Lesson 1 | Circles | 10th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.
G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Expressing Geometric Properties with Equations

G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Foundational Standards

The foundational standards covered in this lesson

G.GPE.B.7

Expressing Geometric Properties with Equations

G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
8.G.B.7

Geometry

8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8

Geometry

8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Criteria for Success

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

Describe the line segment whose endpoints are the center of the circle and a point on the circle as the hypotenuse of a right triangle formed inside the circle.
Derive the equation of a circle centered at the origin as $$x^2+y^2=r^2$$ using repeated reasoning.
Understand and describe the relationship between the equation of a circle at the origin and the Pythagorean Theorem.
Determine the equation of a circle given the center of the circle at the origin and one point on the circle.
Given the equation of a circle centered at the origin, identify the radius and points that lie on the circle.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Many sources on this topic combine writing equations of circles with a center at the origin and writing equations of circles with a center not at the origin. In this unit, these two topics are separated to allow students to have the opportunity to practice the algebraic skills associated with each component.
Students may need to review algebra work with solving equations using square roots and substituting values into an equation with quadratic terms before they can fully access this lesson.
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

Write an equation that models each point’s distance from the center of the circle, point $$A$$.

Guiding Questions

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References

GeoGebra Geometry-8.1 AP1

Problem 2

A circle has a center at the origin and contains the point ($$2, \space \sqrt{5}$$).

Write an equation of the circle.
Identify the radius of this circle.
Name two other points that lie on this circle.

Guiding Questions

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Target Task

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

The circle shown below has a center at the origin.

If a point ($$3, \space y$$) lies on the circle, what is the value of $$y$$?

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 must identify whether an equation forms a circle. For example, "Joe says that the equation $$x^2+y^2=-4$$ forms a circle with radius $$4$$. Why is Joe not correct?"
Include problems where various information is given and ask students to find the radius, one point, or the equation of the circle. Again, only use circles centered at the origin.
EXTENSION: Give a diagram similar to Math Mistakes’ “Circumference and Right Triangles” and ask students to write the equation of the circle.

MARS Formative Assessment Lessons for High School Sorting Equations of Circles 1 — S-1 "Going Round in Circles" Question 1 and S-4 "Going Round in Circles (Revisited)"
Mathematics Vision Project: Secondary Mathematics Two Module 8: Circles and Other Conics — Lesson 1, "Circling Triangles (Or Triangulating Circles)" (Only use problems with circles centered at the origin)
Open Middle Which Circle is Bigger? (High School)
EngageNY Mathematics Geometry > Module 5 > Topic D > Lesson 17 — (Only use problems with circles centered at the origin)

Lesson 2

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

Expressing Geometric Properties with Equations

Expressing Geometric Properties with Equations

Foundational Standards

Expressing Geometric Properties with Equations

Geometry

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

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