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Aitor’s theorem

Xavier Bordoy

Keywords
"why" "funcions" "prova" "teorema" "blog" "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 xx-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 yy of the vertex is positive
  • Thesis: the parabola has not cutting points of xx-axis
General view of the board. You can see zoomed version
Zoom at Aitor’s theorem text. You can see zoomed version

Since then, we have applied this theorem a lot of times for saving us the time to calculate the cutting points of xx-axis and, more important, my students know firsthand why mathematical reasoning is.