YOU SAID:

my teachers pets name is joan, which is also my name, and my teachers name's name

INTO JAPANESE

私の教師のペットの名前は私の名前であるジョアンと私の教師の名前です

BACK INTO ENGLISH

My teacher's pet's name is my name Joan and my teacher's name

INTO JAPANESE

私の先生のペットの名前は私の名前ですジョアンと私の先生の名前

BACK INTO ENGLISH

My teacher's pet's name is my name Joan and my teacher's name