First two weeks of school

It's been a really fun first two weeks of the school year. Yes, exhausting as well, but super exhilarating and exciting too. This year, I started the year a bit differently, focusing more on how I wanted students to work together and think mathematically than on specific content. Because it was the first year I was doing this, I could do the exact same problems with all of my classes. Here's how it went:

Day 1:

• Students came in and saw a seating chart with randomly assigned groups of 3 or 4 and were directed to one of the vertical whiteboards. I wanted to establish this as the norm from the get go.
• Students filled out Google form describing a class in the past they've enjoyed, a class they have not enjoyed, questions that they have about this class, and questions that they have about me. I used the last two prompts as ways to discuss my expectations and structure for the class and to start building some personal relationships.
• Students worked individually for a few minutes and then discussed this problem, which I stole from IMP Year 2. Our first unit will be Statistics for all classes so I thought it would be good to do a fun, but challenging problems, that related to probabilities and ways of counting events.
• Homework was to fill out a Google form asking them about themselves and to keep working on the problem above.

Day 2:

• Students were grouped randomly anew and shared their work on the Tying the Knots problem. We spent the last half of class with group presentations sharing out their progress and practicing how to present and interact with presenters.
• Homework was to write a reflection on themselves as a learner and to start writing up the process and solution for the Tying the Knots problem (I used the Problem of the Week standard categories).

Day 3:

• New random groups, and I used one of @sophgermain's activities for helping students get to know each other. Nothing huge, kids just shared one thing they did over the weekend with their group.
• New problem! This one was incredibly fun. I originally thought that we would take it into proof by induction, but after input from @woutgeo and @hpicciotto decided to stick with a more intuitive visualization of the sequence.
• I basically let students work in their groups without too much guidance from me. Most realized it gave the Fibonacci sequence pretty quickly, but were not able to explain why. Many tried to develop a closed form rule, without much success (surprise, that's actually pretty hard to do). Most groups started trying the extensions, but didn't get super far. I stopped the class a few times and asked various students to explain their group's work. One of my classes this year doesn't have as many whiteboards as I'm used to having, but our desks can be written on so I'm going with that for now.
• Homework was a reflection on their process and feelings when working on these problems and presenting/watching presentations.

Day 4:

• New groups and I answered some more questions about the class and about me. I continued to have them share out a few personal tidbits in their groups as they are still very much getting to know each other (especially the freshmen). Today's questions were about favorite ice cream flavors and favorite movie.
• This was a slightly more structured day. I pushed students to be able to explain why the pattern that was produced matched the Fibonacci sequence. It was helpful to project pictures of their written work and explanations and use that to get more precise and tight in our language. I felt okay adding on to their explanations as needed since there were more extensions to explore (2 by 2 by n case and 3 by n case).
• Homework was to work on the two extensions and to start an integrated review problem set.

Students' work on explaining the derivation of the recursive formula

Some work from the first day on developing a closed form. I was not sure as to whether I should discourage students from going in this direction as finding the closed form rule is extremely challenging. 

More fun student work at the beginning of the exploration.

P.S. @daveinstpaul shared a great follow-up programming project in which students need to write a program to generate all of the possible ways to tile a 2 by n rectangle and then extend it to an m by n rectangle. I'm going to check in with one of the programming teachers tomorrow to see if this might make sense as a posible extension in her class.

P.P.S. A new teacher who I think is going to be amazing visited my class today and I got completely turned around in what I was saying and did not do a great job of moderating the discussion. It's been too long since anyone has observed me, and I just didn't feel comfortable with the kids yet to laugh it off so awkwardness ensued. Bah. We need to be visiting each other's classrooms much more frequently.