Name: Wenwen DU
Lesson plan 2- Inscribed angles
Date: 10-20-2006 Age/Grade Level: 9 th ~10 th
Subject: Geometry # of Students: 30 # of IEP Students: 0
Major Content: Inscribed angles
Unit Title: Analyzing Circles
Length of Lesson: 50 minutes
Goals and Objectives
Students will be able to

Know the definitions of inscribed angle and intercepted arc

Recognize and find measures of inscribed angles

Understand and apply properties of inscribed angles of a circle
Student Prior Knowledge
Students should be able to

Identify and use parts of circles (center, radius, chords, diameter, arcs, chords, Center angle)

Recognize and solve problems by making circle graphs involving these parts (center, radius, chords, diameter, arcs, chords, Center angle)
Connections
**NCTM Standards:
Algebra – Use symbolic algebra to represent and explain mathematical relationships involving inscribed angles of a circle; Write equivalent forms of equations and solve them with fluency- mentally or with paper and pencil.

Geometry – Analyze characteristics and properties of two dimensional geometric shapes
(inscribed angles in a circle) and develop mathematical arguments about the geometric relationships.
Measurement – Find the measurements of inscribed angles and interpreted arcs.
Problem Solving - Build new mathematical knowledge through problem solving with the properties of inscribed angles in a circle.
Reasoning and Proof – Recognize reasoning and proof as fundamental aspects of mathematics; develop and evaluate mathematical arguments and proofs through the properties of inscribed angles of a circle.
Communication - Organize and consolidate their mathematical thinking through communication; communicate their mathematical thinking coherently and clearly to peers, teachers, and others; analyze and evaluate the mathematical thinking and strategies of others; use the language of mathematics to express mathematical ideas precisely.
Connections - Recognize and use connections among mathematics ideas through the attributes and properties of inscribed angles of a circle; understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
**Kentucky Learner Goals and Academic Expectations:
2.10 – Students understand measurement concepts and use measurements appropriately and accurately.
2.12 Students understand mathematical structure concepts including the properties and logic of various mathematical systems.
** Kentucky Core Content for Assessment 4.1 (CCA 4.1):

MA-HS-3.1.6: Student will know the definitions and basic properties of a circle and will use them to prove basic theorems and solve problems
**Kentucky Program of Studies for Secondly Schools
Students will use the definitions and basic properties of a circle (e.g. arcs, chords, central angles, inscribed angles) to prove basic theorems and solve problems
Context
Before this lesson, Students have established the ability to identify parts of circles (center, radius, chords, diameter, arcs, chords, Center angle) and solve the corresponding problems by making circles graphs. Based on the prior knowledge, this lesson is developed to help students to recognize the definitions of inscribed angle and intercepted arc. Students will also be asked to understand and apply properties of inscribed angles of a circle through e xamples and practices.
References:
Burrill, Cummins, Kanold, Boyd, and Malloy (2001), Geometry – Integration, Applications and
Connections
, Teacher’s Edition. Columbus, OH: Glencoe/McGraw-Hill
Resources:

Protractor

Straightedge

Compass
Procedures
I – Warm-up
There are two options for this part. One of them is handing out a bell ringer and students will be given five minutes to finish it. This practice will include four questions (see attachment for detail). Two questions are designed to review prior knowledge on solving problems regarding

arcs and chords in a circle. In the other two questions, the students are asked to simplify polynomials which actually integrated with algebra knowledge. They are nothing to do with the lesson content. However, I think review of prior knowledge is very important in mathematics learning. Students may still have trouble to simplify the polynomials. This short practice may help them recall this part of knowledge. Obviously, I can put some other questions from prior units in geometry instead of this two if I think the students are all fine with Algebra I. It really depends on your students.
The other options for this part could be the review of the previous homework. The time spent on this introduction part will be controlled within 10 minutes.
II-Lesson development
By drawing a circle with center O (See figure 1), I will recall the measure of a minor arc
( PQ ) is the defined as the measure of its corresponding central angle (

PTQ ). Then I will describe the definition of central angles. Therefore, I will introduce them a new type of angles in circles – inscribed angles (which is an angle whose vertex is on the circle and whose sides each contain chords of the circle). The concept of intercepted arc will be followed (which is the arc lies in the interior of an inscribed angle).
Figure 1

Then I will draw some angles in circles (see Fig.2) as examples to let the students distinguish whether they are inscribed angles or not. If they are, point it out what are their corresponding intercepted arcs.
Fig 2
By giving the measure of the center angle ( m PTQ ), Use figure 1 to get the measure of the arc ( mPQ ) and the measure of the inscribed angel ( m PRQ ).
I will encourage the students make a conjecture regarding the relationship between mPQ and m PRQ . Then Theorem 9-4 will be introduced as following,
Theorem 9-4 If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.
To proof this theorem, we need consider three cases.
1.
The center of the circle lies on one of the sides of the angle (which actually has been done with figure 1).
The center of the circle is in the interior of the inscribed angle. 2.
3.
The center of the circle is outside of the inscribed angle.
Case 2 and 3 will not be proven in the class, but could be asked to prove for highachieving students as challenge questions.
Show students an example for the above theorem as in figure 3.

In the following circle (Fig.3), mST

56 , find

SRT

SRT
Fig.3 Fig.4
Add an another inscribed angle (

SPT ) in Fig.3 to obtain Fig.4 and let the students find the measure of

SPT . This example will illustrate the second theorem between inscribed angles and intercepted arcs of equal measure.
Theorem 9-5 If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent.
Give a different example (Fig.5) from the one in Fig. 4 to make sure the students understand the theorem.
In O ,

40 and BC

CD , find m BEC
Fig.5

Before start 3 rd
theorem, an example in figure 6 will be given as following.
Suppose an inscribed angle

PQR intercepts a semicircle at P and R, find m PQR
Fig.6
This example will lead to the 3 rd theorem.
Theorem 9-6 If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent.
Another example as following (figure 7) will be given to ask the students to find the measures of inscribed angles by using the above three theorems. The example is also integrated with algebra.
In A , m 1 6 x

11 , m 2 9 x

19 , m 3 4 y

25 , m 4 3 y

9 , and PQ

RS .
Find m

1, m

2, m

3, and m

4 . (Page 468)
Fig.7

III Closure
A handout will be given in class (see “check for understanding” in attached files) to help the students to review what we have learned in this lesson (definitions of inscribed angles and interpreted arcs, and three corresponding properties). We will lead a short discussion before the class is over.
**If Time Strategies
If a lot of time left (10~15 minutes), I will introduce another theorem regarding the inscribed angles as following by some examples,
Theorem 9-7 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
Or let students work on homework. They can work with their neighbor or ask the teacher for help.
**Modifications (High and Low)
High: Have the students prove case 2 and 3 in theorem 9-4. If time is allowed, also ask them to prove theorem 9-7
Low: Extra periods will be developed to work on the assignments (homework) or quiz.
These practices are necessary to help students to catch the content firmly and solve problems mathematically. More basic information should be designed to avoid low performance students left behind. If necessary, I will tutor students individually. Meanwhile, I may develop particular homework for low-achieving students to help them build up step by step.
Student Assessment

During the lecture time, some students will be asked to give their opinions on what is the next step for the examples on the blackboard.


When they work on practice questions, I will walk around to observe and check their performance.

Homework will be given at the end of the class and will be collected at the beginning of the next class and graded (see attachment for more detail).

Extra periods will be developed as practice on this part of content. Quiz, in class activity, or exam could be additional options besides homework.
Objective/Assessment Plan Organizer
Lesson Learner
Objective
Type of
Assessment
Description of
Assessment
Adaptations and / or
Accommodations
1. Understand the definitions of inscribed angle and intercepted arc.
2. Apply properties of inscribed angles of a circle
Homework
In class activity
Quiz
Exam
Students will be asked to participate
A couple of periods will be designed to during lecture time. work on practice and
Homework, quiz and exam will help students practice. homework. For highachieving students, I may ask them do proof.
For low-achieving students, extra basic information and time will be given to help them to catch up
**Writing Prompt
In the homework (see attachment), Students will be asked to write their comments
(several sentences or one paragraph) on learning inscribed angles.

I. In each figure, O is the center of the circle (page 464 Q38 & 39)
II. Algebra (Page 365 Q55 & 56)
Find
 rs r
2   rs
 r
Find
21
3 5 x y z
7
6 xy z
.

1.
Explain how an intercepted arc and inscribed angle are related.
2.
ABC is inscribed in a circle so that BC is a diameter. What type of triangle is ABC ?
Explain your reasoning.
3.
Compare and contrast inscribed angles and central angles of a circle. If they intercept the same arc, how are they related?
4.
Assess yourself write about and describe three uses of hexagonal nuts, bolts or screws that appear at school or home. Why do your think the hexagon shape is used?

Write down your comments, ideas, and difficulties regarding inscribed angles.