Gregory V. Bard

Associate Professor of Mathematics
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Interactive Web Applications (Interacts)
for Learning Mathematics


Whether you call them applets, apps, interactive webpages, or interacts, these game-like internet resources are great for teaching mathematics to college students. These tools allow students to tweak and manipulate parameters, and therefore teach students how one math object can alter another. Moreover, the visual aspect can be a great boon---a picture is worth a thousand words! Also, interacts are just plain fun, and that helps too.

One of the keys to using interacts in teaching mathematics is: DO NOT ATTEMPT TO REINVENT THE WHEEL. I have on this page a collection of interacts that I've written myself, and further down I have several links to websites that contain dozens and dozens of interacts written by other people.


And now, here are some interacts that I've written myself, for my students.

  • The famous smiley face applet, that shows the role of matrices in computer graphics:
  • The feasible region of a system of inequalities: click here.

  • To visualize the span of two vectors: click here.

  • What is better? 6% compound interest or 8% simple interest? click here.

  • Leontief Input-Output Analysis click here.

  • An interesting problem about a solvent factory and environmental compliance: click here.

  • A simple portfolio balancing problem, risk versus reward: click here.

  • Three Linear Equations in Three Unknowns (unique solution): click here.

  • [Incomplete] The design of the optimal oil barrel, minimizing the metal used: click here.

  • [Incomplete] Step-by-Step, the quadratic equation: click here.

  • [Incomplete] Comparing Contour Plots vs 3D Plots: click here.

  • [Incomplete] The Behavior of Newton's Method on a Polynomial with Multiple Roots: click here.

  • [Incomplete] Visualizing Infinitely Many Solutions in a Linear System of 3 Equations and 3 Unknowns: click here.

  • [Incomplete] How the Complex Numbers are used in Signal Processing: click here.

  • These rather simplistic interacts are the examples that I used in Chapter 6 of my book Sage for Undergraduates:
    • [Incomplete] The Classic Tangent Line Interact: click here.

    • [Incomplete] Visualizing the Definite Integral: click here.

    • [Incomplete] Optimal Aquarium Design: click here.

    • [Incomplete] Amplitude and Phase Shift in Sine Waves: click here.

    • [Incomplete] A Simple Polynomial Graphing Interact: click here.

Here are some collections (treasure-troves, really) of interacts from around the net:


If you'd like to make your own interacts,

  • Chapter Six of my book Sage for Undergraduates, published by the American Mathematical Society in 2014, is an easy-to-understand tutorial for anyone who wants to build an interact using Sage. (Just click "Sage Stuff" on the menu to the left to get a free electronic copy.)

  • The Sage Community Wiki for Interacts has a lot of demos and examples to get you started.

  • There is also a wiki for making applets using JSXGraph, if you already know JavaScript. The syntax is much more difficult than Sage.


Last updated July 21st, 2014.