HP 40gs - Chaos

The program CHAOS is used with The Chaos Game from

Fractals for the classroom: Strategic activities Vol. 1.
New York: Springer-Verlag, 1991.

This publication provides worksheets, a detailed discussion of the mathematics involved and suggests various ways the game can be connected to the secondary school curriculum.

At a simple level the game can be used to show that the outcomes of random events are unpredictable in the short-term, but that in the long-term a predictable pattern emerges.

The game involves

• choosing a starting point within a triangle
• successively choosing one of the three vertices of the triangle at random and marking the point half way between the point you are at and the selected vertex.

One approach is to play the game by hand with dice on a large triangle drawn on paper. No apparent pattern emerges. Then move onto the calculator simulation.

Students are always very surprised by the outcome. An extension can involve investigating different starting points and probabilities.

† is the STO symbol

1†FastRes:
INPUT T;"CHAOS";"T=";"NUMBER OF THROWS";200:
INPUT S;"CHAOS";"S=";"ENTER SIDE LENGTH";4:
INPUT A;"CHAOS";"X=";"START X coordinate";3:
INPUT B;"CHAOS";"Y=";"START Y coordinate";1:
INPUT P; "CHAOS";"P=";"PROBABILITY CORNER A";.3333:
INPUT Q;"CHAOS";"Q=";"PROBABILITY CORNER B";0.3333:
ERASE:
0†Xmin:12†Xmax:
0†Ymin:8†Ymax:
LINE 0;0;S;0:
LINE 0;0;S / 2;S:
LINE S;0;S / 2;S:
A†X:B†Y:
0†C:
WHILE C < T+1 REPEAT C+1†C:
PIXON X;Y:
RANDOM†N:
IF NP THEN 0.5*X†X:0.5*Y†Y:
ELSE IF NP+Q THEN 0.5*(X+S)†X:0.5*Y†Y:
ELSE 0.5(X+.5*S)†X:.5(Y+S)†Y:END:
END:
IF C= =T
THEN FREEZE : END :