The Parabola (Day 2 of 2)

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Objective

SWBAT find or use key features of the graph of a parabola to write equations or sketch graphs.

Big Idea

Why are the directrix and the focus important to understanding parabolas?

Bell Work

10 minutes

After yesterday's lesson, my  students have the Standard Equation for a Parabola on their conics reference sheet. Today's Bell work requires students to think about the information they need to acquire to determine the equation in standard form for a parabola whose vertex and focus are known. I will let the students struggle with this question for a while. As they begin to work I will ask questions like:

  • What equation can you use to describe the graph?
  • Would putting the known points on a graph help? 
  • What do I know about the vertex and the focus of a parabola? 
  • Is this parabola a function or not?

As they move along I will ask:

  • What can you put into the standard form equation from reading the problem statement?
  • What do you need to find? 
  • How can you find the parameter p?

After sufficient time has past for the students to explore the problem fully, I will have a student share a solution. As he/she presents I will return to the above questions to help students clarify their thinking with respect to writing the equation for a parabola.

Where are parabola used?

5 minutes

After finishing the bell work I ask the class, "Why is it important that we learn about parabolas?" Depending on the class I will sometimes say, "Besides doing it to make your life miserable?" This is a comment I use to let the class know that I understand it may be hard for them to find a connection. I want them to be open to looking at the examples that I will share with them.

  1. I use a website called Occurrence of the Conics to demonstrate examples. This site gives students great examples for each conic. 
  2. I also have a 3-D hologram tool called Mirage. I move around the room with the Mirage and ask the students to touch the pig.  It is fun to watch their expressions when the realize that the object is not real. I explain that this "toy" is an example of why satellites are parabolic.  The receiver for the satellite is positions at the focus. When a signal hits the parabola it is projected to the focus.  In the "Mirage" the hologram is at the focus of the bottom parabola and the actual pig is at the focus of the top parabola.  The image is reflected off one parabola to the other by the mirror of the parabolas.

My students become a lot more interested in the conics when I demonstrate their application using these objects.

Another visual I use is a wooden sliced cone.  I explain that the conics are created when a cone is sliced by a plane.  The shape depends on the angle the plane slices the cone.  I will discuss that a parabola is made when the plane is at a slight angle from the side.

 

General form of the equation.

15 minutes

Now that I have the students' attention, it is time for them to do some algebra.  I begin by displaying writing a parabola in general form on the board. My plan is to give all four problems to each group. Each group will be assigned one problem to work and then share their work with the class. The other three problems are to be worked in preparation for a peer group's presentation. I will choose some problems written in terms of x^2 and others that are defined with y^2. I want my students to have the opportunity to notice that a parabola will have only one variable squared in the general form for a conic section.

Next, I will put up several equations on the board for the students to see. I will ask my students to determine if each equation represents a circle or a parabola. As a group we will brainstorm criteria for a rule to determine which type of conic section each equation represents.

Once we establish a way of identifying whether we are working with a circle or a parabola, we will take on of the parabolic equations and work on rewriting the equation in standard form. I will prompt the students by asking, "How can we change the equation to a form that allows us to use some of the algebraic skills that we possess?"  I want the students to come up with the idea that we can complete the square like we did for the circle.

As the students work to convert to standard form, I will observe individual students' progress.  After a student gets the equation, I will ask him/her to sketch the graph that is represented by the equation. I want students to related the general form equation back to the position of the focus and the directrix. I expect that it will be difficult for my students to determine the parameter "p" which is the distance from the vertex to the focus. 

Before moving on, I will lead a discussion wrapping up our work over the last two days. I want to make sure that my students have both forms of a parabolic equation on their Conics Reference Sheet.

 

Writing equations for parabolas

15 minutes

Every year, my students need practice writing equations when given a graph (or information about the graph) of a parabola.

I put a problem from examples for parabola day 2 and give the students a three minutes to work on the problem. Then, I ask a student who is not finding the answer, "What is one question that would help you understand how to write the equation?" I then ask if anyone in the class can answer the question. With rigorous algebra tasks like these, I find that my students often benefit from hearing how other students are thinking.  After a few question cycles, I will give my students a few more minutes to work. If possible, I will select a student to put a solution on the board as they work.

If time allows, we will continue this process for the next 2 examples on the worksheet. Using this instructional practice allow me to identify where confusions exist and plan for upcoming lessons. I am able to my observations of student work, the students questions, and their answers to assess the progress of the class.

Closure

5 minutes

Before class ends, I will assign page 738 #19, 24, 28, 36, 39, 44, 52, 57, 83, 84 from Larson Precalculus with Limits, 2nd ed for homework

I will also give students an exit slip today. I will ask my students to determine which way a parabola will open if the directrix is x=-2 and the vertex is (4, 5). This question will help me to quickly assess whether my students understand how key features determine what the graph may look like.