Geometry
Lesson Plan 1
Unit Title: Circles
Time Estimate: 1 period
Lesson Title: Arcs and Angles
Behavioral Objectives

The student will demonstrate application of arcs and angles by examining patterns in
the measures of central and inscribed angles and their corresponding arcs.

The student will demonstrate analysis of arcs and angles by inferring and applying the
relationships of two inscribed angles that cut off the same arc and an inscribed angle
that cuts off a semicircle.
Materials


Computer and Projector
Geometer’s Sketchpad Module
Lesson Procedures
Motivation:
 Project “Do Now” on SMART Board; have students copy and write answers in
notebook.
 Circulate the room and check student answers.
 Ask for a volunteer to write and explain their answer at the board.
Activities:
1. Seat students in partners and distribute “Angles and Arcs” worksheet. Introduce
central angles and instruct students to draw angles for each type of arc with the vertex
on the center.
2. Display “Angles and Arcs” Geometer’s Sketchpad file. Have students name the
central angle and write down the measures of the central angle and its arc on the
table. Click on “Animate Point C” for a new central angle and arc until they have
completed their table.
3. Click on “Animate Point D” and discuss why the angle measure does not change as
much as when Point C was animated.
4. Instruct students to complete questions 3, 4, and 5 with their partner and circulate the
room to make sure they are on track. Discuss results as a class.
5. Have students name the inscribed angle and write down the measures of the inscribed
angle and its arc on the table. Click on “Animate Point C” for a new inscribed angle
and arc until they have completed their table.
6. Instruct students to complete the remaining questions with their partner and circulate
the room to make sure they are on track.
7. Discuss remaining answers and ask, “What must be true about two inscribed angles
that have different vertices but intercept the same arc?”
8. Assign homework from the textbook.
Closure:
 Distribute “Angles and Arcs Exit Card” for students to complete individually and
collect.
Assessments:
 Class participation
 Completion of worksheet with partner
 Exit Card
 Homework
Do Now
A circle and an angle are drawn in the
same plane. The vertex of the angle is
on the circle.
1. Find and sketch all the possible
ways that the two figures can be
arranged.
2. For each arrangement, give the
number of intersection points.
Name:
Date:
Geometry: Circles – Angles and Arcs
Central angles are angles whose vertex is the center of the circle.
1) Draw and label central angles that cut off a minor arc, a major arc, and a semicircle.
2) Using the Geometer’s Sketchpad module, write down measures of central angles and
the corresponding intercepted arcs.
m<CAB
arc a1
3) Write a conjecture about Central Angles and their Intercepted Arcs:
4) What is the largest measure a central angle can have?
5) Find the measure of each arc if E is the center of the circle:
D
mAC
C
mAD
E
115 °
mBC
98 °
B
A
mDAB
Inscribed angles are angles whose vertex is on the circle and whose sides are segments in
the circle.
6) Draw and label inscribed angles that cut off a minor arc, a major arc, and a semicircle.
7) Using the Geometer’s Sketchpad module, write down measures of inscribed angles
and the corresponding intercepted arcs.
m<CDB
arc a1
8) Write a conjecture about Inscribed Angles and their Intercepted Arcs:
9) What is the measure of an inscribed angle that cuts off a semicircle? Why?
10) Find the measure of each variable:
y°
D
D
120 °
A
40 °
C
E
50 °
F
a°
E
x°
B
b°
G
Geometer’s Sketchpad Module
Name:
Date:
Geometry: Circles: Angles and Arcs Exit Card
1) Draw and label an inscribed angle. Provide one possible set of measurements of the
angle and its intercepted arc.
Find the value of each variable.
2)
3)
D
A
C
x°
F
D
46 °
E
25 °
w°
B
G
Geometry
Lesson Plan 2
Unit Title: Circles
Time Estimate: 2 periods
Lesson Title: Tangent and Secant Segments
Behavioral Objectives

The student will demonstrate evaluation of radii, diameters, and chords by justifying
the relationship and calculating lengths of segments in two chords.

The student will demonstrate evaluation of tangents and secants by justifying the
relationship and calculating lengths of two secant segments that intersect at a point
outside the circle.

The student will demonstrate evaluation of tangents and secants by justifying the
relationship and calculating lengths of a tangent segment and a secant segment that
intersect at a point outside the circle.
Materials



Computer and Projector
Texas Instruments Graphing Calculators
Cabri Jr. software program
Lesson Procedures
Day 1
Motivation:
 Project “Do Now” SAT Question on SMART Board; have students copy and circle
their answer.
 Circulate the room and check student answers.
 Ask for a volunteer to explain their answer.
Activities:
1. Distribute one graphing calculator and the “Tangent and Secant Segments Activity”
worksheet to each student.
2. Project “Cabri Jr. Instructions” on SMART Board for starting the application and
instruct students to follow them.
3. Have students complete the worksheet while walking around and assisting with the
calculator operations.
4. Ask for student conclusions that they discovered from activity.
Closure:
 Display PowerPoint; have students write down the three relationships.
Day 2
Motivation:
 Project “Do Now” Question on SMART Board; have students copy and circle their
answer.
 Circulate the room and check student answers.
 Ask for a volunteer to explain their answer.
Activities:
1. Arrange students in groups of three and distribute the “Tangent and Secant Segments
Practice” worksheet to each student.
2. Instruct students to use their notes from the previous day and apply the theorems in
order to solve the problems as a group.
3. Walk among the groups in order to assist in solving the problems. Check answers
with groups as they work.
4. Assign homework from textbook.
Closure:
 Distribute “Tangent and Secant Segments Exit Card”, have students complete
individually, and collect before end of period.
Assessments:
 Completion of Graphing Calculator Activity
 Class participation in conclusions of Activity
 Completion of Practice Worksheet in group
 Exit Card
 Homework
Do Now
The circle above has center P. Given
segments of the following lengths, which
is the length of the longest one that can
be placed entirely inside this circle?
A. 6.99
B. 7.00
C. 7.99
D. 8.10
E. 14.00
Name:
Date:
Geometry: Circles: Tangent and Secant Segments Activity
Using a graphing calculator and the Cabri Jr. program, follow the steps below to
complete the questions.
Cabri Jr. Instructions






Turn calculator ON
Press the APPS button
Scroll down to the CABRIJR Application
Press ENTER to select
Press ENTER again to load the program
If there are illustrations:
o Press the Y= button
o Scroll down to NEW
o Press ENTER
o If it asks you to save changes:
 Go to the right and select NO
Tangent and Secant Segment PowerPoint
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Do Now
Kim wants to determine the radius of a circular
pool without getting wet. She is located at point K,
which is 4 feet from the pool and 12 feet from the
point of tangency. What is the radius of the pool?
A. 16 ft
B. 32 ft
C. 20 ft
D. 4 10 ft
Name:
Date:
Geometry: Circles: Tangent and Secant Segments Practice
Using the three theorems from the previous activity, solve the following problems with
your group.
1. Chords AB and CD intersect at E. If AE  3, EB  4, CE  x, and ED  x  4,
what is the value of x?
2. A toy truck is located within a circular play area. Alex and Dominic are sitting on
opposite endpoints of a chord that contains the truck. Alex is 4 feet from the truck,
and Dominic is 3 feet from the truck. Maria and Tamara are sitting on opposite
endpoints of another chord containing the truck. Maria is 8 feet from the truck. How
many feet, to the nearest tenth of a foot, is Tamara from the truck? Draw a diagram
to support your answer.
3. Cabins B and G are located on the shore of a circular lake, and cabin L is located
near the lake. Point D is a dock on the lake shore and is collinear with cabins B and L.
The road between cabins G and L is 8 miles long and is tangent to the lake. The path
between cabin L and dock D is 4 miles long. What is the length, in miles, of BD ? ?
4. PA is tangent to circle O at A, PBC is a secant, PB  4, and BC  8. What is the
length of PA ?
5. PA is tangent to circle O at A, secant PBC is drawn, PB = 4, and BC = 12. Find PA.
6. AB is tangent to circle O at B. If AC = 16 and CD = 9, what is the length of AB ?
Name:
Date:
Geometry: Circles: Tangent and Secant Segments Exit Card
1. PC is tangent to circle O at C, PBA is a secant, PB  2, and PC  4. What is the
length of BA ?
2. Find the value of x.
Geometry
Lesson Plan 3
Unit Title: Circles
Time Estimate: 1 period
Lesson Title: Area and Sectors
Behavioral Objectives

The student will demonstrate synthesis of sector area by developing and applying a
process to calculate a portion of the area.
Materials

Computer Lab or Laptop Cart
Lesson Procedures
Previous to Lesson:
 Schedule Computer Lab time
 Post “Sector Area Websites” document on Sharepoint Portal page or Shared drive.
Motivation:
 Display “Computer Lab Activity Instructions” for students to read as they enter.
 Distribute “Sector Area Computer Lab” worksheet to each set of partners.
Activities:
1. Assist students with the Circle Grapher exercises. Check answers as they work.
2. Support each student in creating his or her individual chart using the website and
rubric in order to reflect on their final product.
3. Collect Computer Lab worksheet and graphs.
4. Summarize what they have learned using the “Summary” website. Instruct the
students to click on the website and follow along.
5. Instruct students to click on the “Practice Questions” website and work with their
partner to answer the questions. Inform them that they should write down their work
and be able to explain how they reached their answers.
Closure:
 Assign each of the practice exercises to a set of partners.
 Ask each pair to explain their answers to the entire class.
Assessments:
 Completion of Activity Worksheet with partner
 Completion of Circle Graph with all information labeled
 Participation with partner during computer time
 Explanations to practice exercises
Sector Area Websites
Circle Grapher
http://illuminations.nctm.org/ActivityDetail.aspx?id=60
Summary
http://regentsprep.org/Regents/math/geometry/GP16/CircleSectors.htm
Practice Questions
http://regentsprep.org/Regents/math/geometry/GP16/PracCircleSectorsSegments.htm
Computer Lab Activity Instructions
 Sit with a partner and log on to ONE computer
 Through the district website, log in to Sharepoint
Portal
 Find the Geometry page and open the Word
document titled “Sector Area Websites”
 Click on the first website for the Circle Grapher
and begin answering questions on worksheet.
 Take turns on the computer and reading / writing.
 Due at end of period:
o One Computer Lab worksheet per
partnership
o One Graph per student (individual)
Name:
Date:
Geometry: Circles: Sector Area Computer Lab
Using the “Circle Grapher” website, answer the following questions.
1. In the original circle graph, how many sections are there?
2. Write a fraction that represents the number of hours in a day spent in school. Write
this as a percent.
3. Calculate how many degrees (out of 360) are in the central angle for the section that
represents time spent at school.
4. If the radius of this circle is 4 inches, what would be the area of the entire circle?
5. Using your previous calculations, find the area of just the school section. (this is
called the sector area)
6. Choose one other category and calculate the area of it (if the radius is still 4 inches).
Create a Circle Graph




Individually, create your own circle graph by entering data in the box and clicking
“Update Chart”
Print out the graph, label each section with its category (because we cannot print in
color)
Find the area of each sector (radius is 4) – show all of your work on a separate piece
of paper
Follow the rubric to ensure that you get full credit
Circle Grapher Website
Sector Area Computer Lab – Circle Graph Rubric
Category
Possible Points
Circle Graph has AT
LEAST five sections
1
Circle Graph hours total to
24
1
Circle Graph is properly
labeled with categories
(to replace colors)
2
Calculations for Sector Area
are neatly done on a
separate piece of paper
2
Calculations for Sector Area
are correctly done
4
Points Received
Total Grade:
/ 10
Summary Website
Practice Questions Website
Geometry
Lesson Plan 4
Unit Title: Circles
Time Estimate: 1 period
Lesson Title: Geometric Probability
Behavioral Objectives

The student will demonstrate evaluation of geometric probability by creating a ratio
that compares areas of geometric figures.
Materials

Computer and Projector
Lesson Procedures
Motivation:
Display SAT Do Now Question and ask students to copy and answer. Walk around the
room to check student answers.
Activities:
1. Display Geometry Probability Notes. Discuss with students.
2. Distribute Shaded Regions Worksheet. Students complete in partners.
3. As a class, check answers to worksheet
4. Assign homework.
5. Extension: Segments of Circles
Closure:
Assign exit ticket and collect.
Assessments:
 Class participation
 Completion of Activity Worksheet
 Exit Ticket
 HSPA Open-Ended (Homework)
Do Now
The area of a circle 6 meters in diameter
exceeds the combined areas of a circle 4
meters in diameter and a circle 2 meters
in diameter by how many square meters?
A. 0
B. 3 π
C. 4 π
D. 5 π
Slushies in the cafeteria are made in 8-gallon containers.
Once the level gets below 3, the slushies have a tendency to
be watery.
What is the probability that when you go to the cafeteria for
your afternoon dessert, the slush machine will be below 3
gallons.
What is the probability that when you buy your slushy
today the container will have between 3 and 4 gallons
today?
Click here for illustration.
The probability that there will be between 3 and 4 gallons
of slushy in the container is 1/8. If you can divide the event
into equal parts, it is easier to express the probability.
What is the probability of randomly throwing a dart such
that it hits within the red area, given that the dart will
always land within the boundary of the outer circle?
The probability will be found by finding the area of the
region that is considered a success (the red area) divided by
the sample space which is the region contained by the outer
circle.
If the radius of the circle shaded red is 1 and the radius of
the sample space circle is 5, the Probability of landing in
the red region is
P(Red)= Area of Red/ Area of Outer Circle
In this example, the outer region is a rectangle and the
target area is a circle. To find the probability of randomly
hitting the target area with a dart,
P(blue) = area of circle/area of rectangle
If the radius of the circle is 1 cm, the length of the rectangle
is 5cm, and the width is 2.5 cm.
Find the probability of the blue shaded regions.
1.
2.
3. If you throw a dart randomly at the target shown, what is
the probability that you will hit the shaded area?
Name:
Date:
Geometry: Shaded Regions of Circles
Find the area of each shaded region.
1)
r = 8.7 cm
63 °
2)
d = 6 in
3)
36 m
25 m
4)
4
4
5)
15
12
6)
24 ft
Name:
Date:
Geometry: Shaded Regions of Circles Exit Card
Find the area of the shaded region.
4 cm
6 cm
Name:
Date:
Geometry: Segments of Circles and Concentric Circles
1)
2)
3)
4)
5)
6)
7)
8)