HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Lucas Sequence
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 Mathematical Investigation is shown in the following table.
Level of
Performance
Very good
Good
Fair
Weak
Marks
Mathematical Knowledge and Investigation Skills
The student demonstrates a complete understanding of
the underlying mathematical knowledge and
investigation skills which are relevant to the task, and is
13–16 consistently competent and accurate in applying them in
handling the task. Typically, the student recognises
patterns, and states a conjecture which is justified by a
correct proof.
9–12
The student demonstrates a substantial understanding of
the underlying mathematical knowledge and
investigation skills which are relevant to the task, and is
generally competent and accurate in applying them in
handling the task. Typically, the student recognises
patterns, draws inductive generalisations to arrive at a
conjecture and attempts to provide a proof for the
conjecture.
5–8
The student demonstrates a basic understanding of the
underlying mathematical knowledge and investigation
skills which are relevant to the task, and is occasionally
competent and accurate in applying them in handling
the task. Typically, the student recognises patterns and
attempts to draw inductive generalisations to arrive at a
conjecture.
1–4
The student demonstrates a limited understanding of the
underlying mathematical knowledge and investigation
skills which are relevant to the task, and is rarely
competent and accurate in applying them in handling
the task. Typically, the student manages to recognise
simple patterns and organises results in a way that helps
pattern identifications.
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 forms of
mathematical
presentation
to
present conjectures and proofs.
3
The student is able to communicate
ideas
properly
through
written/verbal accounts, using
appropriate and correct forms of
mathematical presentation such as
algebraic
manipulations
and
geometric deductions.
2
The student is able to communicate
ideas with some appropriate forms
of mathematical presentation such
as
algebraic
formulae
and
geometric facts.
1
The
student
attempts
to
communicate
ideas
using
mathematical
symbols
and
notations,
diagrams,
tables,
calculations etc.

The full mark of a SBA task on Mathematical Investigation submitted should be scaled to 20 marks, of which 16 marks
are awarded for the mathematical knowledge and investigation 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.
Lucas Sequence_Marking Guidelines
2
Marking Guidelines
Solution
Performance
Part A
(a)
Sequence 1
n
1
2
3
4
5
6
7
8
9
10
Tn
0
1
1
2
3
5
8
13
21
34
Evidence:
Numbers in the boxes
Weak:
Less than one-third of the
numbers are correct
Fair:
One-third of the numbers
are correct
Good:
Two-thirds of the numbers
are correct
(b)
Starting
Values
n
10
1
2
3
4
5
6
7
8
9
2
3
5
8
13
21
34
55
89
144
1
1
2
3
5
8
13
21
34
55
Sequence 4
1
2
3
5
8
13
21
34
55
89
Sequence 5
1
0
1
1
2
3
5
8
13
21
Sequence 2
Sequence 3
Very good: All numbers are correct
Tn
(c) For example,
Starting
Values
n
Sequence 6
Sequence 7
Sequence 8
Tn
1
2
3
4
5
6
7
8
9
10
0
2
2
4
6
10
16
26
42
68
2
2
4
6
10
16
26
42
68
110
3
1
4
5
9
14
23
37
60
97
Lucas Sequence_Marking Guidelines
3
Marking Guidelines
Solution
Performance
Part B
(a)
(i)
(ii)
BC  1  x
AB

x
1
Weak:
Incorrect method and
solution

AB
Fair:
Correct method but wrong
solution
Good:
Correct method and part of
the solution being correct
AC
BC
Evidence:
1. Substitution
2. Process of solving the equation
3. Solution of the equation
AB
AC
1 x

x
x
1
Very good: Correct method and
solution
x  1 x
2
2
x  x 1  0
x  0.6180 or  1.618 (rejected)
(b)
Sequence 1
(c)
T1
T2
T2
T3
T3
T4
T4
T5
T5
T6
0
1
0.5
0.6666
0.6
T6
T7
T7
T8
T8
T9
T9
T10
0.625 0.6154 0.6191 0.6176
Sequence 2 0.6666
0.6
0.625 0.6154 0.6191 0.6176 0.6182 0.6180 0.6181
Sequence 3
1
0.5
0.6666
Sequence 4
0.5
0.6666
0.6
Sequence 5
--
0
1
0.5
0.6666
Sequence 6
0
1
0.5
0.6666
0.6
Sequence 7
1
0.5
0.6666
0.6
Sequence 8
3
0.25
0.8
As n increases, the ratio
0.6
0.625 0.6154 0.6191 0.6176 0.6182
Weak:
Incorrect method and
answers
Fair:
Correct method but
wrong answers
Good:
Correct method and some
correct answers
0.625 0.6154 0.6191 0.6176 0.6182 0.6180
0.6
0.625 0.6154 0.6191
0.625 0.6154 0.6191 0.6176
0.625 0.6154 0.6191 0.6176 0.6182
0.5556 0.6429 0.6087 0.6216 0.6167 0.6186
Tn
for each sequence is approaching to 0.6180.
Tn 1
Lucas Sequence_Marking Guidelines
Evidence:
1. Substitution
2. Ratios obtained
4
Very good: Correct method and
answers
Marking Guidelines
Solution
Performance
Part C
(a)
Lucas
Sequence
n
1
2
3
4
5
6
7
8
9
10
Ln
2
1
3
4
7
11
18
29
47
76
Evidence:
Numbers in the boxes
Weak:
Less than one-third of the
numbers are correct
Fair:
One-third of the
numbers are correct
Good:
Two-thirds of the numbers
are correct
Very good: All numbers are correct
(b)
Fibonacci
Sequence
Lucas
Sequence
(i)
n
1
2
3
4
5
6
7
8
9
10
Fn
0
1
1
2
3
5
8
13
21
34
Ln
2
1
3
4
7
11
18
29
47
76
Fn 1  Fn 1  Ln
(ii) For example,
Fn  2  Fn  2  Ln ;
Evidence:
1. The relationship between Fn-1 , Fn+1
and Ln
2. Other relationship found
Weak:
Unable to find any
relationship
Fair:
Able to find less than 3
relevant relationships
Good:
Able to find 3 relevant
relationships
Fn 3  Fn 3  2 Ln ;
Ln 1  Ln 1  5 Fn ; and
Fn  4  Fn 4  3Ln .
Very good: Able to find more than 3
relevant relationships
Lucas Sequence_Marking Guidelines
5