The bell work problem for the day has students use the information from yesterday to identify key features. Students can use their reference sheets and talk with each other.
I move around the room and ask questions:
I have students share the key features and explain how each feature is found. I then ask students how we will find the equations for the asymptotes.
After identifying key concepts we will be writing standard equations for hyperbolas when given information about the hyperbola. I put a problem on the board and the students work on the problem.
I begin with a problem that states the foci and vertices which is more straight forward than my other example. The students work through the problem and share their results.
The second example I give students is harder for the students to solve. The reason is that they immediately assume that the parameters a and b are both 1 because the slope of the asymptotes are 1 and -1. This will confuse students because most determine the value of a to be 2 from the given vertices. The issue is that the students immediately assume that the equation has not been simplified because the formula states the slope as a/b or b/a.
I ask "What do you think was done with the fraction when the asymptotes was written?" Many students then realize it is simplified and they must determine the value of the parameter b so that the slope is 1 and -1.
As with the other conics students need to understand how a conic can be represented in different forms. We have looked at the standard equation and now need to discuss the general form.
I begin this discussion with
I put the bell problem on the board and have the students rewrite the equation in general form. The biggest issue students have is forgetting to distribute the negative. Algebra and computational errors are the main problems. The students sometimes rush through a problem and do not think about the steps they are doing.
The students have a good grasp of rewriting from standard to general so we look at rewriting from general to standard. This can be a struggle if students are just learning procedures. When we did all the other conics the students always added when completing the square. When completing the square on a hyperbola students need to subtract one of the terms. Once students think about the reason we subtract the students are able to do the problem quickly.
As the class ends I want students to compare and think about all the conics sections. I share these questions or students to consider. We look at each question and discuss what the students think. I have my own ideas with these questions but I always get some unusual responses that I have students explain. This is a time that I learn how students are thinking.
I feel that most students will think the parabola is the one that is different. The interesting thing is that we will discuss how the ellipse and hyperbola can be defined by using a directrix.
For practice students are assigned page 758, #10, 16, 24, 28, 30, 36, 42, 50 from Larson, Precalculus with Limits, 2nd ed. These problems allow students to practice identifying key features and writing equations in standard and general form.