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.