Last year at our middle school two of the 6th grade language arts teachers had a competition between their classes for the best spelling scores. The winning class got a pizza party. It really got the kids motivated to study and as the competition heated up, I think it became less about the pizza and more about just proving who was "the best".
I would suggest that you reward both individual and classroom accomplishments so that the children don't lose motivation. Perhaps have a small weekly reward or even a drawing for the students from both classes who have turned in all of their homework on time, then a small party of some kind(cookies, popcorn, etc.) for the classroom who has the best homework turn-in record for the week. You can then throw out the carrot of a big end of quarter/semester/month(whatever time you want)pizza or ice cream party. Maybe even throw in some extra recess time.
Also, don't forget the standard old "pat on the back" reward of just verbally recognizing the students who have achieved their goals. Make announcements and/or hang posters or some other kind of visual that lets the whole school know the great work ethic of these students.
Just wondering if anyone had any ideas for getting kids to do their homework? Something like a competition betwen classes. If the whole class does their homework the class gets rewarded someway??????? Hope this makes sense and thanks for the help.