A Greek letter, the Golden Ratio, and more dancing

Renata's Class:

Following the theme of logic, Renata had the kids playing a game to help them understand the idea of a "fair witness" - that is, one who can only speak things that are true. My main contribution to Renata's class this week was explaining why my tshirt was funny: the symbol in the cat's speech bubble is the Greek letter mu, pronounced "mew".

Susan's Class:

The Golden Ratio is a weird number sort of like pi. Its decimal expansion goes on forever without repeating, and it shows up everywhere. The Golden Ratio is approximately 1.62. It shows up in the proportions of the ideal body according to Leonardo DaVinci, in the Parthenon, and in the shape of credit cards. Sometimes people disagree on whether the Golden Ratio is part of something (such as the dimensions of the Great Pyramid, or the painting of the Mona Lisa).

We started with the idea of DaVinci's ideal dimensions. We paired the kids up and had them measure each other: first the distance from feet to bellybutton, then the distance from bellybutton to head. They divided the first distance by the second, and mostly got numbers between 1.5 and 1.8 - pretty close to the Golden Ratio!

I mentioned that credit cards are in the shape of a Golden Rectangle, and we started looking for other shapes with similar dimensions. The desks? The blackboard? A tissue box? Next time we need to measure other things besides the kids!

Elaine's Class:

More dancing today, with some jobs including the idea of "if and only if". I was impressed by Emi, who really got the idea that "if A then B" means "if A happens you have to do B, but you can also do B if A doesn't happen".

Implication and Dance

In Elaine's class today, we danced. I gave each kid a piece of paper with a job on it. Some of the jobs were

"If someone touches you, fall down."

"Hug people (gently)."

"Act like a Penguin."

Half the class did their jobs, while the other half guessed what the jobs were. Then they switched. We talked about how the rule

"If someone touches you, fall down"

doesn't say anything about what to do if no one touches you. You could fall down or not fall down - either one is ok!

If we wanted you to fall down when someone touched you, and only then, we would have to say so:

"Fall down if and only if someone touches you."

None of the jobs today involved "if and only if", so we'll probably have a future round including some "if and only if" jobs. The kids made suggestions of jobs, so we can use those too.

Truth Tables

On Monday and Tuesday in Renata's class we learned about truth tables.

Since mathematicians are lazy, we abbreviate whole statements by single letters:

A "Maria has a purple pencil."

B "Alec has a blue pencil."

Then we use a truth table to examine what happens if each statement is true or false. The tilde (~) is used to mean "not" or "negation."

A B (A and B) (A or B) ~A

T T T T F

T F F T F

F T F T T

F F F F T

More on Logic

It all began with babies and crocodiles, courtesy of Lewis Carroll.

1. Babies are illogical;
2. Nobody is despised who can manage a crocodile;
3. Illogical persons are despised.

Each of these statements can be written as an implication, a statement of the form "A implies B" or "if A, then B". As mathematicians, we abbreviate such a statement by A -> B. We can replace the statements about babies and crocodiles with implications.

1. baby -> illogical
2. manage crocodile -> not despised
3. illogical -> despised

From the statement A -> B we can conclude that its contrapositive (not A) -> (not B) must also be true, so we can replace our implication for statement 2 with

despised -> cannot manage crocodile.

Then we string the chains of implications together:

baby -> illogical -> despised -> cannot manage crocodile.

Turning this back into English, we conclude that babies cannot manage crocodiles.

After doing several of these puzzles, we had the kids come up with their own. Here are some of their puzzles:

1. All doughnuts are fattening.
2. 0 calorie food is not fattening.
3. Paczkis are big doughnuts.
4. Meijer is selling 0 calorie Paczkis. Can this be true?

1. If you're cool you are a dude.
2. Luke is not a dude.
3. If you're not cool you can't get into the club.

1. All happy people go to Summers-Knoll.
2. All people at Summers-Knoll are smart.
3. Victor does not go to Summers-Knoll.

In the last puzzle, we can conclude that Victor does not go to Summers-Knoll. However, we can't conclude that Victor is not smart. From

SK -> smart

we can conclude the contrapositive:

not smart -> not SK

but we can NOT conclude

not SK -> not smart.

There could be smart people who don't go to Summers Knoll!

Logic Puzzles

Today we worked on a puzzle Renata found in The Man Who Counted. Then I introduced some logic puzzles written by Charles Lutwidge Dodgson, better known as Lewis Carroll. Some of the puzzles may be found here.

Fibonacci Nim

Last week we started playing a game that starts out with two players and a pile of stones. The players take turns taking away handfuls of stones, and whoever takes the last stone wins. The first player may take as many stones as they like, but not the whole pile. After that first move, each player may take up to twice as many stones as the last person took. Passing is not allowed.

A sample game might go like this: we start out with 10 stones. Player A takes 2 stones, leaving 8. Player B may take up to 4 stones, but decides to take only 3, leaving 5. Player A may now take up to 6 stones, so Player A takes the remaining 5 and wins the game.

Depending on the number of stones in the initial pile, either Player A can always win if they play brilliantly enough, or Player B can always win if they play brilliantly enough. After the kids figured out who should win depending on the number of stones, I revealed that the name of the game is Fibonacci Nim.

I won't say more than that, in case someone wants to play you in this game!

Partitions

Today I visited Elaine's class and introduced the kids to partitions. A partition is a way of breaking up a number into a sum of numbers. For example, we can partition 3 in three different ways:

3=3+0

3=2+1

3=1+1+1.

In order to make things fun, we partitioned groups of kids. To partition the number 3, we had a group of 3 kids stand holding hands, and the other kids rearranged those 3 into a pair and a single, or three singles. We got all the way up to partitions of 6!

See Elaine's post for pictures!