This is the write up for the proof and reasoning article:

    I think this section of the book is very true.  I think that students need to be challenged in this way.  The students may not like all the thinking that they have to do on their own, but it will help out in the long run.  I know that when I was in school I wanted the teacher simply to tell me why, not ask me to figure it out.  The examples that were used in this section were a lot of real world examples, which I think helps the students understand more.  Some of the things in this section, like splitting the paper into thirds, I never would have thought of myself.
    Now that I am in college and in my higher-level math classes, I enjoy doing this type of logic.  Giving examples and counterexamples are the main focuses in my current classes.  I am beginning to enjoy this logic, reasoning, and proving, especially when I get the correct answer, it makes me feel a little bit smarter.  In some of my math classes in high school my teachers would give us these types of reasoning and proof questions as a bonus.  There again, the more I got correct the more confidence that I had in myself.  When I teach I want to be able to challenge my students, and give them confidence in themselves as well.
 
 
 
 

After looking at all of the Java Sketches, I realized that I only understood in great detail, the first two Java Sketches.

The first one Median and Centroid Exploration, I understood that if you put the moveable point (G) on the center of the segment (BC), that the two smaller triangles that were formed (the blue and yellow) were egual in area.  In essance the line (AG) bicects angle (BAC) therefore creating two smaller triangles of equal area.



If point (G) is put above the mid of (BC) then the yellow triangle is bigger than the blue, and the angle (BAC) is not bicected.

The opposite holds true if you put the point (G) below the mid (BC), the blue trangle is bigger than the yellow.

 

In the second exploration, Four Centers of a Triangle, I noticed how each of the four centers changed as you moved one of the conners of the triangle.


I also noticed that if you made a right triangle, then two of the centers would lie on the triangle.

 
 
 

It was fun to play around with these explorations, and i think that I learned a little bit from some, but there was also some that I didn't understand.