HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
One Third
Marking Guidelines
教育局 課程發展處 數學教育組
Mathematics Education Section, Curriculum Development Institute
The Education Bureau of the HKSAR
Assessment Scale
The assessment scale for tasks on Problem-solving is shown in the following table.
Level of
performance
Very good
Good
Fair
Weak
Marks
Mathematical Knowledge and Problem-solving Skills
13 – 16
The student demonstrates a complete understanding of
the underlying mathematical knowledge and
problem-solving skills which are relevant to the task,
and is consistently competent and accurate in applying
them in handling the task. Typically, the student is able
to formulate a correct strategy, carry out the strategy
and demonstrate a complete understanding of the
significance and possible limitations of the results
obtained.
9 – 12
The student demonstrates a substantial understanding
of the underlying mathematical knowledge and
problem-solving skills which are relevant to the task,
and is generally competent and accurate in applying
them in handling the task. Typically, the student is able
to formulate a correct strategy and attempts to carry out
the strategy for completing the task.
5–8
The student demonstrates a basic understanding of the
underlying
mathematical
knowledge
and
problem-solving skills which are relevant to the task,
and is occasionally competent and accurate in applying
them in handling the task. Typically, the student has a
basic understanding of the task and is able to formulate
a correct strategy for solving the task.
1–4
The student demonstrates a limited understanding of
the underlying mathematical knowledge and
problem-solving skills which are relevant to the task,
and is rarely competent and accurate in applying them
in handling the task. Typically, the student has a bare
understanding of the task and can only attempt to
complete the simplest part of the task.
Marks
Mathematical Communication Skills
4
The student communicates ideas in a
clear, well organised and logically true
manner
through
coherent
written/verbal
accounts,
using
appropriate and correct mathematical
presentation to express, interpret and
critically review the results obtained.
3
The student is able to communicate
ideas properly through written/verbal
accounts, using appropriate forms of
mathematical presentation such as
mathematical formulae or geometric
facts.
2
The student is able to communicate
basic ideas with limited success in
using appropriate mathematical terms
and terminology.
1
The student attempts to communicate
ideas using some basic forms of
mathematical presentation such as
symbols, notations, diagrams, tables,
graphs etc., but has little success in
doing so.

The full mark of a SBA task on Problem-solving submitted should be scaled to 20 marks, of which 16 marks are awarded for the
mathematical knowledge and problem-solving skills while 4 marks are awarded for the mathematical communication skills.

Teachers should base on the above assessment scale to design SBA tasks for assessing students with different abilities and to
develop the marking guidelines.
One Third_Marking Guidelines
2
Marking Guidelines
Solution
Performance
Part A
1.
As in the figure below, O is the centre of the cake. Cut the cake into 3 pieces along OA,
OB and OC respectively where the angle subtended at centre of each piece is 120.
Evidence:
Method of cutting the cake
Weak:
Attempt to find a method
but the method is wrong
Fair:
Attempt to find a method
but the method is not clearly
spelt out
Good:
Clearly state the method of
cutting the cake, but
insufficient numerical
information is supplied
Very Good: Give clear and complete
method with all relevant
numerical information
provided
2.
The radii and the angles subtended at centre of the 3 pieces of cake are equal.
Therefore, the areas of the 3 pieces of cake are equal.
(Alternative Solution)
Evidence:
Proof for “one-third”
Weak:
The method of proving
the assertion is wrong
Fair:
The method of proving
the assertion is partly
correct
Good:
The method of proving
the assertion is correct
Let r be the radius of the circular cake.
120 
 r 2
360 
1
= r 2
3
1
The ratio of each piece to the whole cake = r 2 : r 2
3
The area of each piece of the cake
=
= 1: 3
Very Good: The method of proving
the assertion is correct and
the presentation is clear
and concise
 Each piece is exactly one-third of the whole cake.
One Third_Marking Guidelines
3
Marking Guidelines
Solution
Performance
Part B
1.
As in the figure below, E and F are points on BC and CD respectively on the two
sides of square such that DF : FC  2 : 1 and BE : EC  2 : 1.
Then, cut the cake along the lines AE and AF.
Evidence:
Method of cutting the cake
Weak:
Attempt to find a method
but the method is wrong
Fair:
Attempt to find a method
but the method is not clearly
spelt out
Good:
Clearly state the method of
cutting the cake, but
insufficient numerical
information is supplied
Very Good: Give clear and complete
method with all relevant
numerical information
provided
2
Let x be the length of AB.
Hence, the lengths of BE and DF are
2
1
x , and that of EC and FC are
x.
3
3
1 2 
x x 
2 3 
1
= x2
3
Since area of  ABE  area of  ADF ,
Area of  ABE =
1 
Area of quadrilateral AECF = x 2  2 x 2 
3 
1 2
= x
3
Weak:
The method of proving
the assertion is wrong
Fair:
The method of proving
the assertion is partly
correct
Good:
The method of proving
the assertion is correct
Very Good: The method of proving
the assertion is correct
and the presentation is
clear and concise
 Each piece is one-third of the whole cake.
Therefore, the 3 pieces of cake are equal in area.
One Third_Marking Guidelines
Evidence:
Proof for “one-third”
4
Marking Guidelines
Solution
Performance
Part C
1.
As in the figure below, select a pair of opposite sides of the quadrilateral, say, DC and
AB . Trisect each of them (i.e. AE = EF = FB and CG = GH = HD). Then, cut the
cake along EH and FG .
C
H
D
A
Evidence:
Method of cutting the cake
Weak:
Attempt to find a method
but the method is wrong
Fair:
Attempt to find a method
but the method is not clearly
spelt out
Good:
Clearly state the method of
cutting the cake, but
insufficient numerical
information is supplied.
Also, point out correctly
which piece is exactly
one-third of the whole cake
G
F
E
B
The quadrilateral EFGH is the piece which is one-third of the whole cake.
Very Good: Give clear and complete
method with all relevant
numerical information
provided and point out
correctly which piece is
exactly one-third of the
whole cake.
2.
As in the figure below, join A and C and trisect AC. Let the two points of trisection be
M and N .
C
H
D
Evidence:
Proof for “one-third”
G
N
Weak:
The method of proving
the assertion is wrong
Fair:
The method of proving
the assertion is partly
correct
Good:
The method of proving
the assertion is correct
M
A
E
F
B
Let S , S1 and S2 be the area of quadrilateral ABCD, ABC and ADC
respectively.
Very Good: The method of proving
the assertion is correct and
the presentation is clear
and concise
In AEM and ABC ,
EAM = BAC (common)

AE AM 1


AB
AC 3
 AEM ~ ABC (2 sides proportional and an included angle)
Area of AEM  AE   1  1

   
Area of ABC  AB   3  9
2
One Third_Marking Guidelines
2
5
Marking Guidelines
Solution
 Area of AEM =
Performance
1
S1
9
FAN = BAC (common)
AF AN 2



AB AC 3
AFN ~ ABC (2 sides proportional and an included angle)
2
2
Area of AFN  AF   2 
4

   
Area of ABC  AB   3  9
4
 Area of AFN = S1
9
Area of EFNM =
4
1
1
S1  S1  S1
9
9
3
Similarly,
1
S2
9
4
Area of CHM = S 2
9
4
1
1
 Area of HGNM = S 2  S 2  S 2
9
9
3
1
1
1
1
 Area of EFNM + Area of HGNM = S1  S 2  (S1  S 2 )  S
3
3
3
3
Area of CGN =
 EM // FN and HM // GN
 EMH  FNG
From similar triangles, we get
1
2
EM  BC , FN  BC
3
3
2
1
HM  AD , GN  AD
3
3
1
EM  HM sin EMH
2
1
= BC  AD sin EMH
9
1
Area of FNG = FN  GN sin FNG
2
1
= BC  AD sin FNG
9
 Area of EMH = Area of FNG
Area of EMH =
Area of EFGH = Area of EFNM + Area of HGNM + Area of FNG – Area of EMH
1
= S
3
 The piece EFGH is exactly one-third of the whole cake.
One Third_Marking Guidelines
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