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 Zacquiri Rutledge. His topic, from Geometry: radius, diameter, and circumference of a circle.

There are many ideas about how to introduce students and have them study the relationships between the radius, diameter and circumference of a circle. However, one of my favorites has always been the month long project assigned to students at the beginning of class. On the very first day of class, the teacher is to assign the students their project. The instructions of this project are for each of the students to find and measure ten different round or circular objects around their home. The students will need to measure the length around the object (the circumference) using a piece of string and a ruler (the teacher might explain to the students or give an example so they know how to do this), the length from one side of the object to the other side passing through the middle (diameter), and the length from the center of the object to the outside (radius). If the students already know what these terms are called that is okay. However, the teacher should avoid explaining these terms until later.

Then a month later, the students are to bring their findings to class. At this point during the class the teacher will have begun her segment of lessons about circles and the various properties of circles. By now the students should have a good idea what the terms radius, diameter, and circumference mean. So the day the students bring in their work, they will be given the following chart, originally designed by the University of Illinois. From here students will slowly begin to fill in their charts with the information they gathered. Once completed students will then begin finding the ratios between diameter-radius and circumference-diameter and recording them. Finally at the bottom, students will find the average of their ratios from the last two columns. Once all of this data is completed, the students should have found that the diameter and radius share a ratio of 2-1 since the diameter is twice the radius. The last column should have produced something close to an average of 3.14159265359 or better known as pi (). Not only will this help students understand that pi is not just a number, but it will also help them to know where it comes from and its importance. From here the teacher would be able to lead into a lesson about some of the other uses of pi and how they all relate back to the relationships between radius, diameter and circumference.

Radius, diameter, and circumference are very important in many topics beyond their definitions. For instance, later on in the geometry course students will talk about the area of circles. Even though the students might have learned how to find the area of simple polygons such as triangles and quadrilaterals, finding the area of a circle is different because of the use of pi. To find the area of a circle, students have to recall the relationships between the radius, diameter, and circumference in order to understand how the area of a circle uses those relationships. Another example of how they are used is in pre-calculus. In pre-calculus students will talk about the unit circle, a circle with a fixed radius of 1 unit. Using the fixed radius of 1, students will discover that the length around (circumference) the unit circle is 2π. This 2π is important because it can be broken into pieces, called radians, and used to help measure Sine, Cosine, and Tangent at certain radians around the circle. Learning about sine, cosine, and tangent opens up even more things for the students, such as trigonometry and calculus. However, no matter how advanced the mathematics become, they always relate back to the simple concepts of the radius, diameter and circumference of a circle and their relationships.

Radius, diameter, and circumference is a topic that has been talked about and used dating back to 2000 B.C. But, what has it actually been used for all this time? How about architecture? Think about massive constructs such as the Theatre of Ephesus in Rome, Italy. Even though the theatre is not a full circle, look at how each of the seats are evenly placed from the stage. This is because when it was designed, the architect likely used the radius and circumference to accurately plot how far each seat needed to be placed in order to be the same exact distance from the stage as everyone else in their row. Even though only half a circle was used for this theatre, the circumference and radius would have been used to find the ratio pi in order to get the area of how much space was allowed for seating.

Another great example of circumference being used is in the invention of the clock. The clock originated as a sun dial, which would use the sun to cast a shadow, which would tell the time of day. These sun dials date back as early as 3500 B.C. However, in 1583 Galileo found a way to use a pendulum to create a clock that always followed the same length of time (Clock). This is important because not long after the first clock was born, so was the circular face of a clock. The face of a clock has the numbers 1-12 on it, each one evenly spaced around the edge of the clock. By using the circumference of any size of circle, the person building the clock would know just how far to space out each of the numbers, giving each hour the same amount of time between them. If even one of the numbers were off on the clock, the time would be off. Also, it can be seen that on modern clocks, the minute hand always stretches the radius of the clock. By stretching out the minute hand on the clock, the designer of the clock can create evenly spaced notches on the face using the circumference, in order to have the minute hand indicate the minute of the hour.

One final example is the use of radius in war, or more specifically the invention of the radar. Radar was originally being experimented with by German physicist Heinrich Hertz in 1887. He had discovered that certain materials allowed radio waves to pass through them, while others reflected them. In 1890, Nikola Tesla realized that large objects could reflect large enough radio waves to be detected. By harnessing this idea, pulse radar would come to be introduced into United States in 1925, and later used in the British Air Force to defend against German air raids during WWII (Science). The reason radar works, however, is because the system has a set radius in which it can detect radio waves. Once the radar system sends out a radio wave, if it does not reflect back within the radius of the detection system, then the radar will not pick up on anything. The system measures the distance by measuring how long it takes for the radio wave to return to the system after it is sent out and comparing that time to radius of detection. This allowed not only military to defend against air attacks, but it was commonly used during naval combat to defend against submarines as Germany used their U-Boats to attack several American and British naval ships during WWII, as well as WWI before the invention of radar.

References:

“Circumference and Pi.” Circumference and Pi. N.p., n.d. Web. 08 Oct. 2015.

“Clock a History – Timekeepers.” Clock a History – Timekeepers. N.p., n.d. Web. 08 Oct. 2015.

“Science Explorations: Journey Into Space: Radar and Sonar | Scholastic.com.” Science Explorations: Journey Into Space: Radar and Sonar | Scholastic.com. N.p., n.d. Web. 08 Oct. 2015.