How can i convice my mom to let me take dance lessons?

Start by doing your research and finding a place to learn that is near you and find a place that isn't too expensive. Also, some places offer a starter package where you can pay for a few lessons just to decide if you like it, they're usually cheap and will give you an idea of what to expect and if you really will stick with it. If you like those then maybe she'll see you really like it. You can also try starting with group cles which are usually cheaper and will also just be an opportunity to try with no obligation. Maybe if you just try it in small doses at first she'll see that you really like it and let you continue more seriously after a while.