spreading the word (activity description for binary encoding/counting game)

[Mike South]

Hi,

on another list I'm on someone was asking about bases other than ten, wanting activities to do with their kids.  Here's what I shared, in case it's of any interest.

mike

Here's a game you can "make" with a deck of cards that introduces binary numbers.  The basic idea here comes from an activity in "Computer Science Unplugged".

Give one kid the ace, two, four, and eight of one suit, and give another the same cards from a different suit (technically the suits have nothing to do with it (which incidentally reminds me a lot of software development (but I digress (parenthetically!)))).

On individual slips of paper, write the numbers from 0 to 15.  

Have each player put their cards in front of them in a row, with the ace on the right, then the 2, the 4, and finally the 8 at the left most.

Mix up the slips of paper and put them in a jar or something, and pick one out.  Whoever is first to arrange their cards (by "arrange" I mean turn each one face up or face down) so that the total that is face up matches the number on the slip gets the slip.  Continue until the slips are gone; count to see who got the most; that's the winner.

So if you pulled out the slip with a three on it, the players would need to have the ace and the two up, and the other cards face down.

You can handicap the faster player by having them say "one-mississippi" with their eyes closed before they see the slip (or longer or shorter word, depending, of course).

You don't need to reset the cards between slips, just pull the next slip out.  Otherwise, when zero comes out it will always be a tie.  Of course if you both start in the same configuration (like all cards face up or all cards face down) you will get a tie on the first draw 1/16th of the time anyway.  You could let the players start with the cards in any configuration hoping their lucky number comes up or whatever.

The tie-in with binary is that what the players are doing (if you look at it the right way) is coming up with the binary representation of a number.  If you think of the face-up cards as ones and the face-down cards as zeros, you can read off the binary number.

You could explain the way it's done in the computer by imagining someone very nearsighted looking at the row of cards.  They can't see well enough to determine how many spots are on each card, but they can tell if the card is face up or face down.  As long as you always keep the cards ordered with the smallest one on the right, the nearsighted person can tell what number you are showing just by knowing that, for example, "face up, face down, face down, face up" is 9.

This is actually a very accurate analogy for why binary is used in computers.  It's easy to reliably determine the difference between "that wire has five volts on it" and "that wire has zero volts on it".  If you were trying to say "that wire has 1 volt, that wire has 6 volts, this one has 7" you would have a much less reliable machine, because the components are going to fluctuate a little and you'll have 6.5 volts and it won't know if you meant 6 or 7.  So instead of ten different "digits" it uses two digits, "on" and "off", which are much more reliable to store and calculate with in a machine.

Another activity you could do is "count" with the cards, starting with zero and going to fifteen.  You can ask which card is the "busiest" and why.  See how fast you can go from zero to fifteen.

You can also pick any number and have them try to make it in more than one way, although the reasoning for why you can't might be a little beyond them, they might be able to appreciate the minimality of it--if you say "I have five showing" you immediately know they have it face down, face up, face down, face up because there's only one way to make five.

It's hard to say for sure, but there could be something of a pedagogical disadvantage to this approach, since the cards in the different places are marked differently, and in an actual base two number the value of the digit is purely determined by the placement.  I wouldn't be too concerned about this, but you might want to consider weaning them off of the numbers (once they are proficient) and just give them cards that are black on one side and white on the other.  If you did it this way I think you actually might get better understanding faster, because it's just really hard for someone used to base ten numbers to see 101 and not think "one hundred and one!".  If you get them used to the idea that the position is defined to be something else, maybe when you come back to using numbers for it it will be easier to make that transition.

As for other bases, we don't really do much in other bases.  You buy eggs in dozens, and a dozen dozens is a gross...but dealing with base 12 is even worse because you have to invent two new symbols.

I like the idea of doing base three but having pictures for the digits rather than numbers (again, to avoid the (12 = twelve!) confusion that is just going to happen).  Like have egg, chick, chicken for 0, 1, 2.  

I have an odometer online that counts in base "tree", but it seems to be having some technical difficulties.  Works ok for me in firefox, something is wacky in chrome. ymmv:

http://leftoverpi.nfshost.com/play/oodometer/

You have to click the "tree" icons at the top left to get it to switch to base "tree".

I have no idea if it will help kids understand base three, I only played with it with my own kids a little.  The reasoning behind the design is that if they get the concept of the odometer, and play with it by clicking the "plus one" button, watching the numbers roll over, etc, they might get the concept of the place value faster this way if you don't have the distraction of "12" looking so much like twelve to their confused little brains.

mike