Fibonacci Piano

Posted on October 26, 2012 by Mr. Chase

Check out this pretty music! [ht: Prisca Chase]

Product Failure

Posted on October 25, 2012 by Mr. Chase

I’ve been taking a grad course in statistics this semester and so I’ve been thinking about all sorts of real world examples of math, including the classic product-failure example that’s a mainstay of most stat classes.

One of the simplest continuous distributions is the exponential distribution which is a pretty decent way to model product failures. The probability of failure $f(t)$ after time $t$ is given by

$f(t)=\frac{1}{\lambda}e^{-t/\lambda}$ .

I read this great article about product failure and testing in Wired this week. I encourage you to check it out. Read the last page of the article especially, where it talks about how cutting-edge companies are modeling minute variations in materials using an electron microscope and some statistics. Instead of actually testing the product over and over again using a fatigue machine, they can create surprisingly accurate models of the materials using computers. Prior to this, the behavior of materials was somewhat unpredictable.

Of course I was excited to see this figure in the article, which shows the Weibull distribution modeling failures of steel bars in a fatigue machine.

The Weibull distribution, unlike the exponential distribution, takes the age of a product into account. If the parameter $k$ is greater than zero, than the rate of product failure increases with time. The probability of failure $f(t)$ after time $t$ is

$f(t)=\frac{k}{\lambda}\left(\frac{t}{\lambda}\right)^{k-1}e^{-(t/\lambda)^k}$ .

The first obvious thing to note is that the exponential distribution is just a special case of the Weibull distribution, with $k=1$ . The next thing to say is that this distribution is single-peaked. So how is the above a Weibull distribution? The article says it is, but I think it might be a linear combination of two Weibull distributions, don’t you? Whatever–normalize, and you’ve got yourself a probability distribution.

[pun warning!]

The real question is, if this is TWO Weibulls, would you settle for the lesser of two Weibulls?

Sorry. I had to.

College majors

Posted on October 24, 2012 by Mr. Chase

Very funny. Though I do like the liberal arts, having gone to a liberal arts college for my undergraduate degree. [ht: Graph Jam]

Percentage Fail

Posted on October 15, 2012 by Mr. Chase

Had to reblog this beauty…

If you don’t follow Math-Fail, you should!

Happy 10-11-12 13:14:15

Posted on October 11, 2012 by Mr. Chase

Thanks to my brother, my dad, and my wife, for pointing this out to me :-):

Happy 10-11-12 13:14:15 !!!

(not as interesting for you European types!)

And here’s a goofy comic (HT: Tim Chase).

Pumpkin Pi

Posted on October 6, 2012 by Mr. Chase

Decorating for fall? Consider creating some beautiful DIY pumpkins with a mathematical flair–perfect for on top of the hearth or next to the porch swing out front :-).

These were created by Messiah College students (my alma mater) and the photo comes by way of Professor Sam Wilcock, my advisor. Go Falcons!

Don’t flip out!

Posted on October 4, 2012 by Mr. Chase

Well, not yet at least. Everyone’s flipping the classroom, but is it really worth it? Yes and no, as NCTM president Linda Gojak explains in her column this week. I don’t always highlight her column, but I especially appreciated the nuanced way in which she approached this trendy subject. There’s something more fundamental that we need to aim for: engaging our students in mathematics and problem solving. Whether we flip or not may be immaterial, as Linda points out.

Here are a few excerpts from her article, which you should check out in full here.

To Flip or Not to Flip: That Is NOT the Question!

By NCTM President Linda M. Gojak NCTM Summing Up, October 3, 2012

…

A recent strategy receiving much attention is the “flipped classroom.” Innovative use of technology to enhance student learning makes flipping possible and motivating for students and teachers.

…

I believe that we need to go further. As we consider effective instruction that leads to student learning, we must remind ourselves of the characteristics of mathematically proficient students.

…

Rich mathematical tasks provide students with opportunities to engage deeply in mathematics as opposed to a lesson in which the teacher demonstrates and explains a procedure and the student attempts make sense of the teacher’s thinking. Communication includes good questions from both teacher and students and discussions that develop in students a deep understanding by wrestling with the mathematical ideas.

…

Although the flipped classroom may be promising, the question is not whether to flip, but rather how to apply the elements of effective instruction to teach students both deep conceptual understanding and procedural fluency.

…

(more)

All that being said, I still DO want to try flipping my classroom on a small scale, one-lesson at a time basis. I promise I’ll try it someday.

Happy Mean Girls day

Posted on October 3, 2012 by Mr. Chase

In my Calculus class today I showed just a short clip from this video (“the limit does not exist!”).

Apparently, completely and totally without my knowledge, I showed this video today, on October 3rd, which just happens to be Mean Girls Day. So..happy Mean Girls Day! (happy/mean…that sounds a little funny)

Kudos to the folks who put these videos together!

Can’t touch this?

Posted on October 1, 2012 by Mr. Chase

Here’s a popular t-shirt design:

But I have a mathematical problem with it. It’s certainly true that THIS particular function never touches its asymptote. I think the t-shirt suggests that this is true of any asymptote, though. As if to say, “hey I’m an asymptote, and as an asymptote, you can’t ever touch me!” However, functions in general CAN touch their asymptotes, sometimes an infinite number of times. (I’ve talked at great length about this issue.)

I also have typesetting-issues with this design (notice the italicized “lim” and the unitalicized variables).

Am I being too picky?