This system is intended to encourage you to participate actively in the course throughout the semester. As you will see, you don't have to do much, but you do have to do it! There will be exactly one problem assigned each class period, due at the beginning of the next class period. If your solution seriously treats the problem, and if it is your own work, that will be considered adequate -- your solution does not have to be absolutely correct, but it should not be careless. Often, as in the first problem, there is no strict right or wrong, just more interesting or less interesting. Your final grade will be determined almost entirely by the number of adequate solutions you have turned in on time. In fact, you have complete control over your final grade. Except for two additional items below, your grade will be determined by the following formula (the numbers refer to problems NOT adequately done on time): 0-3 A-, 4-5 B+, 6-7 B, 8-9 B-, 10-11 C+, 12-13 C, 13-14 C-, 15-16 D+, 17-18 D, 18-19 D-, >19 F.
As you see, you can guarantee yourself an A- by turning everything in on time. To get an A, though, you must do one thing more: write a thoughtful essay of about 5 pages in which you reflect on your experience with the ideas of this course. Especially think about how your view of mathematics has changed, if it has, over the course of the semester. It would be good to work ahead on this assignment, because I will very probably ask you to rewrite it based on the first draft I see. I may not accept the first draft as adequate. An adequate essay will raise every grade on the previous scale by one increment (i.e., not just A- to A, but also C to C+, etc.)
Finally, there will be occasional (optional) "challenge" problems. These will be harder than the usual assignments, and the criteria for adequacy will be higher. Solutions to challenge problems can be turned in any time until the last day of classes. If I think a solution is pretty good, but not good enough, I may ask you to carry it a little further to be considered adequate. Such a solution can make up at any time for a regular assigned problem that you didn't get credit for.
I don't mind your working together, talking about the assignments, etc. In fact, I think this is good. This is how mathematics is typically done! If you get substantial help from someone, give them credit in writing -- that is fine. But you must finally write up your ideas YOURSELF, without reference to what anyone else has written. If two solutions are sufficiently similar, neither of them will get credit. Put enough of yourself into your solution that it will be clearly your own.
The bad news about the grading system is that solutions to problems will not be accepted late FOR ANY REASON, including sickness. This is the unorthodox part. I have tried to build a cushion into the above scheme to allow for those inevitable days when you forget or lose your solution, have a job interview, etc. As you see, this can happen several times and still not jeopardize an A- (but only if you are very regular otherwise). The "challenge" problems are an additional cushion. Maybe a real A should expect to do a challenge problem, but I have not required that.