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Episode 4 Supports

Episode Description

Repeating Your Reasoning: Sasha and Keoni use points on three parabolas that share a y-value to explain why increasing the p-value results in the parabola getting wider on the coordinate grid.

Students’ Conceptual Challenges

Sasha and Keoni are examining the equation 16 = x

^{2}/p when they are asked that happens to x when p increases. They pause as they search for an explanation [4:21].- By using algebra to rewrite the equation so that the variables are expressed in a direct relation with each other, x = 4√p, Sasha and Keoni find a way to respond to the question. The direct representation supports their reasoning on how a change in the p-value impacts the x-value.

- By using algebra to rewrite the equation so that the variables are expressed in a direct relation with each other, x = 4√p, Sasha and Keoni find a way to respond to the question. The direct representation supports their reasoning on how a change in the p-value impacts the x-value.
Focus Questions

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

1. [Pause video at 1:53]. Keoni considers a parabola that goes through the origin and contains the point (4.5. 4). Can someone come up here and draw the rest of the parabola and estimate where the focus would be?

2. [Pause video at 6:44]. How did Keoni and Sasha get the equation x = 4√p?

Supporting Dialogue

Support the opportunity for students to engage in precise language as they articulate mathematical claims:

- As the x-value changes, how does the focus change?
- Restate the claim and reasoning Sasha and Keoni use to argue why the parabola gets wider as the p-value increases.

- As the x-value changes, how does the focus change?
Math Extensions

1. Find the coordinates of points on each of the three parabolas when the y-value is 1.

2. Considering the ordered pairs that you found, what do you notice about the x-values when the p-value increases? How does that impact the width of the parabola as the p-value changes?

Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

Sasha and Keoni use algebraic and geometric thinking to form three arguments that justify why a parabola gets wider on the coordinate grid as the p-value in y = x^{2}/(4p) increases.

Targeted Understandings

This lesson can help students:

- Understand why, when the p-value in y = x
^{2}/(4p) increases the parabolas appear to be getting wider on the same coordinate grid, by advancing three different arguments: - When x is a fixed value (e.g., when x = 2 and y = 1/p, then as p increases, y decreases, which means the parabola will get wider.
- When y is a fixed value (e.g., when y = 4 and 4 = x
^{2}/(4p) or 4√p = x), then as p increases, the square root of p increases. This means that 4√p, which is x, also increases. Thus, as p increases, so does x, which means the parabola will get wider. - A “special point” for a parabola (i.e., a point on the parabola that is aligned horizontally with the focus) can be expressed in general as (2p, p). Thus, as p increases, the special point for the parabola will be growing faster outward than it grows in the upward direction. This means the parabolas will appear to get wider on the same coordinate grid.

Common Core Math Standards

• CCSS.M.HSF.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Sasha and Keoni first notice key features of the graphs of three quadratic functions (all with a vertex at the origin but with different p-values). For example, they notice that when the x-value is the same for all three functions (e.g., when x = 2), then the y-values of the points decrease as the distance between the vertex and focus (the p-value) increases. They observe visually on the graph how decreasing the y-value but keeping the x-value constant forces the parabola to become wider. They then link this observation to the algebraic expression for the parabolas. By fixing the x-value as 2, the equation y = x^{2}/(4p) becomes y = 4/(4p), which is y = 1/p. Sasha and Keoni can see algebraically that as p increases, y decreases. Comparing the algebraic and graphical representations of these quadratic functions provides a way to help understand why increasing the p-value in y = x^{2}/(4p) means the parabolas will appear to be getting wider on the same coordinate grid.

Common Core Math Practices

CCSS.Math.Practice.MP7. Look for and make use of structure.

In this lesson, Sasha and Keoni construct three different viable arguments to explain why increasing the p-value in y = x^{2}/(4p) means the parabolas will appear to be getting wider on the same coordinate grid. Each argument relies on a consideration of different features of the graphs: (a) fixing the x-value in Episodes 2-3; (b) fixing the y-value in Episode 4; and (c) considering special points in Episodes 5-6. Noticing a relationship is much easier than creating an argument to explain why the relationship holds! To accomplish the latter, Sasha and Keoni express their ideas to each other [Episode 2, 2:33 – 3:42], even when it’s difficult to do so. They also justify their conclusions [Episode 2, 5:02 – 5:33] and respond to each others’ explanations [Episode 6, 1:26 – 1:56].