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Episode 3 Supports

  • Episode Description

    Repeating Your Reasoning: Keoni and Sasha look for geometric information of a parabola represented by the equation y = 2(x – 3)2 + 1. They start by finding the vertex and the p-value.

     

  • Students’ Conceptual Challenges

    The challenge is finding the p-value of the parabola. The expression in the last episode was given in standard form. In this algebraic representation, the denominator is not explicit.

     

    • It helps that the teacher asks Sasha and Keoni what they would like y = 2(x – 3)2 + 1 to look like and Keoni is able to say:

  • Focus Questions

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

     

    1. [Pause the video at 2:32]. Explain in your own words why 2 = 1/(½) and why y = 2(x – 3)2 + 1 = y = (x–3)2/(½) + 1.

     

    2. [Pause the video at 3:12]. Is there a way to check that the equation y = 2(x – 3)2 + 1 is the same as the equation y = (x–3)2/(½) + 1?

  • Supporting Dialogue

    Provide opportunities to reflect on previous mathematical strategies using the strategy of revoicing. When student share their ideas with the class, ask another student to revoice the expressed ideas.

     

    Recall the definition of a parabola:

     

    A parabola is the set of points that are equal distance from the focus and directrix.

     

    Discuss with your partner how you can determine the coordinates of some of the points on this parabola using the coordinates of the focus and the equation of the directrix. Prepare your answers to share with the whole class.

     

    Another pair of students working on this problem said that the point (3.25, 1/8) was a special point for this parabola. Add this point to your graph of this parabola. What do you notice? Prepare your answer to share with the whole class.

  • Math Extensions

    1. Compare the equation y = (x + 3)2 – 1 to the equation that Sasha and Keoni worked with in this episode (namely y = 2(x – 3)2 + 1). What is alike? What is different?

    2. Find the h, k, and p-values for the parabola represented by y = (x + 3)2 – 1.

    3. Use the information from Question 2 above to graph y = (x + 3)2 – 1.

     

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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].