Hi, I live under a rock so I am sure you all already know about this, but I just discovered this fabulous collection of open-middle problems, which is my offering to you as today's resource of the day. I was disappointed, however, that there were few problems under the High School category, so I am going to look and try to locate some to add to their collection. I already wrote them about one set of polynomial problems that I had used this year, but I'm not sure whether they'll post it up since the problems (as I noted to them) are not original problems from me, and I had found them on a website that is not operating under Creative Commons.
Anyhow, I should at least write about those problems here, and I'll just link you to them!
During the last weeks of this year, I had taught two methods of polynomial division (long division and the box method) in Algebra 2 and then decided to hand out these "backwards" problems to my students in randomly assigned groups without any hints. I gave each group a big piece of butcher paper, and told them to try to work out the problems, in any order, on that sheet. I'd go around and check them off for the problems that were completely correct, with all work shown on the butcher paper. I also encouraged them to talk to their team mates if they successfully completed a problem and got it checked off. Because they were working off of the same piece of paper, the students were looking at each other's work (especially the problems that I had checked off as being correct) in order to share approaches and to help look for procedural errors. If I do this activity again next year, I'll definitely escalate the level of challenge by asking each group to try to find two different ways of solving each problem.
Since we had just learned and practiced polynomial division, I was surprised and delighted that they came up with really a variety of approaches to these "backwards" problems. Some of them used the idea of a root to set up systems of equations, which provided for a rich discussion afterwards when we compared methods across groups. They also understood that if you know the remainder already, then you can figure out what the exact (perfect) product was during the division process, even though it was something that we had really not talked about explicitly. I loved the creativity they had!
It helped me appreciate that every unit, I should be providing some substantial "backwards" assignment (similar to this one, given with no hints) in order to help them think more flexibly and to help transfer the learning. If you can go forwards and backwards, then that's how you know that you really understand it, right?
Hasta el lunes (si Dios quiere??)!