One way to do this is to make a category for each class that you want a blog for. You can do this by going to Site Admin/posts/categories. As you make posts to your blog, assign a cateogry (i.e. class). When students go to your blog, they can click on their class to see the blogs related to that class.
Also, to make a clean view of the blog, you might want to create a welcome page for your blog and then set it as a static front page by going to settings/reading and then choosing Static page as the front page. Then your classes will have to click on the category to the posts related to them.