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Episode 4 Supports

  • Episode Description

    Repeating Your Reasoning: Keoni and Sasha determine the x-value of a point on the parabola with a y-value of 3.5.

  • Focus Questions

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

     

    1. [Pause video at 1:15]. Sasha and Keoni are finding the x-value of a point on the parabola when the y-value is 3.5. Predict how they are going to solve for x. Is there another method that they could use?

     

    2. [Pause video at 3:40]. Sasha just drew a segment on the x-axis. What does that segment represent?

     

  • Supporting Dialogue

    Focus student attention on precision by asking them:

     

    1. Sasha and Keoni write that b = 3.7, b2= 3.7 and say that x = 3.7. What is 3.7 in this problem? Does is matter what we call it? 

    2. Sasha and Keoni found that when y is 3.5, that b is 3.7. Come up to the graph and show us where you see these values on the graph.

  • Math Extensions

    1. Sometimes equations can have more than one variable. Solve the equation below for x. Solve the equation for y. Which is easier?

     

    x2 + (y – 3)2 = 25

     

    2. Are there some values of (x,y) that you can see that will or will not satisfy the equation without solving for x or y? What are your strategies?

     

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Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

 

Keoni and Sasha create a general method for representing the x-value for any point on a particular parabola, given the y-value of that point. By using their previous results, along with the Pythagorean theorem, they are able to determine the equation for the parabola.

Targeted Understandings

 

This lesson can help students:

 

  • Interpret the meaning and use of an equation that they derive to relate the x-value to the y-value for a general point on a given parabola (x = √(4y)).

  • Conceive of algebraic symbols, such as x and y, as quantities that have infinitely many possible values and that vary together, rather than only as unknowns that have a single numerical value.

  • Express important quantities, such as the distance between a general point on a given parabola and its directrix, by generalizing one’s reasoning from Lesson 2, when students determined the x-value for different points on the parabola when y was 5, 7 and 10.

Common Core Math Standards

• CCSS.M.HSG.GPE.A.2: Derive the equation of a parabola given a focus and directrix.

Sasha and Keoni use the definition of a parabola, along with the Pythagorean theorem, to derive the equation of a given parabola in Lesson 3. They express it in a form that is useful for locating the x-value for any point on the parabola given its y-value (namely, x = √(4y)). In Lesson 4, they re-express and derive the equation as y = x2/4 . Later, they generalize their reasoning to derive the equation for any parabola with vertex (0,0) in Lesson 5 and any parabola with vertex (h,k) in Lesson 9.

 

• CCSS.M.HSA.SSE.A.1.B. Interpret complicated expressions by viewing one or more of their parts as a single entity.  Sasha and Keoni express the distance from a general point (x,y) on a particular parabola to its directrix (y = –1) as y + 1. 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 and 1. Similarly they interpret the expression y – 1 both as an entity and in terms of the distances represented by each of its parts.

 

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.” Sasha and Keoni use the Pythagorean theorem to set up the equation for a given parabola as (y–1)2 + x2 = (y–1)2 and then reason abstractly by performing appropriate algebraic transformations to arrive at the equation x = √(4y). However, they also reason quantitatively by describing the distances represented by each term in both equations: y–1, x, y+1, y, and √4y. They also discuss what the equation x = √(4y) means and how it is useful.