I know that there is a visual demonstration of $a^2+b^2=c^2$ using a smalĺ piece of paper, but there are also a lot of variations.
Which visual or drawing demonstration of the Pythagorean theorem can I show to 14-year-old students?
A number of proofs (43!) can be found at cut-the-knot.org. Some of these are described below.
I don't know how hands on you want the students to get with the visual aspect. I remember doing these proofs around that age. I recall this progression and it made sense to me why Pythagoras's Theorem was true.
Geometrical demonstration: Pythagoras's Theorem can be easily demonstrated. Construct squares on each side of the triangle, as shown below. Then, the large squares can be subdivided into smaller squares, which can be counted to be the same quantity. Also, I recall cutting the $c$ square to show that it overlaps exactly with the $a$ and $b$ squares. This technique can show it is true for particular values of $a,b,c$, but cannot prove the statement is true in general. (Image source: http://www.myastrologybook.com/PythagoreanTheorem16c.gif)
Pythagorean proof: This proof uses squares of equal sizes to show the desired relation. There is some manipulation of the squares required in this proof. (Image source: http://en.wikipedia.org/wiki/File:Pythagore.jpg)
Algebraic proof: This leads naturally from the second proof as a way of confirming what is being seen in the images. A detailed proof is given at mathsisfun.com. I will outline the basic steps below.
Each of the squares in Proof 2 has area $(a+b)^2$. In the left square, the total area is the area of the square plus the area of the four triangles, which is $c^2+4\times\frac12ab$. In the right square, the area is the sum of the two smaller squares plus the area of the for triangles, which is $a^2+b^2+4\times\frac12ab$. Equating both sides and simplifying gives the desired result.
The best explanation I know comes from this answer by Emanuele Paolini (the only thing I did was to redo the pictures, please go and upvote Emanuele's post).
The point is that squares in the usual picture
doesn't have to be squares. The only thing is that we need to use the area (so that is scales with a square of the scale), it could have been pentagons.
But it could also be appropriate right triangles:
It is the similarity of the small triangles that makes it work (and explains why the triangle has to be right for the theorem to work).
I hope this helps $\ddot\smile$
Check out the book "Proofs Without Words" for a bunch of nice visual proofs of the Pythagorean Theorem.
Here is one of my favorite visualizations of the Pythagorean Theorem:
There is also a fun activity here. Have the kids draw two adjacent squares of any size of their choosing, and use the technique shown above to have them cut the squares into 5 pieces that can be arranged to form one larger square. The fun part about this is that the kids get do determine the discrepancy in the sizes of the initial squares, and no matter how large or small the discrepancy, this technique always works.
Here's a link to a Google+ posting about exactly this. In addition to additional links, it has a GIF using water to demonstrate the equalness of squares.
Maybe you can do President Garfield's proof and combine it with a history lesson.
page 161 of the New-England Journal of Education, April 1, 1876 (image from Google Books)
Note: M. C. = Member of Congress.
Here is a Java applet showing Euclid's proof:
This was in my math folder, I don't know its source.
It was shown as 'not needing any further explanation', but of course, for lower level students, I connect the 3 points on the circle's perimeter to form the larger right triangle and remind them of the ratios formed by dropping an altitude (line b).