Fractions and Farey Addition

Benjamin Leis (who posts at Running a Math Club) flagged this video in response to our recent fractions work: Funny Fractions and Ford Circles (Numberphile). 

Ex ante discussion ideas
The video gave me several ideas for possibly interesting conversations with the kids:
(1) Some basic geometry, particularly for J3. Circles that are tangent, nesting pictures, pictures that have fractal qualities.
(2) Comparing Farey addition and regular addition
(3) Well-defined operations on fractions. I always like to discuss whether the operations gives us the same results regardless of the equivalent form we start with? Farey addition is a good example where the choice of representation is important (indeed, Prof Banahon is careful to keep reminding us that he wants the fractions in lowest terms.)
(4) why do we want the fractions in simplest terms? Possibly relate this to the Cat in Numberland (showing rationals are countable).
(5) what happens if we try Farey addition of three fractions in a row: e.g., (1/5) @ (1/3) @ (1/2)? This is one of the few "naturally occurring" non-associative operations I know.
(6) Since associativity doesn't work, surely distribution of multiplication over Farey addition must not work, right? What about commutativity?
(7) Linking back with our comparison game, if a<b, how do a@c and b@c compare? If a@c < b@c, what can we say about a and b?
(8) what if we allow negative numbers? How should we define Farey addition, then?

How the conversation actually went
J2 was especially taken with the picture of the Ford circles and immediately had two requests: he wanted to draw them and he wanted me to create a pencilcode program to draw them.

The former was a great activity with a lot of figuring and fraction practice. Here he is, hard at work:

Along the way, there was lots of discussion about where to position each fraction on the number line (he scaled with 20 cm as the unit distance from 0 to 1), and how big to make each circle. Tangency condition was a nice check on his work. He would see right away when something was wrong (which did happen several times:

We did talk through some of the ideas on my pre-planned list: is Farey addition well-defined on fractions (no! point 3), does associativity work (no! point 5), could we extend to negative numbers (yes, make the numerator negative seems to work best, point 8).  Other areas are still open for future discussion.

Pencilcode result
I wrote a quick program here: FareyFord. You'll notice that it doesn't actually generate Farey sequences. Instead, it creates generations of fractions, starting from 0 and 1 as the original parents. For each new generation, it uses Farey addition to create a new fraction between each adjacent pair in the previous generation.

Here's a picture of the associated Ford circles:

This method raised an interesting question: what is the largest denominator in each generation? If you don't know, it is cute and worth considering.

More Go (miscellaneous)
Note: this part is unrelated to fractions or farey sequences.

J3 wasn't in the mood to play more capture go with me, but I had an idea. I noticed in one of Nick Sibicky's lectures that one of his students was a young girl, roughly around the age of our three kids. I showed that part of the video to J3 and she made the connection: "this is something girls like me do."

We went and played some silly games on very small boards: 1x1, 2x2, 3x3. In the picture below, we set out a blue-green alternating boundary around a 3x3 board. Then, I asked J3 how many different moves were available. She pointed first to the center, then I asked if there were any other spaces that were the same as the center, if we moved the board around or tipped ourselves upside down.

No, so we made the center red. What other moves? She then chose a side square and figured out that there were three other places that were equivalent. Those became yellow. Finally, we figured out that the four corners were also identical, so that gave us the final picture:

Later, I was playing 9x9 with J2. Instead of go stones, we used Banangram tiles for the white stones. At the end of the game, we tried to make words with the captured tiles from the game. Here was one case where we could (sort of?) make a complete scrabble chain with all the captures:

100 board Go

In past posts, we've shown some of the make-shift materials we are using to play/learn Go without a proper set. Over the last two days, we had experiences that reinforce the value of this approach.

Exploring an earlier pattern
First, when playing with J3, she noticed that we could complete a repeating blue-green-yellow-red pattern around the boundary of a 5x5 board. In a follow-up conversation with J1 and J2, we explored this:

J1 explained why it would work, grouping the boundary tiles as 5 for each side of the playing square and 4 in the corners, so 5 x 4 + 4. This made it easy to see that the boundary would be a multiple of 4 and also made it easy to extend to any square board: n x 4 + 4.

J2 had a new idea. He thought we were talking about the pattern continuing as an inward spiral.  That gave us this design:

This also led to discussions about symmetry (the blue and yellow have reflectional symmetries that green and red lack) and further investigation on boards of different sizes. Interestingly, we found that, for some boards, none of the colors have a reflection symmetry.

Trying some tsumego
I set J1 and J2 the following challenge (J3 was watching): are the blue cubes alive or dead in each of these two clusters?

Putting aside the interesting Go discussion that resulted, there are two consequences of doing this on the 100 board: extra unnecessary information and a built-in coordinate system. By unnecessary information, I'm talking about the letters on the white tiles and the numbers on the 100 board. This is information that is entirely orthogonal to solving the life-and-death puzzles. This is a simple toy version of one of the key modern challenges in problem solving: identifying which information is useful and which is a distraction.

On the other hand, for talking about the puzzles, we could say things like "what if white plays a tile on 99?" For J3 who was watching, this offered another little example of the idea that numbers are all around.

Some capture fun

For the last example, I set out some white tiles (some alone, some in groups) and asked J3 to capture them with blue cubes. After we did that, we counted the number of cubes we used to capture by moving them to cells in the 100 board (note that we removed one of the lone white tiles). A fun counting exercise, an opportunity to talk about groups of 10, and more familiarization with the layout of the 100 board.

Teaching math with Go

Recently, I have been insinuating Go playing into my time with the 3 Js. This was initially motivated by a quote I saw on one of the Mathpickle pages (Gamers under Inspired People):

Schools should experiment teaching go* instead of a regular math curriculum for one year to students around the age of 7.  It is my prediction that the strong problem solving skills that this will engender will make superior students than any existing mathematics curriculum.

Now, when we first decided to have kids, my objective was to help them develop into people with whom I would enjoy spending time. In particular, I wanted to be able to play games with them. With that in mind, the Mathpickle idea resonated with another idea from Richard Garfield (via Math Hombre):

play each game so as to increase your chances of winning all games

With these three ideas in mind, I went looking for a way to properly introduce Go to our clan.

Curriculum outline

Not surprisingly this is a question other gaming and math people have asked before. Quickly putting together the ideas I liked the most from other sources, we basically started following the curriculum shown in the Go GO Igo videos with Yoshihara Yukari (Umezawa Yukari at the time of filming):

Yukari Sensei Go Go Igo Part 1

(1) basics

- placing stones

- black vs white

- capturing single stone

(2) capture game

- 6x6 board

- first to capture wins

- etiquette

(3) illegal moves

- playing where your stone will have no liberties

- playing where the stone has no liberties but captures an opponent's stone(s)

(4) expanded capture games

- first to capture 3 stones wins

- infinite capture

- Ko rule

(5) territory

- counting territory at the end of the game

(6) simple capture puzzles

- one move

- two moves

- three moves

Yukari Sensei Go Go Igo Part 2

(7) Etiquette: 

- Nigiri: choosing white vs black

- komi and first player advantage (maybe useful to play some 5x5 or 7x7 games to make the first player advantage clear?)

(8) eyes and false eyes

(9) Scoring

- Dame, 

- kyu, 

- Japanese vs Chinese scoring

- agehama: stones considered captured 

How important is this?

(10) standard patterns

- stair-step (shichou)

- geta (also kosumi? 45 degree cut to capture enemy stones)

Part 3

(11) more puzzles/standard patterns

Part 4

(12) Tsumego

(13) maxims

Some early lessons

Since we don't actually have a Go board or stones, we started with the electronic board CGoban. This works well for J1 and J2. We have also used J1's chess/checkers set as a makeshift 9x9 board (playing on the lines instead of the squares).

For J3, we started playing the simple capture game using the blank side of our 100 board.
For the first lesson, we arranged things like this:

She played the centimeter cubes (which substitute for black stones) and I played the Bananagram tiles (substituting for white stones). I gave her a four stone advantage and we played three games with me starting in different places (center, corner, side) and saw that she could easily capture at least one of my stones without trouble.

Some of J3's observations along the way:

• There are 11 blue tiles forming the boundary
• There are 25 squares in our playing area
• There are five squares along each edge of our playing area
• The placement of the four handicap stones is symmetric in the playing area. There are several symmetries
• Stones in the corner have two directions to live
• Stones on the edge have three directions to live
• Stones inside have four directions to live

For the second lesson, we made the board a little differently based on J3's preference for a blue-green-yellow-red pattern around the border:

This time, J3 made some different observations:

• The pattern continues around the border (at no place, did we have to break the pattern). A more advanced question: will this always happen with our Blue-Green-Yellow-Red pattern around a square board?
• The colors in opposite corners are the same (blue-blue and yellow-yellow)
• There are more than 11 tiles on the border now.
• Still 25 squares on the board and 5 squares along each side

For the third session, J3 was willing to reduce her starting advantage and she wanted to place the handicap stones herself:

This is a losing position (remember, we are still playing where the first to capture at least one stone is the winner):

She didn't take losing especially well, but this is a nice feature of playing these kinds of short games. The kids can make a mistake, they have to deal with failure, but it isn't very costly since each game only takes a couple minutes and the next game starts right away.

Some Go concepts we are still developing
At this stage, we are still working on the basic concepts:

1. once placed, the stones don't move
2. only the main compass directions (north, east, south, west) are liberties. Diagonals don't give life.
3. liberties are shared for a group, not just the individual stones. For example, a stone surrounded by its own color is not dead (if the overall group still has liberties).
4. I need to remember to announce "Atari" when a stone or group has only one remaining liberty.

Observations

From a Go/games perspective, I think it is helps to start playing a lot of low-cost games: fast games where the winning condition is easy to identify and immediate. This allows the kids to make mistakes, see clearly the consequences of those mistakes, and lose, then immediately try again.

From a math perspective, there is a huge amount of elementary math that comes out of the simple games:

1. counting
2. addition
3. patterns
4. some basic multiplication, particularly with the array model

In addition, we had the usual experience with using physical manipulatives: something extra always comes up. For example, using the 100 board inspired J3 to show off to me that she can count to 100 now (using the board as a reference).

I'm looking forward to future sessions.