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Episode 5 Supports

  • Episode Description

    Making Sense: 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).

  • 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.

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Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

 

Keoni and Sasha compare the graphs of y = x2/(4p) for p-values of 1/4, 1/2, and 1. They figure out the effect that changing the value of p has on the graph of the parabola.

Targeted Understandings

 

This lesson can help students:

 

  • Connect geometry with algebra by verifying that a point belongs on a parabola in two ways: (a) using the geometric definition of a parabola, and (b) using algebraic substitution in the equation representing the parabola.

  • Formulate the following relationships across comparable points for a set of parabolas, all with vertex at the origin, but with p-values of 1/4, 1/2, and 1:
       o When the y-value is fixed (y = 4), then the x-value increases as p increases.
       o When the x-value is fixed (x = 2), then the y-value decreases as p increases.

  • Understand the following features of “special points” —points that align horizontally with the focus of a parabola:
       o The y-value of a special point is one-half its x-value.
       o The x-value of a special point is double its y-value.
       o The y-value of a special point is equal to the p-value of the parabola.
       o For a parabola with an unknown p-value, a special point can be expressed as (2p, p).

Common Core Math Standards

 

• CCSS.M.HSF.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

In an elaboration of this standard, the CCSSM learning trajectory for functions (Grade 8 and High School) states, “functions are often studied and understood as families, and students should spend time studying functions within a family, varying parameters to develop an understanding of how the parameters affect the graph of a function and its key features.” In this lesson, Keoni and Sasha graph (by hand) a set of parabolas with increasing p-values of 1/4, 1/2, and 1. They figure out that increasing the p-value in the family of parabolas with vertex at the origin (and as represented by the function y = x2/(4p)) results in the parabola getting wider on the same coordinate grid.

 

 

 

 

 

Common Core Math Practices

 

CCSS.Math.Practice.MP6. Attend to precision.

According to the Common Core’s description of Math Practice 6, “mathematically proficient students try to communicate precisely to others.” In this lesson, Sasha and Keoni improve in their ability to speak with precise mathematical language. For example, in Episode 5, they begin by stating the following inaccurate relationship: “When you change the value of p, the parabola gets wider” [0:38]. They then refine the statement to one that is more accurate: “As p increases, the parabola widens” [0:47]. In a second example, also from Episode 5, Sasha and Keoni are asked to articulate what they notice about the three “special points” that they have identified: (1/2, 1/4), (1, 1/2), and (2,1). At first, Keoni and Sasha report that “the x-value is, it’s half of it, right, it’s 2 and then 1” [5:37]. However, as they continue to work, they are able to more precisely convey that they are halving the x-value to obtain the y-value [6:15], which is not the same as saying that the x-value is one-half the y-value. They also re-express this relationship as the x-value of a special point being double its y-value [7:07]. This precision of language contributes to Sasha and Keoni’s ability to eventually express the special point for a parabola with unknown p-value as (2p, p) [8:02 – 8:20].