K20 LEARN | Center of Attention

Summary

In this geometry lesson, students will recall vocabulary about a circle, use their knowledge of midpoint and distance formulas to write the equation of a circle, and explore the connection between the equation of a circle and the Pythagorean Theorem. This is a multimodality lesson, which means it includes face-to-face, online, and hybrid versions of the lesson. The attachments also include a downloadable Common Cartridge file, which can be imported into a Learning Management System (LMS) such as Canvas or eKadence. The cartridge includes interactive student activities and teacher's notes.

Essential Question(s)

How are triangles and circles related?

Snapshot

Engage

Students recall what they know about circles.

Explore

Given points on a circle, students use their knowledge of the distance formula to find the radius and/or diameter.

Explain

Students complete guided notes with the class and/or watch a video to learn about the properties of a circle. These properties relate to the equation of a circle, how to write the equation of a circle given the endpoints of a diameter, and how to identify the center and radius from a given equation.

Extend

Students write the equation of a circle from a graph and investigate the relationship between the Pythagorean Theorem and the equation of a circle.

Evaluate

Students match equations with circles that have different characteristics.

Instructional Formats

The term "Multimodality" refers to the ability of a lesson to be offered in more than one modality (i.e. face-to-face, online, blended). This lesson has been designed to be offered in multiple formats, while still meeting the same standards and learning objectives. Though fundamentally the same lesson, you will notice that the different modalities may require the lesson to be approached differently. Select the modality that you are interested in to be taken to the section of the course designed for that form of instruction.

Face-to-Face

Materials

Lesson Slides (attached)
Circle Characteristics handout (attached; one per student; printed front only)
Guided Notes handout (attached; one per student; printed front/back)
Guided Notes (Teacher Guide and Model Notes) (attached; for teacher use)
Writing Equations handout (attached; one per student; printed front only)
Exit Ticket handout (attached; one half-sheet per student; printed front only)
Pencils
Coloring utensils (three per student)
Student devices with internet access (for Extend portion)

Online

Materials

Common Cartridge (attached)
Desmos account
Pencils
Paper
Student devices with internet access

Hybrid

Materials

Common Cartridge (attached)
Guided Notes handout (attached; one per student; printed front/back)
Guided Notes (Teacher Guide and Model Notes) (attached; for teacher use)
Writing Equations handout (attached; one per student; printed front only)
Desmos account
Pencils
Coloring utensils (three per student)
Student devices with internet access

Hybrid

Engage

10 Minute(s)

Teacher's Note: Desmos Activity Preparation

To use this Desmos Classroom activity, select the following link: "Center of Attention: Synchronous." Create an account or sign in under the "Activity Sessions" heading. After you log in, the green "Assign" dropdown button will be active. Click the arrow next to the word "Assign," then select "Single Session Code." After making some setting selections, select "Create Invitation Code" and give the session code to students. For more information about previewing and assigning a Desmos Classroom activity, go to https://k20center.ou.edu/externalapps/using -activities/.

For more detailed information about Desmos features and how-to tips, go to https://k20center.ou.edu/externalapps/desmos -home-page/.

To set up the activity’s pacing for students, select "View Dashboard" (next to the session code). In the upper-left corner of your screen, select the icon above the word "Pacing." Desmos Classroom should then prompt you to select the first and last screens that you want students to see. When prompted to set a range, select screens 1 and 5. Select "Restrict to Screens 1–5" to confirm your selection. This allows students to access only screens 1–5 at this time. For more information about teacher pacing, go to https://k20center.ou.edu/externalapps/pacing -activities/.

Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

Introduce the lesson using screens 1–2 of the Desmos activity. Screen 1 displays the lesson’s essential question. Screen 2 identifies the lesson’s learning objectives. Review each of these with students to the extent you feel necessary.

Give students time on screen 3 to learn how to use the Desmos sketch tools.

On screen 4, have students use the Collective Brain Dump strategy to write what they know about circles. Ask students to use numbers and academic vocabulary to label the picture of the circle. Beside the picture, students should write anything else they know about circles that they did not label.

On screen 5, students are prompted to input their answers from the previous screen—specifically, the location of the circle’s center and the lengths of the radius and diameter. As screen 5 has a built-in self-check function, students receive immediate feedback about their answers.

Then, ask students to pair up and compare their lists. Have students volunteer to share what they labeled, numerically and verbally. As students share out, write the circle characteristics on the board for the whole class to see.

Online

Engage

10 Minute(s)

Teacher's Note: Desmos Activity Preparation

Before starting the activity, assign student pairs. In an asynchronous setting, advise students to schedule a time to meet with their partners.

To use this Desmos Classroom activity, select the following link: "Center of Attention: Asynchronous." Create an account or sign in under the "Activity Sessions" heading. After you log in, the green "Assign" dropdown button will be active. Click the arrow next to the word "Assign," then select "Single Session Code." After making some setting selections, select "Create Invitation Code" and give the session code to students. For more information about previewing and assigning a Desmos Classroom activity, go to https://k20center.ou.edu/externalapps/using -activities/.

For more detailed information about Desmos features and how-to tips, go to https://k20center.ou.edu/externalapps/desmos -home-page/.Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

On screen 1, students learn how to use the Desmos sketch tools. On screen 2, students use the Tell Me Everything strategy to label the picture of the circle, both with numbers and with words. Beside the picture, students should write anything else they know about circles that they did not label.

On screen 3, students are prompted to input their answers from the previous screen—specifically, the location of the circle’s center and the lengths of the radius and diameter. As screen 3 has a built-in self-check function, students receive immediate feedback about their answers.

Face-to-Face

Engage

10 Minute(s)

Introduce the lesson using the attached Lesson Slides. Display slide 3 to show students the essential question. Go to slide 4 to share the lesson’s learning objectives. Review each of these slides with students to the extent you feel necessary.

Go to slide 5 and pass out the attached Circle Characteristics handout to each student. Ask students to use the Collective Brain Dump strategy to write what they know about circles on the top part of the handout. Have students use numbers and academic vocabulary to label the picture of the circle. Beside the picture, students should write anything else they know about circles that they did not label.

Then, ask students to pair up and compare their lists. Have students volunteer to share what they labeled, numerically and verbally. As students share out, write the circle characteristics on the board for the whole class to see.

Face-to-Face

Explore

10 Minute(s)

Display slide 6. Ask students to look at the Explore portion of the Circle Characteristics handout and work with their partners to complete it. Have students identify the center, radius, and diameter of a circle that has a center at (3, 5) and passes through (6, 9).

Online

Explore

10 Minute(s)

Screen 4 prompts students to work in pairs. Collaboratively, students must identify the center, radius, and diameter of a circle that has a center at (3, 5) and passes through (6, 9).

As screen 4 has a built-in self-check function, students receive immediate feedback about their answers.

Hybrid

Explore

10 Minute(s)

On the Desmos dashboard, click the orange plus sign to allow students to progress to screen 6. Have students work with their partners to identify the center, radius, and diameter of a circle that has a center at (3, 5) and passes through (6, 9).

As screen 6 has a built-in self-check function, students receive immediate feedback about their answers.

Hybrid

Explain

25 Minute(s)

Discuss students’ strategies for finding the dimensions of a circle without its graph. Come to an agreement about using the distance formula to find the radius.

Then, pass out the attached Guided Notes handout to each student. Complete the handout as a class. Once finished, have students add it to their math notebooks if that is a classroom norm.

Online

Explain

10 Minute(s)

Have students select the link on screen 5 to watch the "Center of Attention" video.

In this video, students learn about the properties of a circle that relate to the equation of a circle, how to write the equation of a circle given the endpoints of a diameter, and how to identify the center and radius from a given equation.

Face-to-Face

Explain

25 Minute(s)

Display slide 7 to provide students with the answers to the Explore portion of the Circle Characteristics handout.

Discuss students’ strategies for finding the dimensions of a circle without its graph. Come to an agreement about using the distance formula to find the radius.

Go to slide 8 and pass out the attached Guided Notes handout to each student. Complete the handout as a class. Once finished, have students add it to their math notebooks if that is a classroom norm.

Face-to-Face

Extend

15 Minute(s)

Display slide 9 and inform students it is time for them to apply what they have learned so far.

Pass out the attached Writing Equations handout to each student. Ask students to work in pairs to find the equation of each provided circle.

Transition through slides 10–11 so students can check their work. Be sure to allow students time to find their mistakes, ask questions, and refine their understanding.

Go to slide 12 and provide students with the link to the GeoGebra activity: geogebra.org/m/a4sm8fc6. Have students read about and interactively explore the relationship between the Pythagorean Theorem and the equation of a circle.

Online

Extend

15 Minute(s)

On screens 6–7, students must find the equations of two different circles, given graphs of each. The graphs are of circles with the centers at lattice points, but the lattice points on each circle are not vertical or horizontal of the center.

Students also are asked to explain their thinking on screens 6–7. As both screens have a built-in self-check function, students receive immediate feedback about their equations.

Screen 8 prompts students to leave the Desmos activity and go to the provided link to the GeoGebra activity: geogebra.org/m/a4sm8fc6. In this activity, students are expected to read about and interactively explore the relationship between the Pythagorean Theorem and the equation of a circle.

When students return to the Desmos activity, they must write what they learned from the GeoGebra activity on screen 8. Students’ responses are shared with the class upon submission.

Hybrid

Extend

15 Minute(s)

On the Desmos dashboard, click the orange "Edit" button and select screens 8 and 9. This allows students to access screens 8–9.

Pass out the attached Writing Equations handout to each student. Have students work with their partners to find the equation of each provided circle and type their final answers into the Desmos activity.

As screens 8–9 have a built-in self-check function, students receive immediate feedback about their equations. Be sure to allow students time to find their mistakes, ask questions, and refine their understanding.

Once students feel confident in their understanding, click the orange plus sign on the Desmos dashboard to allow students to progress to screen 10. This screen prompts students to leave the Desmos activity and go to the provided link to the GeoGebra activity: geogebra.org/m/a4sm8fc6. In this activity, students are expected to read about and interactively explore the relationship between the Pythagorean Theorem and the equation of a circle.

When students return to the Desmos activity, they must write what they learned from the GeoGebra activity on screen 10. Students’ responses are shared with the class upon submission.

Hybrid

Evaluate

5 Minute(s)

On the Desmos dashboard, click the orange "Edit" button and select screen 11, then click "Restrict to Screen 11." On screen 11, have students complete a Card Matching activity to assess their learning individually.

Students are asked to match an equation of a circle with each of the following circles:

A circle with a radius of 2 and a center at (–2, 3).
A circle with a center at (2, –3) and passes though (2, –1).
A circle with a given graph.
A circle with endpoints of a diameter at (2, 7) and (2, –1).

Use student responses to see which misconceptions persist.

Online

Evaluate

5 Minute(s)

On screen 9, students complete a Card Matching activity to assess their learning individually.

Students are asked to match an equation of a circle with each of the following circles:

A circle with a radius of 2 and a center at (–2, 3).
A circle with a center at (2, –3) and passes though (2, –1).
A circle with a given graph.
A circle with endpoints of a diameter at (2, 7) and (2, –1).

Use student responses to see which misconceptions persist.

Face-to-Face

Evaluate

5 Minute(s)

Use the Exit Ticket strategy to assess students’ learning individually.

Go to slide 13 and give each student a half-sheet from the attached Exit Ticket handout. Then, ask students to match each of the following circles with one of the equations from the right-hand column of the Exit Ticket:

A circle with a radius of 2 and a center at (–2, 3).
A circle with a center at (2, –3) and passes though (2, –1).
A circle with a given graph.
A circle with endpoints of a diameter at (2, 7) and (2, –1).

Use student responses to see which misconceptions persist.

Resources

K20 Center. (n.d.). Bell Ringers and Exit Tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Card Matching. Strategies. https://learn.k20center.ou.edu/strategy/1837
K20 Center. (n.d.). Collective Brain Dump. Strategies. https://learn.k20center.ou.edu/strategy/111
K20 Center. (n.d.). Tell Me Everything. Strategies. https://learn.k20center.ou.edu/strategy/107
K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081
K20 Center. (n.d.). GeoGebra. Tech tools. https://learn.k20center.ou.edu/tech-tool/2352
Pixabay. (2016, August 30). Black Ceiling Wall [Photograph]. Pexels. https://www.pexels.com/photo/black-ceiling-wall-161043/

Authentic Lessons for 21st Century Learning

Center of Attention

Writing the Equation of a Circle

Summary

Essential Question(s)

Snapshot

Instructional Formats

Materials

Materials

Materials

Engage

Engage

Engage

Explore

Explore

Explore

Explain

Explain

Explain

Extend

Extend

Extend

Evaluate

Evaluate

Evaluate

Resources

Publisher

Downloads

Standards

Facilitator Resources

Learner Handouts