Many children do not like learning mathematics. They do not find mathematics fun, motivating, and engaging, and they think it is difficult to learn. Computer-based games have the potential and possibility of addressing this problem. This paper proposes a strategy for designing game-based learning environments that takes advantage of the attractiveness of play and games to engage children with mathematical representations in and enjoyable and thoughtful fashion. In this strategy, representations of mathematical concepts are used to mediate between children and playing the game. As the game progresses, the representations become a prominent part of the game. The strategy is demonstrated through an implemented game which takes children from almost no knowledge of transformation geometry to some non-trivial knowledge, involving composite reflections and complex rotations. A study evaluating the effectiveness of the game is presented. The results of the study suggest that, despite the explicitness and difficulty of the mathematical concepts involved, children found the learning process fun and engaging. Furthermore, children exhibited significant improvement in their knowledge of transformation geometry concepts. A number of conclusions with regard to children, design, and learning mathematics are drawn from this research. (Contains 12 figures, 7 tables and 2 notes.)