Here's some of what we've been up to in math recently.

Card Tricks

I start with a deck of cards. I make 6 piles of cards, face up. Then I turn each pile over, so they're face down now. The kids choose 3 piles to keep and give me back the other cards. Then they turn over the top cards on two of the piles. I perform some magic, and tell them the top card on the remaining pile. Of course this trick has to do with math. Maybe I'll reveal the details in a future post.

Measuring Slope

I made a slanted line, and the kids measured my slope. See Renata's post for the story and pictures.

Russell's Paradox

We talked about Russell's Paradox, the short version of which is

"The barber shaves every man who does not shave himself.

Who shaves the barber?"

A more detailed version, with fewer loopholes, is presented here. The kids were great at finding loopholes in the version I gave ("Maybe the barber's a woman!", "The barber is bald!").

Set Theory and Infinity

We talked about sets and about union, intersection, and set difference. Start with two sets,

{1,2,3,4,5} and {2,4,6,8}.

Their union is the set of stuff that's in one or both of the sets:

{1,2,3,4,5,6,8}

Their intersection is the set of stuff that's in BOTH sets:

{2,4}

The set difference {1,2,3,4,5}-{2,4,6,8} is everything in the first set but not the second set:

{1,3,5}

Union, intersection, and set difference can be represented with Venn Diagrams (at least when we're only working with 2 or 3 sets).

We revisited the idea of defining the numbers as sets, only this time we went further.

0={}

1={0}

2={0,1}

10={0,1,...,9}

100={0,1,...,99}

Any whole number can be defined as a set in this way. We can also define infinity as the set of all whole numbers, called w (the Greek letter omega):

w={0,1,2,...,100,...1000,...}

We can add 1 to any whole number, and get "the next" whole number (or set). We can also subtract 1 from any whole number (except 0) and get "the previous" whole number (or set). "Infinity minus 1" doesn't make sense, though, because there is no "previous" set. There is no set that comes immediately before w.