HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Pascal Triangle
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
Instructions
1.
This task consists of FOUR parts, Part A to Part D. Students are required to attempt ALL parts.
2.
Numerical answers should be either exact or correct to 2 decimal places.
Teacher Guidelines
1.
This task requires students to
(a)
carry out simple arithmetic,
(b) investigate relations between numbers,
(c)
make conjectures, and
(d) prove conjectures using relevant mathematical skills.
2.
Students are expected to complete the task within 1.5 hour. However, teachers can exercise their
professional judgment to adjust the time allowed to cater for the ability level of their students.
3.
Teachers may feel free to modify the question and the marking guidelines to cater for the ability level of
their students.
4.
Feedback should be provided to students after marking the task. For instance, teachers can discuss
different approaches of handling each part of the task with students.
Pascal Triangle
2
Part A
A Pascal Triangle is a sequence of numbers given in the following way.
1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
10
10
5
1
1
15
20
15
6
1
35
35
21
7
1
28
8
1
84
36
9
1
7
1
8
28
56
1
9
36
84
126
1

Figure 1
Referring to Figure 1,
(a)
fill in the boxes;
(b) write down the row that follows “1 9 36 … 36 9 1”; and
(c)
find the relationship between the jth term of the ith row and the terms in the (i-1)th row.
Pascal Triangle
3
Part B
1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
56
28
8
1
1
9
36
84
126
126
84
36
9
1

Figure 2
(a)
(i)
Referring to Figure 2, find the sum of numbers in each row. Express your answer in index form.
(ii) What can you conclude?
(b) (i)
Referring to Figure 2, find the sum of numbers in each diagonal from upper right to lower left as
indicated.
(ii) What can you conclude?
(c)
(i)
Referring to Figure 3, find the sum of numbers from the first row to the 8th row within the 2nd
column.
(ii) Referring to Figure 3, find the sum of numbers from the first row to the 8th row within the 3rd
column.
(iii) What can you conclude about the sum of numbers from the first row to the nth row within each
column?
Pascal Triangle
4
1st row
1
2nd row
1
1
3rd row
1
2
1
4th row
1
3
3
1
5th row
1
4
6
4
1
6th row
1
5
10
10
5
1
7th row
1
6
15
20
15
6
1
8th row
1
7
21
35
35
21
7
1
9th row
1
8
28
56
70
56
28
8
1
10th row
1
9
36
84
126
126
84
36
9
1
11th row
1
10
34
110
210
252
210
110
45
10
nd
rd
1
st
column
2
column
3
column
4
th
5
column
th
6
column
th
column
7
th
column
8
th
column
9
th
column
10
1
th
11th
column
Figure 3
Part C
1
1
1
1
1
1
1
1


1
2
3
4
5
6
7
8


1
3
6
10
15
21
28
36


1
4
20
35
56
84
120


1
5
15
35
210
330


1
6
21
56
126
252
462
792


28
84
210
462
924
1716


3432


6435


1
1
8
36
120
330
792
1
9
45
165
495
1287
3003


Figure 4
(a)
Referring to Figure 4, fill in the boxes provided in the array. Explain your answers.
(b) State one special feature of the array in Figure 4.
(Hint: You may make reference to the conclusion in Part B.)
Pascal Triangle
5
column
Part D
Determine a rule and use it to construct your own array. You may make reference to Part A, Part B and Part C.
(a)
Write down the rule determined.
(b)
Construct an array that follows the determined rule in the table below.
(c)
State one special feature of the array.
END OF ASSESSMENT TASK
Pascal Triangle
6