Triangles of pennies

Useless fact of the day: Make a triangle of arbitrary size with pennies, like in the picture below. What is the least number of pennies that have to be moved to make an upside-down triangle?

Turns out the answer is the number of pennies divided by 3, ignoring the remainder. So for the 10-penny triangle above, the answer is 3 moves (since $10/3 = 3 + 1/3$ ). A good formal write-up for why this works, with specific discussion about implementing this in a middle-school classroom, can be found here: http://132.68.98.62/Courses/Algebra_206/Algebra%20-%202005/Articles/2-MTMS-Inquiry%20and%20triangle%20array-2004-9-6.pdf

Image credit: http://www.coolmath4kids.com/math_puzzles/p4-pennytriangle.html

Math emporium

It took some convincing, but I’m now a supporter of this novel way of using technology to teach lower-level mathematics courses at the collegiate level. The results speak for themselves.

http://www.washingtonpost.com/local/education/at-virginia-tech-computers-help-solve-a-math-class-problem/2012/04/22/gIQAmAOmaT_story.html?hpid=z4

http://www.emporium.vt.edu/emporium/home.html

http://www.thencat.org/R2R/AcadPrac/CM/MathEmpFAQ.htm

Infraction

While I can’t take credit for this one-liner, I’m more than happy to share it.

A colleague was explaining his expectations for simplifying expressions such as

$\displaystyle \frac{\displaystyle ~~~\frac{2x}{x^2+1}~~~}{\displaystyle ~~~\frac{x}{x^2-1}~~~}$

Of course, this isn’t yet simplified, but his students were balking about doing the required work. So, on the spur of the moment, he laid down a simple rule:

Not simplifying a fraction in a fraction is an infraction.

Utterly brilliant.