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Episode 6 Supports

  • Episode Description

    Exploring: Sasha and Keoni notice patterns in the equations they have derived for parabolas with the same p-values for different vertices. They predict an equation for a parabola with a p-value of 3 and a vertex at (7, 2).

     

  • Students’ Conceptual Challenges

    Keoni initially thinks the distance from a general point to the directrix (y = –1) is y – 1 [3:48]. Sasha questions this [3:58]. Keoni explains that he was only looking at the label of the directrix when he labeled the distance [4:06].

     

    • Together, they point out the distances of y, 1, and y + 1. Using the coordinate grid, they make sense of the three lengths of the sides of the triangle [4:14-5:54].

  • Focus Questions

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

     

    1. [Pause video at 1:27]. How does Keoni know that his point is a “special point”?

     

    2. [Pause video at 8:11]. Why did Keoni multiply out the (y – 5)2 and the (y +1)2 terms but leave the (x – 7)2 unchanged?

  • Supporting Dialogue

    Provide opportunities to for students to revoice mathematical thinking. Ask a few students to revoice the ideas used in this episode:

     

    • Revoice how you can determine the lengths of the sides of the triangle.

    • Revoice how Sasha and Keoni solved for  y [8:14- 8:59]

  • Math Extensions

    1. Try deriving the equation of another parabola using the methods of this episode. Derive the equation for a parabola with a p-value of 3 and a vertex of (–4, 1).

     

    2. Show your work as you derive this equation. Label your focus and directrix as well as the lengths of the sides of the right triangle.

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Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

 

Sasha and Keoni use a GeoGebra applet to move parabolas to the left, right, up, and down. Then they develop equations for several different parabolas where the vertex is not at the origin.

Targeted Understandings

 

This lesson can help students:

 

  • Decompose the translation of a parabola on a coordinate grid into two directions: horizontal (left/right) and vertical (up/down).

  • Understand that when a parabola is translated left or right, its directrix remains the same, as does the y-value of its focus. The x-value of the focus is adjusted by the direction and distance of the horizontal movement. The length of the horizontal side of the right triangle used to derive the equation will also reflect the horizontal translation (but the length of the vertical side and the hypotenuse remain the same as for the base parabola).

  • Understand that when a parabola is translated up or down, its directrix also moves by the amount of the translation, as does the y-value of its focus. The x-value of the focus remains the same. The lengths of the vertical side and the hypotenuse of the right triangle used to derive the equation will also reflect the vertical translation (but the length of the horizontal side remains the same as for the base parabola).

Common Core Math Standards

 

• CCSS.M.HSF.BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

In this lesson, Sasha and Keoni connect graphical translations of a base parabola (up, down, left, right, and a combination of these movements) with changes in the algebraic expression of the function. For example, shifting a base parabola with p = 3 and vertex at the origin to the right 7 units means that the term x – 7 will be in the numerator for the translated parabola (y = (x – 7)2/12)). When the same base parabola is translated up 2 units, the effect is the addition of 2 units to the equation (which is y = x2/12 for the base parabola and y = x2/12 + 2 for the translated parabola)

Common Core Math Practices

 

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

According to the Common Core’s description of Math Practice 2, mathematically proficient students “know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.” In this lesson, Sasha and Keoni use the visual tool of a function-translation applet in GeoGebra to decompose the translation of a parabola into horizontal and vertical movements [Episode 1, 1:18 – 1:29 and 2:46 – 2:59]. This imagery is powerful for helping Keoni and Sasha conceive of the effect of translation on the lengths of the right triangle that they use to derive the equation of a parabola [see for example, 4:05 – 4:34 in Episode 2].