Rocket Math from the developer Dan Russell-Pinson for $0.99. (There is a free version you can try out.) Personally, I didn't particularly like this one very much at first, but a first grade teacher told me that she's been using it with her class and they all love it. You have to complete multiple choice math problems to earn 'money'. You can choose which operation to work on and the level of difficulty. You use your money to build a rocket, which you then attempt to fly into space on a mission that is also math related. Although the math missions start at an easy level (tap all the even numbers) they cover a wide range of topics. There are 56 different missions. The most advanced ones ask students to do things like identify numbers that are divisible by 3, the equations that have a remainder of 1, or the square root of a given number. This app allows you to save up to 5 profiles for different students. The profile saves the rockets the student has built, along with the info about the maximum height it reached and the maximum score the student achieved on a mission with that rocket.
Neither app has any advertising, neither requires you to be connected to the Internet to play. Both apps have sound effects (which many students like.) Neither has any way for a teacher to check and see what a student has been working on. Students could appear to be working hard but consistently picking the easiest level to work on and not challenging themselves. Still, with free versions of both available right now they're both worth grabbing.