Web-based Practice
Dueling Calculators by NLVM
Course, Subject
Geometry (Next Generation), Math, Science & Technology, Mathematics (NYS P-12 Next Generation)
Virtual Manipulative
Click here for this interactive resource.
Description
This manipulative can be generative for many profitable class discussions. In formal terms, this manipulative illustrates sensitive dependence on initial conditions, a classical definition of chaos. More simply, we have a dramatic illustration of some of the limitations of computing devices.
Students can easily get the impression that calculators are infallible. With a scientific calculator in radian mode, enter a number, say 0.6, and then press Cos. If we repeatedly take the cosine of the displayed answer, eventually the display will approach a fixed number, 0.739085..., which is the root of the equation cos x = x. In a classroom setting, all students performing this same sequence of operations will observe the same behavior, will find the same limit for the iteration of the cosine function, whatever their calculator and however many digits it displays.
For the functions provided in the Dueling Calculator manipulative, you can choose any initial value between 0 and 1. Iterating the function will give essentially the same results on both calculators for the first few iterations. Then the differences begin to build up and the results become wildly different, as may be seen graphically.
Content Provider
The National Library of Virtual Manipulatives is a three-year NSF supported project to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-8 emphasis). The project includes dissemination and extensive internal and external evaluation. For more information, please visit http://nlvm.usu.edu/en/nav/vlibrary.html.
Credits
Principal Investigators
Larry Cannon
Jim Dorward
Bob Heal
Leo Edwards
Java Applet Programming
Ethy Cannon
Joel Duffin
David Stowell
Zeke Susman
Richard Wellman
Jennifer Youngberg
Web Site Programming
Joel Duffin
