Abstract
The traditional presentation of the integers ring generates difficulties for learning the product with its well-known signs rule. Our working hypothesis is that the solely representation of the product of two natural numbers by the area of a rectangle can hinder the vision of the product of two integers. Referring us to the scheme of Mathematical Working Spaces, we need another representation for the semiotic genesis of the integers. We tested didactical resources coming out from a geometrical construction made by Descartes in his geometric supplement of the Discourse on the Method, but we observed that this geometric introduction of the product can also produce difficulties, because it does not easily illustrate the distributive law. We may explain this phenomenon considering the product of numbers as a procept. Thus, the homothetic transformations of the Cartesian plane constitute a convenient geometrical resource that we may propose for teaching of the integers and, more generally, the numbers with a sign.
Multiplication of Numbers; Signs Rule; Distributive Law; 2D Geometry; Mathematical Working Spaces; Semiotic Genesis; Dilation
