[note: sorry for the quick and unedited write up on this one. We are a little pressed for time today.]
Got a neat comment from Jim Propp on one of the Aztec diamond posts last night:
“There are other fun problems about counting tilings. The first one I’d give them is counting domino tilings of a 2-by-n rectangle. Then rhombus tilings of an equiangular hexagon with sides of length a,b,1,a,b,1. If the kids like these, let me know; I can suggest some fun follow-ups.”
I thought about the 2 by N rectangle tiling on the way home last night and thought it would, indeed, make a great project to do with the boys today. Despite a little clumsiness keeping the domino tiles on the screen, it was really fun.
I started by introducing the problem and asking the boys to come up with a strategy to solve it. They had some good thoughts:
Following the strategy they came up with in the first part, we started by looking at some simple cases – 2×1, 2×2, 2×3, and 2×4. These cases allowed us to make a few guesses at the pattern:
In the prior video the kids guessed the tiling patter we were seeing 1, 2, 3, 5 was the Fibonacci numbers. Here we looked at the different ways to tile a 2×5 rectangle to see if there were 8.
Sorry we got a little clumsy with the tiles here – we needed a bit more room.
Now that we were feeling a bit more confident that we were seeing the Fibonacci numbers, the question was . . . why? How could we see the Fibonacci pattern in these tilings?
To understand what was going on, we looked to see if we could find a relationship between the combined tilings of the 2×2 and 2×3 recangles and the tilings of 2×4 rectangle. The relationship wasn’t obvious right away, but they did find it!
Sorry again for the clumsiness with the tiles.
Finally, as a last way of double checking the Fibonacci relationship, we looked to see if the tilings of the 2×5 rectangle related to the combined tilings of the 2×3 and 2×4 rectangles.
So, although we didn’t really do a complete formal proof of the relationship between Fibonacci numbers and the tiling patterns, we did hit on the main ideas in that proof. This activity is a great way to introduce some “mathematical thinking” ideas to kids. Thanks to Jim Propp for suggesting it!
Mike's Math Page 
391b… “It could be 3. Because we went 1, 2… Or it could be 4. We don’t really have enough to go for a pattern.” I wish you could have seen the joy on my face as he said that. Fantastic thinking.
Without having watched all of the videos all the way through… did you ask them about the missing “1” at the beginning of the Fibonacci sequence? The dominoes go 1, 2, 3, 5, 8, … apparently skipping the initial “1”. Why does this happen? Is there a way to word the domino problem so that the initial “1” does appear? (I love this question because it relates to the idea that n choose 0 is equal to 1.) I am going to try this project with my students – really great!