From Fractions to Rational Numbers in their Structure: Constructive Classroom episodes concerning operations and ordering

We trace the guidelines of a research of didactic innovation on the approach to rational numbers from the structural point of view. A key element in the implementation of experimental activities is how to develop a view of rational numbers as a numerical extension of natural numbers by setting students the problem of defining the operations and order in the new set, using their knowledge of natural numbers as a reference point. Here we focus on the part of the research devoted to the comparison between the order of natural and rational numbers, with reference to the question of density. The results obtained show the effectiveness of this approach as far as active participation on behalf of the students in the solving of problems is concerned, as well of tackling the question of motivation. However, the results also show the need for more time to be given to the study of certain issues and their organisation in a more general frame.