We'll begin class by discussing the solution to the Motorboat Problem from the previous lesson. To do this most quickly, I'll stay at the whiteboard and act as the students' "scribe" while they tell me what to do. I'll ask for volunteers to provide just one or two steps at a time, making sure that I get input from a wide variety of students. I'll also make a point of interpreting what the students suggest, rephrasing and explaining as I carry out their instructions. If all goes well, we should be able to get the whole solution laid out in 10 minutes.
There are two important questions I'll try to answer during this process:
Why were we able to treat the motorboat like a projectile? The strength of the quadratic model is that it can be used in a variety of contexts. We should be able to see that this situation is "like" another situation in it's most important mathematical aspects. Here, we have a boat with a certain initial velocity (from the river) and then a constant acceleration (from the motor). Since the acceleration is constant, just like it is for a projectile, we can model this the same way we modeled projectiles. See What Goes Up for specifics.
What does it mean to "look for and make use of structure"? This is closely related to the previous question, but with a broader scope. We were able to identify the most important mathematical features of the situation: the rate of change is changing at a constant rate! We were also able to link this explicitly to the structure of the quadratic equation, identifying the roles of the three coefficients. In this way, we'll be able to solve any number of real-world problems that satisfy these general criteria! (MP 7)
In groups of three, students will solve the Elliptical Area Problem on Modeling Practice and Review 2 by making explicit reference to the Constant Area problem (a.k.a. Tiger, Tiger, Burning Bright). The only significant difference in this case is the presence of pi as a constant multiplier and the use of axes as linear dimensions rather than perimeter (the perimeter of an ellipse is not an easy thing to calculate!)
Initially, students are asked to maximize the area for a given axial length, which requires a straightforward quadratic equation as a model. The final part of the problem, however, asks students to minimize the axial length for a fixed area. This is identical to the problem of the tiger's cage and I will make sure that they draw out the similarities explicitly in small-group discussions.
Since this is a day of practice & review, I will make sure that all groups solve this problem within the 15 minutes provided - providing hints and explanations along the way. See Modeling Practice and Review Solutions for reference.
During the final 20 minutes of class, I'm planning on being flexible. My default plan is for the students to use this time to solve the Wireframe Problem. However, since this problem is similar to the Elliptical Area Problem, I have the option to skip it if it becomes clear that the class needs practice or review in some other area. This is the sort of flexibility that you simply have to have as a teacher!
If you do choose to use the Wireframe Problem, be aware that the solution requires students to find the vertex of a cubic function. Since this is Algebra 2, not Calculus, my expectation is that they will identify the solution graphically or with technology. The value of the problem is not in pinning down the precise solution, but in constructing and interpreting the mathematical model.