This is an experimental applet for constructing piecewise-linear functions. The graph consists of three connected line segments, which can be adjusted by dragging the endpoints. The formula for the function is displayed in the bottom right corner.
Tuesday, September 14, 2010
Friday, September 10, 2010
Multiplication of Inequalities
When we multiply both sides of an inequality by a negative number, the direction of the inequality is reversed, but the direction is preserved when we multiply both sides by a positive number. Why does this happen?
You can use this dynamic worksheet to explore multiplication of inequalities. The values of x, y, and z are changed by dragging the points on the number lines.
Sunday, May 9, 2010
Exterior angles of a polygon
An exterior angle of a polygon is an angle formed by one side and a line extended from an adjacent side. In the figure, the exterior angles of the polygon are shown in green.
What do you think is the sum of the exterior angles of a polygon? Move the slider and find out!
Friday, May 7, 2010
Eratosthenes' measurement of the circumference of the Earth
It is a popular myth that Christopher Columbus set out on his voyage in order to prove that the Earth is round. In fact, this was known long before Columbus. The first accurate measurement of the circumference of the Earth was made by Eratosthenes of Cyrene, circa 240 BC. It is remarkable that this calculation was performed using only high school geometry.
Eratosthenes knew that on the summer solstice at local noon, the sun would be directly overhead in the city of Syene. This is because Syene lies on the Tropic of Cancer. He also knew that at local noon in Alexandria on the summer solstice, the sun was 7 degrees south of the zenith (the highest point in the sky). He used this information, as well as the distance between Syene and Alexandria, to calculate the circumference of the Earth.
In the picture shown above, the sun's rays are nearly parallel lines, because the sun is so far away relative to the size of the Earth. The parallel lines are cut by the transversal AD. Therefore, the sun's angle at D is equal to the central angle shown in the picture. (This angle has been exaggerated for clarity.)
The circumference of the Earth is calculated by solving the following proportion.
(distance)/(circumference) = (central angle)/(360°).
I have created an interactive applet to help in exploring this topic. I borrowed the idea for this applet from Ihor Charischak.
Eratosthenes knew that on the summer solstice at local noon, the sun would be directly overhead in the city of Syene. This is because Syene lies on the Tropic of Cancer. He also knew that at local noon in Alexandria on the summer solstice, the sun was 7 degrees south of the zenith (the highest point in the sky). He used this information, as well as the distance between Syene and Alexandria, to calculate the circumference of the Earth.
In the picture shown above, the sun's rays are nearly parallel lines, because the sun is so far away relative to the size of the Earth. The parallel lines are cut by the transversal AD. Therefore, the sun's angle at D is equal to the central angle shown in the picture. (This angle has been exaggerated for clarity.)
The circumference of the Earth is calculated by solving the following proportion.
(distance)/(circumference) = (central angle)/(360°).
I have created an interactive applet to help in exploring this topic. I borrowed the idea for this applet from Ihor Charischak.
Thursday, May 6, 2010
The Missing Square Puzzle
The Missing Square Puzzle is a classic puzzle that was popularized by Martin Gardner. A right triangle is cut up into four pieces which are rearranged to form an identical triangle, except that a square is missing!
Many words have been written to explain this paradox, but the answer becomes clear if you can actually move the pieces around. Click here to view a GeoGebra worksheet that will help you to explore this intriguing puzzle.
Many words have been written to explain this paradox, but the answer becomes clear if you can actually move the pieces around. Click here to view a GeoGebra worksheet that will help you to explore this intriguing puzzle.
Interactive Proof of the Pythagorean Theorem
I have always been intrigued by proofs without words, and I have been learning how to use GeoGebra to create dynamic geometry applets. Here is a web applet that allows the user to demonstrate the Pythagorean theorem by moving right triangles within a large square. Comments or suggestions are welcome. Click on the picture to launch the applet.
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