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Episode 2 Supports
Episode Description
Exploring: Keoni and Sasha use the Pythagorean Theorem and the definition of a parabola to derive an equation of parabola with a p-value of 3 and a vertex at (7, 0).
Students’ Conceptual Challenges
At [1:31], Sasha whispers, “It looks weird; I don’t get it.” In this new situation where the vertex has shifted over by 7 units, it is challenging to construct a right triangle like they used when the vertex was on the origin..
- While examining what they have drawn so far, Keoni adds an important side of the triangle. He chooses the focus as one of the corners of the triangle. His choice of where to place the side of the triangle results in a useful right triangle. As the episode continues, they figure out how to represent the length of each side of the triangle.
- While examining what they have drawn so far, Keoni adds an important side of the triangle. He chooses the focus as one of the corners of the triangle. His choice of where to place the side of the triangle results in a useful right triangle. As the episode continues, they figure out how to represent the length of each side of the triangle.
Focus Questions
For use in a classroom, pause the video and ask these questions:
1. [Pause video at 0:47]. How does Keoni know where to place the focus and the directrix for this parabola?
2. [Pause video at 6:15]. Looking at the two right triangles, what is the same and what is different?
3. [Pause the video at 9:29]. What is the difference between the representation of (x – 7)2 and x2 – 14x + 49?
Supporting Dialogue
Invite students to engage in stating and justifying mathematical claims with another student:
- Work with a partner to justify the representation of x – 7 for the length of the horizontal side of the triangle that Sasha and Keoni used to derive the equation of a parabola.
- Work with a partner to justify the representation of y – 3 for the length of the vertical side of the triangle that Sasha and Keoni used to derive the equation of a parabola.
- Work with a partner to justify the representation of x – 7 for the length of the horizontal side of the triangle that Sasha and Keoni used to derive the equation of a parabola.
Math Extensions
During the video [1:09], Sasha and Keoni have a brief discussion about choosing what point on the parabola to use to derive the equation for the parabola. They mention the “special point” and the need not to choose it. They state that they need a general point.
- What do they mean by a special point? What are the coordinates for a “special point” on the parabola in this episode?
- What do they mean by a general point?
- What would happen if they used a special point to derive the equation for the parabola?
- What do they mean by a special point? What are the coordinates for a “special point” on the parabola in this episode?
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].
