Geogebra!!!

HI! First, let me say that I LOVE Geogebra. It's free, it's visual, it's powerful. My adivce is to take some time to play with it! You can download and install it on your computer. You can also just run it from any web browser.

Links to the GeoGebra Website:
Open a Geogebra window in your brower (no installation!)
Other downloading and installation options

Second, I'm Dave Sankey. I've taught math for a few years and I've used Geogebra from time to time in the classroom. Here's a selection of the Geogebra Docs that I put together throughout my days.

For these files, you can click "Open." This will take you to a web page with that Geogebra file loaded in the web page. You can use this from any browser and Geogebra does not have to be installed.

You also have the option to download the file for use on your machine. Simply download the file and install Gebgebra to use it on your machine.

Making these pages was not difficult at all. When you are using Geogebra, there is an option to export it as a web page. I just used that option to make all of these pages! So, you can modify an existing file, make your own, and make your own web pages!!!

I tell my student to love their fractions, and my fellow teachers to love their Geogebra.

Crazy Swinging Arms
Open - Download
This is just a cool animation. It's "Geogebra Crack." Push the play button the the bottom left corner. Play around with the sliders and the animation speeds to come up with different designs. To change the animation speeds, right click on the angle sliders and select the properties.
Bezier Curve Constructions
Open - Download - Wikipedia
This shows how a Bezier curve is formed. These curves are used in a large number of applications. They were originally developed for use in the auto industry!
Using Bezier Curves
Open - Download
Use two Bezier Curves to make the letters, numbers, whatever else you can think of to make!!! Hide your control points and see if others can figure out how you did it!
Baseball Projections
Open - Download - Maxima*
When a home run is hit, the distance is calculated based on where the ball would have hit the ground. The MLB simply guesses to find the total length! But, if you also know the amount of time, then you can find the exact parabola. This does not take air resistance into account, but it's still a good exercise. Check it out!!!
The Maxima document is what I used to find the necessary equations. If you want further detail on the calculations, look into it.
Trig Constructions
Open - Download
This shows how the graphs of Trig funtions can be constructed. Sine, Tangent and Secant are calculated from the circle, the others are formed from the inverses. Spin the big dot around and watch it go!
(More) Trig Constructions
Open - Download
This is the same as the one before, but it's a little more simplified.
Pythagorean Theorem
Open - Download - Wikipedia
This is a demonstration of the Pythagorean Theorem.
Fibonicci
Open - Download - Wikipedia
The Fibonicci Sequence is pretty famous, but what happens when you consider what happens between the integers? Some knowledge of complex numbsrs will help!
Taylor Polynomial
Open - Download
This shows a taylor polynomial around a given point. Move the point or adjust the slider to show different polynomials.
Vector Field
Open - Download
Geogebra has some trouble with vector fields, but this makes the point well enough. This is a field for y' = sin(x) y. Move the initial point around to get more solutions.
Polygonal Numbers (Counting)
Open - Download - Maxima*
Every number can be written as three numbers, 4 square numbers, 5 pentogonnal numbers, etc. But what about how many ways each number can be written? This shows the solution for n<=1500. It's kind of insane.
The Mandelbrot Set - Wikipedia - I really like the Mandelbrot set... Here are a few things I've put together about it.
Graphing The Mandelbrot Set
Open - Download - Calculation WS - Graph Paper
The Mandelbrot set is pretty famous, and it is one of the reasons we study fractals today. There are actually only two operations that are repeated to create the infinately complicated designs. There's some info on the following page and a Geogebra window at the bottom that draws the set. I originally designed this sheet with students in mind, so it may be a good place to start if you are curious about the set.
Exploring The Mandelbrot Set
Open - Download - Maxima*
There are certain points that I call "roots" of the Mandelbrot set. These points are difficult to find exactly because they involve some serious maths, including finding the square root of a complex number. While that's not too tricky to do just once it can be a hassle to redo many times, which is true in this set. So, instead of actually doing the math stuff, I just decided to write a Maxima script to find some points by brute force. The result is a number of points that look like the Mandelbrot set. This sheet is a workhorse, displaying thousands of such points and 200 iterations of the indicated point. The stats in the upper right hand corner indicate how close you are to one of the actual roots.
Areas of Particular Iteration in The Mandelbrot Set
Open - Download - Maxima*
The Mandelbrot set converges or diverges... When it converges, it converges to a point or a number of points. Each color indicates a different number of points that the set converges to in that area. The Maxima sheet is my "brute force" method to find out which points are where. Basically, I did about a million iterations, then I compared the next ten to see if any were similar. If number one million matched number one million and 5, then I put it in the set for 5. Rince, lather, repeat.
Different kinds of plots - Geogebra is pretty versatile in the kinds of plots it can handle.
Generic Parametric Plot
Open - Download
Parametric equations can defeat the "vertical line test" because the independent variable is not x or y. You'll study these kinds of equations in later Calc courses. The trace is the slope as you drag the point around.
Parametric Plot Construction
Open - Download
This shows how a parametric plot is actually formed. Watch as the blue matches the blue and the red matches the red. Cool, huh?
Polar Plot Construction
Open - Download
Polar coordinates spin around the origin. So, instead of a point like (x,y), you are concerned with the angle and the distance from the origin. Note that the point on the line is the same, it's just that in polar case the line spins and in the traditional case the line moves forward.
Area under the curve
Area Sliders
Open - Download
This allows you to try to find the area under a curve by moving a number of vertical lines. Do your best to try and find the exact area!
Drawing Rectangles
Open - Download
This is the classic area under the curve illistration. You can vary the finction, the number of recatangles, left, right or midpoint, and then make the rectangles into triangles.
3D Rectangles
Open - Download
This is a demo I did based on a project we did at the AP Calc APSI. I forget what that was, but I still have this file and it looks cool!
Graphing Lines
Standard Form, Given A, B and C
Open - Download
Standard form is: Ax + By = C
Standard Form, Given 2 points
Open - Download
Slope Intercept, Given m and b
Open - Download
Slope intercept from is: y = mx + b
Slope Intercept, Given 2 points
Open - Download
Point Slope Form, Given Point and Slope
Open - Download
Point slope form: y - y0 = m(x - x0)
Point Slope Form, Given 2 Points
Open - Download
Parabola given A, B and C
Open - Download
Parabolas can be written with a number of forms. The most common is: y=ax^2+bx+c
Parabola given 3 points
Open - Download
Parabola given 2 roots and an amplifier
Open - Download
Cubic given 4 points
Open - Download
We don't work with cubics that often... However, I thought this was a really cool effect so ENJOY!!!
Exponential
Open - Download
Exponential functinos can model a lot of different things. When the base becomes negative, you get some complex activity.
Solving Triangles
ASA
Open - Download
There are 4 conditions in which you may attempt to solve a triangle.
  1. A side and any two angles (ASA)
  2. Two sides and the angle between them (SAS)
  3. Two sides and an angle not between them (SSA)
  4. Three sides (SSS)
These sheets can be used to double check your answers, or simply to visualize your answers. It's also helpful to see if the triangle(s) can actually exist.
SAS
Open - Download
SSA
Open - Download
SSS
Open - Download

* Another program I use is Maxima. To get these files, you'll have to right click and save the file to your computer. Like Geogebra, it's free and powerful. Maxima is a CAS that you can use for a number of purposes. If you are not familiar with Maxima or other CASs, it will probably look pretty Greek to you. However, if you'd like to have a program to do some of your math heavy lifting, give it a look!

Word!