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Episode 5 Supports
Episode Description
Reflecting: Sasha and Keoni look back on their work with a parabola with a p-value of 5 and a vertex of (9, 13). By reflecting on their work and the equation y = (x – 9)2/(4p) + 13, they see 13 their graph.
Focus Questions
For use in a classroom, pause the video and ask these questions:
1. [Pause the video at 1:13]. Where is the 8 in the graph? Does any one see the 13 in the graph? Where is it?
2. [Pause the video at 1:43] Can someone show me the 13 in the graph? How do you know that it is 13?
Supporting Dialogue
Invite students to engage in a pair-share activity as they respond to the focus question:
- With your partner, show where you can find 13 on the graph. Prepare your answers to share with the whole class.
- With your partner answer extend your answer to find 13+p on the graph. Prepare your answers to share with the whole class.
- With your partner, show where you can find 13 on the graph. Prepare your answers to share with the whole class.
Math Extensions
1. What expression is easier for you to simplify? How are they different to you? How are they the same?
(y – 13 – p)2
(y – (13 + p))2
((y – 13) – p)2
2. How are each of these expressions present in the Sasha and Keoni’s graph?
Mathematics in this Lesson
Lesson Description
Targeted Understanding
CC Math Standards
CC Math Practices
Lesson Description
Sasha and Keoni develop the vertex form of the equation of a parabola as y = (x–h)2/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.
Targeted Understandings
This lesson can help students:
- Derive the vertex form of the equation of a parabola by using the method developed in previous lessons (which involves the definition of a parabola and the Pythagorean theorem), but generalizing from working with particular vertices to an unknown vertex (h,k) and generalizing from a specific distance from the vertex to the focus to an unknown p-value.
- Interpret algebraic expressions involving parameters (x – h, y – k , and y – k – p) as distances on a coordinate grid.
- Conceive of algebraic expressions, such as x – h, as both a single entity and as a process of subtracting the value of one variable from the value of another.
Common Core Math Standards
• CCSS.M.HSG.GPE.A.2: Derive the equation of a parabola given a focus and directrix.
In this lesson, Sasha and Keoni build upon the method that they developed in Lessons 3 and 4 of using the geometric definition of a parabola and the Pythagorean theorem to derive the equations for particular parabolas with vertex at (0,0). They generalized this method in Lesson 5 for a family of parabolas with vertex at the origin but an unknown distance between the vertex and focus (the p-value). In Lesson 8, they generalized the method to parabolas with a specific non-origin vertex. Finally, in this lesson, they derive the vertex form for a parabola by using parameters (h,k) for the vertex and an unknown p-value.
• CCSS.M.HSA.CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
The students build equations in two variables for different parabolas that capture a relationship between quantities. They represent these relationships on a coordinate grid system.
• CCSS.M.HSA.SSE.A.1.B: Interpret complicated expressions by viewing one or more of their parts as a single entity.
On a parabola with general vertex (h, k) and unknown p-value, Sasha and Keoni express the distance from a general point (x, y) to its directrix as y – k + p. They conceive of this distance as a single entity, which they locate on the graph. They are also able to describe and locate the distances represented by parts of the expression: y, k, p, and y – k.
Common Core Math Practices
CCSS.Math.Practice.MP2: Reason abstractly and quantitatively.
According to the Common Core’s description of Math Practice 2, mathematically proficient students are able to “decontextualize—to abstract a given situation and …manipulate the representing symbols as if they have a life of their own” and to “contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.” In Episode 7, Sasha and Keoni use the Pythagorean theorem to set up the equation for a given parabola as ((y–k)–p)2 + (x–h)2 = ((y–k)–p)2 and then reason abstractly by performing appropriate algebraic transformations to arrive at the equation y = (x–h)2/(4p) + K. However, they also reason quantitatively in Episode 6 by describing each term that they substituted into the Pythagorean theorem (namely, x – h, y – k + p, and y – k – p) as distances on the graph.
