Ok, so maybe this isn't strictly a "math app". The benefits of Minecraft in a math classroom have been shown in multiple articles. I choose this app because it brings Minecraft to students and teachers through a cell phone. Sweet!
This app can break down almost any problem. Do your students need more practice with derivatives? What about integrals? I choose this app because it is a good one specifically for a high school calculus class. It has other functions though as well.
GeoGebra is known for its' high-quality software. As an alternative to standard graphing calculators, this app is free and powerful. Does your geometry classroom have a need for graphing technology? Look no further! You could easily use this app with a large range of high school maths. It's good for teacher use too.
Photomath is absolutely mind-blowing. It's just like it sounds. Take a picture of an equation, and you will get a worked-out version of the problem. This app is a real game-changer when it comes to teaching mathematics. Imagine a world where everyone has access to example problems, just by taking a picture on their phone. I believe that student's mathematical understanding could be greatly enhanced through an app like this.
This is a fun one. For elementary students, this is a must have. I chose this app because it combines a fun, fast-paced game with goals set by Common Core for computation. It comes with multiple game modes that are sure to stimulate those easily-distracted learners. Want to get a kid hooked on math? Get this app!
This app takes your phone and turns it into a super-computer. WolframAlpha can give you the answer to almost any math problem. It also comes with a connection to "the WolframAlpha computational knowledge engine", which houses facts and figures than span across many subject areas and domains. I have chosen this app because it is full of potential. There is no telling what one of your students could do with an app like this!
This free graphing calculator from Desmos does it all. The touch-enabled, user-friendly system sets this app apart from other graphing calculators. I've seen this work in a high school classroom where students used Desmos instead of a traditional graphing calculator. Very useful!
This amazing (free) app blends problem solving and mathematical reasoning. For an educational tool, it is actually pretty addictive and fun. I personally have tried this game and find it to be totally awesome. You could use this app to aid in the teaching of: positive and negative numbers, exponents, PEMDAS, logarithms, inequalities, absolute value, and higher-level problem solving. I would recommend this for students age 10 or higher.