Geometry Unit 9: Circles

Unit Driving Question 

What rules and properties can be found in circle and how can they be applied to real world situations?

Essential Questions 

1. What are the ways we can identify a circle? 
2. What are the parts of a circle?
3. What relationships are formed by line intersecting with, inside and outside the circles? 

Launch Task 

Big Ideas for Development Lessons 

Closure & Assessment 

1 Lesson 

3 Weeks (approximately 1 week per big idea) 

1 Week

Click on the links below to see each Big Idea's Lesson Overview (includes links to teacher notes and student activities) 

1. A circle is uniquely defined in the coordinate plane using its center and radius.  
2. There is a constant proportional relationship between an angle and its arc measures on a circle. 
3. Congruence and similarity prove relationships between segments and figures of a circle. 

1. Formative Assessment 1 (after Big Idea 3) 
2. Re-engagement Activity (Not Provided)
3. Unit 9 Assessment 

Big Idea 1: A circle is uniquely defined in the coordinate plane using it's center and radius. 

OAS-M:  G.C.1.1, G.C.1.3, G. C.1.4

Lessons and Additional Activities

Big Idea 1 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

Describe the characteristics of any circle

• Construct a circle and justify that all radii are equidistant and congruent

• Explain that all points on the circle are equidistant from the center of the circle

• Define radius as a relationship between a circle’s center and points on the circle

• Justify circles are similar using scale factor

• Reason proportionally and connect the measurements of a circle with similarity

• Analyze examples and nonexamples to distinguish radii, chords, tangents, and secants

• Use diagrams to explain the relationship between the radius and a tangent line

• Construct perpendicular bisectors of chords to show they always pass through the circle’s center 

Analyze characteristics of a circle in the coordinate plane

• Graph a circle in the coordinate plane given its center and radius

• Explore whether a point lies on a circle whose radius and center point are known

• Identify and justify possible methods including the use of a compass, straightedge, the Pythagorean Theorem, the distance formula, etc.

• Prove whether a point lies on a circle given the circle’s center and another point on the circle

• Describe characteristics about a circle given its center, radius, points on the circle, etc.

• Example: find the area of a circle whose endpoints of its diameter lie at (2, 3) and (10, -3), or determine the diameter of a circle centered at the origin and passing through (-5, 10)

Determine the standard equation of a circle in the coordinate plane

• Describe connections between a circle’s equation and and its graph in the coordinate plane

• Generate the equation of a circle from its graph

• Generate a circle's equation given information about its center, radius, or points on the circle by using the distance formula and midpoint formulas.   

Big Idea 2: There is a constant proportional relationship between an angle and its arc measures on a circle. 

OAS-M: G.C.1.2

Lessons and Additional Activities

Big Idea 2 Lessons 1-2 Overview (includes links to teacher notes and student activities)

Evidence of Understanding  

 Describe proportional relationships between angles, arcs, and sectors of a circle

• Create visuals to describe a central angle in terms of the fraction of the circle it represents

•  Investigate the constant ratio between an angle’s arc length and a circle’s circumference

• Explain the proportional relationship with the radius

• Use similarity to derive the length of the arc intercepted by an angle

• Explain how sectors are related to area of a circle using similarity relationships

• Derive and use the formula for determining the area of a sector 

Analyze and apply the relationships between the angles and arcs used to describe a circle

• Use examples and nonexamples to distinguish central, inscribed, and circumscribed angles

• Define and classify angles using radii, chords, and tangents

• Investigate relationships between angle and arc measure for a central, inscribed, or circumscribed angle and use them to solve problems

• Prove inscribed angles open to the same arc are congruent (and vice versa)

• Prove parallel chords intercept congruent arcs

• Describe patterns relating the angle and arc measures on a circle by two tangents, two secants, a secant and a tangent, or a chord and a tangent

• Describe patterns relating the angle and arc measures resulting from two intersecting chords

• Calculate the value of an unknown angle or arc measure  

Big Idea 3: Congruence and similarity criteria prove relationship between segments and figures of a circle. 

OAS-M:  G.C.1.2

Lessons and Additional Activities

Big Idea 3 Lessons 1-2 Overview (includes links to teacher notes and student activities)

Evidence of Understanding  

Investigate and describe the relationships that exist between the segments on a circle

• Investigate and describe how the lengths of segments for two intersecting chords, two secants (drawn from the same point), or a tangent and a secant (drawn from the same point) are related

• Prove theorems about the segments of secants, tangents, or intersecting chords

• Recognize congruent chords are equidistant from the center of the circle

• Recognize tangents from a shared point outside the circle are congruent

• Calculate the length of an unknown segment

Use circles in modeling situations and find missing values

• Create diagrams that can be used to represent the situation

• Use markings and notations to create equations that can be used to find missing values