Category: Class Openers

Sometimes I am genuinely surprised when a fun game or idea I was exposed to in elementary school is unknown by my older students. The Towers of Hanoi problem fits the description.  A number of students immediately gravitated towards the game as they entered. And after a quick explanation of the rules, they got down to business:

An online, interaction version of the game was also running on my projector, which generated some class discussion of strategy.  Students were able to solve many of the early challenges, and we began to look for the most efficient methods, sharing our findings on the board.

• 1 disc, 1 move
• 2 discs, 3 moves
• 3 discs, 7 moves
• 4 discs, 15 moves

From here, some students conjectured that 5 disks must take 15 moves….but nothing is quite that easy, and I asked them to prove or show it.  During classwork time, I heard discussion of possible formulas. Since we have been working with both explicit and recursive formulas, this was an effective way to discuss the differences between them, and why we might prefer one over the other.

The explicit formula sure is nice, but the recursive provides a roadmap for solving the problem for any number of disks.

We’re knee deep in arithmetic and geometric sequences right now, looking at explicit and recursive formulas for them. A television ad caught my eye the other day which seemed to fit the theme:

Lots of “cool” reactions from the room led to a discussion of “how did they do that?”.

Guess and Check!

Fortunately, after an evil eye from me, it turns out that student was kidding, and classes understood there had to be some intense planning going on, which is summarized some in this behind-the-scenes video:

But what is the relationship between the dominos? A healthy debate over a possible arithmetic sequence led to one student arguing that we would need many, many more dominos than in the video – if they were in an arithmetic progression. Next, we’ll look at the heights and see if we can come up with a plausible geometric model for the domino progression. A good kick-off to a Friday class!

A famous face met students as they enetered today:

Our friend Gauss…someone on my math Mount Rushmore.  It makes me sad that our country puts politicians on money, while others have used the opportunity to recognize real contributions…end of rant.

Today I offered a challenge whose tone has changed in the face of technology:

Find the sum of the first 10,000 positive integers.

In the past, I have used 1,000 as the ceiling, and observed as students mostly used a “brute force” method to find a sum. Offering candy to the first person to provide a sum, I settled in, knowing this task often takes some time and discussion.

Then….an “I got it” from the back of the room, and a correct answer from a student who Googled the answer.  Sigh…..  Damn you technology and good wifi!  I guess I haven’t given this problem since before I served as a district instructional coach – 3 years makes a big difference here.

In my afternoon class I altered course, providing a clear “no Googling” message with the problem. But I was once again (almost) sabotaged as students used my technology openness against me. Yesterday, we had learned about summation notation, and had a brief tutorial on how to enter summations on graphing calculators. This led to some attempts to simply enter the sum…but with some semi-disastrous results:

So close, but yet so far. And while I’ll save my story of Gauss and his amazing sums for tomorrow, we had a fun debate on notation and communication.