Grading with CCSS?? How does it work?

If science and social studies is integrating with com arts, how do you grade that?  With me teaching com arts and science and social studies to be in different classes, ideas are: do writing together with the other class and both teachers grade it (one for com. arts and one for content) or have a category in the subject area for com arts. The concern with the second is that a student can be really good with science but score poorly because of the com. arts.  The difficulty with the first is having to coordinate the grade to figure out which hour a student has social studies and then having to refile for when they have com. arts.  Key to this would be having students always label a paper with content hour & com arts hour.  After grading one paper, Carrie can refile for me to submit the grade.