Playful Math Education Carnival #119

Welcome to Playful Math Education Carnival #119! Just to be clear, that exclamation is to express excitement, not factorial. Fortunately, you will have a bit of time before there's any danger of confusing this post with the edition (119 factorial).

Anyway, only the very coolest folks get to handle a MTaP edition that can be written with a factorial. And I just realized how close (and yet how far) I was to such glory.

119 Fun facts

• 119 is the number to call for emergency services... in parts of Asia (wiki reference).
• Of course, 119 backwards is 911 which is the US emergency services phone number
• 119 is aspiring, the sequence formed by summing proper factors ends with a perfect number.
• 119 isn't prime, but it almost feels like it
• 119 = 7 x 17. I don't think products of consecutive primes ending in 7 has a name, but maybe it should?

Do you have other fun facts about 119? Please? Please?

Dedication

I'm saddened to note the passing of  Alexander Bogomolny this month, and I dedicate the edition of the  carnival to him. The material he developed and made available on his site

https://www.cut-the-knot.org/ is truly amazing and remains with us for our benefit.

Miscellaneous

Aperiodical has an article from Benjamin Leis on the Big Internet Math-off.

Something James Propp wrote as part of the Big Internet Math-off: A pair of shorts. 

Elementary

I was reminded of the game of Chomp! in Shecky Riemann's linkfest (most of which isn't elementary level, but worth investigating).

Cathy O'Neil tells her mathematician origin story. I hope all our kids can have an empowering math experience like this.

Discussion of a "square dancing" puzzle from Mike Lawler: part 1 and part 2. I think there is a lot more to explore here and hope some of you will write parts 3 and beyond... 

I always love game discussions. Set is a game you probably all know, but in case you don't here's an intro and a deeper analysis in the Aperiodical.

Pat Ballew writes about divisibility rules. Pat also discusses a fun XKCD in prime time fun.

I'm delighted at how this starts with something many take for granted (12 hour vs 24 hour time of day conventions) and then builds a fun exploration.

Middle school

Have you been waiting for someone to write the perfect post giving you an introduction to tons of Desmos activities? Well, Mary Bourrasa has done it for you.

Michael Pershan tweeted a pointer to a nice collection of logic puzzles on puzzling stackexchange.

Denise Gaskins pointed out a past note about factor trees and some cute wordplay from Danica McKellar's book: prime numbers are like monkeys.

This segues directly into a review of two number theory books by Ben Leis (also the author of the Big Internet Math off post above) in which he discusses some other visualizations beyond factor trees: 

High school

Mr Honner: math around us and parallel lines cut congruent arcs on a circle in a nice picture. 

Ben Orlin invents and illustrates a new adage that there are no puddles in mathematics, only oceans in disguise. 

More advanced

Mathematical theorems you had no idea existed because they are false: https://www.facebook.com/BestTheorems/

Have fun finding counterexamples. Also, link disproves the conjecture that there is nothing worthwhile on facebook.

The Scientific American Blog has been running these columns on "my favorite theorem." Go back and take a look (I think this was their first one): Amie Wilkinson's favorite theorem.

A fascinating discussion of the Fields' Medal and some ideas about what it should be supporting. 

What was the score? Maybe the sum of scores was 119?

Math Teachers At Play Carnival #110 Summer Vacation Edition

Hello again math folks! I've been in the middle of a major transition, moving between Asia and North America, so haven't really had time to post recently. Putting together this month's carnival was a nice opportunity to see what everyone else has been writing about and get some new ideas!

As you scan through the links I've highlighted, please don't get too fixated on the grade level splits. These are really approximate and I expect you will find worthwhile activities for all ages in every section.

In Memoriam: Maryam Mirzakhani

On the 14th of July, Maryam Mirzakhani passed away. She was the first woman to win the Fields Medal. It would be wonderful if you could do some exploration in her honor this month. One of her areas of research was on pool tables. Here are some places to get an idea of the way mathematicians have been inspired by this game:

• Elliptical pool (from Numberphile)
• The Illumination Problem (also Numberphile)
• A snooker (problem 3 from the 2013 International Tournament of Young Mathematicians)
• Wolfram MathWorld Discussion of Alhazen's Billiard Problem that indicates the link with optics

If you find other kid-friendly projects related to Mirzakhani's work, please tell me in the comments!

Some 110 facts

This was the best number carnival to be able to host, because:

• 110 = 10 * 11. That means it is pronic, the product of two consecutive integers.
• 110 looks suspiciously like a binary number. Binary 110 = decimal 6. Decimal 110 = Binary 1101110, which I like to read as 110 1 110
• Because it has an odd number of 1s in its binary expansion, 110 is odious
• 110 is a Harshad number because it is divisible by the sum of its digits
• The element with atomic number 110 is Darmstadtium (Ds).
• 110 is the number of millions of dollars spent in March for a Basquiat painting, the highest amount paid at auction for a work by an American artist.

A number talks picture that caught my eye

I'm not sure there is anything especially 110 about this picture, but there are a lot of mathematical questions to ask and things to observe here:

In a related vein, if you and your kids need some mesmerizing math gifs, take a look at Symmetry.

Elementary skills

Denise Gaskins has written a lot to help parents engage playfully and mathematically with their kids. In this blog post, she has collected highlights that are great with young students and worth remembering for older ones, too: How to Talk Math with Your Kids.

I love board games and think there is still tremendous value in the physical games that electronic versions miss. Here's an example from Sasha Fradkin, where cleaning up after playing gives us a chance to think about whether skip counting is just a chant or if the words mean something: Skip counting or word skipping

While she's at it, Sasha Fradkin also has a nice puzzle activity with Numicons. I would think of this as a progression step toward tangram and other dissection puzzles.

Which one doesn't belong is a math meme you should know already. If you don't, ask in the comments and I'll point you in the right direction. Christopher Danielson has recently introduced Which Poster Doesn't Belong? While you are visiting his blog, enjoy his story about The Three Year Old Who is Not a Monster.

Exploding Shapes is a catalyst for notice and wonder from The Math Forum. I really like this because here are many different directions to go and no single "right" answer. Also, let's give a cheer because it looks like this recent set of posts shows the math forum folks have returned to posting nice conversation starters.

Swine on a Line by Jim Propp is a nice game/puzzle that seems a great companion to James Tanton's Exploding Dots. Hmm, maybe July 4th inspired me to look for lots of explosions...?

Middle school(ish)

Rupesh Gesota starts with a nice puzzle and shows us how it was analyzed by several different students: One Puzzle, Many Students, Many Approaches. I particularly like how the introduction to the puzzle encourages us to think of different methods.

Mike Lawler has done a huge number of really great explorations with his kids. Here are some recent projects with books from the Park City Mathematics Institute: Playing Around. If you haven't been following Mike and his kids, I really encourage you to go through his past posts.This blog is fantastic for great projects and connections with other resources.

Curious Cheetah shows us several ways to calculate square roots. I would say, like long division, the value isn't in memorizing the algorithms, but understanding how they work and using them to play with numbers.

Manan Shah has a couple of nice summer explorations. The first is an excursion into the digits of prime numbers: Prime Numbers. The second is a coin flipping and gambling game to ponder during these warm vacation months: Summer Excursion Coin Flipping.

There are other, problem-based, posts on Benjamin Leis's blog, but this one made me jealous of his recent purchase of the A Decade of the Berkeley Math Circle.

High school/more advanced

Thinking Inside the Box, Simon Gregg takes a new look at a familiar shape, the cube. His comment about the exploration really nicely captures something that is beautiful about mathematical exploration: "I came back to a familiar place from an unfamiliar starting place."

A cute absolute value game now appears as a nicely animated game: Absolute Value. I think this is a nice simplification and implementation of the original game.

There's an improv game where the players have to switch between movie genres. Film noire or "hard boiled detective" comes up every time. This TedEd video could introduce fractals and this film genre at the same time.

Michael Pershan puzzles over two measures of steepness in his trigonometry class: When Measures of Steepness Disagree. I really like the questions he raises about how to use two different scales that measure the same concept, but are not linearly related.

Also, Michael links to the New Zealand Avalanche Advisory, with a nice graphic showing a case where the greatest danger of avalanche is in the middle of a slope range:

Dave Richeson breaks down an impressive rainbow photo:

Fair sharing is a really interesting theme to motivate a lot of great math. Tanya Khovanova looks at a couple of fair sharing problems and strategies in Fair share sequences.

Also, check out the sister carnival to this one: The Carnival of Mathematics over at The Aperiodical.

Techniques for teaching

This post is an old classic, but I've been reminded of it because it is used in a workshop that I frequently attend: using student reflections.

Have you visited NRICH recently? No?!?! Go over now (here's the link) and find a really cool activity to do with your kids. Seriously!

Some tips on giving feedback: Effective feedback for deeper learning.

Using Desmos to check your work: Desmos is the new back of the book.

A thought piece on the modern role of teachers: Teachers Sow Thirst for Learning. If you can read Indonesian (which I can't) you may find some other interesting pieces here on math education.

A special announcement

James Tanton is leading a project for a world-wide week of math this fall. Please take a look at the project page Global Math Week

Some simple dice games

J3 and I played several simple games recently that I want to record. One of them, race to the top, is a variation of a more sophisticated game that can be used more generally.

Digits three in a row

Materials: a 100-board, 2d10, colored tiles
Players: 2
Goal: mark three spaces in a row
Basic play:
Two players take turns rolling the two dice. On a player's turn, they form a 2 digit number using the two digits and then claim that space on the 100 board. If the space has previously been claimed, they lose their turn.

The first player to claim three adjacent spaces in a line (horizontal, vertical, or diagonal) is the winner.

Variations:
(1) we used one dice marked 0-9 and another marked 00 - 90 (all multiples of 10) and then added the values to get our two digit numbers. This eliminated any player choice, but helped reinforce the idea that the value in the 10's digit is the number of tens.

(2) the winning condition can be increased to require a line of 4 spaces

(3) the winning condition can be changed to allow any three (or four) spaces that are colinear; these spaces would not have to be adjacent.

Notes:
This is a very simple game, especially the variation we played, but J3 found it fun. It was a useful exercise to practice locating the numbers on the 100 board.

Race to the top

Materials: An 11x 6 grid with one side labelled 2 to 12, 4d6, 3 tokens. Optional: 11 distinct tokens/small objects per player.
Players: 2 - 4
Goal: capture 3 columns.
Basic play:
This game is fairly simple, but has resisted our attempts to succinctly summarize the rules. Here is an explanation as we play two turns.

Here's our playing material, the three green squares are temporary markers:

First player, J1 rolls two ones and two fours. With this roll, J1 could group them into two fives or a two and an eight:

J1 decides on two fives and puts a temporary marker on the second level of the 5's column. After you understand the rules consider whether this choice is better or worse than the 2 and 8.

J1 chooses to continue rolling and gets 1, 1, 4, and 6. J1 groups these as a 5 and a 7, then moves the temporary marker in the 5's column up one level and adds a marker at the bottom of the 7's column.

J1 rolls a third time, getting 1, 1, 2, and 6. The only option is to group these as 3 and 7, so J1 places a temporary marker in the 3's column and advances the marker in the 7's column.

At this point, J1 ends his turn and marks his progress. On his next turn, if he gets a 5, for example, the temporary marker will start at the fourth level of the 5's column (building on his consolidated progress).

D has the next turn. He gets 1, 4, 4, and 5, which he chooses to group as 5 and 9:

D chooses to roll again, getting 4, 5, 5, and 6. This has to be grouped as 9 (advancing in that column) and 11:

D chooses to roll again and gets 2, 3, 5, and 6. This is a lucky roll that can be grouped as 5 and 11, allowing two tokens to advance:

D presses his luck and rolls a fourth time, getting 1, 3, 3 and 5. The dice can't be paired to get a 5, 9 or 11 and there are no more temporary markers available to place, so D loses his progress. J1 will have the next turn.

To be clear about the failure condition: the player must be able to place or advance a temporary token for both pairs of dice. For example, if D had rolled 1, 3, 3, and 6, he still would have lost his progress.

Further rules:

• the first person to end their turn on the sixth level of a column "claims" that column.
• the first person to claim 3 columns wins the game.
• columns that have been claimed by any player are safe values for all players. Players do not need to allocate a temporary token to those columns.
• Players can occupy a square that an other player has marked.

Variations:
(1) Change the winning condition so that the first player to capture a column wins
(2) Change the height of the columns, either fewer than 6 for a faster game or more than 6 for a slower game
(3) Change the failure condition so that only one pair of dice needs to be playable and reduce the temporary tokens to 2.
(4) only allow each player to roll one time. This eliminates the "press-your-luck" aspect of the game and is much more basic.
(5) allow players to jump over a square that has been occupied by another player. This rule particularly fits well if you use objects to record your consolidated progress (which also makes the grid re-usable).

Notes:

I was originally taught this game by Mark Nowacki of Logic Mills.

J3 and I played the variation where each player only rolled one time on their turn.

Good games and bad

Recently, we have been playing the following games:

1. Go (baduk, weiqi, หมากล้อม). For now, we are playing on small boards, usually 5x5 or smaller.
2. Hanabi
3. Cribbage
4. Qwirkle (not regulation play, a form of War invented by J3 and grandma)
5. Munchkin
6. UNO
7. Vanguard

I've ordered these by my own preference. In fact, I would be delighted playing just the first two exclusively and am happy to play cribbage or Qwirkle when asked.

For the other three, I find myself biting my tongue a bit and grudgingly agreeing to be part of the game. I'm in the mood for strategic depth and a moderate (but not large) amount of pure chance. Part of my feeling was echoed in a recent My Little Poppies post: Gateway Games.

However, as in the MLP post, I recognize that my enjoyment of the game is only a part of the reason for the activity. I guess the kids' enjoyment counts, too. 

Beyond that, even the games with limited depth are helping to build habits and skills:

• executive control: assessing the situation, understanding what behavior is appropriate, understanding options and making choices.
• general gaming etiquette: taking turns, use of the game materials
• meta-gaming: helping and encouraging each other, making sure that the littler ones have fun, too
• numeracy and literacy: every time a number or calculation comes up or when something needs to be read, they are reinforcing their observation that math and reading are all around them.
• meta-meta gaming: game choice, consensus building, finding options that interest and are suitable for all the players, knowing when it is time to play and when it isn't.

As a family, and a little team, they are also building a shared set of experiences and jargon as they absorb ideas from each of the games.

All of these are, of course, enough reason to make the effort to be open minded and follow their gaming lead.

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:

Closest neighbor one-on-one

In my last post, I wrote about playing Denise Gaskins' closest neighbor fraction game with our 4th grade class. Yesterday, I spent time with J2 and used the game as a semi-cooperative puzzle.

This activity worked really well and the experience gave me some additional ideas about how to use the core ideas again with the 4th grade class.

Puzzle or game?

First, there were only two of us, one a kid and another an adult, so that background naturally makes the activity very different. As the key modification for play, we played all of our hands open and helped each other find the fraction in each of our hands that was closest to the target for that round. Then, we worked together to determine which of those two "champions" was closest overall.

Some of the consequences:

• the activity was not really competitive (see below)
• J2 had to do a lot more fraction work.

Let me explain the second point here. Because we were looking for the best play, J2 had to consider all of the combinations in his hand (20 choices). Some of those can be rejected quickly with simple analytical strategies depending on the target. Even this is good number sense thinking. Also, some combinations are close competitors and need to be analyzed more carefully.

If we were playing with closed hands, he could choose two cards, play a fraction based on them, and I wouldn't be able to say anything about whether those were his best options or not.

Second, while I write that "we worked together," as a sneaky dad, that means that I pretended to do work, while actually getting J2 to analyze my hand as well as his. Really, the only thing I offered was an alternative comparison strategy, once he had already worked through his own approach.

An example of some strategies

We found that some of the comparisons that arise naturally in this game are quite tricky, even for me. For example, quickly tell me which is closer to 1/3: 1/5 or 4/9?

We found that placing the fractions on a number line was a really helpful strategy for many of the comparisons. We also made very heavy use of the two strategies involving common numerators or common denominators.

Finally, you can see in this example that J2 is comfortable mixing decimals and fractions, for example converting to 1/2 to 3.5/7 to aid some comparison:

Our grid

Through our play, we filled out this grid, taking turns putting in our best results and congratulating each other when our hand was the ultimate champion for that round:

Competition and Strategic thinking

I was particularly pleased by one comment J2 made about this overall game: "this is mostly luck, how well we can play depends on the cards we get." This comment came after one round where he had several duplicate cards in his hand, reducing the number of distinct values he could play. We've discussed elsewhere my goals of helping the kids think about game structure, so I always love it when they bring those ideas up themselves.

Some thoughts about competition. While we played this game non-competitively, I'm not opposed to competition nor do I think that this game always needs to be played non-competitively. Ultimately, my litmus test is how to play in a way that is the most fun. If I were a more serious educator, I suppose I would also consider which way is the most educational, too.

It won't always be obvious what is the best way to play each game. In this case, I got to benefit from the prior experience with the class and my close knowledge of J2. Many times, I'll tell the kids that there are several ways to play and we'll try them out together, then review the experience.

Among other things, this is why I love handicap games like Go. By adjusting the starting advantages, we can create scenarios where it is very competitive and very fun, even though the players have very different levels of experience and current strength in the game. And also, there are things we can do together when we want a non-competitive activity.

Ideas for going back to class

From this time with J2, here are my ideas about taking the game back to the 4th grade class are:

1. Spend a lot of time on fraction comparison strategies before we play
2. Reduce the number of cards dealt to each player
3. play as teams
4. convert to open hands with a lot of talk about why we chose particular plays

An actual puzzle

As a reward for reading down this far, here's an actual puzzle related to the closest neighbors fraction game:

During the round where the target is 1/2, Jay plays 6/6 = 1. Was that her best play? How do we know?

Remember, we are playing our game with a single deck of playing cards and each player is dealt five cards

The target for the next round will be 3/4

Burning 2 identical values might allow Jay to increase the diversity of her hand, especially if she happened to have 3 or 4 sixes.
Hey, life doesn't promise that all puzzles will have solutions that can be wrapped up in a nice neat package, does it?

My Closest Neighbor Fraction game

Denise Gaskins recently flagged a post about a good fraction game: My Closest Neighbor. I tried this out in class today.

A pre-test
First, I wasn't sure whether the level of the game would be right for the kids. I was considering it for the 3rd and 4th graders, but had some alternative activities planned in case. To start, I posed the following questions:

• Which is closest to one-half: 1/3 or 2/5? The third graders really struggled with this, so I left it alone and went to my plan B games. The fourth graders were all confident on this one.
• Which is closest to 3/4: 5/11 or 11/12?  This was a challenge for the fourth graders, but I thought it would be ok to play the game.

In our discussion of the second question, we explored two strategies:

1. making a common denominator
2. comparing with reference numbers

The common denominator is a bit of a pain, since 11 is prime, though at least we have the fact that 4 is a factor of 12. One student soldiered through this approach, but it was difficult for the other kids to follow.

For the second strategy, we made use of some observations that were more elementary for the kids:

(a) 5/11 < 5/10 = 1/2

(b) 3/4 is halfway between 1/2 and 1

(c) 11/12 < 1

Combining these, it was easy for us to draw a rough number line, place 5/11, 1/2, 3/4, 11/12, 1 and see that 11/12 must be closer.

The game
We played three rounds: target 0, target 1/3 and target 1/2. I think this game was very challenging for the kids.  Everyone had to work to figure out the best play from their hand and didn't always make the right (local) choice. For example, whether to choose 5/8 or 5/9 for a 1/2 target.

Once everyone had played, the challenge was still just starting. They had to figure out who was closest. I structured the discussion by helping them figure out which plays were lower than the target and which were higher. For the ones that were lower, they could put them in order and only needed to consider the highest. Then, we worked on the ones higher than the target and got the lowest of those.

In the course of this discussion, we added a third strategy to the ones listed above:

• making a common numerator

Summary thoughts

Fraction comparison like this was still too difficult for the kids to make an engaging game. If I were to do it again, I would change to make it more of a puzzling exercise, removing competition and any sense of time pressure.

Once the kids gain a bit more experience, though, I think this game has some nice features. It is particularly good for practicing fraction sense, and the multiple rounds allow some scope for strategic play.

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.

Leftorvers with 100 game

In Grades 3 and 4, we played a nice game that (I think) we got from Marilyn Burns. Looking for a reference after the fact, I see it explained in her book Lessons for Extending Division.

Basic play

• Start with a target number (we used 100) and collection of available divisors (we used integers 1 to 20)
• Players take turns choosing a divisor from the remaining available options. They divide the current target by that divisor and keep the remainder as their score for the turn. They also subtract the remainder from the target to create a new target for the next player.
• Each divisor gets crossed out when it is used, so it can only be used once.
• The game ends when the target is reduced to 0 or when all available divisors are exhausted.
• We played as a two player game.

Here's an example of a game play:

Player 1 chooses 17. 100 = 17 * 5 + 15, so player 1 scores 15 points, the target is reduced to 85, and 17 is no longer available as a divisor.

Player 2 chooses 20. 85 = 20 * 4 + 5, so player 2 scores 5 points, the target is reduced to 80, and 20 is no longer available as a divisor.

Player 1 chooses 14. 80 = 14 * 5 + 10, so player 1 scores 10 points, the target is reduced to 70, and 14 is no longer available as a divisor.

Player 2 chooses 18. 70 = 18 * 3 + 16, so player 2 scores 16 points, the target is reduced to 54, and 18 is no longer available as a divisor.

Player 1 chooses 19. 54 = 19 * 2 + 16, so player 1 scores 16 points, the target is reduced to 38, and 19 is no longer available as a divisor.

Player 2 chooses 13. 28 = 13 * 2 + 12, so player 2 scores 12 points, the target is reduced to 26, and 13 is no longer available as a divisor.

Player 1 chooses 15. 26 = 15 * 1 + 11, so player 1 scores 11 points, the target is reduced to 15, and 15 is no longer available as a divisor.

Player 2 chooses 16. 15 = 16 * 0 + 15, so player 2 scores 15 points, the target is reduced to 0 and the game ends.

Player one wins 52 to 48.

Our experience

We found this to be a fun, interesting, and engaging game. The practice with dividing and remainders was pretty obvious. In addition, it opened up some opportunities for strategic thinking, particularly at the end-stage of the game. I think there are also several good extension explorations.

Extensions

First, I created a simple pencilcode program for two players to play this game against each other. Here's a playable version (and here's the code).

Second, you'll notice that the first player in our sample game followed a "greedy strategy."  At each stage, that player chose the divisor that would give the most points on that turn. If you look closely, that isn't the best strategy at the end of the game.

So, a natural exploration is to find the best strategy for different starting targets. One specific point of about which we're curious: is it ever desirable to skip your turn (choosing 1 as the divisor is effectively a turn skip)? 

Some other areas for investigation:

• must the game always end on 0 or can we run out of divisors?
• given a target and collection of starting divisors, what is the shortest (number of turns) game possible? What is the longest game (number of turns) that does end at 0?

Block blobs redux

Last December, we played Block Blobs (notes here). This week, we are trying a slightly modified version for two digit multiplication.

The Game

Materials

• 4d6. Two dice are re-labelled 0, 1, 1, 2, 2, 3 (see notes below)
• Graph paper (we are using paper that is roughly 20 cm x 28 cm, lines about 0.5 cm apart)
• colored pencils

Taking a turn

Roll all four dice. Form two 2-digit numbers using the standard dice as ones digits. Then, use your color to outline and shade a rectangle in the grid so that:

1. the side lengths are the 2-digit numbers you formed with the dice
2. At least one unit of the rectangle's border is on the border of your block blob
3. Note: the first player on their first turn must have a corner of their rectangle on the vertex at the center of the grid. The second player has a free play on their first turn.
4. Write down the area of your rectangle

Ending the game

The game ends when one player can't place a rectangle of the required dimensions legally.

When that happens, add up the area of your block blob. Higher value wins.

Notes

"Counting" sides
The side lengths of the rectangles are long enough that counting on the graph paper will be irritating. Instead of counting directly, they can measure the side lengths. For our graph paper, the link between the measurement and the count is nice, since the paper is very nearly 5mm ruled, so they just double the measure. I think this is a really nice measurement and doubling practice, too.

Dice labels

Other labels could be used on the special dice. We chose this arrangement because of the size of the graph paper. Rectangles with sides longer than 40 often won't fit and we think we will even need some single digit rectangles to allow a fun game length. An alternative we are considering is 0, 0, 1, 1, 2, 2.

As an alternative to labeling the dice with new numbers, you could label them with colored dots and give out a mapping table. For example:

blue corresponds to 3
red corresponds to 2
green corresponds to 2
yellow corresponds to 1
black corresponds to 1
white corresponds to 0

This would keep the tens digit dice distinct from the ones digit dice and allow rapid modification if you want to change the allowed tens digits (just tell everyone a new mapping). Alternatively, you could create a mapping using "raw" dice and even allow more strategic flexibility from the players. I have a feeling that this would be confusing to most kids, though.

Reinforcing the distributive law

To facilitate calculating the area and reinforce the distributive law, you might have the students split their rectangles into two (or four or more) pieces and calculate the partial products. You can further decide whether to ask them to split the sides in particular ways or encourage them to find the easiest way to split to help them calculate.

Wimbledon game

We have been playing the game Wimbledon from John Golden. J1 and J2 absolutely love the game and J3 has enjoyed pretending to play as well. Definitely try it out!

Below is a session report sharing some of our experiences with the game, including some alternative rules (aka mis-reading).

Our basic play
For the serve, we allow the following options:
(1) serve a single card from the top of the deck
(2) serve one or two cards from your own hand

We also allow returns that have the same value, if the largest card in the combination is higher. For example, 8 + 2 can be played on top of 7 + 3.

When playing doubles, we followed tennis conventions: one player serves the whole game, return of serve alternates. For the return of serve, the designated player must return on their own. For other returns, either of the partners alone or in combination can play cards for a return.

Having gone back to John's original post, I now see that we played with the inverse rules for aces. On the serve, we counted them as 11, all other times 1. We didn't distinguish between aces played from the hand or served from the deck, those were all 11s. That formed a strong advantage for the servers, while making aces essentially worthless for all other players.

With three players, I had the kids play as partners and I played with a ghost partner. The ghost partner would contribute cards randomly. When the ghost played cards that weren't large enough to be legal plays, we considered that an unforced error and awarded the point to the other team. While the ghost was able to hold serve for one game, it was a big disadvantage. An alternative for three players would be to have the ghost partner with the server and for everyone to take turns serving. This would put the server advantage (with our "house rules" for aces) against the ghost disadvantage.

A modification
In our play, we have found that the 10 value cards and aces (on serve) dominate game play. Here are two ideas to address that:

• Assign face cards values 11 for Jack, 12 for Queen, and 13 for King. Ace, on serve, can have a value 14.
• Allow players to combine as many cards as they like. This would probably work best with our "Further extension" rules below. A possible sub-variant is to only allow gradual escalation where the players can step from single card plays to 2 card plays, from 2 to 3, etc, but could not jump from a single card play to a 3 (or more) card play.

My instinct is that the variation we will like the most is to differentiate the face cards and allow gradual escalation for multi-card plays.

Further extension
Now that we've gotten comfortable with the basic and doubles games, we are considering a more complex version. The idea we are considering is to somehow limit the players' abilities to refresh their hands by redrawing so that burning a lot of cards will have a cost.

This is the rule modification, written for a 2 person game:

1. Create a draw pile for each player with 15 cards.
2. At the start of the game, each player draws 5 cards into their hand.
3. Points are played as in the normal rules
4. At the end of a point, the players refresh their hand up to 5 cards from their draw pile
5. If a player runs out of cards in their draw pile, they cannot draw additional cards to refresh their hand.
6. If both players run out of cards in their draw pile, then shuffle the pile of face-up played cards and give each player a new draw pile with 10 cards.
7. Repeat step 6 as often as needed

The idea of this variation is to thematically mimic the idea that one player could push too hard, too fast, and get tired out relative to the other player. Strategically, our idea is that this will also create a tension between dumping your own low cards and letting the opponent dump.

Evens/odds and a quick update

Early years math seems to put a strange emphasis on even and odd numbers. Recently, a friend asked whether there was a point to this. Maybe it is just one of those little bits of terminology that we are asked to memorize for no reason?

By chance, this was something I had started considering about a month ago. It did seem strange that we spend so much time on this simple way of splitting integers. I wondered if it was worth the attention. From that point, my awareness was raised and I started noticing where it occurs and ways it links with more advanced concepts and future learning. My conclusion is that even/odd is surprisingly deep.

First, it is a simple version of concepts that will be developed further. For example, the alternating (starting with 0) even, odd, even,odd, even... is an illustration of a pattern. They will soon see other alternating patterns, then more complicated patterns and 2d or 3d patterns.

For another example, evens are multiples of 2, odds are numbers with a non-zero remainder when dividing by 2. This leads to understanding other multiple families, division, and division with remainder.

Second, the even/odd distinction is helpful for improved understanding of different calculations. For example, the observations that even+even = even, while odd + odd = even, etc. These can be used to help self-check their calculation and also will form early experiences with algebra. Similarly,
even x odd vs odd x odd reinforce understanding of multiplication. Again, this gets broadened for multiples of 3, 4, 5, etc.

Third, there are a lot of more advanced results that are easiest to prove by parity arguments. Sometimes we are working with a set of things that are even and the key observation is we can pair them up. Other times, we have a set that is odd and the key insight is that, when we pair them, one must be left over.

Recently, with the J1 and J2, we were looking at some constrained ways to put the numbers 1 to 25 on a 5x5 checkerboard. They were able to prove that some versions were impossible simply because 25 is odd, so there are more odd integers in 1 to 25 than even integers.

Lastly, there are techniques in computer science that involve even vs odd. This comes up pretty naturally because of the essential use of binary.

Some games

Recently, J1 and J2 have gotten hooked on some classic card games. In particular, we've been playing a lot of 3-handed cribbage. It is a nice way to do some simple addition practice and build intuition about probability. Probability is now getting even more share of mind: in the last couple of days we started playing poker together. This was actually inspired by some of our reading together.

We are reading the Pushcart War.

In one scene, there is a poker game. Of course, the J's insisted that I explain the game and were eager to try it out. J3 was the huge winner tonight, while I busted out. Oh well.

Secret Numbers (Addition Boomerang variants)

We have been enjoying Mathpickle activities in our math games classes recently. This week, the 3rd and 4th graders will be playing with some of the more advanced Addition Boomerang variations.

During our planning, we came up with one extra pointer to tie the activity more closely with multiplication. Also, we had ideas for variations and wanted to record our notes so we can use them again in the future.

Tie with multiplication

In Gord's video explanation, he sometimes records in the center of each loop how many times that loop has been used. We emphasize this and write some related equations to help draw out the connection between the repeated additions in this activity and multiplication.

The first way we do this is by making tally marks inside the loop every time that branch is chosen. At any time, you can pause and write down an equation for the current total in a form

AxN + BxM = Total

where A and B are the values of the loops, N and M are the number of times each loop has been used.

Alternatively, we can show a "completed" round of throws by simply writing the number of passes for each loop in the middle and, again, write out an equation showing the total as the sum of two products.

Secret Number Variations

We start with a basic addition boomerang lay-out, either with 2 or 4 branches, both players (or teams) share a common set of addends and take turns adding on to a common running total. In our variants, the players choose and write down a secret number that helps inform their target for the game:

• Version A: players pick a number between 70 and 100. This is their target for the game and they win if the common total hits that value, whether the target is reached on their turn or their opponents turn.
• Version B: players pick a number larger than 15. They win if the total hits a multiple of their secret number. For example, if they choose 17 and the running total hits 51 (aka 17 x 3) then they would win. If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.
• Version C: players pick any number. They win if the total hits a multiple of their secret number that is larger than 60 (not equal to 60). If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.

Version B is, I think, the most directly playable.

Possible issues
I'm not sure how to deal with the case where both players choose the same secret number.

For Version A, it will be interesting to see what modification kids can find that will deal with the fact that it is very easy to miss any particular target. In the basic game, once the total is larger than your target, there is no hope of recovery. There are several ways to address this. I would be eager to hear any rule sets that kids create and hear about the experiences.

In Version C, I wonder if choosing 2 as the secret number is too strong a move?