Infinity, parts 3 & 4

Infinity, part 3

We talked about different types of numbers. First we have the counting numbers:

1,2,3,4,5,...

Then we include 0:

0,1,2,3,4,5,...

Then we include negative counting numbers to get the integers:

...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...

This gets us a "numberline" that looks like a bunch of disconnected dots. If we include rational numbers (all fractions), the numberline fills in. Here's the weird thing: although we can't see any holes in the numberline, there are holes nevertheless - infinitely many holes. We use the irrational numbers to fill in those infinitely many holes that we can't see. The real numbers are the rational and irrational numbers.

The integers are discrete: that is, there are big gaps between them. The rational numbers and the real numbers are continuous: between any two rational (or real) numbers, there is another rational (or real) number.

That was quite enough heavy thinking for one day, so we made Mobius strips.

Infinity part 4

An Infinity Bigger than Omega

We finally got to an infinity bigger than omega. First we did some warm-ups. I wrote down three 3-digit numbers:

123

234

235

and asked for a 3-digit number that was different from the first in the first place, from the second in the second place, and from the third in the third place:

123

234

235

One possible answer is

278

because the first digit, 2, is different from 1; the second digit, 7, is different from 3; and the third digit, 8, is different from 5.

If we do this with infinite decimals between 0 and 1, we can argue that it's impossible for these all to "hold hands" with the counting numbers. That means there are uncountably many real numbers - that is, more than omega-many. This is known as Cantor's Diagonal Argument.

Arc Length

Draw a squiggle on a piece of paper. Now measure that squiggle using only a straight ruler! It's hard to get it just right, but by measuring little lines you can make a pretty good guess. If you make the lines shorter, you get an even better guess.

Zeno's Paradox

Start out 10 feet from a wall. Walk halfway to the wall. Walk half of the remaining distance. And again. No matter how many times you walk halfway to the wall, there's still a distance between you and the wall. So you can never get to the wall! But somehow, you can walk over and touch the wall.

Cookies!

Start with a cookie. Cookie monster eats half your cookie. Then he eats half of what's left. If this goes on forever, how much cookie is left at the end of time?

Infinity, part 5

The first thing talked about today was why "infinity minus 1" doesn't make sense. Look at a list of numbers:

1,2,3,4,5,6,7,...

Adding 1 to a number means moving one number to the right in the list. Subtracting 1 from a number means moving one number to the left in the list. If we try to move one number to the left of infinity, where do we end up? We would have to land on a finite number, but which finite number? Gah! "Infinity minus 1" is undefined, because there is no answer that makes sense.

Then we talked about limits. The "limit" of a process can be thought of as what you end up with if you could actually get to the end of time. Here are several examples.

• Take a cookie. Eat half the cookie. Now eat half the cookie that remains. And eat half again. Continue this for all time. The limit is no cookie left!
• Inscribe a triangle in a circle. Then a square. Then a pentagon. Then a hexagon. Continue. The limit is a circle.
• Take a line segment. Draw dots to divide the line segment into thirds. Then erase the middle third (but keep the dots). With the two remaining line segments, draw dots to divide into thirds, then erase the middle third (but keep the dots). Continue. The limit is the Cantor Set.
• This one's easy to see, but hard to explain: the Koch Snowflake. And here's a good edible approximation.
• Draw a square. Divide into ninths, and remove the middle ninth. Divide each remaining square into ninths and remove the middle ninth. Continue. The limit is the Sierpinski Carpet. We didn't make Sierpinski cookies today, but I'm eyeing that for a future project.
• If you take the idea of the Cantor set (1 dimension) and the Sierpinski Carpet (2 dimensions) and go up to 3 dimensions, you get the Menger Sponge. Someone said this made them dizzy - I said it makes me a bit dizzy, too!

My favorite part of the class was when the kids figured out, all on their own, that the Cantor set contains no lines but infinitely many dots. Wow :)

We also had a discussion about how many sides a circle has. 0? 1? infinitely many? 2 (an inside and an outside)?

Infinity, parts 1 & 2

Infinity, part 1

Last week in "Approaching Infinity," we talked about words. "Finite" comes from the root "fin," which means "end" in French. "Finite" means having an end. "INfinite" means not having an end, or being endless. We also brainstormed words meaning "really big": big, large, gigantic, enormous, etc.. Some words that go along with "infinity" are "endless," "forever," and "eternity."

We identified some of the finite things in the room: a paperclip, our bodies, a pencil. We made the infinity symbol out of human bodies:

I told a story about a mountain made of sand and a bird that carries away one grain of sand every thousand years. When the mountain is gone, eternity has barely even started. Here's a fun version of the story as told by Neil Gaiman.

We started set theory. A set is a container that holds things. Some examples:

{1,2,3}

{nose, Peter, tornado}

We learned how to draw the funny "curly brackets" that surround the set. We also looked at the infinite set of all the counting numbers:

{1,2,3,...}

The "..." part means we keep going forever. We listed some other numbers in this set, like 20, and a googleplex, and one hundred, to make sure all the kids understood the "..." part. This set is named "omega," for which I will write w:

w={1,2,3,...}.

I ended class with the following questions to think about for homework:

• Which is bigger, w or {1,2,3,...,w}?
• For infinite sets, what does "bigger" even mean?

Infinity, part 2

A set is a container. Today we made sets - that is, containers that could hold numbers or other things. The sets included a boat, some origami, and several drawings. Sets can contain sets!

Along the way, it was discovered that if you hold paper and an origami crane over the heating vent, the air makes the paper become "magnetic" and stick to the bird:

We looked at the sets

{1,2,3,4,5,...}

{2,4,6,8,10,...}

and agreed that these sets are the same size, since you can make the numbers "hold hands" (Stanley's expression): 1 and 2 hold hands, 2 and 4 hold hands, and so on, until all the numbers get matched up. Similarly, the sets

{1,2,3,4,5,...,w}

{1,2,3,4,5,...}

are the same size: have w hold hands with 1, then 1 holds hands with 2, and so on, until everybody gets matched up!

{w,1,2,3,4,5,...}

{1,2,3,4,5,...}

The blue squiggly line on the board is inspired by Escher's endless stair.

We also tried to make the endless stair out of bodies. There was mixed success.

Scientific Notation

Renata posted in her blog about this fun animation she found about the scale of the universe.

She wanted the kids to understand what was actually going on with the numbers, so we spent most of last week talking about scientific notation and what it really means to multiply by 10.

Using exclamation points as units, here's 1:

!

Here's 10:

!!!!!!!!!!

And here's 100:

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Each time we multiply by 10, we really get a lot more stuff!

Scientific notation uses these facts:

10^0=1

10^1=10

10^2=100

10^(-1)=.1

10^(-2)=.01

With scientific notation, instead of writing a distance as

320 meters

we would write it as

3.2 x 10^2 meters.

This helps us get a sense of how big (or small) things really are, in relation to distances we have some sort of feel for. 2 meters is relatively easy to think about. 20 meters might be a little harder to think about - but 20 meters is really 2 x 10^1 meters, which is 2 meters repeated 10 times.

We had a great time measuring out our own scale of the universe on the sidewalk, and filling in things like the length of our math class (with and without teachers).

Here's Stanley writing the scientific notation:

10^(-1) and 10^(-2) are pretty small!

More Partitions

Last week I visited Mrs. Carpenter's class and we played the partition game I did with Elaine's class in January. If we have 5 kids, what are the different ways we can group them?

5=4+1=3+2=2+2+1=1+1+1+1+1=2+1+1+1=3+1+1

Measuring Elaine's Classroom

On Tuesday, I was with Elaine's class. They've been working on measuring - gallons, quarts, pints, cups, etc.. I decided to do something a little more physical: using parts of ourselves as the measuring tools. We measured tables in handlengths (learning the word "perimeter" as we did so), the length of the room in foot-lengths, and the length of the room in person-lengths!

For the last one, we worked in pairs. One person laid down on the floor, and the other held a ruler to mark where their head had been, so the lying-down person could get up and lie down again with their feet where their head had been. I'm sorry I don't have pictures, but I was sort of busy helping remember numbers and measuring the room myself! For most of the kids, the room was about 7 kid-lengths. The room was only about 5 Jesse-lengths.

Mathematical Bagels

Earlier this year, Joanna sent me a link to a page that explains how to construct a mathematically correct breakfast. With careful cutting, one can turn a bagel into two linked bagel halves!

The activity as described involves drawing on a bagel with a permanent marker. Unfortunately, if you do that you can't eat the bagel. I figured that by now there must be such a thing as edible markers, so I went hunting online and found some. With the aid of food markers, last Thursday we set about bagel-ing.

Most of the kids *almost* got it to work, but had one half that was just a little too thin and broke. We'll try this activity again later in the semester.

The quote of the day, possibly of the year, was from Alec: "The only thing better than food is math that is food."

Step 1: Meet the bagels, and decide which side is the "front."

Step 2: Draw on the bagels.

Step 3: Cut the bagels.

Ta-da!

(This one is mine:)