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Episode 2 Supports

  • Episode Description

    Exploring: Keoni and Sasha create three different methods for constructing points on the parabola.

  • Students’ Conceptual Challenges

    1.The phrase “set of points” in the definition of a parabola may be misinterpreted as “pair of points.” After finding their first point on the parabola (the point halfway between the focus and directrix), Keoni places a second point close to the first one to form a “pair” [2:11].

     

    • Sasha and Keoni test this point using the definition and conclude that it doesn’t work.

     

    2. Students may think that a point on the parabola can also be on the directrix [8:24].

     

    • Because Keoni is concentrating on points that are 3 inches from the focus, he places a point on the directrix. But he quickly realizes that the point will not satisfy the definition of a parabola because it is not also 3 inches from the directrix.

  • Focus Questions

    For use in a classroom, pause the video and ask these questions:

     

    1. [Pause video at 0:48]. Sasha drew a point that she thinks is on the parabola. Keoni suggests that they test her idea. What would be a way to test the point?

     

    2. [Pause video at 4:16]. Sasha seems to have a question about the point Keoni proposed. Can you restate her question?

  • Supporting Dialogue

    When students are working on the task in class, you can support dialogue as follows:

     

    • Ask a student to repeat what another student just said, i.e., “Alex, can you share what you heard Lani say?”

    • Ask students to relate their actions to the definition, i.e., “Do your points fit the definition of a parabola?”

  • Math Extensions

    These questions allow students to extend the concepts and terminology from the episode:

     

    1. With a pencil and a ruler, draw two parallel lines. What are the points that are the same distance from the two lines that you drew? Explain your solution. How do you know those points you found are the same distance from the two given lines that you drew?

     

    2. Draw three points on a piece of paper. Find the points that are the same distance from each of the three points. Explain your solution. How do you know those points you found are the same distance from the three given points that you drew?

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Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

 

Sasha and Keoni create a parabola from its geometric definition. This involves making sense of key terms—like focus, directrix, and equidistant—and figuring out how to measure distances between points and lines.

Targeted Understandings

 

This lesson can help students:

 

  • Internalize the relationship that the distance from a point on the parabola to the focus is the same as the distance from that point to the directrix.

  • Isolate and analyze the constraint that the distance between a point on the parabola and the directrix has to be along a segment that is perpendicular to the directrix.

  • Think of a line parallel to the directrix as consisting of all the points that are the same distance from the directrix.

Common Core Math Standards

• CCSS.M.HSG.GPE.A.2: Derive the equation of a parabola given a focus and directrix.

Lesson 1 provides a foundation for this Common Core Standard. As Sasha and Keoni construct a parabola from its geometric definition, they make sense of the terms focus and directrix, as well as the relationship of points on the parabola to both.  In Lesson 2, they connect the geometry from Lesson 1 with an algebraic coordinate grid, which makes the derivation of an equation of a parabola possible. Sasha and Keoni then derive the equation of:

               o particular parabolas in Lessons 3 and 4;

               o any parabola with vertex (0,0) in Lesson 5; and

               o any parabola with vertex (h,k) in Lesson 9.

 

• CCSS.M.HSN.Q.A.2. Define appropriate quantities for the purpose of descriptive modeling.

The number and quantity strand of the Standards emphasizes reasoning with measurable attributes in situations (such as length).  In this lesson, Sasha and Keoni can construct points on a parabola and reject other potential points by measuring the distance from a potential point to the focus and to the directrix. A crucial part of their ability to create a general method for constructing points on the parabola involves Sasha’s quantitative insight that a line parallel to the directrix consists of a collection of points that are the same distance to the directrix.

Common Core Math Practices

 

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

According to the Common Core’s description of Math Practice 1, “students start by explaining to themselves the meaning of a problem.” Sasha and Keoni begin this lesson by interpreting the meaning of the geometric definition of a parabola. Initially they claim that if a point is to be the same distance from the focus and the directrix, then the focus has to be on the directrix. By analyzing what Math Practice 1 calls “givens, constraints, relationships, and goals,” Sasha and Keoni conjecture and then test different points to see if they fit the definition. Eventually, they realize that if the focus is on the directrix, then the points that fit the definition will form a line rather than a parabola [4:25 in Episode 1].

 

Math Practice 1 also speaks to the importance of persistence in problem solving. The raw footage that was filmed for Episode 1 (making sense of the problem statement) was over 20 minutes long, and the research team began to wonder if the students would succeed! But Sasha and Keoni’s persistence paid off in the clever methods they developed in Episode 2 to construct a parabola. An expression of Sasha’s delight in solving the problem and her subsequent confidence in their ability can be seen at 10:11 in Episode 2.