Constructing an Argument for the Circle Area Formula
Lesson 2 of 6
Objective: SWBAT construct an informal limits argument to explain the circle area formula.
We have just finished a lesson that is very similar to this one. In that lesson, our goal was to show that the circumference of a circle is twice the product of pi and the radius. In this lesson, we'll be showing that the area of a circle is the product of pi and the radius squared. Because the previous lesson gave my students experience with the type of reasoning required in this lesson, I allow students to do more on their own during this lesson.
To get started, I give each student a copy of Argument for Circle Area Formula. I then give the students 3-5 minutes to read the first two paragraphs and pencil in answers for the first six blanks.
Next I have the students discuss their answers with their A-B partners.
Finally, I reveal the answers under the document camera so that students have access to the correct answers before we move on.
Having the correct answers in the blank is not evidence that students have a solid conceptual foundation so I want to make sure that they are clear on the logic we'll be using to make our argument. Toward that end, I explain to students that they will have 5 minutes working with their A-B partners to craft and rehearse a paraphrasing of the logic we'll be using to show that the area of a circle is the product of pi and the radius squared. I explain that they should all be prepared to come to the front of the class and present their paraphrasing as I will be calling on students randomly.
When the rehearsal time is over, I call up a handful of students, one at a time randomly, to present. I'm looking for the following elements in the presentations:
- The area of a regular polygon inscribed in a circle approaches the area of the circle as the number of sides increases.
- The value that the area of the regular polygon approaches is the area of the circle.
- If we can show that the area of the regular polygon approaches pi times r ^2, then we'll, in effect, be showing that the area of a circle is pi times r^2.
My style is basically to carefully echo and provide specific praise when the presentation has included all three elements above, or two constructively point out the missing elements when the presentation has been incomplete. My goal is for later presentations to incorporate the feedback I've given to the earlier presenters.
When I'm ready to move on from the presentations, I have students quickly pencil in their recollection of the formula for the area of a regular polygon. Then I have them compare their formulas with their A-B partners. Finally, I show the formula under the document camera.
Next, I have students complete the bottom half of the first side of the handout, which asks them to study a diagram and describe how the apothem and perimeter of a regular polygon inscribed in a circle behave as the number of sides increases.
When students have had enough time to write their answers, I call on a few students randomly to share their answers with the class. I facilitate the dialogue looking for opportunities to have students critique the reasoning of others. I'm also on the lookout for opportunities to talk about precision. For example, a student might say that the apothem gets longer as n increases. While this is true, it isn't as precise as it could be and, more important, it's not useful. I would push that student to make a more precise statement about the behavior the apothem, particularly one that relates it to the circle since we mean to make a connection between the area of a regular polygon and that of a circle.
As usual, I will show my version of a high quality response under the document camera so that students have an exemplar.
Turning to the other side of the handout, I have students read the first two paragraphs. Then I clarify the directions explaining what I mean by the limits not being specific numerical values and also what I mean when I say to explain the meaning of each limit. I give the students 3 minutes or so to complete this part of the handout, which is essentially a restating of what they were asked at the bottom of the previous page.
As with the last portion of the handout, I'll have randomly selected students show and read their responses on the document camera and we'll discuss these responses as a class before I show my exemplars.
Writing the Final Argument
At this point, students have all of the pieces they need to make a coherent argument. They just need to make sense of it all and synthesize the argument. I will give them 20 minutes to do this. I ask that they spend the first ten minutes working independently. After 10 minutes, I open it up to collaboration so that students can make minor tweaks, if necessary, to their arguments.
Some students will ask if they can assume that C = 2(pi)r. I tell them that they have proven this in the previous lesson so, by all means, use it here if it helps.
At the end of the 20 minutes, I collect the papers and take them home to review, score, and provide feedback. I also select exemplars that I will share with students so that they can have models of high quality work that come from their peers.