Engaging students: Radius, diameter, and circumference of circles.

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 again comes from my former student Nataly Arias. Her topic, from Geometry: the radius, diameter, and circumference of circles.

D3. How did people’s conception of this topic change over time?

In order to calculate the circumference of a circle we must multiply the diameter by . The diameter of a circle is the length of the line through the center and touching two points on its edge. In simpler terms the diameter is two times the radius. To get the circumference of a circle we have to work with its radius or diameter and . So the more important question is, what is and how does it relate to circles? Pi or π is a mathematical constant which represents the ratio of any circle’s circumference to its diameter in Euclidean geometry. It is the same as the ratio of a circle’s area to the square of its radius. This can be seen as far back as 250 BCE in the times of Archimedes. Archimedes wrote several mathematical works including the measurement of a circle. Measurement of the circle is a fragment of a longer work in which is shown to lie between the limits of $3 \frac{10}{71}$ and $3 \frac{1}{7}$ . His approach to determining consisted on inscribing and circumscribing regular polygons with a large number of sides. His approach was followed by everyone until the development of infinite series expansions in India during the 15^th century and the 17^th century in Europe. The circumference of circles was found in the works of Archimedes and is now reflected in our math textbooks. This topic has been seen for many centuries and is still seen today. It has become an important part of math and has become an important part of the mathematics curriculum in schools.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers, Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

If I was teaching a middle school class on radius, diameter, and circumference of circles I would incorporate technology in my lesson. I have seen firsthand how effective technology can be when teaching your students. I showed the class a video clip during the lesson. As soon as I pulled down the screen projector they sat up, their eyes lit up and they were excited. This is why I have chosen this video clip from YouTube for this particular topic. I think it’s important to change it up and not always stick to a particular teaching style. Some students learn more visually and watching a video instead of listening to a lecture might be more entertaining for students. I know that teachers can’t rely on only technology to teach their students but using things like YouTube can certainly help and be beneficial. I chose this clip because I liked that it used and went over several examples and related circles to things students see every day like a pizza, tire, and table. I also like that it went over definitions in a clear and easy to understand way for students. It explained what a radius and diameter is and how to find it. This helpful video discusses the calculation of the circumference and its area. It also explains the relationship between and the circumference. This 8 minute clip could be used as part of your explain section of your lesson or could even be used to help students review the topic before a test or quiz.

D4. What are the contributions of various cultures to this topic?

When dealing with the radius, diameter, and the circumference of circles there is no escaping pi. Pi represents the ratio of any circle’s circumference to its diameter and is one of the most important mathematical constants. It’s used in many formulas from mathematics, engineering, and science. In math we use to solve for the circumference of a circle with formula $C=2\pi r$ . Sometime in early history someone discovered the relationship between the size of the circumference and the diameter of all circles was a constant ratio. This was seen and presented in the earliest recorded mathematical documents of Babylon and Egypt over 2000 years ago. At this time they did not use the symbol that we use today it wasn’t till much later. They had established that the ratio was equal to $\frac{C}{D}$ , where C is the circumference and D is the diameter of any given circle. At this stage, the Egyptian and Babylonian mathematicians came up with numerical approximations to $\frac{C}{D}$ which is the number we now call pi. Their methods are still unclear and unknown today. In their time period there was no modern number system. They didn’t even have pencil and paper. It has been predicted that they used a rope and sticks to draw circles in the sand and that they also used the rope to measure how many diameters made up a circumference of a circle.