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 Avery Fortenberry. His topic, from Algebra: finding the volume and surface area of spheres.

How does this topic extend what your students should have learned in their previous courses?

The topic of volume and surface area of spheres is building upon the students’ knowledge of area and circumference of a circle. A sphere is similar to a circle in that a circle is a closed shape with all points equidistant from the centerpoint (the distance is the radius) and a sphere is a closed object with all points at an equal distance from its centerpoint (the distance is also r). Students will be familiar with the area of a circle formula, which is A=πr^{2} and will be able to easily use and understand the formula for volume of a sphere V=(4/3)πr^{3}. The same is true for circumference in relation with surface area.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Archimedes was the first mathematician to discover the most important ratio in all of mathematics, π. He did this by finding the area of a circle using shapes that were incrementally closer and closer to the same size as that circle. In other words, he would start with a circle and enclose it within a square, then a pentagon, then a hexagon, and so on until he came extremely close to the same shape. He used this same method to find the volume of a sphere by enclosing it within a cylinder of a known volume and cutting out piece by piece and measuring until he found the parabolic segment is 4/3 that of an inscribed triangle.

Source: http://www.storyofmathematics.com/hellenistic_archimedes.html

What are the contributions of various cultures to this topic?

This topic had many cultures contribute to the understanding of it. These contributions came from Greek, Chinese, and Arabic mathematicians. The Greek contribution came mainly from Archimedes, which I discussed in D1. *The Arithmetic Art in Nine Chapters* is a Chinese book written in the 1^{st} century that gave a formula that was close but not exact to finding the volume of a sphere. The author of the book calculated pi as being equal to 25/8 or even as just 3 at times. Ancient Arabic mathematicians submitted very similar ideas to the Chinese in terms of the volume of a sphere. While it is known the Chinese derived some ideas from the Greeks, it is still unclear today how the ideas were spread to the Arabic mathematicians.