Adventures in Origami

I've been on an origami kick recently, and encouraging the habit in the kids.

Origami is useful for a great many things: fine motor control, intuitive understanding of symmetry ("now fold it the same way on the other side"), and three-dimensional visualization ("hold your model in the same orientation as the picture"), to name a few. It's an activity where being careful really does matter if you want to end up with something recognizable, and where practicing makes you better pretty quickly.

Origami EB

During this six-week class we made paper cranes, warblers, and Sonobe Units which can be assembled to form a cube. My favorite class was the one where we made paper cranes out of pieces of paper that started as 5-foot squares!

Three-Headed Dragons

Lukas, Alec, and Max are currently working on the three-headed dragon designed by John Montroll. This is a study in persistence. So far we've spent at least two hours on this beast, and we might be halfway done. The dragons, mine included, currently look like blobs with little tails sticking out.

Area and Volume

Start with a small square piece of paper, side length s. The area of a big square piece of paper with side length 2s is 4 times the area of the small paper, as we can see by putting 4 pieces of origami paper next to each other.

If we make a cube with small pieces of paper (side length s), we get a small cube. If we make a cube with big pieces of paper (side length 2s), we get a big cube. How many small cubes fit inside the big cube? Guesses included 4, 5, 6, 8, and 16. By making a big cube and putting a small cube inside it, we could see that 2 layers of 4 small cubes, for a total of 8, would fit inside.

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:)

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!

Modular Origami and Fibonacci Numbers

On Monday, Renata was out sick and I got to do math all by myself. First we worked in Singapore books, then we did modular origami. I taught those who hadn't learned yet how to make a Sonobe unit (less technically known as a "box side"). Six units can be assembled to make a cube. More units can be assembled to make other things.

We also made sure everyone was familiar with the Fibonacci sequence, which is the sequence of numbers

1,1,2,3,5,8,13,21,34,55,89,...

The first two numbers are predetermined. To get each number from the third number on, we add the previous two numbers. The next number if we continue the sequence above would be

55+89 = 144.

Today, Tuesday, we reviewed how to write numbers in base 3. It can be helpful to break the numbers up first - for example, to write 17 in base 3, first we notice that

17=9+6+2

and then we can write 17 in base 3 as

122.

We also broke numbers up into sums of Fibonacci numbers. The rules:

• Use the biggest Fibonacci numbers you can
• No Fibonacci number may be used more than once.

Some examples:

11=8+3

12=8+3+1

40=34+5+1

We wouldn't write 40=34+3+2+1, because we can write 5 instead of 3+2, and the first rule says we should use the biggest Fibonacci numbers we can. We also wouldn't write 12=8+2+2, because we're not allowed to use any Fibonacci number more than once.

The kids also came up with their own Fibonacci-type sequences, by deciding what they wanted the first two numbers to be, adding those to get the third number, and so on.

Here they are writing their sequences up on the board:

And here are the completed sequences:

Finally, here's Maria's origami "other thing," using 12 units:

Beyond Base 10

This week we looked at numbers written in different bases.

We usually write numbers in base 10. In base 10, the string of digits

321

represents the sum of 3 hundreds, 2 tens, and 1 one. In base 10 the places are

..., ten thousands, thousands, hundreds, tens, ones.

Another way to say this is that the places are powers of ten:

..., 10^4, 10^3, 10^2, 10^1, 10^0.

We could also write numbers in base 2, in which case we're only allowed to use the digits 0 and 1. The places in base 2 are powers of two instead of powers of ten:

..., 2^4, 2^3, 2^2, 2^1, 2^0

or in other words,

..., sixteens, eights, fours, twos, ones

In base 2, the string of digits

1101

represents the sum of 1 eight, 1 four, 0 twos, and 1 one for a total of thirteen.

We could also write numbers in base 3, or base 4, or any base we like. Computers use base 2 a lot. They also use base 16, better known as hexadecimal. In base 16 we use the digits from 0 to 9, but we also use letters:

10 A

11 B

12 C

13 D

14 E

15 F

The place values in hexadecimal are powers of 16:

..., 4096, 256, 16, 1

Thus the string

10AF

represents the sum of 1 four-thousand ninety-six, 0 two-hundred fifty-sixes, 10 sixteens, and 15 ones for a total of 4271.

The class solved the following puzzle:

If only you, I, and DEAD people know hexadecimal, how many people know hexadecimal?

Cryptology

In December we studied codes and properties of the English language. The kids worked in two "competing" groups.

Group 1 came up with this message:

HI STANLEY I LIKE PIZZA THIS CODE IS EASY.

They encoded their message with the classic Caesar Cipher (where each letter is shifted 3, so instead of A we write D, instead of B we write E, and so on). The coded message was:

KL VWDQOHB L OLNH SLCCD WKLV FRGH LV HDVB.

Group 2 broke the code by recognizing that "HI" is a very good word for starting a message, and then seeing that the whole alphabet had been shifted by three characters.

Then Group 2 came up with a message:

HELLO PEOPLE OF EARTH,

THE SKELETAL DEVIL HAS COME OUT OF THE EARTH, FOR I PROSPERO THE MAGICIAN HAVE LET HIM OUT! FOR LUNCH.

P.S. HE WANTS TO EAT YOU ALL.

They encoded this by a random letter substitution, so the coded message was

WEUUI NEINUE IX EGHVW,

VWE YLEUEVGU TERBU WGY QIDE IFV IX VWE EGHVW XIH B NHIYNEHI VWE DGPBQBGZ WGRE UEV WBD IFV XIH UFZQW NY WE KGZVY VI EGV OIF GUU!

Group 1 broke the code using some useful properties of the English language. They knew B should mean either A or I, since those are the only valid one-letter words in English. They also figured out that VWE was THE, since THE is one of the most commonly occurring three-letter words in English. Since the alphabet was all mixed up instead of shifted, this code was pretty tricky to break.

Along the way, I had the kids pick storybooks they were reading and count how many times different letters occurred. This gave everyone a feeling for how common different letters are. For example, T and E show up all the time, but J and Z and Q hardly show up at all. This can be useful in cracking a code - the most frequently occurring symbol probably doesn't mean Z, unless the message is about pizza!