BRONX vs. BROOKLYN (UPTOWN VS. DOWNTOWN SUBWAYS)
(9/30/2014) We talked briefly about this problem from last week. It turns out that everyone did remember and agree on its solution. I told the kids what I had been thinking about during the week – that if there was confusion about the solution, that maybe we could draw or symbolize the passage of time somehow. The question was, “Can you draw time?” Since about half the kids wanted to do this and half did not, I said to think about it and we can try next week.
A SWITCHING PUZZLE
The kids were excited to try Gardner’s “A Switching Puzzle,”1 which I had drawn on the board last week, as shown in the photo. We hadn’t gotten to it last week. But, before telling them what the problem was, they wanted to guess what the question is themselves. Their questions were very appropriate for the drawing; I wish I had kept a list for future use. OTOH, their questions were not what Gardner asked.
I had forgotten my copy of Gardner’s book with the exact wording of the question, so I paraphrased: “Can you use the locomotive to interchange the positions of cars A and B?”
The students worked together to come up with a solution. I noticed that their solution involved the locomotive ending up on the lower track. That reminded me that Gardner’s question required the locomotive to return to its original spot. Back to work.
Four of the six students clustered around the board, taking turns tracing routes with their fingers. They wanted to show me their conjectures, but I couldn’t follow what their fingers were doing in terms of directions and which cars were which, so I gave them dry erase markers to use as train cars. This worked well because the markers can be attached to each other.
The other two students remained at the table, one writing in a notebook, and one seemingly just sitting there doing nothing. They weren’t, however, simply writing or sitting; they were engaged in mathematics. This became apparent when each of them jumped up and announced “I think I’ve got it!” Turns out one was sketching the problem in her notebook, and the other was doing the problem mentally. BTW, “I think I’ve got it,” is the most-uttered phrase so far in the course, followed by, in close second place, “Oh wait, never mind.”
They all ended up at the board, and together came up with the same solution Gardner did. It was a combination of collaborative and individual work, like real world work. Each verbalized conjecture helped everyone in their individual thought process. Once a solution was clear, several students wanted to keep working the problem anyway, but I steered them all to another table as most of the kids were ready for another activity.
[I realize as I write this that Gardner’s original problem stated that only the locomotive can fit through the tunnel – cars A and B cannot. I’ll give this info to the kids next time, even though their solution didn’t involve the tunnel]
I had told the kids earlier, and now I’m telling you, that I just can’t figure out a certain puzzle no matter what I try. So of course, the students were chomping at the bit to try that puzzle. I welcomed the help.
THE FOLDED SHEET
I showed several maps that were, as maps tend to be, easy to unfold and difficult to refold in the same way. Then I showed a puzzle – invented by Henry Ernest Dudeney – from one of Gardner’s books2: “Divide a rectangular sheet of paper into 8 squares and number them on one side only, as shown”(see photo below). “There are 40 different ways that this map can be folded along the ruled lines to form a square packet with has the 1 square face up on top and all the other squares beneath. The problem is to fold this sheet so that the squares are in serial order from 1 to 8, with the 1 face-up on top.”
[NOTE: if you don’t have access to the photo, the numbers in the first row go 1,8,7,4 left to right, and 2,3,6,5 in the second row.]
I explained to the kids that I worked and worked on this puzzle. I read the explanation at the back of the book. I watched a video on youtube3 (which didn’t reveal the solution, but hinted). I was hoping that they could help me.
They tried and tried too. Little progress at first – just 1 then 2 in their proper places. One student made her own “map” out of a new paper with larger squares. Soon everyone could get 1, 2, 3. Another student made a new map with tiny squares. After a while some could get to 4. Then students agreed that there must be some kind of unconventional fold, such as a diagonal, etc. Several tried this and made it to 6. One student even got to 7 but lost 8 in the folds. By then, everyone was frustrated. They decided to take their maps home, and I promised to bring the book with Gardner’s verbal instructions next time.
FOLDING MONEY FUN
I thought it would be a good time to give an “easier” puzzle. (Ha!) Gardner shows in pictures and words how to take a dollar bill and fold it one way and unfold it in apparently same way, but George Washington ends up upside down.4 So I asked the kids to draw Washington’s face on the back of their maps, which for the most part were the size of a dollar bill. I had to specify not to worry about accuracy concerning Washington’s infamous hairstyle, so that we could get right to work.
The question Gardner asks is “Why does the bill turn around?” The question I wish I had asked instead is “can you follow these directions and get the same result.” Gardner says that “If you have followed the illustrations exactly…” But it isn’t so easy to follow the instructions exactly. It’s hard to “fold forward and to the left,” etc. Especially if you’re 10 and have only just recently mastered these directions.
Only two of the students were able to follow the directions. The rest became frustrated. The two who could do it were sitting closest to me and the book. It would have been easier if everyone had their own set of instructions. But then it wouldn’t have been collaborative. I wanted everyone doing the same thing at the same time, and asking each other for help. (This did happen.) I try not to use handouts at all, ever, so that we retain the cooperative nature of the circle. Maybe this just wasn’t a good puzzle for our group. So far in this course, “Bronx vs. Brooklyn” seems the be the most successful puzzle for our group – highest level of engagement, mathematical thinking, collaboration, and NO physical objects to manipulate. “A Switching Problem” has been a great problem too – another one solved more with the mind and the mouth (conversation) than with the hands, for the most part. Hmmm… A thought occurs to me as I write, though: maybe I could make one very large dollar bill for everyone to fold together. An idea for next time. Especially since then I could have fun drawing George’s hair. (It continues to amaze me that often the plan for the next session reveals itself as I write these reports.)
Anyway, the students who did figure out how to turn George upside down did realize the answer to Gardner’s question. The rest struggled with the folding. I fear that I may have turned off at least one student to paper folding today. So my goal is to somehow have success as a group with paper folding before the end of this course. Especially since I have a goal of someday doing a circle on using origami to learn mathematics: “As a craft, origami develops several precious mathematical practices directly useful for geometric construction, topology, integration, and analysis of functions – and indirectly for all of math.”5
We finished off the circle with some riddles and verbal puzzles from Gardner, so everyone left with a smile.
1 Gardner, My Best Mathematical and Logic Puzzles, p22
2 Gardner, My Best Mathematical and Logic Puzzles, p14
3The Folded Sheet: https://www.youtube.com/watch?v=3Rimp-YUnm4
4 Gardner, Perplexing Puzzles and Tantalizing Teasers, p54
5 Droujkova, MobiusNoodles.com, http://www.moebiusnoodles.com/2014/10/fractal-bugfest-future-of-learning-math-storytelling-newsletter-october-6-2014/, October 6, 2014