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Interactive Mathematics on the Web: LiveMath - Live Examples

Carl Spitznagel

[See the Editor's Note at the bottom of page 1 -- the examples on this page are no longer "live".]

As a first example, consider the Power Rule web page, which includes a simple LiveMath notebook designed to allow students to make a guided discovery of the formula for the derivative of the function f (x) = xn. In this web page you can change the exponent in the power function and immediately observe the derivative of the new function. When I use this web page with a class, I give the students a worksheet that provides guidance about the kind of experimentation I expect them to do, and that also suggests some values to try for n (positive constants, negative constants, zero, integers and non-integers, and finally non-constant values, such as x). The point of the exercise is to let students discover for themselves not only the form of the Power Rule, but also the fact that it works for all constant powers, but not for non-constant powers.

My parent web page also includes links to discovery learning notebooks for

  • translations and reflections of functions,
  • the Product Rule,
  • the Quotient Rule,
  • the Chain Rule
  • the Fundamental Theorem of Calculus, and
  • an interactive grapher for functions of two variables.

For an example of the kind of worksheet that I use to guide the students in their interactive discovery process, you can look at a sample Product Rule worksheet. Of course, with any discovery activity, timing is extremely important. There is little point in having students "discover" something after it has already been discussed in class. Also, in certain cases you will need to select very carefully the functions, equations, or expressions that you will ask the students to work with, since LiveMath (or any other CAS) may simplify its results more than you might like, thereby making it difficult for students to make the desired discoveries.

Additional examples of web pages featuring interactive LiveMath notebooks can be found by clicking here. This web page includes

  • links to examples of additional graphing capabilities,
  • interactive pages on
    • the difference quotient,
    • completing the square,
    • the quadratic formula, and
  • a page to explore the geometric effects of various linear transformations.

Carl Spitznagel, "Interactive Mathematics on the Web: LiveMath - Live Examples," Convergence (November 2004)