Regarding the first day, here are two activities. The first is the "classic" handshake problem in ehich the question is: is every person shakes hands with all others in the room, how many handshakes would there be? This is a good chance to look at some counting as well as developing an intuitive recursive formula for solving the problem.
The second is to arrange the desks in a block, 3x4, 5x5, 5x6, or whatever you have, and ask if it is possible for students to sit in each desk and then all change seats by moving to only a horizontally or vertically adjacent desk. Some nice patterns, and properties of hamilton circuits (hint, hint) in an mxn rectangle can emerge here.
Hope this helps.
----- End of forwarded message from Charles Biehl -----
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