During the actual club meeting today I had all the kids talk about their experiences at MathCounts to encourage each other. Again everyone even those who hadn't won anything seemed upbeat. I also gave my long but true speech about focusing on the fun parts of the competition and not the absolute outcomes and finding the joy in the math. Its true but perhaps hard to see in Middle School that the kids who keep going will ultimately benefit regardless of trophies. So as usual, I still worry about the discouraging aspects of these meets but it seems to have gone well.

After this talk, we briefly went over the old problem of the week. I only had one student really work on it so I'm thinking about what to do to refresh. This weeks problem is quite a bit easier and approachable https://mathforlove.com/2018/02/a-mathematician-at-play-puzzle-9/ which may help. I'm also thinking about different types of problems and to remember to talk about participation at the beginning and end of each session for the immediate future, I'm hoping to get back to near half of the kids working on this.

For the main task of the day I chose a topic from the recent Math Teacher's Circle magazine

https://www.mathteacherscircle.org/news/mtc-magazine/s2018/touching-infinity/, counting the set of rationals. In addition to looking interesting, this tied in well with 2 weeks ago on Farey Sequences: http://mymathclub.blogspot.com/2018/02/213-farey-sequences.html

Before starting though I wanted to warm up a bit with a small problem I'd seen on twitter. So I had all the kids work on the whiteboard with a number line to find where to place 1/4 between 1/3 and 1/5. I didn't supply any hash marks or much more than a simple explanation of the problem. I was gratified this time that almost everyone came up with an accurate answer. Universally kids chose to convert the fractions into the GCD denominator of 60 and place 1/4 = 15/60 between 20/60 and 12/60. On review as a group, I also asked since 1/4 is not there what is the number in the exact middle which was a good followup question. Note: for some reason when I did it myself beforehand I chose to calculate the difference between each endpoint and 1/4 and then find the ratio of the two distances which no one else did.

With that covered we dove in and I described the hyperbinary system. It uses the binary place values but in addition to 0 and 1 you may also use 2 in every digit. i.e. 27 = 0x11011 AND 0x2211 This took a few examples to make clear. From there I had everyone start to make a chart of the first 15 numbers in hyperbinary and how many representations each had.

The next step was to get everyone to find a pattern in the chart. This was only partly successful. The kids eventually identified the left hand rule where

*b*(

*n*) =

*b*(2

*n*+1) but finding

*b*(

*n*) +

*b*(2

*n*+1) =

*b*(2

*n*+2) proved more difficult. So in the interest of time I showed them this one. I'm still brainstorming how to do this a bit better in the future (and I really want to repeat this again since the whole activity is fascinating). One note here: its super important to keep these 2 rules in mind yourself and practicing write beforehand is very helpful.

We then moved onto the big discovery the Calkin-Wolf Tree

Again I described the rules for the tree and had everyone generate them on the whiteboard. I asked kids who finished to see if they could find a relationship between the tree and the previous hyperbinary numbers chart. We had just enough time for one student to discover they were identical and wrap up a bit as a group. So I pointed out a few more interesting facts about the tree we didn't have time to work on like the presence of every rational reduced fraction.

Overall, this went well but I could definitely improve the experience. I think I had between 50-75% engagement over the whole session which was a bit too low to my taste. The sustained effort by the time we were working on the tree was definitely at the limit for some students. On the bright side those who stayed engaged were very excited. I think what might work better would be to do both parts simultaneously and let the kids move between white board stations. At the end we could then look for the patterns between the two parts as a group.

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