Today we celebrate the third Permutation Day of the year! I call days like today permutation days because the digits of the day and the month can be rearranged to form the year.
Celebrate Permutation Day by mixing things up! Try doing things in a different order today. Just remember, for some operations, order definitely matters!
We have been having fun with our SumBlox, which recently arrived. The number blocks are cleverly designed so that the height of each is proportional to its value.
Here we have the ten block
and here we have multiple mathematical ways to achieve the same height as the ten block: five + five, three + seven, and nine + one.
So far, playing with SumBlox seems like a fun way to build number sense and explore basic properties of addition like equivalence and commutativity. But there does seem to be one problem: I think they got the height of this block wrong!
I recently participated in a workshop hosted by the Museum of Mathematics about the Rosenthal Prize for Innovation in Math Teaching. The Rosenthal Prize invites classroom teachers to submit outstanding, fun, creative, and engaging math lessons: the author of the best lesson receives $25,000, and other noteworthy submissions are honored as well.
The purpose of the workshop was to help prospective applicants understand the submission, revision, and judging process for the prize. The workshop panel included the directors of the museum, past judges, and three former winners of the Rosenthal Prize (including myself).
The video is embedded below, or you can watch on YouTube here.
Please spread the word about the Rosenthal Prize: it’s rare to have such incentive to build and share creative, engaging mathematics lessons!
Futility Closet recently posted a nice puzzle about the sum of the angles in the “points” of a star polygon.
It’s easy to show that the five acute angles in the points of a regular star, like the one at left, total 180°.
Can you show that the sum of these angles in an irregular star, like the one at right, is also 180°? (link)
A clever proof is shown, but what I would consider the standard proof is clever, simple, and beautifully generalizable.
Consider the star pentagon below.
By the exterior angle theorem, we have
So
where the last equality follows from the triangle angle sum formula.
Do this for each of the five “points” and sum the equations
Now, the sum of the interior angles of any pentagon, regular or not, is 
And so, a little arithmetic gives us
For stars of this type, where the points are formed by intersecting two sides of an n-gon that are separated by exactly one side, this method generalizes beautifully. The above equation becomes
which simplifies to
In particular, we see that when n = 5, we have that 
Now, there are other types of star polygons. For example, if you start with an octagon, extend the sides, and consider the intersections of two sides that themselves are separated by exactly two sides of the octagon, you get something that looks like this.
While the formula above doesn’t apply to this star, a similar technique does. The big difference is that, instead of the star’s points being attached an an n-gon (a pentagon, in the first example), this star’s points are attached to another star polygon! There are lots of fun directions to go with this exploration.
I’ve been having a lot of fun exploring mathematics through 3D printing.
Recently, I had the idea to 3D-print a “cube frame”, that is, the edges of a cube. My first mathematical task was to figure out an equation whose graph was such a cube frame.
It took a little work, but I ended up with this.
I exported the graph into the appropriate file format and successfully printed my cube frame!
Just as fun as the mathematical challenges of producing the graph are the engineering challenges, and mysteries, of the 3D printing process itself. For example, notice the scaffolding that the software automatically adds in order to print the top square of the cube.
There are some interesting mathematical and structural consequences of the scaffold-building algorithm, but even more amazing was that one “face” of the cube contained no scaffolding at all. This meant the 3D printer printed one edge of the top square into thin air! And it succeeded!
I’m excited and inspired by 3D printing, and I’m looking forward to finding more ways to integrate into our math classrooms.
