I base this warm-up directly off of the previous day’s investigation. Again, I didn’t use levels because I felt that this was a day where everyone could start at the same place. Some students spent almost the entire time working on the first problem, while other students were able to skip this problem entirely. I just want my students to choose one or two problems to work on for the first 20 minutes of class—often my students want to get directly started on their work from yesterday, but I find it helpful to ask them to do a bit of additional practice each day. When they tell me, “Can we just work on the problem set from yesterday because it is the same as the warm-up?” I tell them that I want them to do at least one or two additional problems to practice and think more deeply about the ideas. For these kinds of conversations, I try to have an individualized approach, rather than setting down a rigid whole class expectation. I tell some students who are struggling to just do one or two of the more basic problems, while I encourage other students to do one or two more challenging problems.
This lesson is also a great opportunity to allow students to use computers. The best part about having some computers available is that students can check their own work as they find functions to fit data tables. This way students can get their own immediate feedback as they work.
After yesterday’s investigation, I realized that we could spend much more time just on the data tables. I wanted to have further discussions with students about questions like these:
I created two different levels of the assessment. More Surface Area Volume Data Tables is th more basic level and it provides information about the length, width and height of the prisms, and also indicates which column is surface area and which is volume. With all this information, students can approach these problems in several different ways. Some of my students actually built the prisms while others looked for patterns in the length, width and height columns and were able to use these patterns to find function rules. Cubic and Quadratic Data Tables involves only 4 data tables, but much less information is provided, so it offers a higher level of challenge. You can give students a choice about which level to try, but I prefer to distribute what I think are the appropriate levels first and then tell them that they can change levels if they prefer. I find this strategy a bit more effective, because it prevents students from just choosing the easier level out of laziness.
Students can finish yesterday’s investigation first, but I want my students to explore these problems in order to gain more experience with data tables before getting to work on the graphs.
The overall purpose of these problems is for students to start making generalizations about the behavior of polynomial functions with different degrees--by comparing quadratic and cubic functions, students can start to identify different ways that having a higher degree in a polynomial affects its behavior.