Help with checking understanding

Hi all,
I need help with a couple of English homework questions. I am supposed to explain the meaning of the underlined language, then explain (very) briefly how I would check if someone understood my explanation.
the example given is:
"Last night, we didn't turn in until very late", and the explanation is that 'turn in' means go to bed. Meaning conveyance is checked by asking the student "What time did you turn it last night?".
So, my questions:

"Yesterday I was painting the flat" vs. "Yesterday I painted the flat".
I have written that 'was painting' indicates what you were doing, while 'painted' indicates that you finished the job. Is this correct, and how would I check that someone understood?

"He didn't manage to finish the assignment".
I'm not sure how to explain what 'manage to' means in this sentence. I've tentatively said that 'manage to' is similar to saying 'achieve' or 'accomplish', but again, do not know how to check another's understanding. I could ask "How did you manage to get your groceries home", and use the answer to determine if they understood?

Thank you for any help,
Sam.