Run the Country Activity
This is an activity I have used in the UK in looking at how Government works, though it could easily be adapted for US Government. Each class member is assigned a department and then must jostle for the ear of the Prime Minister/ President. Obviously the Chancellor is a key player and you may want to select mathematical geniuses for this role, or your most devious student. It's an excellent activity to get students thinking about political accountability, particularly when they cannot agree on a resolution. It is best used across at least 2 lessons to have a full impact.