Activity Overview
The investigation deepens as students continue their quest to uncover the connections between the Pythagoras’s theorem, the Pythagorean Circle and infamous Fibonacci sequence that will appear centuries later. In this episode students discover irrational secrets, numbers that can’t be tamed or fully expressed. While the Babylonians provide a numerical entry point, the hauntingly elusive square roots threaten to expand the number system.
Objectives
Students complete several calculations around Pythagoras's theorem, some involving integers, some with rational numbers, then finally irrational. The focus moves to understadning irrational numbers with the final clue being released following a calculation that demosntrates that calculator rounding hides the non-exact result of approximated values. Students are required to explain how the geometric solution in the final clue results in an area of exactly 2 units, but the use of the approximated hypotenuse produces a different result, with the aim being for students to appreciate the use of surds.
Vocabulary
- Pythagoras
- Proof
- Rational and Irrational
- Babylonian Technique
- Square-root (surds)
- Exact vs Approximate
About the Lesson
In this episode, students use the Pythagorean theorem, established in episode 1, to unveil irrational numbers. Students use the remarkably powerful Babylonian technique for calculating the square-root of a number, developed centuries before Pythagoras, to compute the length of the hypotenuse only to discover that the result does not work, exactly.
While students use Pythagoras’s theorem, the focus in this episode is on the struggle with irrational numbers, something which the theorem exposes. The theorem itself is a useful tool for many practical situations, however it provides a massive conceptual problem for the mathematical purist as it demands an expansion of the number system held at the time.