In this paper, we offer general and specific strategies for using activities based on origami to invoke mathematical thinking. Origami refers to the Japanese art of paper folding. Background is given on origami, including the development and critical aspects of the practice of origami in the United States. Next, we indicate how origami can relate to the current goals of K -12 mathematical education. General strategies are described that can be applied to any origami model. The basic strategy is for students to do the teaching. Then we describe in detail a variety of models. The examples include a variety of different computer-based techniques that middle and high school students can use to produce supporting materials for teaching.