##
* *Reflection: Developing a Conceptual Understanding
Equations of Circles - Section 2: Equations of Circles

When I first taught this lesson, I thought I did a fair job of explaining the conceptual side of completing the square. But I walked away feeling that I could do a better job of it. I thought my explanation, though thorough, was overly clinical and too teacher-centered. I wanted to capture students' imaginations and involve them more in the process of sense-making. I also wanted to incorporate a piece that would hold them accountable for demonstrating their conceptual understanding of completing the square.

This is what spawned the fictional conversation between Dr. Blabbendorf and his 5 year-old granddaughter, Samantha.

I decided to make Samantha 5 years-old so that she would have license to ask totally naive questions that students themselves might relate to. At the same time, though, I didn't want students to relate to Samantha as a peer. I wanted students to see themselves as more experienced and more mathematically sophisticated than Samantha so that they would expect a deeper understanding from themselves than that which Samantha exhibited. So while it was perfectly ok for Samantha to dismiss all of her Grandfather's jargon and move on to wanting to play, I wanted to compel students to make sense of what Dr. Blabbendorf has said and demonstrate that understanding by responding to the writing prompt.

In terms of having the students role play, I think having them role play Samantha allows them to acknowledge the aspect of themselves that simply wants to dismiss things that appear to be too complicated and having them role play Dr. Blabbendorf allows them to take the perspective of a more mature expert. After this whole dramatic experience, I'm hoping that they will land themselves somewhere between those two extremes and leave having a conceptual understanding of completing the square.

*Using drama to develop conceptual understanding*

*Developing a Conceptual Understanding: Using drama to develop conceptual understanding*

# Equations of Circles

Lesson 4 of 7

## Objective: SWBAT complete the square to find the center and radius of a circle.

#### Activating Prior Knowledge

*20 min*

In this section, I will be guiding students through Activate Prior Knowledge_Equations of Circles.

In the first portion of the handout, students practice squaring simple binomials. I ask students not to rely on distributing (a.k.a "FOILing"), but instead to recognize the general pattern that is at play when we square a binomial. I explicitly forbid them to distribute on #6 and #7. When students have had time to finish, I will quickly reveal the answers to #1-#5 and then discuss the thought process (MP8) that leads to the answers for #6 and #7

The second portion of the handout gives students practice factoring quadratic trinomials and recognizing perfect square binomials as special cases of these. When students have had adequate time to finish, I quickly reveal the answers to #8-#13 and then discuss the answers #14 and #15 at a deeper level.

The third portion of the handout informally eases into the practice of completing the square, giving students practice filling in the missing linear or constant term to make a perfect square quadratic trinomial. In #22, students are basically finding the general constant, c, that must be added to complete the square given x^2+bx. At this point, I don't formally introduce completing the square. I choose instead to focus on the pattern recognition aspect (MP8).

Next I ask students to pair share on #23, discussing their answers and rationale. Then I go over it in order to prime students for the way we use addition property of equality when we complete the square to write equations of circles.

Finally, I have students complete #24-#27 and I follow up with a quick explanation of the answers.

*expand content*

#### Equations of Circles

*25 min*

In this section, I will have students working through Equations of Circles.

The first portion of the handout deals with the general purpose of completing the square. I truly believe that students get so caught up in the procedure of completing the square that they don't pay much attention to the reason for completing the square or what it accomplishes. This first portion is in the form of a short skit between a young girl and her grandfather, Dr. Blabbendorf. I will either have students role play the dialogue in pairs or I will have two volunteer students perform the dialogue for the entire class.

Next, I have the students write to demonstrate their understanding of what Dr. Blabbendorf has said. While I can understand why 5-year old Samantha doesn't understand her grandfather, I expect more from my students.

On page two of the handout, I provide students with a worked example of solving a quadratic equation by completing the square in order to illustrate the strategic purpose of completing the square. Although the steps are laid out, I still provide commentary to really emphasize the what's and why's of completing the square.

After that, students read the bottom of page 2 carefully and then respond to the Check for Understanding. I will have my students exchange papers to read each others' responses and I will also randomly call on non-volunteers to share what they have written.

Finally students get practice writing the equations of circles in standard form. The first three exercises already contain perfect square trinomials in x and y so students do not need to complete the square. These exercises are juxtaposed with #5-#7 so that students hopefully come to see completing the square as a tool we use when we need to create perfect squares. As I demonstrate the answers to #5-#7, I am basically doing direct instruction on completing the square to write the equation of a circle in standard form. This will be the main input I provide for students to learn this technique so I am careful to model it well and address any questions that may arise.

p.s. #4 is just to make sure students are staying on their toes, analyzing structure, and not just having knee-jerk reactions.

*expand content*

#### Create a Card Sort

*25 min*

In Creating Circle Equation Problems, students get to see the completing the square process for writing circle equations from a different perspective.Through the paradigm of reverse engineering, students get to see how teachers like me make up problems. My hope is that this will give them more insight into the process of completing the square to write equations of circles.

When students have understood this process, it is their turn to play the role of teacher and create problems of their own. I will have the students work in pairs to complete the back side of the handout. Before they start, we read through the directions and constraints as a class. I also let them know at this time that they will be competing with other groups so they should make their task challenging.

Once the pairs have finished, they will trade with another pair and see which pair can match the others' first...showing all work to complete the square and write the equation in standard form, of course.

Alternatively, if I want this to be a more independent activity, I will have each student solve their own problems, showing all work to complete the square and write the equation of the circle in standard form.

*expand content*

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: Prove Slope Criteria for Parallel and Perpendicular lines
- LESSON 2: Algebraic Proof of the Perpendicular Bisector Theorem using Coordinate Geometry
- LESSON 3: Loci and Analtyic Geometry
- LESSON 4: Equations of Circles
- LESSON 5: Proving the Medians in a Triangle Meet at a Point
- LESSON 6: Partitioning Segments in the Coordinate Plane
- LESSON 7: Prove Triangle Midsegment Theorem using Analytic Geometry