show her how responsible you are. try to hint at it sometimes. like- oh, i'd love to go to my friend's house but how will you know when to pick me up cause i can't call you! Good luck! (it might take a while...)
To convince your mom to let you buy a phone, you can talk to them about the dangers out there and if you are in an emergency you can call them. You should also show them that you are responsible by doing your homework, chores, and cleaning your room.