How do I work out a problem with a teacher who loses the assignments I turn in and then accuses me of not doing the homework?
What a frustrating situation for you both! Start off by calmly explaining to your teacher that getting credit for your homework is important to you and that you want to work together to find a solution. Find a time to talk when you both have a few minutes without distractions, such as between classes or right after school. Try your best not to point a finger at your teacher and expect her to take all of the blame, but do be honest about your concerns and then offer a practical solution.
If you don't already have one, get a small notebook that you use specifically for recording homework assignments and their due dates. The next time you turn in an assignment, stick around after class for a minute to have your teacher write her initials and the date next to the assignment listed in your book. If there's any question later about whether or not the assignment was turned in, you'll have a written record. The key to making this work is for you to be diligent and get a signature for every single assignment (even if you turn in something late once in a while) for at least the next few weeks.
For particularly important assignments that may have been handwritten, you might want to go one step further and make a copy of your work before you turn in the original. That way, just in case the original copy goes missing, you won't have to redo any of your work.
After several weeks, check in with your teacher to talk about how your arrangement has been going. Be sure to thank her for working with you to make this situation better and talk about whether or not you should continue to get her sign-off on your assignments.