Algebra & Beyond

Sketchpad 3 introduces many tools for integrating dynamic geometry with dynamic algebra. The sketches in this gallery illustrate some possibilities, from the equation for a line, to integration of a cubic. They all make use of dynamic plotting of points from measured or calculated quanitities, and the ones that plot curves make use of constructed loci.

To get started constructing your own, try a Primer on Plotting.

The sketches on this page are available individually (below) or can be downloaded as a package.

Slope and Intercept
Use this sketch as an electronic blackboard or exploratory environment for looking at the relationship between the equation for a line, the angle the line makes with the x-axis, and the tangent of that angle.
A cubic equation of the form y = a (x - p) (x - q) (x - r) is plotted. The curve responds dynamically to changes in the coefficients a, p, q, and r. Two limits of integration, x1 and x2, are given, and between these limits, eight rectangles are constructed to approximate the area under the curve. The total integral is computed both by adding the areas of the rectangles and by an exact computation. It is interesting to compare the results of the two computations.
Sine Graph
This sketch provides an interactive demonstration of the effect of varying the amplitude, period, and phase of a sinusoidal curve. Dragging points within the sketch changes the coefficients.
Complex Numbers
Two complex numbers and their product are shown in the complex plane. As the given numbers are changed by dragging, their product changes accordingly.

Some questions to investigate include: What number when squared equals -1? Is this the only number? What is the geometric result of multiplying a number by i? By -i? Can you find a number which, when squared, equals i?

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