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Episode 2 Supports

  • Episode Description

    Exploring: Sasha and Keoni use the Pythagorean theorem to justify why (4,4) is a point on the parabola.

  • Students’ Conceptual Challenges

    1. Students may be confused about what measuring system to use with a coordinate grid, especially after having used a ruler to measure similar distances in Lesson 1  [1:14].

     

    • Sasha wants to use a ruler to measure the distance from a point to the focus, but Keoni convinces her that it is not needed.

    2. Sasha and Keoni are stumped by how to measure the distance of a diagonal line segment from (4,4) to the focus [1:46].

     

    • Taking stock of everything they know and identifying what they were trying to find prepares Sasha and Keoni to be able to use the Pythagorean Theorem as a tool after the teacher suggests its use [1:56 and 2:52].

  • Focus Questions

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

     

    1. [Pause video at 0:54].  How does Sasha know that the distance from (4, 4) to the focus is 5 units?

     

    2. [Pause video at 2:51].What are Sasha and Keoni trying to figure out? What have they figured out so far? Does anyone have any ideas of what might help them solve their problem?

  • Supporting Dialogue

    Invite students to reflect on measuring between points by asking them these questions:

     

    • How can you measure the distance between a point on a parabola and the directrix?

    • How can you measure the distance between a point on the parabola and a point on the focus? How is this measurement different from finding the distance from a point to the directrix?

  • Math Extensions

    1. Pythagorean Triples are the positive integers that satisfy the Pythagorean theorem. Which of the following are Pythagorean Triples? Can you find three triples add to the list?

     

    • 8, 15, 17
    • 11, 60, 62
    • 28, 45, 53
    • 24, 143, 145
    • 36, 77, 85

     

    2. The Pythagorean theorem is a famous and useful result from mathematics. How do we know it is true? There are many different mathematical demonstrations that show why the Pythagorean theorem works. Even President Garfield created his own unique argument of why it works.  Use the Internet to locate proofs of the Pythagorean theorem. Which one makes the most sense to you?

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Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

 

Keoni and Sasha work with a parabola on the coordinate grid. They use the properties of the grid and the Pythagorean theorem to determine if the coordinates of a point are on a given parabola. They apply these methods to find the missing x-value of a point on the parabola for a given y-value.

Targeted Understandings

 

This lesson can help students:

 

  • Identify and apply key elements of the geometric definition of a parabola in the context of the Cartesian coordinate grid.

  • Conceive of a point on a Cartesian coordinate grid, not only as a location, but also as representing distances in 2-dimensional space.

  • Apply the Pythagorean Theorem in a new setting to measure distance.

Common Core Math Standards

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

Lesson 2 connects the geometric definition of a parabola 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.HSG.GPE.B.4. Use coordinates to prove simple geometric theorems algebraically.

Sasha and Keoni use the coordinates on an algebraic Cartesian grid, along with the definition of a parabola, to validate that three points are on the parabola.

 

• CCSS.M.8.G.B.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Keoni and Sasha use the Pythagorean theorem, along with the coordinate system and the definition of a parabola, to determine the x-value for a point on the parabola given its y-value.

Common Core Math Practices

 

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

In this lesson, Sasha and Keoni use an important mathematical tool— the Pythagorean theorem.  A discussion in the lesson models an important habit of mind related to tool use. Specifically, when Keoni and Sasha are stumped about how to measure the length of a diagonal line segment from a point on the parabola to the focus [1:46, Episode 2], their teacher encourages them to write down everything they know [1:56, Episode 2] and articulate what they are trying to find [2:53, Episode 2]. In the process, a right triangle emerges on the grid, with two sides of known length and one of unknown length.  This practice of analyzing the situation prepares Sasha and Keoni to strategically apply the Pythagorean theorem once the teacher suggests its use [3:56, Episode 2].