Here’s a recent xkcd which as a math educator I found particularly funny. Some questions for my readers:
- What numbers besides 3 and 9 would “work” here?
- Do you have any particularly funny or interesting stories of students getting something right for the wrong reason? Post them in a comment (or, better yet, on your own blog if you have one!).
Also, many parts of my optimal change-carrying challenge remain unanswered (although this is likely due in part to the unfortunate downtime I experienced shortly after publishing it). If you haven’t already, give it a try, you’re sure to find some aspect of the problem that interests you!
The Math Less Traveled 
1. 2 and 4
2. Simplify the fractions:
a) 16/64
b) 19/95
Strike out the common ‘6’ or the common ‘9’ and you end up with the correct answer.
Update to my previous comment:
For the first question, any two numbers of the form ‘x’, ‘x^2’ would work?
In 10th grade, Algebra 2, I still couldn’t graph better than “a rough idea” – too sloppy, no real interest in the skill.
So I solved systems of equations algebraically, graphed the solution, and graphed the y-intercepts, and made it seem like I crossed the lines to find the point of intersection….
Raj: Yes, I think you are right about x and x^2. I like your other example too. =)
Jonathan: well, that was hardly the WRONG reason. Maybe “not the reason the teacher was looking for”… =)
I agree on the n and n² answer.
If we just think of the numbers as a x b, you end up with:
ab = a√(b²) –> b²/a = ab
Simplifying:
b² = a²b (multiply both sides by a, where a ≠ 0)
b = a² (divide both sides by b, where b ≠ 0)
So the numbers are a and a² (for non-zero values of a).
Glad TMLT is still alive and well! I’ve got a dead link when I’ve clicked a couple of times recently.
Lots of examples of getting the right answer for the wrong reason in engineering (in fact the skeptical might say that describes just about the whole of engineering).
Maybe the most significant was Galileo’s theory of beam bending, which was out by a factor of 3 (on the un-safe side!), but continued to be used for over 100 years after Euler and Bernoulli got the right answer, presumably because they fudged the tensile strength of the materials they used to make it work.
You might be interested in a series I’ve started on the Hole Through the Middle of the Earth (starting with a simple derivation of the period of a body moving with simple harmonic motion).
http://newtonexcelbach.wordpress.com/2010/06/14/elegant-solutions-simple-harmonic-motion-and-the-hole-through-the-middle-of-the-earth/
Thanks for the link to your series! Strangely enough, I was just talking about the Hole Through the Middle of the Earth problem last night with some friends. =)
Oh my gosh, xkcd is great! Thanks for the link!