Explain to your students that you are going to keep working on the Car Talk problem that we started the other day. Right now would be a good time to bring up MP1 and the importance of perseverance. This task is not easy - students will have to persevere in order to get the correct answer and make sense of the problem.
Like we did with the Maximizing Volume activity, students are going to turn in a final copy of their work with explanations and diagrams. Attached is the rubric that I will use to grade it. Give this to students today so if a group finishes the task, they can begin work on their final draft. Stress that students should be writing their answer as if they were explaining it to a member of another precalculus class. Their submission should be a stand-alone explanation of the problem that someone could easily follow.
Like I mentioned in the first day that we worked on the Car Talk problem, you only want to intervene in their work if it is absolutely necessary. This is a very difficult decision to make as a teacher, but you want to pay particular attention to their progress and make the decision if they need help. I will usually give help if one of these things is occurring:
1. Students are going down a deep path that is completely incorrect. If this happens, you might want to pose a good question to help them discover a flaw in their argument.
2. Students have a major misconception in the set up of the problem. If it is not conceptual in nature, such as an incorrect measurement, you may want to remind them about the given information. If it is conceptual in nature, you can always ask a question to think about it.
3. Students don't know what tool to use. In this problem, students may try to solve algebraically when it is necessary to use a graphing calculator. Let them struggle for a little while, but then ask if they can think of a more effective tool to use.
In this video are some more thoughts about when and how to help students.
Here are a few other hints that are continuations of the Explore section of the first day of this lesson. Hopefully you will not have to give all of these hints - it will definitely depend on the level of your students. But they are available if you feel it in necessary.
We know that the area of the shaded region is 25pi. How could we make an expression to represent the area of the shaded region?
Adding there lines may help students find a way to think about the area of the shaded region. You may want to ask them what shapes they see. Students may see a triangle and a sector. Thus, to find the area of the shaded region, they will have to find the area of the sector and subtract the area of triangle, and that expression will be equal to 25pi.
Students may need to be reminded how to find the area of a sector. Give them a diagram like shown below to get them thinking about it.
Finally, then will arrive at a really complicated equation where they have inverse trig, square roots, and non-like terms that is equal to 25pi. And they may have no idea how to algebraically solve for d. Brainstorm some other ways to solve the equation. How could they solve something like x^2 - 3x + 7 = 80 if they did not want to use the Quadratic Formula?
During the last few minutes of class, I reiterate my expectations for the final write-up of the problem. Then, I ask students to list some of the mathematical topics that were needed to solve this problem. This is a nice opportunity for students to see that they have drawn on topics from many years of mathematics. They should be proud of the work they have done!
This problem is rich and challenging, so I make sure to acknowledge the fact that they solved the problem, or at least made a lot of progress. Working on one problem for a day and a half may still be fairly new for students, but hopefully they will be getting used to it.