd d

ays o

f a w

eek

How

ever

, som

e of

them

faile

d to

find

the

num

ber o

f day

s of

an

activ

ity f

rom

the

cal

enda

r, ac

cord

ing

to g

iven

co

nditi

ons.

• M

ost

stude

nts

wer

e ca

pabl

e of

tel

ling

the

time

on a

clo

ck f

ace

and

a di

gita

l cl

ock.

M

any

stude

nts

coul

d us

e th

e ‘2

4-ho

ur

time’

to

expr

ess t

he ti

me

appr

opria

tely

. •

For

the

leng

th o

f ob

ject

s, m

ost

stude

nts

wer

e ca

pabl

e of

usin

g su

itabl

e to

ols

to m

easu

re a

nd

appr

opria

te u

nits

to re

cord

.

• Th

e m

ajor

ity

of

stude

nts

wer

e ca

pabl

e of

m

easu

ring

the

wei

ght o

f obj

ects

usin

g ‘g

ram

’ (g)

or

‘ki

logr

am’

(kg)

, an

d sh

owed

im

prov

emen

t w

hen

com

parin

g w

eigh

ts of

obj

ects.

•

T he

maj

ority

of

stude

nts

coul

d m

easu

re a

nd

com

pare

the

cap

acity

of

cont

aine

rs,

but

few

co

uld

not u

se a

ppro

pria

te u

nits.

• P.

6 stu

dent

s co

uld

writ

e th

e co

rrect

dat

es a

nd

appl

y th

e ‘2

4-ho

ur ti

me’

. •

P.6

stude

nts

wer

e ca

pabl

e of

re

cord

ing

the

leng

th,

wei

ght

and

capa

city

with

app

ropr

iate

un

its.

• P.

6 stu

dent

s w

ere

capa

ble

of m

easu

ring

and

com

parin

g th

e ca

paci

ty o

f con

tain

ers.

• P.

6 stu

dent

s co

uld

calc

ulat

e th

e pe

rimet

ers

and

area

s of s

impl

e 2-

D sh

apes

as w

ell a

s the

vol

ume

of so

lids.

• P .

6 stu

dent

s w

ere

capa

ble

of

appl

ying

th

e fo

rmul

a of

circ

umfe

renc

e.

• P.

6 stu

dent

s cou

ld so

lve

simpl

e pr

oble

ms r

elat

ed

to sp

eed.

• St

uden

ts co

uld

choo

se a

n ap

prop

riate

uni

t an

d th

e de

gree

of

ac

cura

cy

for

real

-life

m

easu

rem

ents,

how

ever

, th

ey w

ere

not

able

to

choo

se th

e ap

prop

riate

mea

surin

g te

chni

que.

•

Stud

ents

coul

d ca

lcul

ate

arc

leng

ths,

area

s of

se

ctor

s, vo

lum

es o

f sp

here

s an

d cu

rved

sur

face

ar

eas o

f con

es.

•

Perfo

rman

ce

of

stude

nts

was

qu

ite

good

in

ca

lcul

atin

g ci

rcum

fere

nces

and

are

as o

f ci

rcle

s. Th

ey c

ould

also

fin

d th

e to

tal

surfa

ce a

reas

of

trian

gula

r pris

ms a

nd v

olum

es o

f cyl

inde

rs.

• Q

uite

a

num

ber

of

stude

nts

coul

d es

timat

e m

easu

res a

nd g

ive

reas

onab

le e

xpla

natio

ns.

• St

uden

ts w

ere

wea

k in

abs

tract

con

cept

s (su

ch a

s us

ing

rela

tions

hip

of s

imila

r fig

ures

to

find

mea

sure

s).

L

evel

D

i men

sion

P.

3 P.

6 S.

3

Shap

e an

d Sp

ace

• P.

3 stu

dent

s co

uld

reco

gniz

e co

nes/p

yram

ids,

cylin

ders

/pris

ms a

nd sp

here

s. •

P.3

stude

nts w

ere

capa

ble

of id

entif

ying

diff

eren

t 2-

D sh

apes

. •

P.3

stude

nts

wer

e ca

pabl

e of

iden

tifyi

ng s

traig

ht

lines

, cu

rves

, pa

ralle

l lin

es a

nd p

erpe

ndic

ular

lin

es.

•

P.3

stude

nts

coul

d re

cogn

ize

right

ang

les

and

com

pare

the

size

of a

ngle

s. •

P.3

stude

nts

coul

d re

cogn

ize

the

four

dire

ctio

ns,

nam

ely

north

, eas

t, so

uth

and

wes

t.

• P.

6 stu

dent

s co

uld

reco

gniz

e th

e ch

arac

teris

tics

of

cone

s, py

ram

ids,

cylin

ders

, pr

isms

and

s phe

res.

• P .

6 stu

dent

s w

ere

capa

ble

of i

dent

ifyin

g 2-

D

shap

es in

diff

eren

t orie

ntat

ions

. •

P.6

stude

nts

coul

d re

cogn

ize

the

eigh

t m

ain

dire

ctio

ns a

nd a

pply

the

know

ledg

e of

dire

ctio

n to

solv

e pr

oble

ms.

• St

uden

ts co

uld

not

dem

onstr

ate

reco

gniti

on o

f so

me

com

mon

ter

ms

in g

eom

etry

(e.

g. r

egul

ar

poly

gons

, adj

acen

t ang

les)

. •

S tud

ents

coul

d id

entif

y th

e re

latio

n be

twee

n s im

ple

3-D

sol

ids

and

thei

r co

rresp

ondi

ng 2

-D

figur

es. T

hey

coul

d al

so sk

etch

sim

ple

solid

s. •

Stud

ents

coul

d de

al w

ith s

impl

e sy

mm

etry

and

tra

nsfo

rmat

ion.

•

S om

e stu

dent

s co

nfus

ed

the

r eas

ons

for

cong

ruen

t tri

angl

es

with

th

at

for

simila

r tri

angl

es.

• S t

uden

ts ha

d go

od k

now

ledg

e of

the

rect

angu

lar

coor

dina

te s

yste

m.

How

ever

, th

eir p

erfo

rman

ce

was

fai

r on

ly w

hen

they

had

to

find

area

s of

sim

ple

figur

es.

• St

uden

ts pe

rform

ed

satis

fact

orily

in

so

lvin

g pr

oble

ms b

y us

ing

trigo

nom

etry

. •

Som

e stu

dent

s co

uld

not

iden

tify

t he

angl

e be

twee

n a

line

and

a pl

ane.

• St

uden

ts pe

rform

ed q

uite

wel

l in

writ

ing

proo

fs

by

usin

g th

e co

nditi

ons

for

cong

ruen

t an

d s im

ilar

trian

gles

. W

hen

angl

es a

nd l

ines

wer

e re

late

d,

man

y stu

dent

s co

uld

not

corre

ctly

pr

ovid

e re

ason

s or c

ompl

ete

the

proo

fs.

S3MATHEMATICS

379

L

evel

D

imen

sion

P.

3 P.

6 S.

3

Shap

e an

d Sp

ace

• P.

3 s tu

dent

s co

uld

reco

gniz

e co

nes/p

yram

ids,

cylin

ders

/pris

ms a

nd sp

here

s. •

P.3

stude

nts w

ere

capa

ble

of id

entif

ying

diff

eren

t 2-

D sh

apes

. •

P.3

stude

nts

wer

e ca

pabl

e of

iden

tifyi

ng s

traig

ht

lines

, cu

rves

, pa

ralle

l lin

es a

nd p

erpe

ndic

ular

lin

es.

•

P.3

stude

nts

coul

d re

cogn

ize

right

ang

les

and

com

pare

the

size

of a

ngle

s. •

P.3

s tude

nts

coul

d re

cogn

ize

the

four

dire

ctio

ns,

nam

ely

north

, eas

t, so

uth

and

wes

t.

• P .

6 stu

dent

s co

uld

reco

gniz

e th

e ch

arac

teris

tics

of

cone

s, py

ram

ids,

cylin

ders

, pr

isms

and

sphe

res.

• P.

6 stu

dent

s w

ere

capa

ble

of i

dent

ifyin

g 2-

D

shap

es in

diff

eren

t orie

ntat

ions

. •

P .6

stude

nts

coul

d re

cogn

ize

the

eigh

t m

ain

dire

ctio

ns a

nd a

pply

the

know

ledg

e of

dire

ctio

n to

solv

e pr

oble

ms.

• S t

uden

ts co

uld

not

dem

onstr

ate

reco

gniti

on o

f so

me

com

mon

ter

ms

in g

eom

etry

(e.

g. r

egul

ar

poly

gons

, adj

acen

t ang

les)

. •

Stud

ents

coul

d id

entif

y th

e re

latio

n be

twee

n sim

ple

3-D

sol

ids

and

thei

r co

rresp

ondi

ng 2

-D

figur

es. T

hey

coul

d al

so sk

etch

sim

ple

solid

s. •

S tud

ents

coul

d de

al w

ith s

impl

e sy

mm

etry

and

tra

nsfo

rmat

ion.

•

Som

e stu

dent

s co

nfus

ed

the

reas

ons

for

cong

ruen

t tri

angl

es

with

th

at

for

simila

r tri

angl

es.

• S t

uden

ts ha

d go

od k

now

ledg

e of

the

rect

angu

lar

coor

dina

te s

yste

m.

How

ever

, th

eir p

erfo

rman

ce

was

fai

r on

ly w

hen

they

had

to

find

area

s of

sim

ple

figur

es.

• St

uden

ts pe

rform

ed

satis

fact

orily

in

so

lvin

g pr

oble

ms b

y us

ing

trigo

nom

etry

. •

Som

e stu

dent

s co

uld

not

iden

tify

the

angl

e be

twee

n a

line

and

a pl

ane.

• S t

uden

ts pe

rform

ed q

uite

wel

l in

writ

ing

proo

fs

by

usin

g th

e co

nditi

ons

for

cong

ruen

t an

d sim

ilar

trian

gles

. W

hen

angl

es a

nd l

ines

wer

e re

late

d,

man

y stu

dent

s co

uld

not

corre

ctly

pr

ovid

e re

ason

s or c

ompl

ete

the

proo

fs.

S3 MATHEMATICS

380

L

evel

D

imen

sion

P .

3 P .

6 S .

3

Dat

a H

andl

ing

• P .

3 stu

dent

s pe

rform

ed w

ell

at r

eadi

ng s

impl

e pi

ctog

ram

s w

ith o

ne-to

-one

repr

esen

tatio

n. T

hey

coul

d an

swer

str

aigh

tforw

ard

ques

tions

by

re

triev

ing

data

giv

en in

the

pict

ogra

m.

• M

any

stude

nts

wer

e ab

le t

o in

terp

ret

the

data

gi

ven

in

pict

ogra

ms

whe

n an

swer

ing

open

-end

ed q

uesti

ons.

• P.

3 stu

dent

s co

uld

cons

truct

pic

togr

ams

usin

g on

e-to

-one

rep

rese

ntat

ion

and

give

a s

uita

ble

title

. •

A s m

all

prop

ortio

n of

P.3

stu

dent

s m

istak

enly

ad

ded

a ‘fr

eque

ncy

axis’

to a

pic

togr

am.

• P .

6 stu

dent

s pe

rform

ed

wel

l at

re

adin

g pi

ctog

ram

s an

d ba

r ch

arts,

inc

ludi

ng t

hose

of

grea

ter f

requ

ency

cou

nts.

•

P.6

stude

nts

coul

d ex

tract

th

e re

leva

nt

info

rmat

ion

from

giv

en st

atist

ical

gra

phs.

• P.

6 stu

dent

s co

uld

cons

truct

sta

tistic

al g

raph

s fro

m t

abul

ated

dat

a, th

ough

a s

mal

l nu

mbe

r of

th

em c

onfu

sed

pict

ogra

ms

with

bar

cha

rts o

r m

istak

enly

ad

ded

a ‘fr

eque

ncy

axis’

to

a

pict

ogra

m.

• P .

6 stu

dent

s wer

e ca

pabl

e of

find

ing

the

aver

age

of a

gro

up o

f dat

a an

d so

lvin

g sim

ple

prob

lem

s of

ave

rage

s.

• S t

uden

ts re

cogn

ized

the

bas

ic p

roce

dure

s of

sta

tistic

al w

ork

and

the

data

col

lect

ion

met

hods

.

• So

me

stude

nts c

ould

not

dist

ingu

ish d

iscre

te a

nd

cont

inuo

us d

ata.

• S t

uden

ts co

uld

inte

rpre

t sim

ple

statis

tical

cha

rts

and

choo

se a

ppro

pria

te d

iagr

ams

/ gr

aphs

to

pres

ent a

set o

f dat

a. •

Som

e stu

dent

s cou

ld n

ot c

onstr

uct s

tem

-and

-leaf

di

agra

ms

and

com

pare

the

pre

sent

atio

ns o

f th

e sa

me

set o

f dat

a by

usin

g sta

tistic

al c

harts

. •

S om

e stu

dent

s co

uld

not

iden

tify

sour

ces

of

dece

ptio

n in

misl

eadi

ng g

raph

s or

in

case

s of

m

isuse

of a

vera

ges.

• S o

me

stude

nts

coul

d no

t ca

lcul

ate

mea

n fro

m

grou

ped

data

, bu

t th

ey p

erfo

rmed

wel

l in

the

ca

ses o

f ung

roup

ed d

ata.

•

Stud

ents

coul

d ca

lcul

ate

the

theo

retic

al

prob

abili

ty b

y lis

ting.

How

ever

, per

form

ance

in

calc

ulat

ing

the

empi

rical

pro

babi

lity

was

fair.

secondary 3 assessment design - hkeaa

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