Fibonacci Sequence – http://nlvm.usu.edu/en/nav/frames_asid_315_g_4_t_1.html
The purpose of this manipulative is to allow students to explore the Fibonacci sequence, the golden ratio, and the relationship between these two things.
The applet itself is very simple. One can use it without entering any data, or can enter two starting digits to see how this changes the sequence. It works very well.
Accompanying the applet are some instructions for use, information on the NCTM standards which the applet meets, and a brief article explaining some background and making some suggestions as to how to use the applet in a classroom setting.
Buttons on the applet allow you to scroll through numbers in the sequence. Along with each number is the ratio of the quotient of the number and the one just higher to it. This allows one to see that as you progress through the sequence, this quotient approaches the golden ratio, which is conveniently included.
Taken by itself, I think this applet is interesting, and might be a fun way to change pace in a class. I could also see using this as part of a lesson in the history of math, and also as a way to augment lessons on functions. Combining it with other applets, such as the Golden Rectangle that follows, could yield a more powerful lesson.
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Golden Rectangle – http://nlvm.usu.edu/en/nav/frames_asid_133_g_4_t_1.html
When you run the Golden Rectangle applet, a spiral is drawn inside a rectangle in such a way that each time the spiral intersects the side of the rectangle, a perpendicular is drawn.
This perpendicular creates a new rectangle of the same proportion as the original. All the rectangles created a proportioned to the Golden Ratio. Like the Fibonacci Sequence applet above, this tool includes instructions, information on NCTM standards, and some background and suggested uses.
The Golden Rectangle provides a visual representation of the Golden Ratio illustrated in the Fibonacci Sequence applet. Using these two tools together could help students get a sense of the relationship between mathematics and the real world, especially if done in the context of a lesson on where the concepts of Golden Ratio and Fibonacci Sequence first came from, and how they’ve been significant over time. I think one of the things that makes it difficult for some students to “get” math is that in addition to being abstract, it probably seems to be rather arbitrary. Teaching more how these ideas came into being, what sort of problems gave rise to them, and how they’re used today might make the subject much more concrete for many.
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Turtle Geometry – http://nlvm.usu.edu/en/nav/frames_asid_178_g_4_t_1.html?open=instructions&from=vlibrary.html
Turtle geometry is like a simplified game version of Geometer’s Sketchpad that uses only lines and angles. By entering a series of commands, one can program the turtle (cursor) to move geometrically.
With the right combination of moves, one could draw various shapes. To do this successful requires some thought about which angles and lengths would be appropriate.
In addition to a blank screen, one could choose to navigate a maze, follow a drawing, or walk through randomly placed rocks. Each type of exercise requires students to think about angle and length/distance, which are important concepts for geometry. This applet can also help students begin to think of some of the ways that mathematics might play a role in things like computer programming and building. Making these connections would likely require some discussion.
I am not sure, though, that this manipulative would be worth using in the classroom. I can think of more active ways of teaching the same concepts that might involve getting outside the classroom and building an actual maze or writing out instructions for a geometric treasure map.
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Linear Function Machine – http://www.shodor.org/interactivate/activities/LinearFunctMachine/
One of the clearest metaphors I encountered as a student was to think of a function as a machine. This function machine applet has students input x values, receive y values, and then work out the linear relationship between the two.
The applet begins with blank input boxes. One can enter multiple values, one at a time, in order to get a sense of the pattern the function follows.
The applet page includes four tabs: Learner, Activity, Help, and Instructor. Activity is open by default and includes the applet. Learner provides some basic definitions and background, and a list of links to related resources. Help contains instructions for using the applet. Instructor outlines some basic information useful to teachers, including how the applet aligns with various standards.
This is a very slick and useful tool. The same group (shodor.org) has a simplified version that helps to introduce the concept of functions. I could see using both applets in the classroom. Last spring I observed a lesson where students were first being introduced to functions. The idea seemed very foreign to them, almost like it was a major departure from the math that had come before. This might have been because of the new notation. Being able to have a tool that’s both fun and challenging and provides a concrete metaphor for what’s happening with the numbers would, I hope, make the transition easier.
Aside from being a dynamic play-around-with-mathy-stuff kind of tool, 
