Perhaps you haven't paid it any attention, but there is a relatively new feature visible in the Google Classroom Stream. Off to the left, you will see a small section that says "Upcoming." At a glance, it shows you what assignments are due soon. It appears the same from both teacher and student perspective, but it will operate differently in each.
I use this feature weekly when I attempt to keep up with student work. This saves me time of having to look through and open each assignment in Classwork. In addition, I can categorize assignments as "reviewed" and only see ones that still need my attention. On the student side, they are able to see their assignments as Assigned, Missing or Done. This simple tool can be very valuable for both teachers and students when managing time and workflow.
Below is a series of screenshots that shows you how to get started using Google Classroom's "Upcoming & To Review" features.
When you open a Google Classroom, on the Stream tab, you will see small box titled Upcoming. In this box, there will be displayed some assignments that are due soon. If you click on any of the assignments in the box, it will take directly to the assignment to evaluate student work. If you click "View all", you'll be taken to the "To Review" page for that particular Google Classroom.