Year 2012 in Review - The Most Popular Posts

It's the end of the year and it's time to look back your favorite posts. Below are the most popular posts for 2012 in terms of traffic (Source: Google Analytics).

1. GeoGebraTube: 12000+ applets and counting
2. Pythagorean Theorem Proof
3. Animation Using Bezier Curves
4. Math and Multimedia Carnival 20
5. The Arithmetic-Mean Geometric-Mean Inequality
6. Deriving the Equation of the Parablola
7. Fibonacci Number and the Fibonacci Spiral
8. Tangram Puzzle: Constructing a Square
9. Systems of Linear Equations
10. The Euler Line
11. Bezier Curves - How Recursive Algorithm Works
12. Squaring a Triangle

Have a prosperous new year everyone!

Square Rectangle Tessellation

The applet below is another tessellation. It is composed of rectangles and equilateral triangles. It is one of the demiregular tessellations.
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

GeoGebra 4.2 Released

GeoGebra 4.2, the newest version of GeoGebra, is now available for download.  Here are the links to know more about the new features.

• Overview of the New Features
• The Official Release Notes
• A more detailed explanation of the Release Notes
• GeoGebra 4.2 Sneak Peek Series

Please visit  my GeoGebra Tutorials page to learn about GeoGebra. I will update them to the newest version as soon as possible. 

Hexagon Triangle Tessellation

Move Points A and B to explore the figure. Explain why the figure tessellates. 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 tessellation above is one of the 8 semi-regular tessellations. Semi regular tessellations are regular tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular tessellations, or sometimes Archimedean tessellations.

Weisstein, Eric W. "Semiregular Tessellation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SemiregularTessellation.html

GeoGebra 4.2 Release Candidate

The Release Candidate of GeoGebra 4.2 is now available. Several of its new features are explained by Balazs Koren in the official GeoGebra blog. I have also written a Sneak Peek on its new features. The Sneak Peek includes the following topics:

1. Sneak Peek 1:  The GeoGebra Window
2. Sneak Peek 2: The Object Properties
3. Sneak Peek 3: Graphics and Layout
4. Sneak Peek 4: Defaults and Advanced Sections
5. Sneak Peek 5: Ten New Improvements

GeoGebra 4.2 will be released in December 1. If you want to learn about this amazing software, you may want to check my GeoGebra Tutorial Series, a collection of more than 50 tutorials from beginner to advanced. 

Slope of a Line

The line below can be changed by dragging the blue points to a different location or by dragging the entire line.  Points C and D can be moved along the line, but they cannot change the line.

• Move C and D along the line and calculate the ratio
  m= ∆y/∆x=(y₂- y₁)/(x₂- x₁) for several different positions of the points. 
• Show the equation of the line and look for a relationship between the coefficients of the equation and  the slope and the y-intercept of the line.

Irina Boyadzhiev, Created with GeoGebra

Download the applet

Tessellation 2: Square, Hexagon, Triangle

The figure below shows that a plane can be tessellated using 3 regular polygons: squares, hexagons, and triangles. 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 GeoGebra 4.2 Sneak Peek Series

I have started writing about the new features of GeoGebra 4.2 in the GeoGebra 4.2 Sneak Peek Series. The series will cover the major changes and new features of GeoGebra. As of this writing, there are already three articles in the series. I will update the series from time to time.

If all goes well, GeoGebra 4.2 will be released within the year.

The GeoGebra for iPad Kickstarter Project has reached US$10,000 but is still accepting donations. Aside from the iPad application, an Android application is also being developed. 

New GeoGebra Institute Website

The GeoGebra Institute of Metro Manila has a new website. We have moved some of the applets and have also created a new GeoGebra Primer. We will be accepting contributors from Filipino GeoGebra users  starting next month. If you are interested, you may contact me at nismedmultimedia@gmail.com.

The GeoGebra Institute of Metro Manila is the first GeoGebra Institute in the Philippines. The Institute is hosted by UP NISMED. 

Demiregular Tessellation 1

Move point A and point B to explore the tessellation. 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 There is no clear definition of what demi-regular tessellations are. Some sources say that there are 14 of them, but give different types. Some also say that there are 20 of them.

A journey along the graph of a function - Increasing and Decreasing Functions

Click the Start button to start the journey of the Tortoise along the graph of the function. When the Tortoise stops, click on him to start the next section of the trip. Irina Boyadzhiev, Created with GeoGebra

This applet can be used for illustration of “increasing” and “decreasing” intervals for a function. The students with some knowledge of calculus can look at the graph of the derivative and how it relates to the intervals of increasing/decreasing.

Download the applet

Increasing and Decreasing Functions

Use the check boxes to show/hide the objects and use the slider to move the points.
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 In an increasing function, the value of y-increases as the value of x-increases. On the other hand, in a decreasing function, the value of y-decreases as the value of x-increases.

The Euler Line

In the figure below, the green, red, and violet 'points' are the orthocenter, centroid, and circumcenter of triangle ABC respectively. Move the points A, B, and C and notice what happens. Use these points to create different types of triangles. 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
The blue line above is called the Euler line. The Euler line passes through the three important points determine by a non-equilateral triangle. It also passes through the center of the nine point circle.

GeoGebra for the iPad Project Needs Your Support

The International GeoGebra Institute has started a Kickstarter project for the GeoGebra the iPad. The project aims to create an iPad application for GeoGebra. If the project is completed, the GeoGebra iPad app will be available for download (for free) at the Apple Store. Please help us raise $10,000 for the project.

For more information about the project, click here. 

GeoGebraTube: 12000+ applets and counting

GeoGebraTube, the official applet repository site of GeoGebra, has now more than 12000 GeoGebra applets after a year it has gone live — thanks to GeoGebraists all over the world.

Applets in GeoGebraTube can be downloaded (including the GGB file), embedded in blogs or websites, or shared to colleagues and friends using social media sites.

Recently, the site has also allowed Google, Facebook, and Twitter login making it even more accessible to a lot of users. Of course,  the old GeoGebra Forum login name and password can still be used.

Note: This is a repost from Mathematics and Multimedia

Sums and Product

Move the green point to a desired location, and then move the red point 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

Questions

1.) What do the numbers on the side of the rectangle represent? 

2.) What does the number inside the rectangle represent?

3.) What is the relationship among the numbers on the green point and the other numbers?

4.) What is the minimum/maximum number that can be obtained inside the rectangle? 


5.) What does the geometric and algebraic interpretation of the problem above?

GeoGebra launches official blog

GeoGebra recently launched the official  GeoGebra blog  . Theblog will contain the latest news, tips, tricks, and everything about GeoGebra. According to the administrator, The blog will also feature post contributions fro GeoGebra bloggers all over the world. 

Recently, GeoGebra is also starting to integrate theorem proving.

One month leave ends on June 22nd

You are probably wondering why I stopped updating this blog for a month. Well, I am quite busy, so I thought I needed a month off. We have a new curriculum here in the Philippines,  so our institute is working on the new teaching guides and learning packages.  This takes most of my time at work, and for the past month, I am bringing some of my work home.

In addition, I am also growing The Mathematical Palette, my new blog. The blog is not about the 'academics' of mathematics but highlights the 'math is fun' mantra.  Most of my posts there are about the appreciation of the beauty and wonders of mathematics. 

Of course, you have probably seen Mathematics and Multimedia, my most active blog. I post  articles there everyday which is also one of the reasons why I have not updated this blog lately. 

With all that said, vacation is over. I will resume GeoGebra applet blogging this weekend. 

By the way, you are all invited to post your applets here. If you are interested to be a contributor to this blog, you can reach me at 

mathandmultimedia@gmail.com.

The Arithmetic Mean Geometric Mean Inequality

Move A along the circumference of the semi-circle. What can you say about the relationship between AD and the radius of the circle?

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 applet above shows the geometric proof of the arithmetic mean geometric mean inequality. The diameter of the circle is a + b and the radius is (a + b)/2. The length AD is less than or equal to the radius of the circle.  The complete explanation of the inequality can be found here. 

Bézier Curves - How the Recursive Algorithm Works

A set of n+1 points will create a Bézier curve of n-th order.

• Drag the first slider to select the number of points. 
• Drag the point on the slider "Recursion Level" to 1 to see the first new set of points. 
• Continue dragging the Recursion Level slider to the end to see all recursive steps and how the last point, generating the Bézier curve, is constructed.
•  Hit the Play button to follow the generating point on the Bézier Curve. 

Irina Boyadzhiev, Created with GeoGebra

More about the construction and other Bezier curve constructions at: MyGeoGebra

Download Applet

Angle Bisector of a Right Triangle

Move point P.  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

1. What can you say about the diagonal of the outer square in relation to its angles? in relation to the right triangles formed by the square? 
2. What can you say about the diagonal of the outer square in relation to the inner square?
3. What conjecture can you make from your observation in 2?