Who knew there was a National Library of Virtual Manipulatives?

The National Library of Virtual Manipulatives includes a handy graphing tool under Algebra (available for grades 3-5, 6-8, and 9-12). This manipulative will help students visualize different functions and the relationship between them. It’s a flexible tool in that you can test up to 3 variable constants with sliders and define up to 3 different functions and graph them simultaneously.

The right sidebar lists the information for how to use the tool in clear and concise language. The main input window is in the upper left of the screen and includes shortcuts for how to write a function. Each of the buttons (parameters, window, and functions) leads to an inset window where you can change the parameters (constants within the equation that are connected to sliders), window size (zoom or reset), and functions (add up to 3 different functions to compare). All of these features are self-explanatory and are easy to use.

The goal of this manipulative is to help students visualize how a function f(x) changes as the input (x) changes. It enables students to trace up to 3 functions at once and determine intercepts. Overall, it reaches that goal. However, there was an issue I found with the Trace feature: the steps in x are preset, so that I could not find the exact intercept as cos(x) crossed the x-axis. As you can see in the screenshot, I traced three functions: f(x) = sin(x), g(x) = xf(x), and h(x) = 3f’(x). When x is approximately pi/2, f(x) should be 1, g(x) should pi/2, and h(x) should be 0. Instead, h(x) = 0.06, and the next step forward in x results in a negative value of h(x). Even when I zoomed in (available under window), the x steps were the same size, too large for my purpose.

When I tutored a high school junior on trigonometric functions, I would have loved to be able to use this manipulative. Every time he came to another change in period, amplitude, or phase shift, he wanted to start from scratch to solve the equation. If I could have showed him this tool with f(x) = a (sinbx +c), then it would have helped him internalize those parameters. The graphing tool is an enhancement for learning because it allows the student the ability to see how functions change in an interactive way, with the student leading the process.