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.

Questions

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.

Questions

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

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 www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
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

Snapshot

Parabola as Locus of Points

Move point C and observe what happens.

The green trace generated by point D is a parabola.  A parabola is the set of points equidistant from a given point (focus) and from a given point (directrix). In the applet above, point B is the locus and the red line is the directrix.

Tangram Puzzle 1

Use the tangrams below to construct a square.  Drag the interior of the polygons to move them and drag the point to rotate.

Tracing the Sine Function

Move point C about the circle. Observe what happens to point P.

Questions

1. What can you say about the lengths of segment CD and segment PE?
2. What can you say about the distance of E from the origin? What is it equal to?
3. What do x and y represent in P(x,y)?
4. What do the traces of P represent?

Dice Rolling Simulation

The applet below is a simulation of rolling of two dice. Click the Roll Dice button to roll the dice. Do this 30 times and record the sum. What do you observe?

Locating Pi on the Number Line

Move slider t to the extreme right and observe what happens.
This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
The circumference of a circle with diameter 1 is approximately equal to 3.1416 or $\pi$.

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 $\alpha$ and $\beta$ 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.

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