Download Relationships of Arcs and Angles Lessonplan by Dave

David Johnson
MTH5500 Summer09
Assignment: Lesson Plan
Relationships of angles and arcs with circles
NCTM Objectives:
Instructional programs from prekindergarten through grade 12 should enable all students to-
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analyze characteristics and properties of two- and three-dimensional geometric shapes
and develop mathematical arguments about geometric relationships;
specify locations and describe spatial relationships using coordinate geometry and other
representational systems
use visualization, spatial reasoning, and geometric modeling to solve problems.
Illinois Standards:
Goal 9/Geometry
A. Demonstrate and apply geometric concepts involving points, lines, planes and space.
B. Identify, describe, classify and compare relationships using points, lines, planes and
solids.
C. Construct convincing arguments and proofs to solve problems.
Lesson Objectives:
 Use angles formed by tangents and chords to solve problems in geometry
 Use angles formed by lines that intersect a circle to solve problems
Materials Needed:
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Geometry textbook (McDougal Littell series textbook)
Computers loaded with Geometry Sketchpad
Computer for teacher with projector
Instructional Worksheets: Worksheet1-inside angles, Worksheet2-outside angles
Class Breakdown:
**Note this lesson plan is based on a 90minute class period.
 Work on daily warm-up problems
 Go over daily homework problems and questions from previous lessons
 Introduce lesson goals by first talking about previous discussions/lessons of how central
angles and inscribed angles are formed in a circle.
 The teacher will have students pair up. Each group will be given a handout either
worksheet1 or worksheet2. It should hopefully work out to where half of the groups
have worksheet1 and half have worksheet2. Students will work together using Geometry
Sketchpad and the worksheet assigned to their group to complete the indicated
construction. They will have to try to identify a pattern from the angle measures and arcs
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measures found as indicated on their worksheet and formulate an equation from their
identified pattern.
After about twenty minutes or so, students will take and pair up with another group (at
this point students will be in groups of four) working on the same worksheet and compare
their results.
After about ten minutes or so of confirming answers, students will be paired with another
student so that in each pair there is someone who is from a group that worked on
worksheet1 and one that worked on worksheet2. Students will then present their
particular constructions and results to one another.
Once students have presented their results to their partner, we will then gather back as a
class and discuss their conclusions and equations to make a final conclusion for the
lesson.
Assignment out of book pages624-625, 627: 8-24even, 26-35all, 40, 42, 43.
Relationships of angles and arcs with circles
(Worksheet1)
Constructing Chords and Measuring the Angles their Intercepted Arcs
There are other relationships formed with angles and arcs within circles besides central angles
and inscribed angles. In this activity, you’ll investigate relationships among angles inside a
circle and the arcs they intercept, which are formed by two chords.
1. Construct a circle. Select and label the center point as point C and the point on the circle as
point R. Thus segment CR is the radius of circle C.
Angles inside a circle
Step 1 Construct a circle with
center C
C
2. Construct a chord with the endpoints A and B.
Angles inside a circle
Step 2 Construct a chord with
endpoints A and B
A
C
B
3. Construct a point on segment AB. Label the point D. Place another point on the circle, call
it E. Construct line DE.
Angles inside a circle
Step 3 Construct a point on
segment AB. Label D .
Place another point on
circle. Lable E.
Construct line D E
A
E
C
D
B
4. Label the point where line DE intersects the circle, call it F. Construct segment EF. Hide
line DE.
Angles inside a circle
Step 4 Label t he point where the circle and line D E
intersect F.
Construct segment EF.
Hide line DE.
A
E
D
C
B
F
5. Construct arcs AE and FB by highlighting C then E and A. Follow the same directions for
arc FB.
Angles inside a circle
Step 5 Construct arcs EA and FB by higlighting C
then E then A. Follow the same directions f or F
and B.
A
E
D
C
B
F
6. Find the measures of arcs EA and FB. Then find the measure of all the angles in the circle.
Angles inside a circle
Step 6 Find the measures of arcs EA and FB.
Find the measure of angle ADE.
A
m FB = 33.88
m EA = 40.78
E
mADE = 37.33
D
C
B
F
** Look at worksheet f or f urther instructions.
Exploration: Now take a moment to explore and develop some ideas about your observations of
your construction by completing the following questions.
1. Observe the relationships between ADE and EA and FB as you drag point B around the circle.
Record the measurements in the table as you drag point B to new positions along the circle.
ADE
EA
FB
2. Write a conjecture and develop an equation about the measurements you recorded in the table
from problem 1.
3.
Verify you conjecture, using GSP calculate function.
Relationships of angles and arcs with circles
(Worksheet2)
Constructing Tangents and Secants and Measuring Angles outside the circle and their
Intercepted Arcs
Relationships between angles and arcs can also be formed outside circles. In this activity, you’ll
investigate the relationships among angles outside circles and the arcs they intercept, which can
be formed by the intersection of two secants, two tangents, or a combination of the two.
1. Construct a circle. Select and label the center point as point C and the point on the circle as
point R. Thus segment CR is the radius of circle C.
Angles outside the circle
Step 1: C onstruct a circle
with center C
C
2. Construct two secants(rays) AB and AD that intersect at A. B and D are the points on the
circle. Label the intersection of AB and circle C as point E. Label the intersection point AD
and circle C as point F.
A
Angles outside the circle
Step 2: Construct
secants(ray s) AB and
AD that intersect at A. B
and D are points on the
circle.
Label the intersection
of AB and circle C , E.
Label the intersection
point of AD and circle
C, F
E
F
B
C
D
3. Construct arcs BD and EF by highlighting C then B and D. Follow the same directions for
arc EF.
A
Angles outside the circle
Step 3: Construct arcs BD
and EF by highlighting C
then B and D. Follow the
same directions f or arc EF.
E
F
B
C
D
4. Measure arcs BD and EF. Also measure angle BAD.
Angles outside t he circle
A
Step 4: Measure arcs
FE and BC. Also
measure angle BAD
E
m FE = 29.51
m BD = 77.00
mBAD = 23.75
F
B
C
** Look at worksheet f or f urther instructions.
D
Exploration: Now take a moment to explore and develop some ideas about your observations
of your construction by completing the following questions.
1. Observe the relationships between BAD and BD and FE as you drag point B around the
circle. Record the measurements in the table as you drag point B to new positions along the
circle.
BAD
BD
EF
2. Write a conjecture about the measurements you recorded in the table from problem 1.
3. Verify you conjecture, using GSP calculate function