I talked to my son about how magical Pl is. How it appears in so many places in nature. It was observed. Ancients measured the circumference of the circle and it’s ratio with the diameter. That remained constant irrespective of the circle – PI. It also shows up in so many places in Physics.
How many people tried calculating PI including 5th century mathematicians in China / India, Archimedes and Srinivasa Ramanujam. We talked about how hard it is to calculate PI. How it is an irrational number and no repeatable patter. Then I said we will beat Archimedes today… in one afternoon.
Creating a sense of mystery and adventure I feel is key to getting kids excited about learning any thing new. They always want to play. Why not?
Then we talked about the intuition behind the Archimedes method . If your draw a polygon (multi-sided shape like a square with 4 sides) in a circle and then keep increasing the number of sides, eventually it will come very close to a circle. Archimedes had a simple way to calculate the perimeter of the polygon and thus approximate the circumference of the circle.
Archimedes stopped at a 96 sided polygon. Using simple excel or a more sophisticated program in python we can go to MNs of sides to estimate the perimeter easily.
The interesting thing about using the Archimedes method is that it requires application of Pythagoras theorem, infinite series and basic geometry.
Learning happens in the mind of the kid. Only when they reconstruct pieces and learn to assemble seemingly unconnected pieces (Pythagoras theorem interacting with infinite series) will real learning happen. This is the essence of Constructionism.