## Geogebra Examples

- Geometry
Triangles

Pythagorean Theorem - Demonstrates the Pythagorean Theorem by repositioning four a-by-b right triangles within an (a+b)-sided square.

Angle and Opposite Side - Is an angle and the opposite side length enough to guarantee triangle congruency?

Angle - Side - Angle - Are two angles and the side length between them enough to guarantee triangle congruency?

Side - Side - Side - Are three side lengths enough to guarantee triange congruency?

- Circles
Circle Area - Motivate the formula for area of a circle based on areas of regular polygons.

- Solids
The Cavalieri Principle - Prisms - Compare the volumes of prisms by base area and cross-section.

Prism Scaling - What happens when we scale the height of a prism? The same number of 3d units, but of different size. Would this apply to solids other than prisms??

The Cavalieri Principle - Pyramids - Compare the volumes of pyramids by base area and cross-section.

The Cavalieri Principle - Slant - Does a slanted prism have the same volume as a right prism? Check the cross-sections.

Pyramid Volume - Justify the formula for pyramid volume by dividing a cube into three congruent pyramids.

Sphere Volume - Justify the formula for sphere volume using Cavalieri's Principle.

Sphere Section - The calculations behind the cone vs. sphere construction.

Fill the bowl - Raise and lower the water level in the bowl and guess how full it is. Then show the RatioText to check your guess (Notes: Sorry about the leaky bowl. This one's a little slow to update.)

- Trigonometry
Sun Elevation Graph - Using a sine function to model sun elevations during the year in various cities.

Length of day - Using a sine function to model length of day during the year in various cities.

Globe - Demonstrate the effect of the earth's tilt through the seasons.

Sun Elevation - Demonstrate the elevation of the sun at various latitudes and seasons.

- Calculus
The Derivative - Input (and optionally hide) a function, then explore. Options include color-coding the function for increasing/decreasing or for concavity. The tangent line at a point can be viewed, as well as graphs of the first and second derivatives.

Continuity - Explore the epsilon-delta definition of continuity. Move a point on the graph, set your epsilon, and find a sufficiently small delta.

- Miscellaneous
Gerrymandering - Pack 'n' Crack - See the difference that drawing district boundaries can make on representation. Set the level of support for "red" and "green", then pack and crack to achieve the best results for your side.

The Gallows Problem - Explore a curious puzzle, with some relation to geometry and slope.

A treasure map of a desert island, tells you to start at the old gallows, then count your steps as you walk to the big tree on the east of the island, turn right and measure off the same distance. Pin a marker in the ground at the spot. Now go back to the gallows and count your steps to the big tree on the west, turn left and measure off that distance. Pin a marker at the second spot. The treasure will be halfway between the two markers.

Sadly, when you go to the island, the two trees are still there but the gallows has blown away. Can you still find the treasure? This Geogebra construction shows that the position of the gallows doesn't really matter. Can you figure out why?