Constructing an Argument for the Circumference Formula

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Objective

SWBAT construct a semi-formal argument to establish that the circumference of a circle is 2(pi)r

Big Idea

Circular Argument? In this lesson, students use structural analysis, trigonometry, and limits to show that the circumference of a circle is twice the product of pi and the radius.

Scaffolded Argument

30 minutes

To begin, I give each student a copy of Argument for Circumference Formula.

The first three paragraphs of the handout deal with (1) identifying the goal of the argument we'll be making and (2)devising a plan. I explain this to students. Then I have them read the first three paragraphs on their own and think about how they'll fill in the first four blanks only. Next, I have them discuss their ideas for filling in these first four blanks with their seat partners.

When they have had adequate time to discuss, I use the Completed Argument for Circumference Formula to show the correct answers for the first four blanks only. Next, I pause to really synthesize and lock in what has happened so far. Without this step, many of my students would be content just to have the blanks filled in and they wouldn't have the slightest clue as to what it all meant.

So on the board (or under the document camera), I write the logical basis we've established:

  1. We take the position of knowing, but not assuming, that the circumference is 2(pi)r... i.e., we don't take it as a given.
  2. We know that the perimeter of the regular polygon approaches the circumference of the circle in which it is inscribed. In other words, the value the perimeter approaches is the circumference.
  3. If we can show that the value the perimeter approaches is 2(pi)r, then we will have shown that the circumference is 2(pi)r.

It is key for students to understand this logic so I stop for a pair-share to give students the opportunity to summarize the logic without looking at their notes.

 

Next, I have students read the fourth paragraph on the handout, which explains why we'll be deriving an expression for the perimeter of the regular polygon in terms of the radius. After students have read it, I take a moment to clarify what has been said and emphasize this as a decision point. I want to make sure that they understand that the perimeter could be expressed in terms of s, apothem, or radius and that we are choosing to express it in terms of radius because of our goal to show that C = 2(pi)r.

Once we're clear on that, I give the students time to derive the expression for perimeter in terms of radius. They have performed an analogous task in a previous lesson so I feel comfortable letting them work independently on this. When they have finished, or even before, they can flip the paper over to see the desired expression.  I also show the process quickly once everyone has had time to finish, again using the Completed Argument for Circumference Formula.

 

After that I have students brainstorm with their seat partners around the questions posed in the first paragraph on side two of the handout. Once they have begun to think on their own, I have them proceed to filling in the next seven blanks on the handout. When they have had adequate time to do this, I reveal the correct answers, giving a thorough rationale for each one. Students will have to be able to make this and similar arguments in the near future so I want to make sure they have had the proper input before being asked to do so.

Finally, in preparation for the technology demonstration, I recap the logic outlined in 1-3 above, and make the connection between that logic and our trying to show that the expression nsin(180/n) approaches pi as n approaches infinity. The main idea here is to make sure my students understand what we should expect from the technology demonstration before I start it.

Once that's taken care of, I demonstrate that nsin(180/n) approaches pi as n approaches infinity. That demonstration is the focus of the following video:

 

Once I've given the demonstration, I want to give students a chance to make sense of what they've just seen and how it fits into the lesson. During this time, I have students write their final sketches and reflections at the bottom of page two of the handout. 

Transfer Task

20 minutes

While I do want my students to be able to reproduce the argument we made in this lesson, I also want to make sure that they have learned transferrable skills during the lesson. To give them the opportunity to demonstrate this, I give them the Circumference Formula Transfer Task

This task requires logic and structural analysis similar to what was required for this lesson. 

I give students 20 minutes to work on it. Before they get started, I emphasize the importance of the explanation of why the graph makes sense. This is how I will know if students are really analyzing structure and if they understand the concept of limit.

When the 20 minutes have elapsed, I collect the papers and take them home to score and provide feedback. While I'm scoring, I look for exemplars that I will show to the class when I pass the papers back.