# Day 26: Emperor’s Banquet

It is always a good week when I have back to back lessons from Fawn’s site. Geometry presented and wrapped up Mrs. Murphy’s Laundry with group presentations. As the period went on, the presentations continued to develop, which was great to see.

In Algebra, students worked through The Emperor’s Banquet. I put the problem up on the projector and had students read it then work by themselves for five minutes to make sense of the problem. After, they discussed their progress with their neighbors.

I saw a lot of students taking this direction. A few went up and shared how they decided where to sit if there were 5 guests at the banquet.

In my eyes this image shows a great example of first step in developing algebraic thinking on the student’s end. After this point students slowly added mathematical framework to the problem. It was extremely difficult for me to step back and just let things happened. But after a while numbers started appearing.

What happened in green was awesome; a huge shift in structure, which makes communication much easier on the student’s end.

More patience on my end paid off and tables started emerging; another big step, this time into abstraction.

Conjectures started coming out and we created a table on the board. After about 10 minutes the table reached 17 guests and most of the conjectures had counterexamples. So we started searching for patterns different ways (in green).

At this point I felt very stretched in my ability to provide further scaffolding. In a few periods, students noticed linear patterns between 3, 6, and 12 guests. Which was extended to answer almost every case (excluding primes) ie. where should you sit if there are 18 guests? For 9 guests the last seat is 2, so double both of those, 18 guests, the last seat is 4. One co-worker suggested going in the direction of piecewise, and helping students see that certain rules work for certain number of guests; interesting!

I went into today’s lesson with the hope that students would look deeper into patterns, but what came out of today was much richer on both my end and the student’s.

**This problem is also great for Algebra 2 and even pre-calc… I believe it can be modeled by a logarithmic composed with a floor function **