Keoni and Sasha use the applet to explore the graphs of parabolas with a vertex at (9, 13) and an unknown p-value. Sasha and Keoni determine how to represent the coordinates of the focus and the equation of the directrix when p can take on any value.
Episode Supports
Students’ Conceptual Challenges
Sasha and Keoni struggle with the equation for the directrix when the p-value is unknown [4:00-4:27]. Keoni thinks it’s y=p. Both Sasha and Keoni seem unsure about how to determine the equation. Part of the difficulty is that this is first time they need to use the variable p in the equation.
➤ It helps them resolve the difficulty by identifying every distance that they do know. Adding the line y=13 to the graph also supports their reasoning. Testing a conjectured equation for the directrix with a previous result helps them resolve their uncertainty [6:23].
Focus Questions
For use in a classroom, pause the video and ask these questions:
1. [Pause the video at 1:30]. Can the p-value be more than 6? What will happen?
2. [Pause the video at 7:43]. What is your conjecture for the coordinates of the focus?
Supporting Dialogue
Provide opportunities to revoice the mathematical ideas of others:
1. Sasha and Keoni noticed that they could represent several distances. What are those distances? Come up here to show us.
2. Revoice what Sasha and Keoni said about why there is a –7 in the equation instead of a +7. How about someone else? Revoice what Sasha and Keoni and saying.
Math Extensions
1. Consider a parabola of any value of p with a vertex at (–9, 13). Find the coordinates and the equation of the directrix for this parabola. Explain your thinking.
2. Consider a parabola of any value of p with a vertex at (9, –13). Find the coordinates and the equation of the directrix for this parabola. Explain your thinking.
