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.
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).
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?
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.
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.