Lesson 17- Basic Proofs

Everybody needs a mouth:

If we didn’t have a mouth, we wouldn’t be able to eat.

If we weren’t able to eat, we wouldn’t get energy.

If we didn’t have any energy, we would die.

Therefore, everybody needs a mouth.

Does this make sense? Is it correct?

Actually, for this problem, I asked the kids to prove why we need a mouth instead of telling them myself. It’s always better to have the kids doing something or figuring something out instead of just sitting there and listening (or in some cases, not listening).

How can you prove that you breathe with your nose and not your fingers?

They couldn’t quite figure out how to prove it at first, but then they closed their noses and said that they couldn’t breathe, and then they made fists and squeezed their fingers and they could still breathe.

Circular Reasoning- Do these make sense?

1.       

“This is a dinosaur bone!”

“How do you know?”

“Because it was stuck in this billion year old sand.”

“How do you know the sand was a billion years old?”

“Because there was a dinosaur bone inside it.”

At first, they thought that this made perfect sense, but then I drew them a diagram, and they realized that the reasoning was going in circles.

2.       

“I am not a monster. I don’t even know what a monster is.”

“How do you know that you are not a monster if you don’t know what a monster is?”

“Because if I was a monster, I would know.”

I prompted them with a few hints which were actually questions AFTER I LET THEM THINK A LITTLE BIT ON THEIR OWN. I asked them questions like “Can somebody know that they are something (or in this case, not something) if they don’t even know what that thing is?” Soon they realized that was I was saying was correct. I gave them a couple more variations of this problem after they solved just to make sure that they understood. For example, I said “If you were an apple, and somebody came up to you and said, “you are an apple” and you said “ I am not an apple. I don’t even know what an apple is”, would that make sense?”

Chairs:

In a rectangular room, put 8 chairs around the walls so that there are 3 chairs at each wall.

We did a few easier examples before this one. For example, I asked them to put 12 chairs around a square room so that there would be an equal number of chairs at each side. Then I asked them to figure out a strategy for placing an equal number of chairs at each wall to add up to a certain number (the solution was to keep adding one chair to each wall in a circle until there is enough chairs). Then, after a few tries, most of the kids figured out the answer to the problem and THE KIDS THAT FIGURED IT OUT ALREADY EXPLAINED IT TO THE KIDS THAT DIDN’T UNDERSTAND.

Estimate:

a)         How many people are in the house

b)         How many poker chips I am holding

c)         How many poker chips are in this bag

d)         How many people live on this street

e)         How many apples equal the weight of you

The answers varied wildly. Some people said that there were a thousand poker chips in the bag, and some said that there were 5 (there were actually about 50 of them). Also, before we did “how many apples equal the weight of you” we did “how many apples equal the weight of a cat” because that was an easier variation.

General Comments:

I was planning to fit in a lot more things in this lesson, but we spent a lot of time on each problem (which is actually a good thing, because it’s better to have all of the kids completely understand only a couple of problems instead of having the kids have no clue what’s going on in many problems).