Assigning odds allows the students to check to see if they have the correct answer to every single problem that they attempt. This is important because I do not want them practicing 10 homework problems incorrectly.
Scenario One – Student gets a correct answer
If they do the first problem and see that they have it correct, they gain encouragement and confidently move on to the next problem.
Scenario Two – Student gets an incorrect answer
If they do the first problem and then see that it is incorrect, they know they did something wrong. Hopefully they will not continue to do the remainder of the homework incorrectly. A student that does an entire homework assignment incorrectly has not only gained nothing, but they have committed incorrect practices to memory.
My hope is that once they discover a problem is done incorrectly, they will research how to do it right. They can read their book, look back into notes, search the net, call a friend, ask a parent, or email me. I would rather a student not do the homework than do 10 problems incorrectly.
I will look at the students’ homework each day. I am not looking for the correct answers because they already have these. I am looking for the correct procedures. I want to make sure they each know how to get the correct answer.
I believe that homework should be practice. But for homework to be productive practice, students need to do homework correctly. This means having the answers to all of their homework problems while they are working them.
I Speak Math 
Having correct answers can be useful if students don’t get too strong a hint from the correct answer. This would not work for homework exercises on proofs, for example, where the solution is the procedure.
Just another reason I am sooooo glad I don’t teach Geometry. ; )
As a parent and a teacher I think some homework is valuable but if one does not grade the work what is the point. How can the child correct errors if the work is not graded? I have smart children but I have to prepare myself for homework almost every night because it can be very stressful. Trying to help more than one child can be monumental if they are all struggling to complete work. I have received many packets of homework at weeks end or weeks and weeks later without a mark. Why should I spend this valuable time stressing over something that will not be corrected?