Turning a Circle into a Rectangle

After reading Steven Strogatz's column Take It to the Limit, I developed this animation (using GeoGebra) of Strogatz's diagram which shows why the formula for the area of circle is what it is, and also provides an approximation for π.

Riemann Sums

A GeoGebra applet from The Lawrenceville School's Equation Plotter collection which draws and calculates area approximations for any function you enter.

Pythagorean Proof

Jim Morey created this elegant interactive geometric proof of the Pythagorean Theorem.

Estimate the Correlation Coefficient

This is an applet which provides you with a set of data points and asks you to guess the correlation coefficient. You can also create your own set of points and guess its correlation coefficient.

Law of Sines and Law of Cosines

This applet can be used to provide convincing evidence for the Laws of Sines and Cosines and, once the laws are established, to provide practice problems. I like to begin by showing just the measure of an angle and its opposite side and turning on the help to see the ratio of the sine of the angle to the length of the opposite side. Repeating this for the other two angle-side pairs and dragging the vertices around to create new triangles suggests that these three ratios are always equal for a given triangle. Similarly, sliding the dot along the line reveals the expressions and calculations associated with the Law of Cosines. To provide practice problems, turn off the help and choose any three of the six possible pieces of information. Drag a vertex (or two or three) to create a new triangle. Check answers by revealing the remaining angle and side measures.

Vectors in the Ocean

This applet provides an introduction to the vector equation of a line in the context of a boat traveling in the ocean. You specify the position vector and the velocity vector and then watch how the boat moves and the equation changes as the time changes. (Making the ocean visible sets the mood, but turning it off makes it easier to see what's happening!)

Derivative Plotter

This applet from the Flash Mathlets collection by Barbara Kaskosz invites you to draw (with your mouse) a derivative of any of the included example functions or one you enter yourself. You can check your graph by dragging a slider to have the applet draw the actual graph of the derivative.

Accumulated Rate of Change and Antiderivatives

This applet from the Flash Mathlets collection by Barbara Kaskosz includes graphs of a bunch of sample rate of change functions. For each, as you drag a slider, you see what the associated population (i.e., antiderivative) would look like. You can also input your own rate of change function and initial condition. What's more, you can draw your own  function, specify an initial condition, draw what you think the associated antiderivative would look like, and then check your antiderivative by moving the slider to see the applet draw the antiderivative.

Definite Integral as an Area Accumulator

This clear, elegant applet from the Flash Mathlets collection by Barbara Kaskosz shows the integral as an area accumulator for a few simple functions.

WolframAlpha

Type in, for example, an equation you want to solve, a function you want to graph or the names of two cities. You will want to keep playing. This video by Robert Talbert gives some ideas about how a math teacher or student might begin exploring.

Vertical Motion Simulation

This GeoGebra applet by Linda Fahlberg-Stojanovska simulates the motion of a projectile fired either straight up or straight down on Earth in the absence of air resistance. The user sets the initial height and the initial velocity and indicates whether units of measure should be feet or meters. You see the motion of the ball along the vertical axis, while the height as a function of time is plotted for you.

Graph of a quartic

In this GeoGebra applet, you graph a quartic by setting the values of the four zeros and the leading coefficient. The coordinates of all relative extrema are shown.

Visualizing the dot product

Mystery Vector Function is a GeoGebra applet I created to introduce students to the dot product. In this applet, students observe a rectangle in which the length of one side is equal to the magnitude of vector b and the length of other side is equal to the projection of vector a onto b. They discover that the signed area of this rectangle is equal to the product of the magnitude of a, the magnitude of b, and the cosine of the angle between them. They can then use the angle difference identity for cosine to show that this is equal to the sum of the product of the x-components and the product of the y-components.