Download Relationships between Central Angles and Inscribed Angles

Lesson Title: Investigating the Relationship Between the Inscribed Angle and Central Angle
Course: CC Geometry, Unit 3 Date: ____
Teacher(s): _________ Start/end times: ______________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
G.C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. (Include the relationship
between central and inscribed angles and inscribed angles on a diameter are right angles.)
MP1:
MP3:
MP4:
MP5:
MP8:
Make sense of problems and persevere in solving them.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Look for and express regularity in repeated reasoning.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Divide students into groups of 6-8 students and provide
them with string. Explain that today students will be
creating a human circle, while representing the circle
vocabulary below:
 Chord
 Radius
 Central angle
 Inscribed angle
Give each group time to present their diagrams or have
each group guess how each vocabulary word is
represented in the human diagrams. Clear up any
misconceptions as groups present.
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Have students participate in a High Five Activity, where
the students travel around the room until they find a
partner to “high five” and then share a relationship they
investigated today. Have them “high five” at least 3
people.
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Explain that today students will be investigating the relationship between a central angle and an inscribed angle
in a circle that share the same arc.
2. Assign students pairs to use dynamic geometry software, such as, Cabri, Jr., Geometer’s Sketchpad, or
GeoGebra and provide them with a copy of the Investigating Relationships between Central Angles and
Inscribed Angles Activity. (Teachers can refer to the example video: G.C.A.2 Sketchpad Inscribed Angles.)
3. Give students time to work through the problem and explore the relationships in Sketchpad. (Look for evidence
of MP1, MP3, MP5, and MP8.)
4. Then, as a class come back together and discuss the relationships they found. Then, pose the following question:
“What is the relationship between the measure of the intercepted arc and the inscribed angle?” (Students should
be able to make the connection that the measure of the intercepted arc is equal to the central angle, therefore, the
inscribed angle is one half the intercepted arc.)
5. Next, pose students with the following question: What conjectures can you make about two inscribed angles that
intercept the same arc? Provide students with time to use dynamic geometry software to prove/disprove their
conjectures. (Look for evidence of MP3 and MP4.)
6. As a class, discuss what students proved or disproved and clear up any misconceptions.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Investigating the Relationship Between the Inscribed Angle and Central Angle
Course: CC Geometry, Unit 3 Date: ____
Teacher(s): _________ Start/end times: ______________
After completing the investigation, students should be able to form a conjecture that the inscribed angle is always
one half of the shared central angle or intercepted arc. Students should also notice that when the central angle is a
straight angle, the central angle is also the diameter of the triangle. Students may also see that this shape is also a
right triangle, so when a right triangle is inscribed in a circle, the hypotenuse is the diameter of a circle.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Vocabulary: chord, inscribed angle, central angle, intercepted arc
Connections: After completing the investigation, students will hopefully make connections to the fact that when a
right triangle is inscribed in a circle, the hypotenuse will always be the diameter of the circle.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Investigating Relationships between Central Angles
and Inscribed Angles Activity
G.C.A.2 Sketchpad Inscribed Angles Video
Dynamic Geometry Software
String
For homework, ask students to write a more formal proof
of the inscribed angle theorem by creating their own
diagram.
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Were students able to articulate that the measure of an inscribed angle is one half the measure of the central angle
when they share the same intercepted arc?
Were students able to recognize that when a right triangle is inscribed in a triangle, the hypotenuse is the diameter?
Were students able to set up a diagram and prove their conjectures about the relationship between two inscribed
angles that intercept the same arc?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.