Where Is Pythagoras Hiding?
An animated investigation on the island of Samos that finds the Pythagorean theorem hiding inside the distance formula and the equation of a circle.
In class, the formulas for the distance between two points and the equation of a circle are generally given no more than as formulas.
It is generally easy for students to memorize and apply these formulas, but they are not always offered the chance to understand where they come from, what they mean, and why they are connected.
This animation was designed so that the learner could follow a number of careful steps and thus work out the formulas for themselves. By starting with a simple example, such as calculating the distance between two points on an island, the learner quickly realizes that the Pythagorean theorem is contained in the distance formula. It is for this reason that the animation has been given its name and also explains the scenario of the investigation.
I didn't want the distance formula to appear at the end of the story since, as the circle includes all points that are at a fixed distance from the centre, the same idea is then used to derive the equation of a circle; what could have been given as a separate and unrelated formula instead follows directly from something the learner has already grasped.
Finally, I would like the learner to understand both formulas and their connection to Pythagoras and to each other. Instead of having to remember the two expressions separately, they should be able to associate them with a geometric concept which they are already familiar with.
In my opinion, it is far more valuable than just memorizing them since even if the exact formulas are forgotten at a later stage, the basic structure gives the learner a way of reconstructing them.