Geometry : Unit #12 - Unit Circle Trigonmetry : Lesson #1

Deriving Formulas for Sector Area and Arc Length

Objective: SWBAT derive the formulas for the area of a sector and the length of an arc and explain why they make sense.
Standards: HSG-C.B.5
Subject(s): Math
60 minutes
1 Activating Prior Knowledge - 25 minutes

The essential understanding for this lesson is that proportional reasoning can be used to determine the lengths of arcs and the areas of sectors. To warm up to this idea (and at the same time work some Statistics into the mix) I start students off with some work on creating pie charts. Each student will receive APK_Deriving Sector Area Formula. I will have them work in groups of 2-4 to create pie charts using compass and protractor.

When students are done, I'll call students up to present their work. As they present, I'll be sure to emphasize our use of proportional reasoning.

2 Deriving the Formulas - 25 minutes

The formulas students will be deriving in this lesson are basically designed to calculate portions of circle area and circumferences. I start this section by giving students some specific cases where the portion of the circle is pretty obvious (e.g., a 90 degree arc represents 1/4 of a circle.). There are four such cases on the Deriving Formulas for Sector Areas and Arc Lengths resource. Each case is slightly less obvious than the ones preceding.

I have students complete the first four cases and then I call non-volunteers to come and present their solutions under the document camera. The main thing I emphasize is how students are determining the fraction of the circle that is represented by the arc or central angle measure.

After that, students will work on the general case. Then I'll call on non-volunteers again to make sure that we get the correct ideas out to all students. 

Finally, students will work independently on the Reflection section of the handout to make sure that each student has understood how we derived the formulas.


3 Complex Problem Task - 25 minutes

While the formula for finding sector areas is fairly simple, the problem students will be doing in this section will provide plenty of challenge. I've found that this is a very good problem to make sure students really understand and are able to apply the formula.

Students will be working on the Complex Sector Area Problem (Solution on page 2). The problem requires some thought and students will definitely need to practice MP1: Making Sense of Problems and Persevering in Solving Them.

See the following screencast for an explanation of the solution.



The Power of a Well-Represented Problem

I had assigned the complex problem task as a homework problem as this was one of the recommended assignments from the textbook authors. This is my first time using this particular text so I was not familiar with the problem. So when I sat down at home to do the problem, I felt as if I was having the same experience as the students have. I had to make sense of the problem, draw pictures, and get started with a plan. As I went through this process, I tried to be keenly aware of what misconceptions students might have. I was also very aware of my own thought process in solving the problem, so I tried my best to document my thinking so that I could relate it to students. When it was finally time for me to present the solution to the class, they really appreciated the careful way in which I had represented the problem and laid out its solution. I hope that the take-home message was that when we organize all of our information nicely, solutions that weren't apparent start to emerge.