##
* *Reflection: Developing a Conceptual Understanding
The Parabola (Day 1 of 2) - Section 3: Developing the standard equation

This year I had an interesting comment come from some students that I had not expected. When we discussed the directrix some students asked if the directrix was an asymptote. At first I was not sure where this idea was coming from but knew that I needed to clear this misconception.

After thinking I realized that students realized I had stated that that the parabola would not cross the directrix. Many students have a misconception that an asymptote is never crossed. To help students understand the difference I began by discussing the asymptote of an exponential function we talked about how the function gets closer as closer to the asymptote as x goes to infinity (or negative infinity). I then asked if the parabola gets closer to the directrix at the ends (we were using an x^2 example)

Understanding that the directrix is different than an asymptote is important since all conics can be defined using a directrix. It is also important for students to realize how a graph of a functions will get closer to the asymptote as x goes to infinity or as x approaches the vertical asymptote.

*Directrix vs. Asymptote*

*Developing a Conceptual Understanding: Directrix vs. Asymptote*

# The Parabola (Day 1 of 2)

Lesson 3 of 13

## Objective: SWBAT use different key features of a parabola to solve problems.

*40 minutes*

#### Bell work

*10 min*

Today we are ready to do Activity 2 from the Wax Paper Activity packet. After working on the circle, my students are more receptive to doing this activity on their own. However, I expect that Step 2 will be difficult for some students. I be very precise with my explanation of the process in the packet, but I will not be surprised if students ask how to complete this step.

Like with the circle folding activity, I give students about 10 minutes to work and we then look at the design created by the fold lines. Most students will recognize the parabolic shape since they have learned about these curves in Algebra 2.

As we discuss, I will show several different designs (parabola 1 and Parabola 2). I will ask questions like, "Why is one design more narrow than another?" I want students to begin to reflect on how the distance between the point F and the line determine the shape of the parabola. Discussing ways to change the shape of the design is a great way to prepare students for the formal definition of this conic section.

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Once students understand that there is definitely a relationship between the point F and the line from our design, I share the definition of parabola with them. I ask a student to read the definition. Then, as a class, we start break down the definition to understand what it is saying.

When I break this down I use the literacy technique **thinking aloud** suggested in Cris Tovani's I Read it But I Don't Get It. I model this strategy first. As I read the definition, I will ask myself questions (outloud), draw a diagram, and label the diagram using the information in the definition.

- I will draw a line and label it as a directrix.
- I will label a point and call it the focus.
- I then ask myself the question, "What does it mean to be equidistant?"
- I now get specific with points. I say, "What points are equidistant? Equidistant to what?"
- Then, I will draw lines to represent the distances.
- Finally I say,"How do I find the distance from a point to a line?"

I have learned that these are questions that most students will have as they consider the definition in light of their experience with the folding activity. But, there is more to consider which is part of the fun and part of the challenge.

- I think aloud "Does it say all the distances are equal? No! Only the distance from each point to the focus and to the directrix.
- I ask myself, "How can I write the distance from the point to the line using algebra?"
- Then, "How can I write an equation to represent the distance from the focus to the point using algebra?"

If this sequence is progressing well, I will share the proof of parabola formula from Larson's Precalculus with Limits (p. 805). We read the proof and discuss the steps shown in the proof.

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#### Using the standard equation

*10 min*

Students have seen the standard equation and how it is proved. I have students put standard equations of a parabola on their reference sheet.

Students are usually confused with the 2 different versions of the equation. I start helping students analyze the equations by asking "Which form of the equation is a function? Why is it a function? If the equation represents a function what do you know? How will the graph look if it is a function? What will it look like if it is not a function? At the bottom of the slide I use the diagrams to explain what is meant by "p is a directed distance."

I give students some information about a parabola and ask them to write an equation. "How will you know which equation to use? Would a sketch help?" "How can you find the parameter p?"

After writing the equation for the example I give the students another example (page 2). I make sure my examples show both a vertical and a horizontal directrix so students can see how to determine the structure or the equation.

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#### Closure

*5 min*

As class comes to an end I show students the stanadard equations for a circle and a parbola. Students compare the standard equations and then predict how the general equation will look if it is representing a parabola.

This activity allows me to assess what students are understanding with the equations. I want students to notice that only one variable is squared for a parabola and the equation is not solved for a constant. I am interested in seeing how many students predict that only one variable the x or the y will be squared on the general equation.

We will continue with parabolas the next day since we have not had time to see applications or practice changing between general and standard form.

#### Resources

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: The Circle (Day 1 of 2)
- LESSON 2: The Circle (Day 2 of 2)
- LESSON 3: The Parabola (Day 1 of 2)
- LESSON 4: The Parabola (Day 2 of 2)
- LESSON 5: The Ellipse (Day 1 of 3)
- LESSON 6: The Ellipse (Day 2 of 3)
- LESSON 7: The Ellipse (Day 3 of 3)
- LESSON 8: The Hyperbola (Day 1 of 2)
- LESSON 9: The Hyperbola (Day 2 of 2)
- LESSON 10: Graphing Conics
- LESSON 11: Wax Paper Activity (Day 1 of 2)
- LESSON 12: Wax Paper Activity (Day 2 of 2)
- LESSON 13: Individual portion of the Assessment