You need some money to take the bus. You ask a stranger (request).
Could/Would you lend me some money to take the bus, please?
You don't find your keys. You need help (possibility).
Can you help me find my keys?
You are in an exam. You need to go to the restroom (permission).
May I go to the restroom?
You missed a class. You need to ask a classmate for his notes (request).
Could/Would you share your notes with me, please?
You don't understand the instructions of the homework. You ask your teacher (request).
Could/Would you explain the instructions again, please?
You have an exam and don't have a pen. You ask a classmate (possibility).
Can I borrow a pen? / Can you lend me a pen?
You need to study. Your friend is listening to loud music (willingness).
Will you turn down the music, please?
Your classmate is very smart. The exam is in 2 weeks and you don't understand (willingness).
Will you help me study?
It's hot. Your classmate has the door closed (willingness).
Will you open the door, please?
You need more time to finish an exam. You ask the teacher for more time (request).
Could/Would you give me more time to finish, please?
You need to leave class early. You ask the teacher (permission).
May I leave early today?
You want to come inside a house (permission).
May I come in, please?