Try out the free online Desmos calculator (available as a web app for laptops and a mobile app for phones and tablets). Try playing with the sliders in the face I drew below to change this fellow's expression.
Desmos supports the work of math teachers everywhere by curating lesson ideas that make use of visually captivating ways for students to learn math. Check out their lessons designed by teachers for teachers.
Geogebra is another free "must have" in your digital learning toolbox. It's a website and an app that's available on all mobile platforms. As the name suggests, it helps students make the link between geometry and algebra in a very visually tactile way. There are over 150 000 interactive materials you can use right now, such as the Pythagorean Theorem Proof Without Words or the Area of a Circle - Wedge Demo below.
The real power in using either of these tools comes from students exploring, creating, and sharing their own materials on a class blog, wiki, or other shared online space.
You might just be looking for a graphing calculator app. Free Graphing Calculator is popular on iOS and Algeo is popular on Android.
Another strong math app for learning is Math 42 (still in development for android). It's works like a private tutor suggesting possible nest steps, showing annotated solutions when needed, and targeted skills training.
For younger kids, from early through middle years, they can learn math tactilely using any of the Motion Math Games apps. They're not free but they do offer a fun and engaging way to learning about fractions, decimals, percents, place value, estimation, and all four basic operations.
Wolfram Alpha is another online tool worth exploring. It looks like a search engine but it's not; it's a computation engine. It will compute everything it can for any entry you type in the search bar. Wolfram Alpha not only brings high powered mathematics to the web, browsing through their many examples will help make connections between math and many other subjects; it shows how there really is math in everything.
Finally, if you're looking for rich math tasks to explore with your students, with or without digital tools, browse through the NRICH Mathematics collections of resources. My favourite colleaction are the Developing Mathematical Habits of Mind high school resources; problems for Being Curious, Thoughtful, Collaborative, and Determined. They also have teaching resources for all levels of mathematics.
If you use any of these suggestions in your classes let us know how it went over in the comments. Keep in mind, no app can ever replace the impact of artful questions a thoughtful teacher asks while looking over a students' shoulder.
photo credits: creative commons licensed (BY) flickr photo by Steppschuh, creative commons licensed (BY) flickr photo by ashkyd