Since my original post, it was decided that our district would issue a second Google Apps for Education account to all of our teachers in our student domain (we have one domain exclusively for faculty/staff and one for students). Therefore, it is my recommendation that our teachers create their digital assignment dropboxes in the student domain. There are two main advantages of doing this. First is that the teachers can now require their students to be signed into their Google Apps accounts in order to submit their work instead of leaving the form open to the public. Second, the teachers can automatically collect the usernames of the students filling out the form as an additional validation check. To accomplish these goals, the teachers should create their forms in the student domain and check the appropriate boxes in the upper left corner of the form editing window:
Here is my original screencast with more information and the instructions for how to create the form. Be sure to change the playback quality to 720p HD for the best clarity.