Lesson 2 | Circles | 10th Grade Mathematics

Objective

Given a circle with a center translated from the origin, write the equation of the circle and describe its features.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.C.A.1 — Prove that all circles are similar.

Circles

G.C.A.1 — Prove that all circles are similar.

G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Congruence

G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G.GPE.A.1 — Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Expressing Geometric Properties with Equations

G.GPE.A.1 — Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Using given information about a circle or a description of a transformation, write the equation of a circle.
Write the equation of a circle not centered at the origin in standard form, $$(x-h)^2+(y-k)^2=r^2$$, where $$(h, k)$$ is the center of the circle.
Explain the standard form of the equation of a circle through transformations as well as the distance between points and the Pythagorean Theorem.
Determine the necessary features of a circle required to write the equation of the circle.
Identify transformations of circles in terms of translations, based on the coordinates of the center, and dilations, based on the size of the radius.
Prove that all circles are similar.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

A student may ask if it is possible for $$x^2$$ and $$y^2$$ to have a coefficient. As a response, you can show them the L that is formed when the coefficients are not one.
Circle dilation is not a major concept, but students should have exposure to this idea.
The following resource may be helpful to show students the rationale of using the Pythagorean Theorem to derive the equation of the circle: Illustrative Mathematics, “Explaining the Equation for a Circle."
Desmos' Standard Form of a Circle may be helpful to show how the $$h$$, $$k$$, and $$r$$ in the standard form of a circle affect the graph of a circle.
ADDITIONAL REVIEW: Because this lesson has a lot of conceptual components, in order to prepare for Lesson 3, teachers should take the opportunity through the independent practice to review completing the square to write quadratic equations in vertex form.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Circles $$A$$ and $$C$$ are shown below.

Write an equation that describes each of the circles.

Guiding Questions

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References

GeoGebra Geometry - 8.2 AP1

Problem 2

Write the equation of a circle whose diameter has endpoints of $$(-8, 2)$$ and $$ (2, 2)$$.

Guiding Questions

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Problem 3

Describe how you could prove that circle $$A$$ is similar to circle $$B$$.

Guiding Questions

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References

Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 2, "A Develop Understanding Task"

Module 7: Circles a Geometric Perspective from Geometry: A Learning Cycle Approach made available by Mathematics Vision Project under the CC BY 4.0 license. © 2016 Mathematics Vision Project. Accessed Oct. 19, 2017, 2:58 p.m..

Modified by Fishtank Learning, Inc.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

Identify the radius and the center of the circle given by the equation $$(x-7)^2+(y-8)^2=9$$.

References

EngageNY Mathematics Geometry > Module 5 > Topic D > Lesson 17 — Exit Ticket, Question #1

Geometry > Module 5 > Topic D > Lesson 17 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Write the equation for a circle with center (0, -4) and radius 8.

References

EngageNY Mathematics Geometry > Module 5 > Topic D > Lesson 17 — Exit Ticket, Question #2

Problem 3

Write the equation of the circle below.

Problem 4

A circle has diameter with endpoints (6, 5) and (8, 5). Write the equation for the circle.

References

EngageNY Mathematics Geometry > Module 5 > Topic D > Lesson 17 — Exit Ticket, Question #4

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students are given the center and radius and asked to write the equation.
Include problems where students are given the graph of a circle not at the origin and asked to write the equation.
Include problems where students are given the equation and asked to identify the center and radius.
Include problems where students are given the endpoints of the diameter of the circle.
Include problems where students are given the dilation of a circle and its center and asked to write the new equation of the circle.
Include problems where students must identify the dilations of a circle based on the equation of the circle.
Include problems where students describe which components of the standard form of a circle determine the dilation and which determine the translation.
EXTENSION: Include problems such as “What features are the same/different between the two circles given by the equations: $$x^2+y^2=16$$ and $$2x^2+2y^2=16$$? Show your reasoning algebraically.”
EXTENSION: Include problems with systems of equations between two circles, which is discussed in Algebra 2.
EXTENSION: Include problems such as “What are the $$x$$-intercepts of the circle?”

MARS Summative Assessment Tasks for High School Circle Patterns
GeoGebra CSS High School: Geometry (Circles) — "Similar Circles?"
Illustrative Mathematics Explaining the Equation for a Circle
MARS Formative Assessment Lessons for High School Sorting Equations of Circles 1 — S-2, S-3, and T-9 for solutions
Mathematics Vision Project: Secondary Mathematics Two Module 8: Circles and Other Conics — Lesson 2, "Getting Centered" (Choose questions that align to the Criteria for Success)
Mathematics Vision Project: Secondary Mathematics Two Module 8: Circles and Other Conics — Lesson 3, "Circle Challenges" (Choose questions that align to the Criteria for Success)
EngageNY Mathematics Geometry > Module 5 > Topic D > Lesson 17

Lesson 1

Lesson 3

Lesson Map

Topic A: Equations of Circles

Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.

G.GPE.A.1 G.GPE.B.4

Given a circle with a center translated from the origin, write the equation of the circle and describe its features.

G.C.A.1 G.CO.A.5 G.GPE.A.1

Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle.

G.CO.A.5 G.GPE.A.1

Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem.

G.C.A.2

Describe the relationship between inscribed and central angles in terms of their intercepted arc.

G.C.A.2

Determine the angle and length relationships between intersecting chords.

G.C.A.2

Prove properties of angles in a quadrilateral inscribed in a circle.

G.C.A.3

Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.

G.C.A.2 G.C.A.3

Construct tangent lines to a circle to define and describe the circumscribed angle.

G.C.A.2 G.C.A.4

Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems.

G.C.A.2

Topic C: Arc Length, Radians, and Sector Area

Define, describe, and calculate arc length.

G.C.B.5

Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians.

G.C.B.5

Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius.

G.C.B.5

Use sector area of circles to calculate the composite area of figures.

G.C.B.5 N.Q.A.2 N.Q.A.3

Circles

Objective

Common Core Standards

Core Standards

Circles

Congruence

Expressing Geometric Properties with Equations

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Target Task

Problem 1

References

Problem 2

References

Problem 3

Problem 4

References

Additional Practice

Lesson Map

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