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Episode 5 Supports

  • Episode Description

    Reflecting: Sasha and Keoni examine yet another equation of a parabola that is not in vertex form, y = x2 – 4x + 5. They start out by seeking a method to re-express the equation in vertex form.

     

  • Students’ Conceptual Challenges

    Re-expressing this equation in vertex form is challenging. Many students have experiences re-expressing quadratic expressions in factored form. Keoni writes (x – 5)(x + 1).  While this is true, it does not support their problem solving goal.

     

    • After looking for geometric information, Keoni erases his factored representation. Sasha and Keoni keep working to find a way to re-express the equation in vertex form.

  • Focus Questions

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

     

    1. [Pause the video at 1:38].  Why did Keoni erase (x – 5)(x + 1)? What’s wrong?

     

    2. [Pause the video at 3:00]. What do think of Sasha’s expression x(x – 4) + 5? Is it correct? Does it help?

  • Supporting Dialogue

    Invite students to reflect on problem solving as a whole class. Elicit multiple answers from the class:

     

    • At 2:23, Keoni says “let’s not give up.” What do you do when you get stuck?

    • It seems that Sasha and Keoni make progress when the teacher asks how the equation y = x2 – 4x + 5 is different from y = x2 – 4x + 4.  How does that question help them? How can they check that y = x2 – 4x + 5 is the same as y = (x – 2)2 + 1?

  • Math Extensions

    1. Rewrite the equation y = x2 – 4x + 7 in vertex form. How do you know the two equations are equivalent?

    2. Rewrite the equation y = x2 – 4x + 3 in vertex form. How do you know the two equations are equivalent?

     

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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].