One of the least logical fairy tales I know is the one called The Little Speckled Hen (Курочка Ряба). In it, a chicken lays a golden egg and her owners take turns trying to crack it. They fail at it and yet a tiny mouse runs by, hits the egg with its tail, the egg falls down and cracks. The part that drives me crazy every time is not that the owners of the chicken cry over the cracked egg (shouldn’t they thank the mouse instead?!), but that they cheer up when the hen promises to lay only normal eggs in the future. Really?!
There were only three kids today – V and M, who are regulars, and Little M, who wanted to join us for the experiment part, but ended up contributing heavily to the math exploration part as well.
We started today’s circle discussing this exact point. And then it was time to solve the first problem:
The Hen starts laying regular eggs. Some are white and some are speckled. She lays 9 eggs. The number of speckled eggs is only half of the number of white eggs. How many speckled eggs are there? How many white eggs?
Both M and V stumbled on this problem because both thought that it said “The number of speckled eggs is half of the total number of eggs”. Then V drew 9 eggs and quickly figured out the answer. M is always reluctant to draw anything. But once he sketched the problem, he was able to solve it by testing 1, 2 and 3 speckled eggs.
We moved on to the next problem. In it, the owners of the Hen take those 9 eggs (3 speckled and 6 white) and play a game. They put all their eggs into one basket, close their eyes and take turns picking up a few eggs. The goal is to get at least one egg of each kind. What is the minimum number of eggs they have to pick up to win the game? We modeled this one with the plastic beads.
V’s first idea was two eggs. He tested it and pulled two different colored beads. But when I tried, I pulled 2 blue beads. “Ah, but you have to be lucky!” said V. I asked if there was a way to ensure the win even for an otherwise unlucky player. M, on the other hand, suggested that 8 eggs had to be picked up. It seemed to work, but was this the least number of eggs?
V suggested 3 eggs. M suggested 7 eggs. Neither was able to explain the why. So we worked on it some more. I asked if 4 eggs would ensure the win. It was at this point that V said that since there are 6 white eggs, someone who’s really unlucky can pick 4 and all 4 can be white. After that both V and M were able to reason through the answer.
We then talked about how the Hen was like a machine – she ate food and it got transformed inside her into eggs (ok, with 3 boys in the circle, of course there was also the “and poop” part). We then talked about machines we know – washer, dryer, dishwasher – and how something goes in and something different comes out. Can we put dirty dishes into a washer? Can we run these machines backwards? What happens inside a machine if I know that when I input 1, I get back 0; when I input 2, I get back 0; when I input 3, I get back 0. V suggested that the machine subtracted the number from itself. Big M suggested that the machine multiplied by 0.
Then I gave them a couple of examples of function machines where they had to figure out the functions inside. After that they were eager to try inventing their own machines. This part is always fun. Little M invented a machine that worked completely randomly. You put any number of plastic beads into it and just about anything could come out (a ball, a pepper, a word “mama”). Big M’s machine took any 2D or 3D shape and created a fractal based on that shape. If only one shape went in, it would Droste it, creating nested shapes. But if more than one shape went into the machine (i.e. two triangles, three circles), it created more sophisticated structures (Sierpinski triangle, something that looked like an Apollonian gasket, etc). V created a machine that took whatever it was and turned it into the same object.
They had so much fun inventing their function machines! But it was getting late. Time to do the experiment. Since the fairy tale was about a chicken, it made sense to experiment with eggs. Are eggshells fragile? How come a hen doesn’t squish or crack the eggs when she’s sitting on them? What happens if we try standing on eggs?
Then we tried to break the eggs with our hands. First, we tried doing it by wrapping one hand around an egg and squeezing as hard as we could. None of the eggs broke! Then how come a tiny baby-chick can crack the egg with its tiny beak? Is it because they are tapping from the inside? Then how come I can crack an egg tapping it lightly on the outside?
Sure, a few eggs were broken, but as the saying goes… Well, no math circle next week (holiday), so we’ll be meeting again in 2 weeks.