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# Counting Collections and Dots and Boxes, fractional version

I just rewrote our write up of Counting Collections, and reclassified it on our Lessons page as a Foundational Activity. The reclassification was motivated in part by a conversation with a Kindergarten teacher, who mentioned that she had been having the kids in her room count collections every Friday. “I help kids who need it,” she said, “but honestly, they’re so focused and engaged, I could probably disappear and the class would still run all right without me.” When lessons work this fluidly, it is easy to feel that you aren’t doing enough as a teacher. In fact, it’s the opposite: the necessity of heroics, theatrics, and great feats of charismatic teaching can be a sign that the activity you’re pushing at the kids might not be a good fit. The easy path can sometimes be the best one, especially when the teacher is underworked because the students are doing the heavy thinking.

Speaking of student inspiration, I was playing Dots and Boxes with a student last week, and he made a play I hadn’t seen before: instead of connecting two dots with one edge, he made two half edges. Suddenly, it was a new game. I made four quarter edges; he played three third-edges. Then we started connecting using multiple fractions: 1/2, 1/3, and 1/6. This fractional version of Dots and Boxes goes like this. On your turn, you have to add a total length of 1 unit to the board. If you complete a box, you get a point, and an extra unit length to add before your turn is over. Whoever completes the most boxes wins.

So after 8 turns from Blue and 7 from Red, a game might look like this.

It’s Red’s turn. What move can Red make? Any full edge will give Blue a box to complete. But what about adding half-edges?

Now where can Red go? Adding halves will give Blue a box, but Red can add three thirds!

Does Blue have a move that won’t cost a box? I’ll let you figure it out. This game is totally new to me, but it’s clear that it will end (every turn adds a unit length of line segment to the board, after all), and this particular game won’t end in a tie either. I think it’s more likely to end in a blowout for the winning player.

Try out the game and let me know how it goes!

1. Joshua Greene

Conceptually, a very nice variation. For almost all of my students, there would be no practical way to distinguish between their 3rds and halves (or any non-zero segment smaller than a full side, for that matter). When we play, the most flexibility we can add are either using full or half segments.

Implementing this game could be a nice coding project.