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Algebra Class Openers

Class Opener – Day 29 – Geometric Series

Aren’t infinite geometric series cool?  If you just shouted “yes”, then you are potentially as geeky as I am. A “proof without words” from MathFail kicked off today’s discussion:

Proof
I wasn’t quite sure what sort of observations I would receive from my class. But just enough ideas were generated to get us going:

There are an infinite number of triangles down the right side.

All those triangles on the right add up to the half-triangle on the left.

Both are great starts for what I hope my students will learn today. A video I made in my driveway continued the ideas of geometric series and their infinite terms.

A few students wanted to argue that the sequence in the video was arithmetic, but some meaningful debate yielded agreement that geometric made more sense.  Groups then worked through a similar problem involving a Superball being dropped, leading to terms and total distance traveled.

seriesMany groups employed a “brute force” method to find their answers. Using the Desmos calculator (many students chose to use the iPhone app), we found value in developing the equation and using tables and summation symbols to find solutions. This was my first time usign Desmos with this particular lesson, and it was an awesome addition, which added value to the need for writing a clear function to define your situation.

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Class Openers

Class Opener – Day 28 – Towers of Hanoi

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:

hanoi

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.

formulas

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

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Class Openers

Class Opener – Day 27 – Let the Dominos Fall!

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!