Interesting Mathematical Fallacies I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I thought, it's a small group of kids that have strong math skills and seem to have an endless desire for knowledge. I have been able to keep them reasonably in check intellectually, but I'm nearing the end of my bag of tricks. 
For better or worse, they love mathematical fallacies. Of course they were familiar with showing $0=1$ (all numbers equal all other numbers fallacy), and after I showed them $1=-1$ through complex numbers I was out of ideas. I knew of the, all triangles are isosceles triangles fallacy, but couldn't remember it at the time. Does anyone have any interesting fallacies that I can show them? Their math knowledge goes through most of calculus but fallacies outside of that are fine too.
If this is a ridiculous question, I apologize. I figured there were more fallacies out there than what I could find on wikipedia and a few of the other top search results.
Thanks in advanced.
Note: I have another 3 weeks with them so no major rush.
 A: $$\int\frac{dx}{x}=\int(1)\Bigl (\frac{1}{x}\Bigr)\,dx=x\Bigl(\frac{1}{x}\Bigr)-\int x\Bigl(\frac{-1}{x^2}\Bigr)\,dx=1+\int\frac{dx}{x}\quad\hbox{so}\quad 0=1\ .$$
A: Consider the following image:

I've drawn two circles, and the two diagonal lines are the diameters of the circles.
The two angles in the skinny triangle are both right angles, because they are inscribed in a semicircle.
Practice misdrawing this figure before you try presenting this live: it wouldn't do to accidentally draw it correctly, nor to make it too obvious you're distorting things. (or more precisely, that the distortion makes a significant qualitative difference to the shape)
A: When I work with a particularly motivated group of students, I don't feel it's my responsibility to do all of the heavy lifting.
Encourage this group of kids to take what you've shown them and allow them to create their own fallacies and evaluate one another's work.
Have them sign up for their own Stack Exchange accounts. Then they'll have this resource to use when they go on to college.
If you're interested in growing as a teacher, I highly recommend you take about an hour to go through the following resources:
Watch Dan Meyer's 12-minute TED talk here: https://www.ted.com/talks/dan_meyer_math_curriculum_makeover
Then browse and play around with Dan Meyer's bank of 3-Act problems, here: https://docs.google.com/spreadsheet/ccc?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE#gid=0
You have an incredible opportunity. Your students will love being pushed and challenged to justify their mathematical reasoning. Good luck!
