At the beginning of class, give students 3-4 minutes to talk with their partner about the work that they did yesterday. They should be summarizing the main ideas of the problem and the strategies they used to find a solution. As they are working, get a sense of the progress of each group so you can select who you want to share. Take a look at Lesson Progress Summary and Preview to hear my thoughts about managing this two-day lesson.
After the Opening. ask some groups to share the gist of the problem and to summarize their current solution strategy. It is always a good idea to pre-screen the responses you get. (The opening provided a good opportunity to choose a couple of groups.) I recommend that you do not choose students who have the entire thing figured out - you don't want to spoil it for the rest of the students.
For the class discussion, try to sequence the responses so that they reveal a little bit of the solution at a time. Ask one group to talk about why the volume of each box can change. Ask another to talk about how they found the length, width, and height of the box based on the cutouts. Finally, a group can talk about how they used the dimensions to find the volume as a function of the length of the cutouts. The phrase "as a function of" may be new to your students, so spend a little time talking about what that means.
At this point the class should understand that we have a function that gives us the volume of the box as a function of the length of the cutouts ( V(x)=x(24-2x)(24-4x) ). Now we want to think how the function can be used to find the maximum volume of the box. Ask for some suggestions. My students will typically use guess and check, the table function on their graphing calculator, or a graph. Again, try to sequence the responses so that you know what you will get before you even ask the question. I would start with guess and check and work my way up to the graph. Once the strategies are presented, ask them which strategy is the most efficient.
While any of these strategies could be used to find the solution to this problem, graphing is the most efficient and sophisticated method. At this point I always ask students to draw a sketch of the graph without using the graphing calculator. I want them to see the general shape of the graph and to notice that the maximum volume occurs in the range of 0 < x < 12. Ask them what is special about this interval of values. Why don't we care about other points on the graph that will clearly produce a larger volume? This is a great springboard to talk about the relevant domain for a specific context. While the function has a domain of all real numbers, only the x values from 0 to 12 make sense in the context of this problem. Go over the process of finding the maximum point using the graphing calculator so that all students can use the technology.
Push a little further and you can talk about the maximum point on the portion of the graph from 0 to 12. Is it the maximum point for the entire graph? What is it the maximum of? Is it important? These questions will get students thinking about relative extrema and why they are important. After the class makes sense of this concept, then you can to introduce the vocabulary associated with it.
Defining relative extrema can be difficult. I find that they are easy for students to identify, but it is tough to put the definition into words. Ask a few students to explain it in their own words to get a few representations. Some may say that it is the highest or lowest point in a specific region of the graph. Others may say it occurs when the graph changes from increasing to decreasing. Get a few different interpretations because you never know which one will stick with a student.
Finally, the attached worksheet will get students thinking about the concepts we went over. This serves as a good assignment for them to work on individually for homework.