Do teachers want students to add their attempts on solving their own question in an email?
Yes absolutely include it. That's the only way for the instructor to see where the gaps in your knowledge are. If you think it makes the email too long, then put a concise summary at the top and then the details below.

Edit: consider also that in the above post, you attempted to answer your own question. Look how much context that gave to the question. Imagine if you'd posted the title and the bolded statement only. We would not have understood your thought process. You might not have received many useful answers, just comments asking you to clarify what you were asking