Sunday, February 7, 2016

Surreal numbers and whole body integers

Two unrelated activities to note:
(1) Exploring checker stacks and surreal numbers with J1 and J2
(2) A whole body numbers game with J3

Surreal Numbers and Kids

If, like me from one month ago, you don't know about surreal numbers, I think you'll find they can be a really engaging exploration with kids. The main attraction is the appearance of infinities and infinitesimals, both of which really seem to resonate with young mathematicians. In addition, there's fantastic icing on this cake, too: you can explore by playing a simple (to learn) game with a lot of depth.

Credits: this exploration is strongly inspired by Mike Lawler's recent posts about surreal numbers and the Jim Propp post that inspired him. If you are interested in doing this type of exploration with your kids, I strongly suggest going through all of their posts on the subject.

Note: since we used black and red checkers, while the convention in the other posts is blue and red, I will abbreviate B and R so you can naturally substitute your own preferred color scheme.

How we got started
Using a set of regular, stackable, checkers (black and red), I showed each of the older J's the position RB + BR (a stack with red on the bottom, black on the top and a stack with black on the bottom, red on top) and explained that the basic moves.

This was a good initial example because it let us talk about each of the major scenarios:

  • We will investigate cases where B moves first and others where R moves first
  • Each one can only take stacks above one of their own color checkers
  • If the colors allow, they can take a top checker and leave the rest of a stack undisturbed
  • If their color is the bottom of a stack, they can remove the whole stack
  • Usually, they will have choices about which stacks to remove
Then, I explained the losing condition: if you don't have any more moves, you lose. They quickly realized this was the same as when they no longer had any checkers of their own color on the board.

Next, we quickly played a set of simple games:
  1. Single B checker
  2. Single R checker
  3. B + R
  4. RB + BR
  5. RB + B
  6. B+B+B+B +R+R (and similar)
  7. BBB+R+R (and similar)
Connecting with numbers
I told them that one amazing thing is that we can give each game position a value. Then went through:
  1. Single B checker is +1
  2. What do they guess a single R checker is? explain -1
  3. What about B+R? eventually get to 0
  4. Explain the fundamental trichotomy: positive value means winning strategy for B exists, negative for R, what about 0?
Powers of 1/2
The first really juicy bit came when I asked what they thought the value of a BR stack would be. This established a common sequence of investigations:
  1. If alone, can we see a winning strategy for either B or R? In this case, obviously B. Thus, the value is positive
  2. Compare with 1 by playing the game BR + R. They were really fast about seeing this link, for others this is worth writing out and spending some time discussing. In this case, they saw BR + R must be negative, so the value of BR is between 0 and 1.
  3. Guess a value; they've got enough experience to think that often the answers are "nice," so 1/2 was a natural guess.
  4. Test. In this case, it meant checking BR + BR + R
Next stop was RB. I wanted to make sure negatives weren't left out and to reinforce the symmetry in the colors, so would ask them to swap the colors and get a value as a quick follow-up.

Next, I asked them to see if they could find a configuration with value 1/4. This took a long time and there were lots of false starts. I didn't think this would be easy and didn't help them shortcut the exploration.

Once they got 1/4, though, they had a fast guess about 1/8. We checked it and then they made a conjecture about further powers of 1/2.

Deep blue and red
Seeing powers of 2 in some form is always fun, but both knew that they had been promised infinity and wanted to see it. Using our plastic dinosaurs, I introduced the deep checkers deep blue (represented by a blue mini) and deep red (represented by an orange mini).

Our sequence for these was similar to the powers of 1/2. The value of deep blue is positive, so they compared to 1 by playing the game BBBB.....+R. Each had a sparkle of insight, but quickly played BBB.....+R+R+R just to check, then announced that the value would be bigger than any integer.
I gave them the name omega, and we checked the deep red: RRRR.....

At this point, one of them put the deep red on a B checker and asked what that would be. Again, followed the previous recipe to realize that it must be positive, but smaller than 1, then smaller than 1/2. At that point, one of them made this arrangement:

Ok, I know that mastadons aren't dinosaurs

If it isn't clear, the realization was that the game sequence B, BR, BRR, BRRR, ... would have values
1, 1/2, 1/4, 1/8, .... and BRRRR..... would be at the end of that sequence with value ... larger than 0, but smaller than every power of 1/2. You can see that they made a similar connection with the negative values.

A little bit about deep purple
The two kids were totally satisfied now, having gotten omega and epsilon (along with omega +1, epsilon - 1, omega + epsilon, etc). To give them something to chew on for later (and because we had purple dinosaurs in our set) I introduced the deep purple, BRBRBRBR.... Immediately, they designated the yellow dinosaurs as the inverse of deep purple (RBRBRBRB.....)

So, what's the value of deep purple? What they've gotten so far:
  • positive
  • smaller than 1 (by playing BRBRBRBR..... + R)
  • Bigger than 1/2 (by playing BRBRBRBR..... + RB)
  • guess 2/3. I don't know where this came from, but I confirmed that is the value
  • working on finding winning strategies for the second player in R+R+3(BRBRBR.....)
Wrap up videos
As a round-up in the evening, we watched the videos from the first post in Mike Lawler's sequence. I paused frequently to let my two shout out their answers and explanations for where Mike's boys were in their exploration. This seemed to be a very effective way to underline their experience for the day.

Number match on the number stairs

A game for J3. This is a simple game with some variations that makes use of our stairway "number line" and a three year old's natural enthusiasm for running up and jumping down stairs.

As pictured previously, we have labeled the stairs in our house from 0 to 36 (more to come). We have a set of cards with numbers on them. P mixes them up, then gives them, one at a time, to J3 to put on the corresponding step, and they sing count up and down:

Some other variations:

  • using playing cards instead of number cards
  • Using cards with dot or shape patterns
  • Using cards with number words ("one" instead of "1")
  • child sends the parent to a particular step, checks if the parent got it right

Pillow forts

In case you missed it, pillow forts have been in fashion recently. Here is an example:

1 comment:

  1. Finally got around to going through this more carefully today and it is fantastic! Can't wait to try this all out with the 4th and 5th graders this Thursday.