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Episode 2 Supports

  • Episode Description

    Exploring: Sasha and Keoni graph the equation y = (x–2.4)2/6. They determine the coordinates of the focus and the equation of the directrix from the geometric information in the equation.

     

  • Students’ Conceptual Challenges

    The placement of the focus, which is no longer on the y-axis, is challenging for students. Keoni initially misplaces it at (0, 1.5). He apparently considers the value of p (which is 1.5) but doesn’t move the focus to (2.4, 1.5).

     

    • When Keoni examines his graph, he notices that the placement of the focus “doesn’t look right.” He adjusts the placement of the focus.

  • Focus Questions

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

     

    1. [Pause the video at 0:52]. The upper left window shows Keoni’s work. Where has he placed the focus and directrix? What would justify the placement there?

     

    2. [Pause the video at 3:25]. Look at the expression that Keoni wrote above the graph, y = (0–2.4)2/6. What geometric information can he get out of that expression?

  • Supporting Dialogue

    Invite students to generate multiple strategies to find points on the parabola by asking:

     

    • With your partner, reflect on what strategies Sasha and Keoni use to find points on the parabola. What are other strategies that you can use to locate and determine the coordinates of points on a parabola? Prepare to share your strategies with the whole class.

  • Math Extensions

    Special points are points on the parabola that are horizontally level with the focus. What are the coordinates of the special points of the parabola represented by the equation y = (x–2.4)2/6? Explain how you know.

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Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

 

Given the equation of a parabola in any form, Sasha and Keoni find geometric information (such as the focus, directrix, p-value, and vertex) about the parabola.

Targeted Understandings

 

This lesson can help students:

 

  • Understand that when the equation representing a parabola is written in vertex form [y = (x–h)2/(4p) + k], then the vertex can be easily determined as (h ,k) and the p-value can located in the denominator and can be used, along with the vertex, to determine the focus and directrix of the parabola.

  • Understand that geometric information (e.g., the vertex, p-value, focus, and directrix) can also be located from a parabola that is given in a different form, by first re-expressing the equation in an equivalent vertex form.

Common Core Math Standards

 

• CCSS.M.HSF.IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

In this lesson, Sasha and Keoni re-express quadratic functions like y = 2(x – 3)2 + 1 and y = x2 – 4x + 4 in vertex form [y = (x–h)2/(4p) + k] in order to locate geometric information about the graphs (such as the vertex, p-value, focus and directrix). To rewrite the function y = 2(x – 3)2 + 1 in vertex form, they think of 2 as 1 divided by ½ [see 2:20 – 2:31 in Episode 3] and ½ as 4 multiplied by 1/8 [3:21 – 3:43, Episode 3]. To rewrite the function y = x2 – 4x + 4 in vertex form, Sasha and Keoni figure out that they need to factor, but that not every way of factoring is helpful. For example, factoring out the x from x2 – 4x yields y = x(x – 4) + 4 [1:06 – 1:30, Episode 4], which doesn’t help them. However, factoring the entire trinomial yields y = (x – 2)2, which is very close to being in vertex form [3:14 – 2:52, Episode 4]

 

Common Core Math Practices

 

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

An important aspect of Math Practice 1 is not giving up, even when several attempts have not been fruitful. In Episode 5, Sasha and Keoni face a challenging task of rewriting y = x2 – 4x + 5 in vertex form. They begin with a false start, as Keoni incorrectly factors x2 – 4x + 5 as (x – 5)(x + 1) [1:08 – 1:44, Episode 5]. Instead of getting discouraged, Keoni tells Sasha, “Let’s not give up” [2:19 – 2:26]. And they don’t! Sasha tries factoring out the x [2:53 – 3:07], which doesn’t end up helping. However, when they explain the correspondences between y = x2 – 4x + 5 and the function they worked with in Episode 4 [y = x2 – 4x + 4], they are able to successfully rewrite the given equation as y = (x – 2)2 + 1 [4:25 – 4:46].