Brazil is a freely downloadable executable program that uses iterated function systems to create fractals. The graphical interface is very easy and fun to use.
The program can be used to help the user make the connection between the affine transformations in an IFS and its attractor.
The program can be also used just for fun by younger students. Even when used in this way, it is likely that the students will develop an enhanced feel for geometry and transformations.
Technical Requirements:
Brazil runs on Windows-based platforms.
Identify Major Learning Goals:
At one level, Brazil can be used as a fun math game is which users manipulate geometric figures and try to predict the consequences.
At another level, Brazil is designed to allow the user to discover the relation between the affine transformations in an IFS and its attractor.
Target Student Population:
High school and college math students studying fractals. Younger students may enjoy using Brazil also.
Prerequisite Knowledge or Skills:
No real prerequisite knowledge is necessary since the program may be used as a strictly graphical fractal building game, but if the user is familiar with affine transformations of the plane and iteration of a function or process the scope of what may be learned from this applet increases significantly.
Content Quality
Rating:
Strengths:
Brazil is a very nice piece of IFS (iterated function system) software. In it, the affine transformations of the plane that constitute an IFS are defined entirely graphically/geometrically. An affine transformation of the plane maps an image I1 (i.e., subset of the plane) to another image I2. Brazil lets the user graphically manipulate a starting image I1 by compressing/expanding, translating, shearing, rotating, and/or reflecting it to produce an image I2. Brazil then places the affine transformation that maps I1 to I2 into the list of functions that constitute the user?s IFS. When all the transformations have been created, a button click generates the (usually fractal) attractor of the IFS.
All of this works very smoothly. The user carries out his image transformations by means of slider and/or button clicks. Graphic feedback on this process is continuous and immediate. This interface lends itself to discovery activities that ask the user to find the IFS that will generate a given fractal.
It doesn?t hurt that using Brazil is a great deal of fun. Getting students to sit down and work with Brazil is no problem. Getting them to stop is another thing!
Concerns:
The mathematical content of Brazil would be strengthened if the user could see the explicit mathematical expressions for the affine transformations of the plane that are being generated by the user?s graphical/geometrical manipulations. Some of this information is available on Brazil?s ?Construction? tab, but a separate optionally-opened window where the affine transformations appear and are continuously updated would be much more useful.
There is not much in the way of explanation of IFS here. The instructor will need to supply this.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
Brazil is fun and easy to use, which enhances learning and makes it easier to get students to stay on task long enough to absorb more difficult concepts.
Because Brazil makes it so easy to vary the functions in an IFS and quickly observe the impact on the attractor, it is a very effective way for students to develop a feel for this IFS functions ? IFS attractor relation.
Brazil is great at developing geometric intuition in students, but less so at developing algebraic intuition (see concerns).
Concerns:
The effectiveness of Brazil would be improved if the user could see the explicit mathematical expressions for the affine transformations of the plane that are being generated by the user?s graphical/geometrical manipulations.
The steps in the iteration process are not explicitly carried out; rather the (usually fractal) attractor of the IFS appears fully formed using the ?chaos game? probabilistic method for approximating IFS attractors. There is nothing wrong with this mathematically, but it is less effective at driving home the IFS mechanism for generating fractals than is explicit step-by-step iteration.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
It would be difficult to imagine an interface easier to use than this one. Parameters are adjusted using convenient sliders, with the effects being immediately visible in the design window. The proper use of most of the controls is self-evident; help is available when necessary.
Concerns:
Brazil does require downloading and installation. Although this process does run smoothly, it is slightly more involved than using applets.
Creative Commons:
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