Sasha and Keoni use the equation y = x2/(4p) to plot a parabola for p = 1/4. They make a conjecture for how the shape of the parabola will change as gets larger.
Students’ Conceptual Challenges
After using the equation to plot points on the parabola y = x2 with a p-value of 1/4, Keoni struggles to locate coordinates of the focus of the parabola [5:06-5:24]. He first places the focus at (0,1) [5:35-5:59]. Keoni states that (0,1) is a “general place” to put the focus.
➤Sasha and Keoni notice a conflict when asked to state the p-value for the parabola when the focus is one unit above the origin. They restate that p is the distance between the focus and the vertex. Keoni notices that they are currently working with a p-value of 1/4. Consequently, Sasha and Keoni adjust the focus location [6:02-6:19].
For use in a classroom, pause the video and ask these questions:
1. [Pause video at 5:05]. What are some other points that you know are on the parabola because of the geometry of the parabola?
2. [Pause video at 9:16]. What are the coordinates of the red point that Keoni says is on the parabola?
Invite students attend to the reasoning of others while reflecting on multiple strategies:
1. Stop the video at [10:00]. Ask one student to present one method for checking to see if the point (½, ¼) is on the parabola. Ask a second student to use the first student’s method to check a different point, say (1,1).
2. Repeat the process for a new method of checking.
1. What happens when the focus is below the vertex? Graph the parabola with a focus at (0, –¼) and vertex at (0, 0). Label the focus, the directrix, and several points on the parabola.