When dealing with elementary algorithms in the classroom, we almost always limit ourselves to considering the purely procedural aspects. In so doing, we run the risk of providing students with a distorted and reductive vision of mathematics, which could foster an attitude of disaffection towards the discipline among students. This work aims to show that we can overturn this situation by changing our approach and thus make the study of some elementary algorithms an opportunity to increase interest in mathematics and encourage the development of skills and competences in students. Our idea is to use the inquiry-based learning methodology in the study of algorithms in order to re-evaluate their role in school mathematics classes. Below we will describe an experiment on the teaching of elementary algorithms conducted between 2016 and 2018 as part of the 'Math High School' project of the University of Salerno and based on thorough historical and scientific research. The project involved a group of 120 first-year high school students and the skills that the students were able to develop suggest that the experiment should be extended to the use of IBL methodology in handling other algorithms. The inquiry-based activities proposed can be applied in the classroom by teachers and can offer practical strategies and tools for teaching algorithms.