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# Aitor’s theorem

Xavier Bordoy

Category

blog

Keywords

"why" "funcions" "prova" "teorema" "raonar" "funció de segon grau" "paràbola" "vèrtex" "concavitat" "demostrar"

Abstract

Sometimes, good moments of reasoning appear in classroom

Sometimes magic happens in class: good moments of reasoning appear in classroom instantly, without any planification. Sometimes one student asks you for a reasoning validation and then you are marvelled. This happened to me two weeks ago. We were revising homework: sketching graph of quadratic functions with their concavity, their vertex and the cutting points to axes… then, Aitor said:

But it’s needless. We don’t need to calculate the cutting points to $x$-axis because the vertex is “positive”

What?, I said. Can you explain it with more detail?

And then, we put in blackboard what we have known as Aitor’s theorem. We wrote in blackboard its hypothesis and its thesis:

• Hypothesis: The parabola is concave and the $y$ of the vertex is positive
• Thesis: the parabola has not cutting points of $x$-axis
Since then, we have applied this theorem a lot of times for saving us the time to calculate the cutting points of $x$-axis and, more important, my students know firsthand why mathematical reasoning is.