One night my family ordered Chili’s to-go and my son’s food container was square and my wife’s was octogonal. My son asked, “I wonder why the container is an octagon?”
I thought about it for a moment and toyed with the idea that is had something to do with the cost.
I asked my son which container would hold more and which had more material. I then asked if there could be a square container with the same volume as the octogonal container. He said there couldn’t be. 🙂
At this point, I simplified the problem to areas and not volumes.
It took a moment for my son to envision a square a bit “smaller” than an octagon might have the same area. I asked if the areas are the same, how about the perimeters? I posited that the more sides, the less perimeter for the same amount of area. I then asked which shape would ‘contain’ the most area with the least perimeter.
He thought about it for a while…at first said something about the sides would be infinite.
Then he had an aha moment that you could see on his face…”a circle!”
I led him through the math of calculating the area of an octagon with a side of one, setting it equal to pi*r*r, and solving for r. I then asked him for the formula of the “perimeter” of a circle. He said, “you mean circumference?” Yes! We calculated that the circumference was 7.7 and the perimeter of the octagon was 8. I asked about a square and had him guess if it was more or less than the octagon. He guessed more and we calculated it…8.7. He said we should take our scratch paper to the Chili’s folks and tell them how they could save money on packaging material if all their containers were circular.
Some of his unsolicited quotes after our discussion:
“Who knew math could be fun.”
“It wasn’t like you were teaching it…it was like I was discovering it.”
I hope I have lots of moments like this when I’m teaching.