Product Rule in a Thinking Classroom

Today was day 3 of running my classroom as a thinking classroom - visible random groups (of 3) working on vertical non-permanent surfaces. We started with a quick demonstration to show that the derivative of a product is not the product of the derivatives. Students then worked through a product rule "discovery" activity that I have been using for years. You can find it here. I wish I could give proper credit for it, but I do not remember who shared it with me. 

Here is the setup:

Students worked out expressions for length, width and area and for their rates of change before completing the following table.

Their job was to work with the numbers they came up with in the table to figure out a pattern that worked for each row - that would be the product rule.

It is always interesting to see who comes up with it quickly and who takes a little longer (often because they are trying really complicated things!).

Once I felt like the majority of students had found the pattern, I sent them off to their VNPS to work through today's sequence of questions. 

The last of these asked them to come up with the product rule for three terms. I loved what some of the groups did. They extended the introductory activity to 3-D, added height and worked through the numbers again! (sorry that the picture quality is terrible)

Here's a group that came up with a conjecture for the product rule with three terms and tested it out. I would like to say they did this instead of asking me if they were right, but they did jump right to it when I said they should check it for themselves.

It was really exciting to see such fantastic work, at such a high level from all my students. I love how they trust that they will be able to tackle all the questions I give them and believe in themselves enough to try.

P.S. All of my planning is on one getting-bigger-by-the-day Word document. I'll post the whole thing at the end of the unit.