Thursday, February 16, 2017

More Man Who Counted (gaps and notes)

As previously mentioned, we have been reading The Man Who Counted. While the story is good and there are nice math puzzles, we've found some of our best conversations have come from errors or weaknesses in the book. Here are three examples:

How old was Diophantus?

In chapter 24, we encounter a puzzle to figure out how old Diophantus was when he died. In summary, the clues are:

  1. he was a child for 1/6 of his life
  2. he was an adolescent for 1/12 of his life. (J1: "what's that?" J0: "a teenager")
  3. childless marriage for 1/7 of his life
  4. Five more years passed, then had a child
  5. The child got to half its father's age, then died.
  6. Diophantus lived for four more years
Perhaps we are wrong about our interpretation of the clues, but we noticed two things:
(a) the answer is not a whole number of years.
(b) the answer given in the book doesn't fit the clues.

For the first part, it seems a natural assumption of these types of puzzles that we are only working with whole number years. Sometimes, this is an interesting assumption to directly challenge.
Here, since the clues involve a second person (Diophantus's child) we felt whole numbers were a strong assumption. Also, the name Diophantus, you know?

Each clue required some discussion for us to agree on the interpretation. The one that seems most open is the fifth clue. In particular, did the child live until its age was half of the age of its father at the time of birth or to the point that, contemporaneously, it was half its father's age?

For completeness, I'd note that neither interpretation matches the book's answer. The first interpretation does allow a whole number answer, but it doesn't give whole numbers for all the listed segments of Diophantus's life.

Just so you can check for yourself, the solution given in the book is 84 years old.

How do you fix it?
We discussed several possible fixes:

  • accept answers that aren't whole numbers or require whole number segments for each clue. This allows us to take the alternative interpretation of the fifth clue (though that still isn't satisfying) or to accept the clues and just take a new answer. This isn't satisfactory because... Diophantus.
  • Change clue 4 or clue 5 to match the book's answer. This approach seemed to fix the puzzle without distorting it or changing the mathematics required to analyze it.
  • Change clue 1, 2, or 3. While possible, these seemed to open the possibility of changing the character of the puzzle. Also, these fractions were plausible based on our own experience of human life spans.
Of course, an even more satisfying answer would be to introduce a further variable and make the puzzle into one that makes heavy(ier) use of the integer restriction.

Clever Suitors

In chapter 31, Beremiz is confronted by a nice logic puzzle. Three suitors are put to a test, each is blindfolded and has disc strapped to his back. The background of the discs: other than color, the discs are all identical, there are five to choose from, 2 black and 3 white.

The first suitor is allowed to see the colors of the discs on the backs of his two competitors, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.

The second suitor is allowed to see the disc on the back of the third suitor, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.

Finally, the third suitor is required to identify the color of his own disc and explain his reasoning.  He succeeds.

Weakness 1
As a logic puzzle, we enjoyed this. Our problems came from the context in the story. This challenge was set to the three suitors as a way of fairly judging between them by finding the most clever suitor. However, this process was clearly unfair. In fact, it is inherent in the solution that it was impossible for the first and second suitors to determine the color of their own discs.

This led to a nice discussion about who really held the power in this process: the person who structured the problem by deciding what color disc should be on which suitor and what order they would be allowed to give their answers.

Extensions:
  • consider all arrangements of discs. Are there any arrangements where none of the suitors can answer correctly?
  • What is the winning fraction for each suitor? If you were a suitor, would you prefer to answer first, second, or third?

Weakness 2
Our second objection was non-mathematical, but again related to the story context. The fundamental problem wasn't how to choose a suitor. The fundamental problem was how the king could remain peacefully friendly toward all the suitors' home nations through this process.

For this discussion, we went back to the story of Helen of Sparta, which we'd read a long time ago in the D'Aulaire's Book of Greek Myths. Of course, that also led to discussion of the division of the golden apples, another puzzle we all felt surely could have been solved more effectively with some mathematical reasoning...

The Last Matter of Love

The last puzzle of the book is in chapter 33. It is another logic puzzle, again intended to test the merit of a suitor in marriage. The test:

  • there are five people
  • two have black eyes and always tell the truth
  • three have blue eyes and always lie
  • the suitor is permitted to ask three of them, in turn, a "simple" question each.
  • the suitor must determine the eye color of all five people
As a logic puzzle, we readers get some extra information:
  1. The first person is asked: "what are the color of your eyes?" The answer is unintelligible.
  2. The second is asked: "What did the first person say?" The answer is "blue eyes."
  3. The third is asked: "What are the eye colors of the first and second people?" The answers are "the first has black eyes and the second has blue eyes."
Simple questions
Our first objection was the part about asking "simple" questions. Having developed our taste for these types of puzzles through the knights and knaves examples of Raymond Smullyan (RIP, we loved your work!!!), the third question really bothered us. If you're going to go that far, why not ask the third person for the color eyes of all five people?

Personally, I would prefer that the puzzle require us to ask each person a single yes/no question.

As an extension: can you solve the puzzle with that restriction? 

Getting lucky
Again, we felt that this puzzle didn't meet the requirements of the context: to prove the worthiness of the suitor. Putting aside the question of whether this is really an appropriate way to decide whether two people should be allowed to marry, the hero here got lucky.

Extension: what eye color for the third person would have caused the suitor to fail?
Extension: what answer from the third person would have caused the suitor to fail?
Extension: for what arrangement of eye colors would the questions asked by the suitor guarantee success?
Extension: what was the suitors' probability of success, given those were the three questions asked?

Waste
Our final objection was the simple waste in the first question. From a narrative perspective, this is justified and even seems made to serve the purposes of the suitor. However, it opens another idea:
can you solve the puzzle, regardless of eye color arrangement, with only two questions?

Feel free to test this with yes/no questions only or your own suitable definition of a "simple" question.

The power of...

As a final thought, let me say that I think errors and ambiguity in a text are a feature, not a bug. It is another great opportunity for us to emphasize that mathematics is about the power of reasoning, not the power of authority.

Tuesday, February 14, 2017

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.

Monday, February 13, 2017

NRICH 5 Steps to 50

A quick note about the game we played in first grade today: 5 Steps to 50.

This is an NRICH activity that I've had on my radar for a while. I even made a pencilcode program to explore the activity in reverse. True to their other activities (check them out!!!) 5 steps to 50 requires very little explanation, is accessible to students with limited background, but has depth and richness.

Our lesson outline
I explained the basic activity and did an example at the board. To get my starting value, I had one student roll for the 10s digit and one for the 1s digit. Then we talked through together as we added 10s and 1s.

I then distributed dice and had the kids try 3 rounds. As they worked, I confirmed several rules:

  1. the only operations allowed are +1, -1, +10, -10
  2. we must use exactly five steps (I note that this is ambiguous on the NRICH description, they say "you can then make 5 jumps")
  3. we are allowed to do the operations in any order
  4. we can mix addition and subtraction operations
After everyone had been through 3 rounds, we regrouped to summarize our findings:

  • Which starting numbers can jump to 50?
  • Which starting numbers cannot jump to 50?
We helped the kids resolve disagreements and then posed the following:
  • What is the smallest number that can jump to 50?
  • What is the largest number that can jump to 50?
For those to challenges, we kept the restriction that the numbers must be possible to generate from 2d6.

Basic level
To engage with the activity, some of the kids just started trying operations without much planning. This quickly reinforced the basic points about addition and place-value and commutativity of addition.

For these kids, it was helpful to ask a couple of prompting questions:

  • What do you notice? This is a standard that never gets old!
  • If this path doesn't get to 50, does that mean there is no path to 50?
This second question, particularly, raises the interesting observation that it is easy to show when a number can jump to 50 (just show a path) but to show that no path is possible requires a different type of thinking.

Getting more advanced
The next level of sophistication was really about noticing that the key consideration is the distance to 50. In particular, this identified a symmetry, where n could jump to 50 if 2*50 - n can jump to 50. Of course, the kids didn't phrase this relationship in this way....

The next major step is thinking about a way to systematically write down the paths.

Wednesday, February 1, 2017

Perfect Play for My closest neighbor

Joe Schwartz at Exit10a wrote a fraction comparison post that prompted me to write up more of my experience and thoughts on this game.

Let's find perfect play
This week, I intended to use the game one last time with the 4th graders as an extended warm-up to our class. The challenge I presented:

If we got super lucky and were given perfect cards for each round of the game, what are the best possible plays?

My intention was to spend about 20 minutes on this. Depending on how quickly it went and the kids' reactions, I considered giving them a follow-up for a short homework: what are the best plays if we include all cards A (1) through K (13)?

How did it go?
In the end, the basic activity took the whole class. These comparisons were difficult for the kids, so we spent time talking about each different strategy for comparison:

  1. common denominators
  2. common numerators
  3. distance to 1
  4. relationship to another benchmark number. Like 1/2 in Joe's 4/6 and 8/18 example, a benchmark is a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark. 

For visualization, drawing on a number line seemed to work best.

I did not assign the full deck challenge as homework. Instead, we gave them some more work with fractions of pies and bars.

What have I learned?
This game is really effective at distinguishing levels of understanding:
(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.

(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.

(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.

(3) proficient: have at least one consistent strategy they can work through to make a comparison

(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.

What would I do differently?
Generally, I think it is valuable to spend more time and more models directed at the basic understanding of what fractions mean. The kids who were at or close to stage 4 have, over the years, been seeing diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. Oh, and also actual pies (mostly pizza), cakes, cookies, and chocolate bars discussed using fractional language.

More locally, for this game in a class of mixed levels, I would

  • lean toward doing this more as a cooperative puzzle
  • re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 (note: I don't have strong feelings about where 2 fits in this sequence)
  • I also would consider allowing equivalent fractions to the target as winning plays

Impassable Din Daeng (BKK intersections 3)

It has been a while since I've done one of these.  For your topological and civil engineering pleasure, I present  Din Daeng intersection (แยกดินแดง):

Notice how the expressway obscures key details?

Magnified view. We now see a tunnel, but where does it emerge?

This intersection has a special place in my heart. Last month, J1 and J2 played in a squash tournament at the Thai-Japan Youth Centre. It isn't visible on either map I've included, but is just to the northeast of the intersection.  Since we live on the west of the intersection, we needed to cross somehow.

After three failed attempts (following the driving directions on google maps) to get through from west to east, I ended up parking our car in one of the small side streets and we just walked. The walk took about 30 minutes...

ideas for upcoming classes

warm-ups for all

WODB: (1) shapes book (2) wodb.ca
any: Traffic lights/inverse tic tac toe/faces game
good options here, mostly grades 1/2: some games
dots & boxes (maybe with an arithmetic component)
loop-de-loops

Grades 1 and 2

close to 100 game: 
Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.
Procedure: 
- Deal out 6 cards to each player
- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers.  The goal is to get the sum of the two numbers as close to 100 as possible but cannot exceed 100.


Grades 3 and 4

Factor finding game (maybe warm-up?)
Factors and Multiples game
Contig for 3 and 4 (explanation).
Times tic-tac-toe: review for Grades 3 and 4
Fraction war for grade 4 (smallest card is numerator)
Multiplicaton models: worth making for grades 3-4 for solidifying concepts? Associated games
d 2





card on head game


Pico Fermi Bagel

Magic triangle puzzles

damult dice



(1) Dice game perudo
Equipment
- multi-player, 2-5
- Everyone gets the same number of 6 sided dice (full game they get 5, I would start with 3)
- Everyone has a cup to shake and conceal their dice

Basic Play
- Simultaneously, players shake their cups and turn them over on the ground or a table. They peak in to look at their own dice, but keep them concealed from the other players.
- starting randomly (or from the person who lost a dice in the last round), players make bids, for example: two 3s. 
This bid signifies that the player has 2 (or more dice) showing the value 3.
- the next player has two choices: 
  • call/doubt the previous player's bid: if they do this, all players show their dice. If there are enough to meet the bid, the caller loses a die. If not, then the bidder loses a die.
  • raise, either the number of dice or the value or both get increased 
Advanced rules
- Ones are wild, they count as any number toward the target bid
- If someone drops to their last dice, they start the next round. On that round, only the number of dice can be increased in the bid, not the value. Ones are not wild on this round
- After someone bids, the next player has a third option, to call "exact." If the bid is exactly matched by the dice, then the bidder loses a die and the caller gets an extra one. If the actual dice show either more or less than the bid, the caller loses a die.

remainder jump
we played this game before, but we could give them blank boards and let them create. See the last page here: http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf



(1) double digit and double dollar:
We've done something like this, but I think there could be a good variation done trying to make 1000 baht, using 1, 2, 5, 10, 20, 50, and 100 baht units.


(2) biggest rectangle. This could be used as a warm-up. For the older kids, they will probably have seen something like this, but I like the inclusion of perimeters that are even but not divisible by 4 and odd perimeters and the question about "smallest area" (here are 5 questions).


(3) some of these games are promising:



Tuesday, January 31, 2017

Quadratic Friends (The Man Who Counted)

J1, J2, and I are currently reading The Man Who Counted. Here are some quick thoughts:

Quadratic Friends
The book is a great entry point for mathematical discussions. In fact, it makes it questionable as bedtime reading, since I have to be careful to find a more narrative section to close the evening. Otherwise, we would just continue talking and they'd never get to sleep.

Fortunately, the J's are willing to extend some of these conversations over to the next day, so we're not obligated to wrap up everything in one evening.

Here is an example discussion: in one of the early chapters, the protagonist Beremiz talks about the special relationship between 13 and 16. Namely:
13 * 13 = 169
1 + 6 + 9 = 16
16 * 16 = 256
2 + 5 + 6 = 13
Finding more
We wondered: what other pairs of numbers share this property?

Our first instinct was to gather data, so we started calculating some examples. We began with 0 and worked up, squaring, adding the digits, repeating. We found a couple of cases that flowed into the 13-16 relationship, for example 7. This gives a feeling that 7 is very fond of 13, but 13 only has eyes for 16.  Not the usual way people think about numbers, I guess.

Along the way, we made some interesting observations about this iterative process. I won't spoil the surprise, but would encourage you to explore yourself.

I'd note that J1 did the calculations up to 30 in his head, while I was a bit lazy and wrote a pencilcode program.

An extension
This conversation branched in an interesting way. Squaring is a natural thing to do with numbers, but summing the digits is a bit artificial. It depends on a choice of base. So, a natural follow-up question:
what quadratic friends exist in other bases?  This is an exploration for another day.