The Value of Late Assignments

So, I've taken in my classes to having students come to each class with short reaction papers to the day's reading. They have to find a paragraph, quote it, and respond to it. Not only does it make sure they read, but it also means they have to come prepared to start an interesting conversation and once it is started, it gives them something to say. We can spend less time working through the readings and more time doing something with them -- analyzing, weighing objections, seeing how to extend them...

But it also does two other things. First, it keeps them writing. The more you do it, the better you get. Second, since these are check/zero grades, it gives them a chance to take a risk without fear of it affecting their grade. My students tend to be risk averse, don't swing for the fences if it might mean striking out. Mediocrity is safe. They are encouraged to be provocative in these papers and by in large it works.

The question, then, is about late papers. Should I accept them? On the one hand, once we've discussed the topic, half the point is gone. It means that the work trying to wrestle with a hard text has been done for them. It means they already have others' ideas to use. Further, it means they won't be prepared to contribute to the classroom. On the other hand, if I accept no late papers, then if they miss one because it was triaged out, they write it off and won't bother going back to read it. This way they still have to be in the text, even if it doesn't help me in the classroom, it does keep them engaged. Are these pedagogical advantages for slackers worth the cost?