No captions Captions Stop the video above first if it is playing.
No captions Captions Stop the video above first if it is playing.
Sasha and Keoni reflect on their graphs of y = x2/(4p) with p-values of 1/4, 1/2, and 1. They consider the effect of increasing and decreasing the p-value on the graph of the parabola.
Episode Supports
Students’ Conceptual Challenges
Seeing and expressing a general relationship for all “special points” is challenging. Initially Keoni and Sasha see halving occurring (e.g., half of 2 is 1) but struggle to use the language of x- and y-values to express this relationship [5:37].
➤ The teacher encourages Sasha and Keoni to also relate the special points to the p-values of their three parabolas and to try any conjectures they make. Sasha sees that the p-values and the y-values are the same for all the special points [6:30]. In the process of verifying this relationship, Sasha and Keoni also determine that the x-value is always double the y- or p-value [7:10], which they express as (2p, p).
Focus Questions
For use in a classroom, pause the video and ask these questions:
1. [Pause video at 0:42]. What is the evidence that supports Keoni’s claim that the graph gets wider?
2. [Pause video at 2:40]. Sasha compares the x-values for the points on the three parabolas that all have a y-value of 4. What is happening?
3. [Pause video at 3:52]. Keoni highlights the points on the three parabolas that all have an x-value of 2. What do you notice about the y-values for these points? How is this related to the change in the p-values?
Supporting Dialogue
Encourage the students to reflect on the precision of language when constructing a mathematical claim:
1. Compare these statements: “When p changes, the graph gets wider” and “As p increases, the graph gets wider.” Do they mean the same thing? Why or why not?
2. Write a claim, in your own words, that describes the effect of changing the p-value on the graph of the parabola. Be sure to use precise language that accurately conveys what you mean.
Math Extensions
1. Graph the parabola represented by y = x2/4. You can use ideas from Sasha and Keoni. Plot a few more points near the vertex: Find the y-values when the x-value is 1, –1, 1/2, –1/2, 0.1, and -0.1. What do you notice about the shape of the parabola near the vertex?
2. On each parabola, y = x2, y = x2/2, y = x2/4, plot the point where the x-value is 1. Explain what happens to the y-values of these points on the three parabolas as the p-value increases.
Captions
