Sasha and Keoni notice patterns in the equations they have derived for parabolas with the same p-values for different vertices. They predict an equation for a parabola with a p-value of 3 and a vertex at (7, 2).
Students’ Conceptual Challenges
Keoni initially thinks the distance from a general point to the directrix (y = –1) is y – 1 [3:48]. Sasha questions this [3:58]. Keoni explains that he was only looking at the label of the directrix when he labeled the distance [4:06].
➤ Together, they point out the distances of y, 1, and y + 1. Using the coordinate grid, they make sense of the three lengths of the sides of the triangle [4:14-5:54].
For use in a classroom, pause the video and ask these questions:
1. [Pause video at 1:27]. How does Keoni know that his point is a “special point”?
2. [Pause video at 8:11]. Why did Keoni multiply out the (y – 5)2 and the (y +1)2 terms but leave the (x – 7)2 unchanged?
Provide opportunities to for students to revoice mathematical thinking. Ask a few students to revoice the ideas used in this episode:
1. Revoice how you can determine the lengths of the sides of the triangle.
2. Revoice how Sasha and Keoni solved for y [8:14- 8:59].
1. Try deriving the equation of another parabola using the methods of this episode. Derive the equation for a parabola with a p-value of 3 and a vertex of (–4, 1).
2. Show your work as you derive this equation. Label your focus and directrix as well as the lengths of the sides of the right triangle.