HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Figurate Numbers
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
13 – 16
The student demonstrates a complete understanding of
the underlying mathematical knowledge and
investigation skills which are relevant to the task, and is
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 mathematical communication
skills.
‧
Teachers should base on the above assessment scale to design SBA tasks and the marking guidelines for assessing students with
different abilities.
Figurate Numbers_Marking Guidelines
2
Marking Guidelines
Solution
Performance
Part A
1.
Evidence: The answers
1st term : Number of dots =
1
2nd term : Number of dots =
6
3rd term : Number of dots =
15
4th term : Number of dots =
28
Weak:
Give one or no correct
answer
Fair:
Give only two correct
answers
Good:
Give any three of the correct
answers
Very good: All answers are correct
2.
Method 1:
The number of dots in the 10th term of the hexagonal numbers
Evidence:
1. The pattern of figurate numbers
2. The answer
= 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37 = 190
Weak:
Cannot recognise pattern of
figurate numbers
Method 2:
The number of dots in the 10th term of the hexagonal numbers
Fair:
= (1 +2 +3 + 4 + …+1 0 ) × 2 – 1 + ( 1 +3 + 5 + 7 + … +1 7 )
10(10  1)
9(17  1)
=
 2 1 
2
2
= 190
Have some idea about
pattern of figurate numbers
but cannot form correct
series
Good:
Can recognise the pattern of
figurate numbers but cannot
give correct number of dots
Method 3:
The number of dots in the 10th term of the hexagonal numbers
Very good: Can recognise the pattern of
figurate numbers and give
correct number of dots
= 1 + (1 ×4 + 1 ) + ( 2 ×4 + 1 ) + ( 3 ×4 +1 ) + … +(9 × 4 + 1 )
9(9  1)
= 10 
4
2
= 190
3.
Evidence:
1. The pattern of figurate numbers
2. The general term and the sum of
arithmetic sequence
3. The answer
Method 1 :
The number of dots in nth term of the hexagonal numbers
= 1 + 5 + 9 + 1 3 + 1 7 + …… .+ ( 4 n 3 )
n(1  4n  3)
=
2
n(4n  2)
=
= n(2n  1)
2
Weak:
Method 2 :
The number of dots in nth term of the hexagonal numbers
= (1 +2 +3 +4 + …+ n) × 2 – 1 + ( 1 + 3 + 5 + … +(2 n - 3 ) )
n(n  1)
(n  1)(2n  3  1)
=
 2 1 
2
2
= n(n  1)  1  (n  1)(n  1)
= n(2n  1)
Figurate Numbers_Marking Guidelines
3
Fail to identify the pattern of
figurate numbers
Fair:
Only recognise pattern of
figurate numbers but cannot
give correct general term of
arithmetic sequence
Good:
Can recognise the pattern of
figurate numbers and give
correct general term of
arithmetic sequence but cannot
get correct answer
Very good: Can recognise the pattern of
figurate numbers, give
correct general term of
arithmetic sequence and
answer
Marking Guidelines
Solution
Performance
Method 3 :
The number of dots in nth term of the hexagonal numbers
= 1 + (1×4 + 1) + (2×4 + 1) + (3×4+1) + …+((n1)×4 + 1)
(n  1)( n  1  1)
= n
4
2
= n  2n(n  1)
= n(2n  1)
Part B
Evidence: The solutions
1.
The height of the model = 110 × 1.2cm = 132 cm (or 1.32 m)
2.
n
3.
110
 22
5
Weak:
Attempt the questions but
the answers are 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
The number of levels from top to the 86 th level =110-86+1 = 25
The nth term of hexagonal numbers: n = 25÷5 = 5
k = the 5th term of hexagonal numbers, T(5)= 5(10-1) = 45
Very good: All answers are correct
4.
Evidence:
1. The pattern of figurate numbers
2. The answer
Method 1 :
Total number of spheres
= [T(1) + T(2) + T(3) + ……+ T(22) ]×5
= [1 + 6 + 15 + 28 + 45 + 66 + 91 + 120+ 153 + 190 + 231 + 276 + 325 + 378 + 435
+ 496 + 561 + 630 + 703 +780 + 861 + 946]×5
Weak:
Fair:
= 36,685
Fail to identify the pattern of
figurate numbers
Only recognize the pattern
of figurate numbers but
cannot give the correct
series
Able to recognise the pattern
of figurate numbers and give
the correct series but the
answer is incorrect
Method 2 :
Total number of spheres
= [ T(1) + T(2) + T(3) + ……+ T(22) ]×5
Good:
= [ (2 12  1)  (2  22  2)  (2  32  3)  (2  222  22) ] ×5
Very good: Able to recognise the pattern
of figurate numbers and give
the correct series and answer
= [ 2  (12  22  32    222 )  (1  2  3    22) ]×5
= [2 
22(22  1)( 2  22  1) 22(22  1)

] 5
6
2
= 36,685
Figurate Numbers_Marking Guidelines
4
Marking Guidelines
Solution
Performance
Part C
Evidence: The solutions
5 + 6 + 7 + + (n+4) = 110
n(n  9)
 110
2
n 2  9n  220  0
n = 11 or -20 (reject)
1.
 n = 11
2.
(a)
Weak:
Attempt the questions but
the answers are incorrect
Fair:
Some mistakes are found in
the working and not all
answers are correct
Good:
All answers are correct but
minor mistakes are found in
the working
m= k + 4
(b) Total number of spheres = T ( k) × ( k +4 )
Very good: All answers are correct
 (2k 2  k )( k  4)
 2k 3  7 k 2  4k
3.
Evidence:
1. The pattern of figurate numbers
2. The number of spheres required to
build the model
3. The conclusion
Disagree
Explanation:
Method 1 :
Total number of spheres required
= T ( 1 ) ×5 + T (2 ) × 6 + T ( 3 ) × 7 +  + T ( 11 ) × 1 5
Weak:
= 1 ×5 + 6 ×6 + 1 5 ×7 + 2 8 × 8 + 4 5 ×9 + 6 6 ×1 0 + 9 1 × 11 + 1 2 0 ×1 2
+ 1 5 3 ×1 3 + 1 9 0 ×1 4 + 2 3 1 ×1 5
Fair:
= 11,990 < 36,685
Good:
Therefore, the total number of spheres required to build the model in Part C is less
than that in Part B.
Method 2 :
Total number of spheres required
Very good: Able to recognize the
pattern of figurate numbers;
give correct series, the
number of spheres required
and conclusion
= T ( 1 ) ×5 + T (2 ) × 6 + T ( 3 ) × 7 +  + T ( 11 ) × 1 5
=
=
(2  13  7  12  4  1)  (2  23  7  2 2  4  2) 
(2  33  7  32  4  3)    (2  113  7  112  4  11)
2(13  23  33    113 )  7(12  22  32    112 )  4(1  2  3    11)
= 2
112 (11  1) 2
11(11  1)( 22  1)
11(11  12 )
 7
 4
4
6
2
= 11,990 < 36,685
Therefore, the total number of spheres required to build the model in Part C is less
than that in Part B.
Figurate Numbers_Marking Guidelines
Fail to identify the pattern of
figurate numbers
Only recognize the pattern
of figurate numbers but
cannot write down correct
series
Able to recognize the pattern
of figurate numbers and give
correct series but unable to
calculate correctly the number
of spheres required
5