The second assignment was handed out as group work in class, with the students to write up individual answers. The purpose of this form of assignment (which I used throughout the term) is to get the students talking mathematics with each other, but have them understand the individual responsibility of understanding that goes along with this. At the end of the year interview, many of the students discussed the importance of "bouncing ideas" off of each other in their projects, and the group work was to encourage this understanding.
The particular problems in this assignment broke into three pieces. The first third of the assignment was for the students to make a conjecture about the relation of the period of the decimal form of 1/n and the number n. The idea was for them to learn to conjecture and prove the conjecture. Also, I wanted them to understand how the division algorithm naturally leads to the result using the pigeonhole principle. For the most part, students performed well on this portion of the assignment, although writing the proof was difficult for about half of the students.
The second part of the assignment is to show that there exists a rational number such that when it is raised to an irrational power you get an irrational number. This can be done using countability arguments, but the project outlines an easier non-constructive method for showing this. The problem gets at the heart of the creation of a non-constructive proof and how you use the technique when you have to. The problem is really about giving the students even more exposure to irrational numbers. Student work on this problem was all over the board, ranging from no work at all to a perfect paper (see the work by John).
The third problem was again letting the students see Simpson's paradox for themselves, using an example from baseball that was given in a Monthly article. The students did well on this last problem.
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