In order to ensure that you aren't just haphazardly moving tiles back and forth between the sides, each level gives you a certain number of moves you can use to complete your objective. Also, there's a reservoir of tiles that can be used in your problem solving.
The simplest operation that can be performed in DragonBox+ is addition. For example, if you have a lizard tile on the same side as your box, you can grab a "negative" version of the lizard and "add" it to both sides, which will help to isolate your box. You can also divide by tiles and multiply by tiles in later chapters. But of course, the beauty of the game is that these complex-sounding mathematical operations are disguised as simple game rules.
One thing I would add to DragonBox+ is more explicit explanation of the mathematical concepts being taught. While the game does slowly mix real numbers in with the pictured tiles, which is brilliant, I still think an explanation or a pictureless math chapter at the end of the game would go a long way in helping a student to apply the game-learned rules to the real world of algebra. That said, there is a fine line that shouldn't be crossed with such an explanation, otherwise it may turn a child off. Also, I think it would be great if you could "level up" your avatar by completing chapters or challenges.
Dragonbox+ is also available on iOS, Windows, and Mac.