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 Madison duPont. Her topic, from Geometry: finding the volume and surface area of prisms and cylinders.
How could you as a teacher create an activity or project that involves your topic?
A couple activities from my lesson plan attached below were activities I found to be helpful and interesting for my students when introducing surface area and deriving the formulas themselves from area and circumference formulas they already knew. The first activity I’d like to highlight, though it’s simple, was successful in introducing the concept of surface area to students. In our engage, students were shown pictures of cat posts, cylinders, prism-shaped presents and so on and asked how they could determine the amount of materials needed to cover the surface. They seemed familiar with the concept, but not necessarily the mathematical term or procedure for doing such. After getting their pre-conception-based suggestions and asking them the difference between that and the space the shape takes up (volume), my partner and I were able to see light bulbs go off in their minds and we were able to provide them the answer by introducing the concept of the lesson, surface area. The remaining lesson was an activity where they found the areas of the shapes connected in a cylinder’s net in order to find the total area. After the explore, we had them build the cylinder and then try to determine the area using other formulas. During class discussion, we had students present answers and solidify the reason behind the concept of the formula they found emphasizing the use of circumference being multiplied by length (like length x width of a rectangle but the circumference is the “width”) and that we needed to multiply the area of the circle by two because there were two bases on top and bottom. The student-lead activity of the lesson can be extended to deriving the formula of a surface area of a prism using a prism net, constructing the 3D shape, and then determining the areas of each with different strategies. Once surface area is completed with the two shapes volume exploration could be performed in a similar matter and after all is said and done, the differences between volume and surface area could be compared and contrasted using a chart or Venn-diagram. The activities used and extended from this lesson plan seemed beneficial and better than simply giving the student formulas to memorize and explain because the students physically create the surfaces and see the transition for 2D to 3D and respective use formulas they know to conceptually understand a method of finding the surface area or volume in addition to seeing the formulas. This will help students remember formulas and extend surface area and volume of prisms and cylinders to future topics.
How can this topic be used in your students’ future courses in mathematics or science?
Concepts of volume and surface area of cylinders and prisms will be used in several different courses and topics. The first example is in more advanced math topics such as Pre-calculus and Calculus when they are solving word problems such as determining optimized surface areas for companies to use production materials or the volume of water in a cylindrical tank as water is increasing or decreasing within. Another advanced math course that will utilize the concepts of surface area and volume are the higher calculus courses during which you are expected to find volume (integral of 3D figure) and surface area (using double integrals and partial derivatives) of shapes and also when using cylindrical shell, washer, and disk methods to solve integral problems. The formulas for these methods are largely based off of the concept of surface area and volume. In addition to mathematics, surface area will be discussed in sciences in a more conceptual way. In chemistry, surface area is relevant to chemical kinetics as the rate of a reaction is directly related to the surface area of a substance. In other words, as you increase the substance’s surface area, the rate of the reaction is also increased. Additionally, biology uses surface area concepts when considering the size of an organism and how its surface area affects its body temperature or digestion compared to an organism with a different surface area and volume. Lastly, biology relates to these concepts when learning about the surface area to volume ratio of a cell. This ratio bounds the viable size of a cell as the cell’s volume increases faster than the surface area (Surface Area, Wikipedia, 2016). With the knowledge of what is to be built off of these concepts, understanding surface area and volume of 3D shapes such as cylinders and prisms beyond memorized formulas becomes evidently imperative.
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.)
Archimedes, a Greek mathematician, considered his work with cylinders and spheres to be his “most beautiful achievements” as he was able to discover the volume and surface area of these shapes and even wanted his monument to involve a sphere and cylinder (MathPages). He did so by first exploring the area of a circle, which he did by bounding the upper and lower bounds of the circle according to circumference and radius and inscribed/circumscribed n-sided polygons. He then progressed to exploration of the sphere and derived surface area and then the surface area of a cylinder. After, he considered the volume of each shape using what he discovered from surface area with inscribed/circumscribed shapes. According to Mustafa Mawaldi, Archimedes published findings in a book called The Sphere and Cylinder. The more recent history of surface area occurred at the turn of the twentieth century when Henri Lebesgue and Herman Minkowski used the concepts of surface area to develop the geometric measure theory. This theory studies surface area of any dimensions that make up an irregular object (Surface Area, Wikipedia, 2016). Though this is not a comprehensive timeline of the development of surface area and volume, these facts demonstrate that surface area and volume was relevant even in Ancient Greek times and still allows for exploration today, making the topic more relevant and interesting.