Learning about solids of revolution in Calculus can be confusing, especially if you have learned two different methods (often called the disk and shell methods).  Below is a concept map created in Bubbl.us to help students understand which would be easiest to use. You will have to zoom in to see all of the writing. You can use your mouse to move around the content.

Pythagorean Theorem

Go on a quest to explore the Pythagorean Theorem 
and learn that Pythagorus isn't the only one who discovered the beautiful connection between the sides of a right triangle.

Watch on Khan Academy

Read on Wikipedia

Explore on pbs

Read about different cultures and the theorem

Chinese Proof

Hint

visualizing 1

visualizing 2

visualizing 3

pbs ladder and baseball problem

braining camp

History of Math
Timeline of a section of history using Time Toast http://www.timetoast.com/

Slope Intercept form of a Line

Web 2.0 Evaluation of GeoGebra

GeoGebra is a free program that integrates geometry, algebra, and calculus.  It runs on any platform that supports java.  GeoGebra can be used for teacher presentations, simulations, and both student and teacher authoring.  GeoGebra allows for students to explore a concept in depth and with a visual approach.  Time is reduced in modeling and exploring, and students are allowed to discover patterns and connections.  Students and teachers are able to use this a collaboration tool and can share what they have learned, although embedding a project onto websites or in blogs requires more technology knowledge.  GeoGebra is a powerful program and moderately easy to use.

For teachers, GeoGebra can be used to decrease modeling time, increase motivation, and engage them in critical thinking.  Teachers can show geometric constructions, calculating the area under a curve, and manipulation of parent graphs with ease rather than relying on tedious pencil and paper methods.  Students are able to receive immediate feedback as the equation changes (by changing the slope of a line or the vertex of a parabola.) Using visual representations gives teachers another method to reach students who are visual learners, and for others, it encourages full understanding of a concept.  Not only are students able to see the visual representation, but also they are able to discover patterns by manipulating variables and receiving immediate visual conformation.   

GeoGebra can also be used a student authoring tool.  GeoGebra allows students to explore real life applications and discover patterns in self-directed projects.  Since it easily models equations, calculations involving long decimals and complex calculations are not a deterrent.  The interactive and student led nature of the program increases the control the student has over their own learning as well as the motivation.

Since GeoGebra runs on a Java application it has a tendency to lag.  However, it can be downloaded to your computer and use it without connecting to the Internet.  GeoGebra is moderately easy to use and is fairly intuitive using buttons to represent actions (the text buttons have ABC written over them and the move button is represented as an arrow.)  Any questions on how the program works can be answered in the clear tutorials on the website as well as on YouTube.

Teachers and students can use created applications to share their knowledge.  GeoGebra has a site where projects can be downloaded and used by the online community.  The site is easy to use with a vast source of material available for a variety of math concepts.  If teachers or students want to embed a project into their websites or blogs, the process is more complicated, but achievable.

Overall, GeoGebra is a helpful tool when teaching mathematics.  It gives teachers a way to easily model concepts.  Students are able to be part of discovering patterns in mathematics, as well as receiving immediate visual feedback.  This tool will be especially useful with the problem solving emphasis of the Common Core Standards.  Both students and teachers can use the program to create their own projects with relative ease.  They are also able to access a variety of resources on the website using this program. 

I have created a new applet using Geogebra so that students can use sliders to change the slope and intercept of a line and receive immediate feedback. This applet requires that java is enabled. Use the sliders to change the slope and/or intercept of the line to see how it changes.

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 Sum of the Angles in a Triangle is 180 degrees

All of the triangles add up to 180 degrees.  We can do a rigorous proof of this (see below), but if you just want to illustrate this in a visual way, you can check out my newest You Tube video!

Before doing a rigorous proof of the triangle angle sum theorem (the sum of the angles in a triangle is 180 degrees) some background knowledge of geometry is needed.

1) A straight angle is defined as 180 degrees.

 

2) Congruent means the same size.  Also Congruent Angles have the same measure of degrees (or radians).  When we refer to the actual the degree measure, we use an m in front of the angle sign.

3) Parallel lines are lines are ALWAYS equidistant (equal distance) from each other.

The red segments show that the distance is equal.  

The red segments show the distance is not equal.

4) Understanding substitution:  If we say a variable x = z and we also know that y = z, then where ever we see an x we can substitute a y (since they both equal z.)  We can also substitute y in for x.

 5. When you have a line and a point not that line there is only one line you can draw through that point that is parallel to the original line.  This is called EUCLID'S PARALLEL POSTULATE. 

A line and a point not on the line.

Only one parallel line can be made through the point.
The pink lines are NOT parallel to the blue.

6. Vertical Angles are congruent.

7. When parallel lines are intersected by the same line or segment (called a TRANSVERSAL) the angles created are related to each other.

Adding another transversal to our parallel lines and using that vertical angles are congruent we can show numerous angle congruences.

For this particular proof, we only need to use the alternate interior angles (you can also construct a proof using corresponding angles.)

Okay, now we are close to being ready to construct a formal proof.  It is best to go through the steps informally first, and then write them down.   Here is a review of the background information we will use.

Draw a triangle.

Add a line parallel to AB.

Extend line AC so that you can recognize the transversal.

Mark the alternate interior angles as congruent.

Extend CB and mark those alternate interior angles congruent.

Now all the angles line up so they must be 180 degrees.
We will have to use substitution when we get to this part in our formal proof.

 Now, on to the formal Proof!

Given: Any triangle ABC

Prove: The angle sum is 180 degrees

Auxiliary line just means one that was not given in the diagram.

Angles are defined a a point between two rays.

I added a pink color so that < 3 stood out.

Remember if we actually want it to equal 180 degrees, we need to use measure (m) 

Beautiful!

Watch the video by clicking the link below.

Proof Video