Tributes to Escher's Relativity

I came across this version of Escher's Relativity and had to snap it up while shipping was free.  I recalled an awesome Lego version by Andrew Lipson awhile back :



And then there's a 3D-printed version which might be taking things too far:



 

the camel problem

Monday, Monday...

Pi Day falls on a Monday this year.  But you knew that by now, since the Doomsday for 2011 is Monday. This means that, in 2011, the following dates are all Mondays:

• 4/4


• 6/6


• 8/8


• 10/10


• 12/12

Think "I work 9 to 5 at the 7-11" and you'll remember these Mondays:

• 9/5


• 7/11

...and their opposites:

• 5/9


• 11/7

There's Pi Day (or Einstein's birthday, if you prefer), the Fourth of July, Halloween, and the day after Christmas:

• 3/14


• 7/4


• 10/31


• 12/26

So if someone asks, "What day of the week is September 20," you think, "I know the 5th is a Monday, so the 12th and the 19th are Mondays... Tuesday."

Of course you'll need a nice pi shirt for Pi Day.  Here's a cool one:



And here are a couple other math shirts from Woot.  I got the counting sheep.  Number 3 is pooping.



 

northwest indian art

my setup

funny little woman

dy/dx song revisited

By popular demand, a side-by-side comparison between "dy/dx" (a derivatives song I made last year) and "Aicha" (a 10-year old viral video by a lovestruck teen):

Rotating cube

Here's an investigation opportunity that came up as a student was playing with my new favorite toy.

A cube is rotating about its main diagonal:

I came up with a few questions to challenge students at different levels (What is the length of the main diagonal? What type of curve appears in the silhouette? What is the volume of the rotated solid?) but wondered what else we could ask about this. I haven't solved the latter two - maybe you can.

the notebook

Lifelong learning?

In an interview with BLDGBLOG, film editor and sound designer Walter Murch discusses his personal interests in mathematics and astronomy.  He discovered that the spacing of rings in Copernicus' illustration of a heliocentric universe closely match those in the dome of the Pantheon, built 1400 years earlier.





Murch then relates his work with Bode's Law, an exponential function that yields the relative sizes of the real orbits of the planets...more or less.  The excitement that Murch describes as he uses ratios to satisfy his own curiosity is something math teachers try to foster but rarely achieve:

I came up with a formula that generated the same set of ratios, yet was different from the original – and that really made the hair on the back of my neck stand up!

I suspect that events such as this are what fueled Murch's success as a lifelong learner, a concept much talked about but systemically discouraged .

What are the conditions necessary to encourage lifelong learning, what is the cost, and what is the value?

What is this thing?

I made my first truncated icosahedron (a.k.a "soccer ball") out of the magnetic balls.  I just built little rings of 5 and 6 balls, and stuck them together so that each vertex had two hexagons and one pentagon.  This resulted in little squares being formed where the faces met, and little triangles around what I considered the vertices (of the truncated icosahedron).  I wondered what this polyhedron would be called if each ball were a vertex.  A quick Google search reminded me that I know very little about this subject, and I wasn't able to find the answer.

Can you name this polyhedron?

The best math manipulative?

What's your favorite math manipulative?
I'm excited to get a few sets of magnetic balls (variously called BuckyBalls, Nanodots, Neocube, etc) for our student math team to explore. They're addictive to play with and can spawn discussion on a range of topics: platonic solids, crystalline structure, surface area and volume, number theory, nanotechnology, tessellations, and more that I haven't thought of. Plus, for motivation there's already a ton of videos of people creating cool things out of these. Here's a cool one to bring up recursion or fractals:

Valentine's Day heart graphs

Besides the cardioid, whose name comes from it's heart-like shape, there are some cool graphs out there that people have made to look like hearts.

I think this is the best-looking one:

Here's a picture to help students practice plotting points (and help start a discussion of symmetry), which I made with my Graph Art program from awhile back.

The Points:
(-2,2)
(-2,4)
(1,6)
(4,4)
(5,0)
(5,-5)
(0,-5)
(-4,-4)
(-6,-1)
(-4,2)
(-2,2)
--break--
(5,5)
(7,6)
(6,7)
(5,5)
(2,2)
--break--
(-2,-2)
(-6,-6)
(-6,-5)
(-7,-7)
(-5,-6)
(-6,-6)

But, if your mother doesn't appreciate graphs as much as flowers, here are a few Valentine's Day gift ideas

Ethiopian math in context

Another teacher and I recently took our student math team on a field trip to the museum, followed by an Ethiopian dinner.  What better place to discuss Ethiopian multiplication?


I learned that this is also called Russian multiplication - another reason to eat out...

Pimpact on student learning

I enjoyed Freakonomics and have been recently loving the illustrated version of Superfreakonomics.



The authors have made statistical methods interesting by way of the content under study.  Does a prostitute do better to work with a pimp or solo?  What data and measures would we need to understand this?  Reading this I was struck by three things:

• The relevance upon which we place such import may really be a safer way to say excitement.  Of course our lessons should be relevant, but only insomuch as they increase students' desire to learn.  Balancing a checkbook is relevant but not interesting; detecting fraud in sumo wrestling scores is irrelevant but exciting.


• The content and applications we present must be inherently exciting.  We may feel that the Pythagorean Theorem is beautiful in and of itself, and we can do our best to convey that aspect, but we will fail half of our students if this is the extent to which we attempt to pique interest.


• Statistical methods are under-emphasized in high school, relative to their importance in the world.  Decisions are being increasingly data-driven, and demand is high for competent analysts of all stripes.  As much as I hate to admit it, statistics should probably supersede calculus in the effort to prepare students for the real world.

I do not think we need to teach students about the pros and cons of working with a pimp per se, but making our applications more surprising, varied, and spicy could draw in the students who would otherwise miss the point.

How to make students love square roots

Final exam scores in my calculus class were lower than I would have liked, with a few high-scores and a lot of medium-to-low scores. Reducing the number of possible points would bump the high scores over 100%, which is undesirable.  Since the scores are not normally distributed,  Another teacher in my department had a better idea: adjusting each score by taking the square root of its percentage.  Thus, a 0.36 becomes a 0.6, a 0.49 becomes a 0.70, etc.  Students were underwhelmed when I passed back the exams with only raw scores shown.  After discussing the problems they encountered, I had students "adjust" their scores.  Needless to say, they now love square roots.

In a traditional grading system, below 60% is considered failing:


Here's how the square root affects grade assignments:


This idea gives rise to a discussion about the square root function itself (why is x² > x if 0 < x < 1?), and other possible functions to adjust scores.  For example, if the square root (x^0.5)  is too extreme, what about x^0.9?  Perhaps you choose whatever power will bring the mean to 75%?