The second kind of activity is class-wide (or relatively large-group) collaboration on the construction of HTML tables. On the one hand, the contents of these tables support some other aspect of the mathematical activity of the class and provide data on which students can test conjectures. On the other, they give students a first experience with studying and editing the HTML source code for a webpage, using some of the basic HTML tag syntax, and managing whatever logistics are necessary to post the edited page to the web. To get this kind of activity started, a template is posted for the students to extend. It is important to include an "author" column for each row of the table.
Some diligence is required on the part of the instructor here. As the tables grow, occasionally there are students who forget to use a simple text editor and use instead some standard Web authoring tool such as Microsoft’s Front Page. This is easy to detect, because the size of the posted file typically jumps by a factor of more than three. It is important that this be caught and corrected before other students download the "corrupted" file. Of course, the other aspect that must be diligently checked -- and to some extent the students will do this themselves if they are encouraged -- is the correctness of the data in the tables.
The first "Phase IB" table I assign supports the number theory thread in the course. It is simply a table of prime factorizations, lists of divisors, number D(n) of divisors, and sum S(n) of divisors for each integer. Each student is responsible for adding at least 3 or 4 more rows to the table, depending on the size of the class. The object is to get more than a hundred rows in the table so that there is enough data for the students to detect patterns, e.g.,
Here is the template for this table:
Number |
Prime |
Divisors |
D(n) |
S(n) |
Author |
1 |
1 |
1 |
1 |
1 |
Dr. D |
2 |
2 |
1, 2 |
2 |
3 |
Dr. D |
3 |
3 |
1, 3 |
2 |
4 |
Dr. D |
4 |
2^{2} |
1, 2, 4 |
3 |
7 |
Dr. D |
5 |
5 |
1, 5 |
2 |
6 |
Dr. D |
6 |
2 * 3 |
1, 2, 3, 6 |
4 |
12 |
Dr. D |
7 |
7 |
1, 7 |
2 |
8 |
Dr. D |
8 |
2^{3} |
1, 2, 4, 8 |
4 |
15 |
Dr. D |
9 |
3^{2} |
1, 3, 9 |
3 |
13 |
Dr. D |
10 |
2 * 5 |
1, 2, 5, 10 |
4 |
18 |
Dr. D |
11 |
11 |
1, 11 |
2 |
12 |
Dr. D |
12 |
2^{2} * 3 |
1, 2, 3, 4, 6, 12 |
6 |
28 |
Dr. D |
Later in the semester, I use a table concerning "Egyptian fractions" (reciprocals of integers) -- the problem is, "For each n, find two distinct positive integers x and y so that 1/n = 1/x + 1/y." Of course, this problem can be solved pretty easily by partial fractions, but it provides more practice with HTML -- and students begin to get a little more creative with fonts, etc. Also, the little bit of algebra required to prove that the problem can always be solved is a good review.
Another interesting problem, for which the students can collect data in tabular form, is "Which (positive) integers can be expressed as the sum of two or more consecutive (positive) integers?" Data are collected in a table that begins
n |
Situation |
Author |
1 |
Cannot be done |
Dr. D |
2 |
Cannot be done |
Dr. D |
3 |
1+2 |
Dr. D |
4 |
Cannot be done |
Dr. D |
5 |
2+3 |
Dr. D |
6 |
1+2+3 |
Dr. D |
Students work collaboratively on this problem in an electronic threaded discussion, as well as by constructing the table. They quickly realize that the odd numbers are easy, and begin to come to grips with the others. It is difficult to prove that it can be done for all numbers except the powers of two, but there have always been a couple of students who work very hard and get at least very close to the complete proof.