Wednesday, January 18, 2017

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?

Tuesday, January 17, 2017

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.

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.

Thursday, January 12, 2017

Compass only non-collapsing compass (Euclidea Series)

Since I've branched into this topic, I want to include some notes on additional references I've found and some results I've been able to get. First, an admission:

I'm still having trouble finding the intersection points of a circle and a line when the circle's center is on the line!
Another resource
James King (UW home page) has a nice session outline working through compass only constructions. Be warned, there are some spoilers for some of the Euclidea challenges in that material. Unfortunately, I can't tell from write-up when and where this was used. Maybe this NWMI meeting?

In Professor King's notes, he describes the construction I'm struggling with as "important and difficult," so I will have to redouble my efforts.

Non-collapsing compass from collapsing compass
As an aside, perhaps you have noticed that everyone else refers to the two types of compasses as collapsible and non-collapsible. Doesn't that terminology strike you as wrong, too? The point isn't that one type of compass is able to collapse, but rather that it always collapses when not drawing a given circle. Also, a common form of "non-collapsible" compass is able to collapse!

Just squeeze the legs together to collapse

Anyway, I was able to figure out the construction of a non-collapsing compass, just in time for Euclidea 13.2.

I'll give a construction below. For a more mild hint, the key is the ability to reflect points through a line we've already "constructed" (where we have two points on the line).

I've chosen to do this in GeoGebra so I can show a sequence, explain some of the reasoning, and grey out earlier parts of the construction that we don't need anymore. Check my work to make sure I didn't slip in any subtle straightedge moves!

Tuesday, January 10, 2017

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):

(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

(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)

(11) more puzzles/standard patterns

(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.


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.

Beginning Restricted Constructions (Euclidea Series)

Note: This is a little bit out of order, but reflects the challenge on which we're currently working.

Choices of Construction Rules
The classic construction problems involve specific, restricted tools:

  1. unmarked straight-edge: this tool is only able to determine a line that contains two existing points (previously given or already constructed). It has no length markings and only one side.
  2. collapsing compass: this tool is only able to determine a circle given the center and a point on the circle. Again, both center and an included point must already be given earlier in the construction. Like the straight-edge, the compass has no way to indicate the length of the radius. (side note: when I was in HS geometry, we didn't require the compasses to be collapsing).
  3. A plane as a writing surface and infinitely fine/precise pens: these allow us to identify the intersection points of the constructed objects (circles and lines).

I, and maybe you, for years considered these rules to be natural. However, there are several other construction rules that are at least as natural:

  • Origami: moves based on folding paper. This is really a fascinating topic. Take a look at this Numberphile video for a taste.
  • Straight-edge only constructions: for those unfortunate to have left their compass at home.
  • Compass-only constructions: easy to imagine how ancients made a good compass, but how would they have gotten a really straight edge anyway?!

Several of the Euclidea challenges impose either straight-edge only or compass-only restrictions, which got me interested in this general topic again.

Examples in Euclidea
Unless I've missed some, the restriction challenges start officially in Theta pack:

  • Drop a perpendicular (8.4): we are only given the straight-edge, but we are also given a circle.
  • Mid-point (8.5): find the midpoint of a segment with the straight-edge and a parallel line.
  • Segment trisection (9.7): same restrictions as 8.5
  • Midpoint (13.1): this time, we've only got the compass, no straight-edge
  • Some I've not yet unlocked: Tangent to Circle (13.5), Drop a Perpendicular (13.7), Line-Circle Intersection (13.8)
In addition, Zeta pack has a some challenges that, while not marked as restrictions, are good warm-ups: Point Reflection (6.1), Line Reflection (6.2), and Translate Segment (6.6).

Humans can forget things!
If you've made it to Zeta pack, you know that the collapsing compass is able to replicate the function of a non-collapsing compass, so that particular restriction doesn't change what is theoretically constructible (but it sure reduces a lot of extra steps and auxiliary objects in real constructions.)

It turns out that the straight-edge is also unnecessary! This result is the Mohr-Mascheroni Theorem. One really fascinating/disturbing fact about this result is that it was first proven, as far as we know, in 1672 by Mohr, but his proof was lost for over 250 years. Mascheroni independently discovered and proved the result about 120 years after Mohr.

I think this is an important lesson that, with search-empowered internet, is easy to forget: human knowledge accumulation isn't always steady or certain.

Compass-only program
Cut-the-Knot (a generally excellent resource) has a good discussion of restricted constructions and outlines a program for proving the sufficiency of compass-only constructions.

I have started to work through this list. Generally, I won't post solutions since Cut the Knot already has solutions. However, I was concerned about whether the compass they were using was collapsing or not. The solution for 6 (given three points, find a fourth point that makes them vertices of a parallelogram) clearly uses a non-collapsing compass. So, that leaves an open challenge: how to build a non-collapsing compass from just a collapsing compass.

I'm currently stuck on this. There are two ways I could break through:
(a) find the point of intersection of two lines. If I could do this, I would use the ability to find perpendicular lines to move distances around.
(b) Find the point of intersection of a line and a circle with the center on the line.

Note, this second one is very similar to Cut the Knot's third challenge, but that assumes the center of the circle is not on the line segment. Amazingly, this slight change makes the challenge much harder, at least for me.

Running, rates, rounding

My running session this morning gave me an idea for a kind of 3-act math discussion with J1 and J2. I will discuss this with them when they come back from camp and see what they think. I expect the last questions will be hard for them and I would like to see how much progress they can make working together.

First Act

Today, I went running and recorded some information on my GPS. For five laps, I ran moderately fast. Here is the data:
Time           Rate         Distance
3:00          12.7 kph         635 m
3:00          12.9 kph         647 m
3:00          12.6 kph         633 m
3:00          12.7 kph         637 m
3:00          12.8 kph         645 m

What do you notice?
What do you wonder?

Second Act

My target was actually to run 12 kph for each of these three minute segments. After the first lap, I knew that I could run more slowly and still hit my target. I wondered, how much less than 635m could I run and still hit my target?
If I compare two laps, both rates and distances, can I figure out the distance I get for each 0.1 kph? Is there another way to calculate the difference in distances for each 0.1 kph?

Third act

For some reason, this made me think about rounding that J1 had recently been studying. He is a bit disturbed about what to do with values that are halfway between the rounded levels, for example whether 15 should round up or down to the nearest ten. Since this investigation of running data involved calculations with measured values and rounding, I though it would be instructive to explore a couple of calculations:

  • I have two distances, rounded to the nearest 10 cm of 20 cm and 10 cm. What is a reasonable range for the difference of those distances?
  • My GPS measured a time of 3 minutes (3:00, rounded to the nearest second) and speed of 12 kph (12.0 kph rounded to the nearest tenth of a kilometer per hour). What distance did I run? What is a reasonable range for that distance?