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This lesson provided by:
Author: Cathy Jones
System:Alabama Department of Education
School:Alabama Department of Education
Lesson Plan ID: 26412
Title:

Arc Lengths and Sector Areas

Overview/Annotation:

Students will work in cooperative groups to discover the relationships between arc length, central angle measure, and circumference.  They will also discover the relationship between circle area, central angle measures, and sector area.  Students will share their discoveries and create formulas for calculating arc length and sector area.  Then they will practice these skills in a journal entry and solving a real-life extension.

This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.

Content Standard(s):
MA2013(9-12) Geometry28. Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. [G-C5]
Local/National Standards:

Geometry 9-12

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

  • explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.
Primary Learning Objective(s):

Students will find the area of a sector given its arc length and radius.

Students will find the arc length of a sector given its area and radius.

Students will find the area and/or arc length of a circle given the measure of the central angle and radius.

Additional Learning Objective(s):

Students will review computing the area and circumference of circles.

Students will review constructing and measuring angles.

Students will use inductive reasoning to compare their measurements and calculations and to generalize their results.

 

Approximate Duration of the Lesson: 91 to 120 Minutes
Materials and Equipment:

Materials in AMSTI Kit for Geometry:   natural cotton string, protractors, rulers, safety compasses, calculators, measuring tapes 

Other Materials:  Approximately 8 cans of various sizes (example:  soup, vegetable, peanut, coke, potato chip)

Technology Resources Needed:

The websites below offer step by step directions and/or online calculators for computing the area of a sector or arc length of a circle.  TeacherTube tutorial links are also included.  Web links are not necessary for teaching the content, but can be helpful as you prepare to teach the lesson and can provide a mechanism for students to review lesson content.

Calculators (AMSTI kit), CTX projector (optional), document camera or overhead projector

http://www.analyzemath.com/Geometry_calculators/arc_length_area_sector.html

http://www.math-prof.com/Geom/Geom_Ch_32.asp

http://www.easycalculation.com/area/circle.php

http://www.easycalculation.com/area/learn-circle.php

http://www.teachertube.com/viewVideo.php?video_id=1423&title=Geo_Screencast__Sector_Area

http://www.youtube.com/watch?v=anV3HI-1vyY&feature=related

Background/Preparation:

  • Students should be able to calculate the area and circumference of a circle.
  • Students should be familiar with vocabulary related to a circle (arc, central angle, radius, diameter, chord, area, circumference).
  • Students should know how to find an angle’s measure.
Procedures/Activities:

Step 1 - Engage

Students should be in cooperative groups of four students per group.  Groups should be assigned to be heterogeneous, ideally consisting of one high achieving student, two average students, and one struggling student.  The teacher should make group assignments prior to class.  A good way to assign groups is to write the four names on a sticky note, place the note on the assigned groups of desks or table, and have students to find their group as they enter the room.  Naturally, the teacher should keep a list of group assignments in case students decide to change groups without permission.

Show the picture of the Ferris wheel and pose each question individually, allowing for students to respond.  Take special care to discuss the length of the cables between the seats.  While this may represent the distance between the seats, tell them that this would be an approximation of the arc length of the circle that is represented.  This circle would have a radius that is the distance from the center of the Ferris wheel to the place that the seat is attached to the ride.  Tell students that they will be investigating arc length as one of the day’s activities.

Similarly, spend time discussing the “slice” of the Ferris wheel that is represented by the area between two seats.  This would be similar to a slice of pizza.  While the students are not expected to give the area at this point, some may give ideas of how it might be calculated.  Listen to their ideas, and tell them that today’s investigation will also involve calculating sector area.

Step 2 - Explore

Distribute the materials listed in the Materials and Equipment section of this plan, the handout “Arc Length and Sector Area”, and one can to each group.  If you do not have a document camera, distribute transparencies and overhead markers to each group so that they can plan for their presentation to the class.

 

Discuss the instructions on the sheet.  Make sure you answer students’ initial questions about the activity, but do not provide specific guidance as to how the students are to complete the activity.  This is an exploration, so the students should devise their own means of solving the problem.  Tell the students that they will explain their answers/reasoning to the class once they have solved the problem.  Allow 20 minutes for students to complete this activity.

 

Note:  Each group should get a different can to use so that each group’s answers will be different.  When they share their information at the end of the activity, they will look for patterns that are consistent for all of the cans. 

 

As students complete the exploration, circulate among groups to listen to their discussions and, if needed, to answer questions that may arise.  It may be necessary for you to give hints on how to get started. 

 

Questions that you might need to ask students to prod their thinking:

 

Are you sure that the circumference is written in centimeters?  How can you tell?

How do you use the length of the diameter or the radius to find the circumference of a circle?

What measure other than the diameter or radius is used to compute the circumference of a circle?

Where is the center of the circle that you constructed?

How are you sure that you have divided the circle into four equal parts?

How is arc measure related to arc length?

Is the answer what you expected?

What surprises you about that answer?

How will you explain this to the class?

 

If students are to use the overhead projector for presentation, allow an extra 5-10 minutes for them to prepare their transparencies.

Step 3 - Explain

After 20 minutes (25-30 if using overhead projector), begin group presentations.  Allow approximately 5 minutes per group and encourage all members of the group to participate in the discussion.  Have a student use a blank Arc Length and Sector Area handout to document each group’s answers on one sheet. 

 

You may wish to plan the order of the presentations.  One way to do this is to keep notes as you circulate among the groups.  If a group has struggled with the assignment, you might wish for that group to go last.  In doing so, you give that group of students the opportunity to find their mistake(s) and make corrections prior to their presentation.  This allows the group to be confident in their answers.  Otherwise, groups may volunteer to go or the teacher may choose groups at random to present.

 

Questions that need to be answered as groups present:

How did your group arrive at the answer to #___ (any question)?

What mistakes did your group make before arriving at a correct answer?

Have you found an easier way to find the answer to #___ (any question)?

What information helped you find the answer to #___ (any question)?

Remember that this is the students’ time to explain their reasoning and to construct their understanding of the concept.  In very rare circumstances, the teacher will have to step in and explain.  Let the students explain and summarize.

Step 4 - Evaluate

After the presentations have been completed, assign the handout Arc Length and Sector Area Journal Activity to be completed as a journal entry.  If time is short, this can be a homework assignment.

 

Practice problems from the students’ textbook could also be assigned.

Step 5 - Extend

In cooperative groups, assign Consider this Ferris Wheel Assignment activity.  This assignment is a possible real-life extension of using arc length and sector area.

Circulate among groups as they work through this activity.  If necessary, ask questions to prompt student thought.

 


Attachments:**Some files will display in a new window. Others will prompt you to download. ConsidertheFerrisWheelExtend.rtf
ArcLengthandSectorAreaJournalActivity.rtf
ConsidertheFerrisWheel.rtf
ArcLengthandSectorArea.rtf
Assessment Strategies:

The student presentations and journal entries can be used as informal assessment to determine students’ understanding of the concept.  Based upon the misconceptions discovered during the informal assessment, the teacher can plan for additional instruction targeted to improving understanding of a particular aspect of the concept. Practice problems from the student texts can be assigned as practice. The activity in step 5 can be a culmination evaluation of this objective.

Extension:

Students who already know the content can skip the investigation and go directly to the activity in step 5.

Remediation:

Not needed.  The grouping strategy is conducive to encouraging participation and building background knowledge of the struggling student.

Each area below is a direct link to general teaching strategies/classroom accommodations for students with identified learning and/or behavior problems such as: reading or math performance below grade level; test or classroom assignments/quizzes at a failing level; failure to complete assignments independently; difficulty with short-term memory, abstract concepts, staying on task, or following directions; poor peer interaction or temper tantrums, and other learning or behavior problems.

Presentation of Material Environment
Time Demands Materials
Attention Using Groups and Peers
Assisting the Reluctant Starter Dealing with Inappropriate Behavior

Be sure to check the student's IEP for specific accommodations.
Variations Submitted by ALEX Users:
Alabama Virtual Library
Alabama Virtual Library

Hosted by Alabama Supercomputer Authority
The University of Alabama at Birmingham
The University of Alabama at Birmingham
The Malone Family Foundation
The Malone Family Foundation
Thinkfinity
Thinkfinity
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