Proof, as a central and integral part of mathematics, is an essential component of mathematical education and is considered as the basic procedure for revealing the truth of mathematical propositions and for teaching productive reasoning as part of human civilization. Is there therefore room for conjectures in mathematics? In this paper after discussing at a theoretical level the concepts of proof and conjecture, both in a paper-and-pencil environment and in a dynamic geometry environment (DGE) as well as how school practice affects them, we fully explain a task involving various mathematical disciplines, which we tackle using elementary mathematics, in a mathematics education context. On the occasion of the Greek educational system we refer to some parameters of the teaching of geometry in school and we propose an activity, within a DGE, that could enable students to be guided in the formulation and exploration of conjectures. Finally, we discuss the teaching implications of this activity and make some suggestions.