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* *Reflection: Complex Tasks
Conics Sections with Princess the Dog - Section 1: Launch

The most challenging aspect of this task was for students to understand exactly what was going on in the second scenario where Princess could move freely along a leash that is attached at both ends. Students had a difficult time visualizing what this would look like. I am glad that they were talking things over and I think having a model of this would be too leading, but I wish I had a physical model to show at the end of the lesson. I had to rely on a Geogebra animation where I could move a point along an ellipse.

In this photo a student had the general shape of the ellipse but was confused about how to geometrically find some of the points on the ellipse. This student was also confused about the premise and was thinking in terms of a polygon instead of a curved boundary.

*Complex Tasks: Student Struggles*

# Conics Sections with Princess the Dog

Lesson 1 of 10

## Objective: SWBAT use a context to explore circles and ellipses.

#### Launch

*15 min*

My students learned about conic sections and their equations in Algebra 2. However, I do not assume that this is something they remember well. There is so much associated vocabulary and conics can be viewed through so many different lenses (as a cross section, locus, and algebraic equation) that it is a topic that is confusing to students. The purpose of today’s lesson is for students to review a little about conics and for me as a teacher to **formatively assess what they remember. **

I give students this task worksheet and have them work on it with their tables. I give them about 10 minutes or so to get as far as they can. I don’t expect most students to recall the equation of an ellipse off the top of their head, but every student can usually remember at least one aspect.

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#### Share

*15 min*

When it is time to share as a class, I can usually fly through question #1 with no problems – students are usually in agreement that the boundary is a circle and they remember the equation of the circle.

For question #2, students know that it is an ellipse, **but often have difficulty explaining why**. It can be helpful to build a physical model with two nails in a piece of wood and then you can attach Princess’s leash to both stakes. A pencil can be used to model the path. I want my students to know that for every point on an ellipse, the sum of the distances to the foci is a constant (8 feet in this case). So I will build this concept if no student can articulate it.

There is usually some **disagreement about the ordered pairs** for the vertices of the major axis of the ellipse. Some students think that Princess can travel 5 feet from the origin while others think that she can only go 4 feet. I explain about this in the video below using the diagrams that my students wrote on the board.

When we are in agreement, I see if students can piece together the equation for the ellipse. I don’t worry if they can’t because I know that we will review all of this in the coming days.

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#### Math Graffiti

*20 min*

After this introductory activity, I ask students what they think the next unit will be about and at this point they know it is conic sections. We are going to use **Math Graffiti** to summarize what students remember about conic sections. This is a great way for you as a teacher to find out what they already know and to inform your lesson planning over the next few days. This formative assessment tool is very powerful and can be used in many different ways.

I put four posters around the room and title them “parabolas,” “hyperbolas,” “ellipses,” and “conic sections.” I explain that they will **write down anything they remember about these topics **– it can be vocabulary, equations, graphs, etc. They can write questions they have about these topics or about what other people wrote. They can also correct comments if they think someone made a mistake. For the “conic sections” poster, they should write general statements about common threads between all of these topics. Here are examples of the parabola, ellipse, and hyperbola posters from one of my classes.

This is an awesome activity! It gives me so much information about what they know. I usually find that I am short-changing my students and they know more than I give them credit for. Even I will join in and write comments or questions about what I see. Students are especially eager to help me when I pose a question. This **process brings out much deeper thinking** than if I stood at the front of the room and asked students what they remember about these topics.

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Conics Sections with Princess the Dog
- LESSON 2: Exploring Ellipses
- LESSON 3: Hashing Out Hyperbolas
- LESSON 4: Pondering Parabolas
- LESSON 5: Classifying Conics
- LESSON 6: Rotated Conic Sections - Day 1 of 2
- LESSON 7: Rotated Conic Sections - Day 2 of 2
- LESSON 8: Unit Review: Conic Sections
- LESSON 9: Review Game: Lingo
- LESSON 10: Unit Assessment: Conic Sections