The purpose of this study is to clarify the elements which enable students to connect between inductive activity and deductive one in the process of teaching Pythagorean Theorem. For the purpose, we designed the learning trajectory of Pythagorean Theorem based on mainly two points of view: H. Steinbring's idea of "Epistemological Triangle" and the viewpoint of "Mathematics as the science of patterns" known for the name of Eric Ch. Wittmann. Three lessons were done according to the learning trajectory in our research project "School Support Project", which was conducted in a junior high school for about four months from August, 2011 to December. The lessons were recorded with video cameras and the detailed transcripts were prepared for the analysis. We analyzed the process of the lessons and tried to find the important elements which supported the students mathematical activity, especially ones which made connection between discovery and demonstration of Pythagorean Theorem. The important elements clarified from the analysis were summarized as follows: (1) It is important to develop the students own construction, which are various and individual one, in the stage of inductive activity. The students could use the constructions as referent context in the stage of deductive activity spontaneously. (2) It is also important to start the lessons from the specialization of Pythagorean Theorem, especially the case of the base length is 1, and to develop based on the viewpoint of "Mathematics as the science of patterns". The students could develop the specialized idea more generalized one by using "What if not?" strategy automatically. The typical patterns of inductive activity which had been done in the previous stage would be repeated in the next stage independently.
