You need to leave class early. You ask the teacher (permission).
May I leave early today?
You need to study. Your friend is listening to loud music (willingness).
Will you turn down the music, please?
You missed a class. You need to ask a classmate for his notes (request).
Could/Would you share your notes with me, please?
You are in an exam. You need to go to the restroom (permission).
May I go to the restroom?
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 don't find your keys. You need help (possibility).
Can you help me find my keys?
You want to come inside a house (permission).
May I come in, please?
You don't understand the instructions of the homework. You ask your teacher (request).
Could/Would you explain the instructions again, please?
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 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?