The Exact Chaos Spreadsheet Game

John D. Cook points out the following interesting fact:

Pick a number x between 0 and 1. Then repeatedly replace x with 4x(1-x). For almost all starting values of x, the result exhibits chaos. Two people could play this game with starting values very close together, and eventually their sequences will diverge.

Cook's on to something here. This does have the makings of a simple game that ties in to math and computing concepts like functions, iterations, spreadsheets and chaos. Here are the rules:

The first player picks a to number from 1 to 99 with the only condition being that the number can't have a five in it (with a little trial and error you should be able to figure out why).

The second player can either steal the first player's number or come up with one of his or her own. If the number is stolen, the first player has to come up with another number.

Both numbers are divided by one hundred and put into the first row of a spreadsheet. Under the first type


=4*A1*(1-A1)


Under the second

=4*B1*(1-B1)



Now copy the formulas going down twenty rows. The copy function should produce an iterative function, with =4*A1*(1-A1) followed by =4*A2*(1-A2) and so on.

Have the students make a line graph of the two series. The winner is the one who ends up with the largest number.

The Number-line Code

This one grew out of some time I spent recently helping a family first-grader with her subtraction homework. I made up some sheets with numbers and letters. My original thought was just to make the practice sheets more interesting by having the answer spell out a word, but I noticed my tutoree (who naturally has very good math genes) had, without prompting, started using the code key as a number line to figure out the answers.

The more I thought about it, the more I liked the idea of using this approach for a wide range of problems. For the very early grades, it teaches numbers and letters. It has great appeal for kids (particularly if presented with the right air of mystery -- I'd suggest a pirate motif). It reinforces the number line concept and the essential idea of looking things up in tables.

Here are some sample problems

For those learning to read, present it as a straightforward code:

PIRATES LOVE 7-15-12-4

For slightly more advanced students, replace the numbers with problems


PIRATES LOVE

10 - 3

10 + 5

6 + 6

6 - 2

For even more advanced students, make it a more explicit number line exercise

PIRATES LOVE


7

+8

-3

-8


You can even play with functions by using the old SAT trick of using a circle to represent x+1 and a triangle to represent x-1 (or any other functions you can think of).

triangle 8
circle 14
circle 11
triangle 5

I'd suggest using this technique frequently enough to keep the familiarity high. It also offers extensive opportunities for teaching across the curriculum.

Anyone have any other ideas?