HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Houses in Different Shapes
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 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.
Houses in Different Shapes_Marking Guidelines
2
Marking Guidelines
Solution
Performance
Part A
1.
(a)
(b)
(c)
The area of DEF
1
 x  x sin 60
2
3 2 2

x m
4
Evidence:
The solutions
In order to make the area of DEF greatest, E must be the mid-point of AB and F
must be the mid-point of CB. Then, EF = 5 m.
The capacity of the building DEFGHI
3 2
 5   3

4
75 3 3

m
4
Weak:
Attempt the questions but
the working is incorrect
Fair:
Some mistakes are found
in the working and not all
the answers are correct
Good:
All answers are correct but
minor mistakes are found in
the working
Very good: All answers and working
are correct
___________________________________________________________________________________________________________
Part B
2.
Evidence:
1. Locations of D, E, F and G
2. The shape of the figure
B
(a)
Some locations and the
shape of the figure are
incorrect.
Fair:
All locations are correct
but inaccurate shape of
the figure
Good:
Correct locations of D,
E, F and G and accurate
shape of the figure, but
wrong labelling
F
E
A
Weak:
C
G
D
Very good: Correct locations of D, E,
F and G and accurate
shape of the figure
___________________________________________________________________________________________________________
(b)
Let DE be x m. Then AD 
10  x
m.
2
Evidence:
The solutions
x
10  x
2


3 10  x  2 x
tan 60 
Weak:
Attempt the question but
the working is incorrect
Fair:
Some mistakes are found
in the working and the
answer is incorrect
Good:
The answer is correct but
minor mistakes are found in
the working
10 3
x
2 3
 10 3  2  3 
Hence, the length of each side of DEFG = 10 3  2  3  m
Very good: The answer and working
are correct
Houses in Different Shapes_Marking Guidelines
3
Marking Guidelines
Solution
(c)
Performance
The capacity of the house with the greatest square base DEFG
 10 3  2  3   3
Evidence:
1.
The comparison result
2.
The explanation
 900  7  4 3  m3
Weak:
Wrong comparison result
Fair:
Correct comparison result
but with little explanation
Good:
Correct comparison result
with some explanation
2
 900  4  4 3  3
From Question 1(c), the capacity of the house with the greatest equilateral
75 3 3
triangular base DEF 
m
4
Since 900  7  4 3  
75 3
, the capacity of the house with the greatest square
4
base is greater than the one with the greatest equilateral triangular base.
Very good: Correct comparison result
with thorough explanation
___________________________________________________________________________________________________________
Evidence:
1.
Location of the figure
2.
The shape of the figure
Part C
3.
B
(a)
Weak:
Incorrect location of the
figure and inaccurate
shape of the figure
Fair:
Correct location of the
figure but inaccurate
shape of the figure
Good:
Incorrect location of the
figure but accurate shape
of the figure
Very good:
Correct location and
accurate shape of the
figure
C
A
___________________________________________________________________________________________________________
(b)
B
X
Y
O
A
Houses in Different Shapes_Marking Guidelines
C
Z
4
Marking Guidelines
Solution
From (a), the required circle is the in-circle of ABC.
Let O be the in-centre and X, Y and Z be the points of contact of AB, BC and
CA respectively.
Then, OX  OY  OZ  the radius of the in-circle.
Let r m be the radius of the in-circle.
Performance
Evidence:
The solution
Weak:
Attempt the question but
the working is incorrect
Fair:
Some mistakes are found
in the working and the
answer is incorrect
Good:
The answer is correct but
minor mistakes are found in
the working
The area of ABC

1
1
1
 OX  AB   OY  BC   OZ  CA
2
2
2
1
3 1
2
 10  
 10r  10r  10r 
2
2 2
30r  50 3
r
5 3
3
Very good: The answer and working
are correct
5 3
The radius of the circle 
m
3
(c)
The capacity of the building with the circular base in (b)
2
5 3
  
  3
 3 
Evidence:
The proof
Weak:
 25 m3
Since 25  900  7  4 3  
75 3
, the capacity of the building with the
4
circular base in (b) is greater than the capacity of the building with the
equilateral triangular base in Question 1(c) or with square base in Question
2(c).
Fair:
Good:
Attempt to do the proof
but the working is
irrelevant
Attempt to do the proof
but the proof is not
complete
The proof is correct but
minor mistakes are found in
the working
Very good: The proof is correct and
complete
___________________________________________________________________________________________________________
(d)
No. It is because the amount of materials required to produce the surface of the
building is also the main concern of the property developer.
Evidence:
The explanation
For the equilateral triangular base, the total surface area without the base
1 2 3
 5  3  3   5 

2
 2 
 55 .8 m2
Weak:
Irrelevant explanation
Fair:
Relevant explanation, but
no supporting calculation
is given
Good:
Relevant explanation with
some supporting calculation
For the square base, the total surface area without the base
 10 3  2  3   4  3  10 3  2  3  
2
 77 .2 m2
For the circular base, the total surface area without the base
5 3
5 3
 2 
  3  

 3 
 3 
2
80.6 m2
The amount of materials required is the greatest for the building with circular
base, though the capacity of the house produced is the largest. It would be up to
the user to consider whether he wants a spacious environment or to save
money.
Houses in Different Shapes_Marking Guidelines
5
Very good: Relevant explanation with
detailed supporting
calculation
Marking Guidelines
Solution
Performance
Part D
4.
(a)
Since ABC is equilateral, all the smaller triangles are congruent.
Let x m be the radius of each in-circle.
1
1
1
x  AB  x  BO  x  OA
2
2
2



1 1
3
1
3 2 
2
 10  
  2
  x 10  10 
3 2
 2  2 
2 3 
5
x
2 

3 1 

3

The area of ABO 
Evidence:
The solution
Weak:
Attempt the question but
the working is incorrect
Fair:
Some mistakes are found
in the working and the
answer is incorrect
Good:
The answer is correct but
minor mistakes are found in
the working
Area of the shaded region
2
2
 
5
1 
5
 360  60

 

 3   2 
 sin 60


2 
2

360
  3 1  2  
3
1






  
3  
3  


 

  17.2127 m 2



Very good: The answer and working
are correct
Hence, the total capacity of three houses
= 17.2127  3
= 51.6 m2 (to 3 sig fig)
(b)
Evidence:
1. Location of the figure
2. The shape of the figure
(i)
B
O
A
Houses in Different Shapes_Marking Guidelines
C
6
Weak:
Incorrect location of the
figure and inaccurate
shape of the figure
Fair:
Correct location of the
figure but inaccurate
shape of the figure
Good:
Incorrect location of the
figure but accurate shape
of the figure
Very good:
Correct location and
accurate shape of the
figure
Marking Guidelines
Solution
(b)
Performance
Evidence:
1.
Location of the figure
2.
The shape of the figure
(ii)
B
O
Weak:
Incorrect location of the
figure and inaccurate
shape of the figure
Fair:
Correct location of the
figure but inaccurate
shape of the figure
Good:
Incorrect location of the
figure but accurate shape
of the figure
Very good:
Correct location and
accurate shape of the
figure
C
A
___________________________________________________________________________________________________________
(c)
The area of shaded region in (b)(i)
Evidence:
1.
The comparison result
2.
The explanation
1
1
 102  sin 60
2
4
The capacity of the building with this base
1
1
 102  sin 60  3
2
4
 32.5 m3
Weak:
Incorrect result and
irrelevant explanation
Fair:
Correct result and relevant
explanation, but no
supporting calculation is
given
Good:
Correct result and relevant
explanation with some
supporting calculation
For the area of shaded region in (b) (ii),
let x m be the length of the side of the square.
tan 60 
5
x
2
x
10
x
2 3 1
Very good: Correct result and relevant
explanation with detailed
supporting calculation
The capacity of the building with this base
= The area of shaded region in (b) (ii) 3
2
 10  2

1  10 
 
 3 2 
  sin 60   3
 2 3 1
 2 3  1 

 51.7 m2
Hence, the capacity of building with base in (b)(ii)
> the capacity of building with base in (a)
> the capacity of building with base in (b)(i).
[Note that the capacity of building with base in (b)(ii) and the capacity of
building with base in (a) are almost the same.]
Houses in Different Shapes_Marking Guidelines
7