Engaging students: Finding the circumference of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Chais Price. His topic, from Geometry: finding the circumference of a circle.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

We are familiar with the formula for the circumference of a circle as C=2πr. But where does this come from? What is unique about this formula? The formula for finding the area of a circle is similar. What do the formulas for finding the circumference and area or a circle have in common? It turns out that the number pi is included in both of these formulas and makes it unique. Why is it unique you aks? Because of what pi represents. Pi is an irrational number that is seen throughout various math classes. Since pi is irrational, our calculations for circumference and area of a circle are approximations. So how did this symbol end up in these formulas for a circle? It turns out that there has been a long history of pi beginning with the Ancient Babylonians. These ancient tablets are dated somewhere between 1900-1680 BC. The babylonians used a base 60 system and had no place value. Based on these tablets the circumference is 3 and the ratio is 45/60. By using this ratio = to the area we have 45/60=9/4 pi which places a value of 3 on the number pi.

Now lets shift forward about 1650 years to the time period of the famous mathematician Archimedes of Syracuse ( 287-212 BC). What Archimedes discovered was that there was a ratio relationship that existed amongst circles and circular objects. This ratio was uncovered through a tedious and time consuming process of inscribing and circumscribing regular polygons by hand. Then he found the area of the polygons and concluded that since the circle lies between the polygons, the area would be an approximation. He doubled the sided of his polygons all the way up to 96 sides. It turns out that this number known as pi is the ratio of the circumference of a circle and its diameter. This lesson is not on pi but it is important to understand the formula for the circumference and area of a circle and where it comes from.

This is actually an octagon and not a hexagon. But as you can imagine, the more sides you add, the more the regular polygon resembles a circle. I actually inscribed a 96 sided polygon inside a circle and you could not even tell from the naked eye that it had any straight edges. It looked just like a circle. So the idea is that the more sides you use in a calculation involving pi, the more accurate the solution will be. We can conclude with the notion that pi = the ratio of the circumference and the diameter of a circle.

http://illuminations.nctm.org/Activity.aspx?id=3548 This website visually illustrates how the more sides you have with a polygon, the closer you get to the actual figure of a circle.

How could you as a teacher create an activity or project that involves your topic?

For this lesson we are going to use several circular objects that the teacher brings in. These objects can include things such as tape, frisbee, can, etc. Along with these objects the students will be given string and have access to a ruler, markers, or anything else useful for this activity. I will also have a handout which includes a table to write in the objects measured as well as the measurements themselves. The students are to measure the objects circumference and diameter and then fill in the table. This lesson is designed to have the student discover pi as the ratio between the circumference and diameter of an object where pi= c/d. Then solving for c which is the circumference, we get the circumference formula: c=d*pi or c=2*pi*r.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This video acts as an engagement for students relating a pretty accurate approximation to the circumference of the Earth.