unblock me

Just discovered a new puzzle game: Unblock Me. Combining elements of the Rubik's cube, the 15-game, and puzzles, Unblock Me requires deft spatial awareness, keen executive function, or just plain luck. The goal? Shift a bunch of blocks around so that the target (red) block can escape from the box. Seems like an easy premise but the difficulty level ramps up quickly! Even the game's creator has tagged the game: "Unblock is a puzzle game that makes you think."

I'm playing this game and thinking about my current read, Daniel Pink's Drive. Pink writes about motivation and the human need for mastery, autonomy, and purpose. No rewards await my completion of the puzzle, yet I tackle each level with interest and desire for success. For those levels I finish by simply stumbling upon the solution, I go back and attempt to find the solution, challenging myself to use as few moves as possible.

Who defines mastery here? And what defines mastery? In this case, it's up to me to decide, but far from the picture are the tangible rewards that we offer to students in the classroom each day. Candy, stars, grades all confound the learning process and inhibit the ultimate goal that we, as educators, should have: a love of learning for its own sake. So how do we reintroduce that in the classroom? As someone who works with middle and high school aged students, how do I undo the years of extrinsically motivated education that my students have received?

In terms of this game and my own teaching, what a cool way to engage my students in spatial reasoning and planning. Definitely on the list of games to play this year!

rate-time-distance problems

Most of the rate-time-distance problems that students find in their Algebra textbooks are contrived and pointless. There's no reason why students should care about how long it would take for two trains separated by 200 miles, one traveling at 45 miles per hour and the other traveling at 65 miles per hour, to pass one another on parallel tracks.

However, current storms and tornado warnings suggest very relevant and much more interesting problems to consider. Meteorologists, or their underpaid underlings, are currently making many similar calculations every minute:

A storm traveling east at 50 miles per hour is currently 32 miles away. How many minutes will it take for the storm to reach your town?

Assuming the storm picks up speed at a rate of 3mph every five minutes, how long will it take to reach your town?

Two storms, one 54 miles to the north and traveling at 52 miles per hour, the other 38 miles to the west and traveling at 45 miles per hour, are headed in your direction. Which will arrive first?

And the list goes on...

a geometric observation

I've been spending a lot of time thinking about centers of triangles recently -- they're fascinating. Here's a recent observation about circumcenters (the point of concurrency of the perpendicular bisectors of a triangle):

The measure of the angle formed by constructing segments from two vertices of a triangle to its circumcenter is twice the measure of the angle at the third vertex.

In other words, given ∆ABC with circumcenter H, m∠AHB = 2(m∠ACB).

A proof? Think about circumscribing a circle about the triangle; the circumcenter of the triangle is the center of that circle...

chocolate oranges

Every year around this time, I open up a chocolate orange and begin to wonder the same thing. As I remove wedges of the chocolate treat, is there any point at which the surface area of the partially-eaten chocolate orange is greater than the surface area of the intact orange? In other words, given a piece of foil that just covers the full orange (with no foil left over), will removing wedges render the foil insufficient to cover the remaining orange?

some sums

Had trouble falling asleep last night, so I began adding up numbers in my head. "Discovered" this interesting property of integers, though I'm sure it already exists more formally in the mathematics world:

Let n be a positive integer. Let S be the sum of all integers from 1 to n, inclusive.

If n is odd, then S is a multiple of n. However, the same property does not exist for even numbers.

It will take me some time to try to even wrap my head around this, and who knows if I even have the ability to do so. But pretty cool nonetheless. And a bit frightening that this is what goes through my head in the wee hours of the morning...

correlation?

I have noticed more white hairs on my head since getting married...

counting birthdays

happy (26th) birthday, nick!

Because math, in so many ways, in grounded in counting, I spend a lot of time thinking about how we quantify things in our lives. I've also been thinking a lot about birthdays recently, as I find myself surrounded by many friends and family members who were born in this tenth month.

What has always bothered me most is that a person can truly have only one birthday, to be celebrated on the day of one's birth. Everything that comes after represents an anniversary of one's birth day. But this is a minor point, an argument that I concede I will never win.

The argument I will win, however, is the following: We almost always incorrectly count birthdays. For example, the date on which one turns 25 is actually one's 26th birthday! But look at the Hallmark cards that arrive in the mailbox and at the writing on the cake and they will all say "Happy 25th Birthday!" Has anyone ever looked at these and said, "Hm... These are a year late."