Reflecting the Exponential Function

The red point is the reflection of the green point on the blue line. Move the green point and observe what happens.


  1. What is the relationship between the coordinates of the points?
  2. If the coordinates of the green point are (x,y), what will be the coordinates of the reflected point?
  3. If the green point and red point are connected with a line, what can you say about that line and the line y = x?

Exploring Triangles

Triangle ACD and triangle BCD are equilateral triangles. 


  1. What can you say about triangle BCD and the red triangle? Move the slider confirm your answer.
  2. Change the shape of the triangle by moving point C, and use the slider to compare the triangles. Does your observation still hold?
  3. Make a conjecture about triangle ACE and triangle BCD
  4. Prove your conjecture. 

Circle Area Approximation

Move slider r to change the length of the radius and move slider n to the extreme right. What do you observe? 
This is a Java Applet created using GeoGebra from – it looks like you don’t have Java installed, please go to
Linda Fahlberg Stojanovska, August 2011, Created with GeoGebra

The applet above shows that as the number of divisions of a circle with radius r increases, the sectors can be arranged into a rectangle with base πr and base r


Area Under a Curve

The applet below shows the relationship among the area under the curve, the lower sum, and the upper sum. As the number of rectangles n increases, the sum of the areas of the rectangles get closer and closer to the actual area under the curve.
The approximation above is called the Riemann Sum. It was named after Bernhard Riemann, a German mathematician. 

Angle Sum Theorem

Move points A, B, and C to choose the triangle size and shape you want.  Move sliders and to the extreme right.

  1. What do you observe about angles 1, 2 and 3
  2. What is the sum of angles 1, 2, and 3? Justify your answer. 
  3. Based on your observations above, what can you say about the sum of the interior angles of a triangle? Explain your answer. 
1 2