Before learning Scratch, Sploder could be a good introduction to game programming for students. There are 5 types of games that can be created, and one, Retro Arcade, has a version that runs on iOS, Android, and Kindle Fire devices.
To save and share games, students will need to create an account. The site is pretty adamant on not collecting any personal information from students under the age of 13, requiring a parent email address. It is not clear whether the teacher can use their own email address or not.