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Episode 6 Supports
Episode Description
Making Sense: Sasha and Keoni build on what they have learned in the previous episodes to begin to develop the general equation for any parabola with vertex (h, k) and the distance p from the vertex to the focus.
Students’ Conceptual Challenges
Sasha found the distance from the focus and the x-axis, which is k+p [1:59]. But she then equated this distance to the focus rather than identifying the coordinate pair that represents the focus.
- After the teacher asks whether k+p represents the x-value or the y-value of the focus, Sasha and Keoni represented the focus with a coordinate pair [2:27].
- After the teacher asks whether k+p represents the x-value or the y-value of the focus, Sasha and Keoni represented the focus with a coordinate pair [2:27].
Focus Questions
For use in a classroom, pause the video and ask these questions:
1. [Pause the video at 1:34]. Where is the length k on the graph?
2. [Pause the video at 4:18]. What distance are Sasha and Keoni trying to find here? What is its significance?
3. [Pause the video at 7:25]. How can you represent the length that Sasha just circled?
Supporting Dialogue
Provide opportunities to for students to revoice a 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 right triangle.
- Revoice how the lengths in the expressions can be seen on the graph.
- Revoice how you can determine the lengths of the sides of the right triangle.
Math Extensions
- Why is the vertex labeled (h, k)? A general point on the graph is labeled (x, y). Why the difference?
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.
