Visually understanding the arithmetic progression formula

a didactic experience through figurate numbers, geometry, and the History of Mathematics

Abstract

Arithmetic Progression (AP) is a mathematical topic pertinent to High School education that is typically taught and demonstrated algebraically, often reducing the content to formula memorization. The only commonly made historical consideration about this topic is the anecdote of Gauss, recounting his childhood genius - an elitist narrative that contrasts with the trajectory of the average student. With such issues in mind, a didactic project was developed that articulates the History of Mathematics, manipulative materials, and problem-solving to teach AP in a contextualized manner. The goal of this article is to describe this approach so that other Math teachers can also benefit from it. Given that sequences were extensively studied by the Pythagoreans in Antiquity using the visual resource of figurate numbers, such a context was chosen to situate the study of AP. Through investigative activities with colored chips, it was demonstrated in the classroom how the formulas of AP can be visually deduced from triangular numbers - a type of figurate number of great historical importance. Finally, using this manipulative material, it was pictorially demonstrated why the formula for the sum of AP terms is identical to the formula for the area of a trapezoid. Through a final evaluative activity, students demonstrated having developed a pictorial and algebraic understanding of APs, successfully solving complex problems. The didactic experience proved to be a success, considering the students' positive perception and the articulation of diverse contents developed and executed successfully.