# Engaging students: Finding the volume and surface area of prisms and cylinders

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 Angelica Albarracin. 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?

For finding the surface area of prisms and cylinders, I as the teacher would create an activity centered around using the nets of these figures to better visualize this concept. In my experience, many students do not struggle with the computational aspect of finding the surface area of prisms and cylinders, but rather, they tend to forget to calculate the area of all the faces of such figures. When a student views these three-dimensional figures on paper, it can be easy to forget some faces as not all of them can be illustrated, requiring the student to have an accurate depiction of the figure already in mind. By having students work with nets, they will have some guidance in calculating the surface area of prisms and cylinders. Additionally, having the students construct each intended figure with the net can also help students develop a better understanding of the composition of prisms and cylinders.

A project I could use as a teacher in order to help students understand volume of prisms and cylinders would be to have the students create their own drink company. I could provide the students with several models of different styles of cans they could use and have them find the volume of their selected can as a requirement. I think this would be a fun way to not only allow to students some creative freedom but also provide practice calculating the volumes of various prisms and cylinders. Students would have to consider aspects such as how much liquid one container holds over another, how portable the shape is, and how will others drink from it. Students could also find the surface area of their drink cans in order to see how much material would be needed to print a label that would fit around each can.

How can this topic be used in your students’ future courses in mathematics or science?

Finding the volume and surface area of prisms and cylinders provides a basic background for students to start exploring more complex shapes such as spheres, cones, and pyramids. However, in Calculus I, this topic is taken further with the introduction of integrals and the concept of finding the area under irregular curves. Later down the line, students will also learn about volumes of solids of revolution. For rounded curves, an approximation for such solids is comprised of taking the sum of the volume of many cylinders; the more cylinders there are, the closer the approximation will be to the true volume. An image of this is shown below:

Continuing with the theme of solids of revolutions, Calculus II is when students must find the surface area of these solids. To approximate the surface area, we take the surface area of frustums that can be formed under the curve. Frustums are similar to cones as they both have circular bases, but instead of coming to a point, a frustum also has a circular top. As before, the greater the amount of frustums used in the approximation, the closer the calculated value is to the true surface area. The formula for the surface area of a frustum is $A = 2\pi r h$ A = where $r =(r_1+r_2)/2$. Frustums are unique in that both circular bases are different. In the case that the bases are the same, the formula for $r$ becomes $r =(2r_1)/2 = r_1$,  in which case the formula for surface area becomes $A = 2\pi r h$  which is exactly the formula for the surface area of a cylinder. Below is an image of the surface area approximation of a solid formed by revolution:

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.)

The ancient Greeks are responsible for naming many of the figures and solids we commonly see in Geometry. For example, the word “prism” comes from the Greek word meaning “to saw”, which comes from the fact the cross sections (or cuts) of a prism are congruent. The word “cylinder” also comes from Greek, specifically from the word that means “to roll”. In addition, the Greeks were also “the first to systematically investigate the areas and volumes of plan figures and solids”. One of the most famous of these Greeks is the mathematician Archimedes who is directly responsible for the approximation of the area of a circle, the approximation of pi, the formulas for the volume and surface area of a sphere, and a technique called the “method of exhaustion”, which was used to find areas and volumes of figures in a manner similar to that of modern calculus. Archimedes viewed his discovery of the formula for the surface area of a sphere as his greatest mathematical achievement and even instructed that it be remembered on his gravestone as a sphere within a cylinder.

Another mathematician who developed techniques that bore similarities to modern calculus was Italian mathematician Bonaventura Francesco Cavalieri. While his discoveries pertained to finding the volume of objects, he was able to use are of cross sections to show that “two objects have the same volume if the areas of their corresponding cross-sections are equal in all cases”. This came to be known as Cavalieri’s Principle, but it is important to note that Chinese mathematician Zu Gengzhi had previously discovered this principle hundreds of years before Cavalieri. The next biggest advancement in this topic is attributed to integrals and making sense of the idea of finding the area under a curve. An approximate method for finding the area of a figure with an irregular boundary was developed known as Simpson’s Rule which had previously been known by Cavalieri but was rediscovered in the 1600s.

References:

https://amsi.org.au/teacher_modules/area_volume_surface_area.html

https://tutorial.math.lamar.edu/classes/calci/Area_Volume_Formulas.aspx

https://tutorial.math.lamar.edu/classes/calcii/surfacearea.aspx

https://en.wikipedia.org/wiki/Surface_area

# Engaging students: Defining the terms prism, cylinder, cone, pyramid, and sphere

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 Alejandro Rivas. His topic, from Geometry: defining the terms prism, cylinder, cone, pyramid, and sphere.

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

I as a teacher can create a research activity or project with prisms, cylinders, cones, pyramids, and spheres. The activity would entail having the students do some research over a particular building or structure of their choice. Once the students have decided on which building or structure I will ask them to identify all of the prisms, cylinders, cones, pyramids, and spheres the building or structure contain. The students will have to count the quantity of each, figure out a way that all of the 3-dimensional figures hold the building or structure together, have a picture, and  to present to the class. After the students have presented their projects, I will then explain how prisms, cylinders, cones, pyramids, and spheres are involved in our everyday lives. I will tie it in and explain that certain professions use these 3-dimensional figures such as Engineering, Architecture, Art, Graphic Design, etc.

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

This topic extends what my students should have learned in previous courses by them being able to identify simple shapes that form prisms, cylinders, cones, pyramids, and spheres. For examples, the most common and referred prism is the rectangular prism. The prior knowledge of the shapes the students need to have are rectangles and squares. To expand my student’s knowledge from previous courses I will have them build prisms, cylinders, cones, pyramids, and spheres out construction paper. Before they cut out and form the 3-dimensional figures the students will have to identify each shape. I will split the students up into different groups. Once the groups have been formed I will let the students choose between a prism, cylinder, cone, pyramid, and sphere. Once they choose the 3-dimensional figure they will create a poster that must contain the shapes that are being used in order to form the 3-dimensional shape, and the steps the students took to get the end result.

How can technology be used to effectively engage students with the topic?

A way that technology can be used to effectively engage students with defining the terms prism, cylinder, cone, pyramid, and sphere is by playing a game of Kahoot! I would begin the class with giving the students the definitions of the different 3-dimensional figures. Once they know the definitions I will break the students off into groups of 2 or 3 depending on the class size and have them come up with a team name. The Kahoot! will have different questions pertaining to the definition of prism, cylinder, cone, pyramid, and sphere. This should be able help me, the instructor, gauge how much the students know about prisms, cylinders, cones, pyramids, and spheres. This will also give me an opportunity to help the students understand major differences between the 3-dimensional figures. This will allow me to go into detail about the bases of certain 3-dimensional figures and how that ties into the reasoning behind their specific name.

# Engaging students: Finding the volume and surface area of prisms and cylinders

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.

https://en.wikipedia.org/wiki/Surface_area

http://www.mathpages.com/home/kmath343/kmath343.htm

http://www.muslimheritage.com/article/volume-sphere-arabic-mathematics-historical-and-analytical-survey

https://en.wikipedia.org/wiki/Surface_area

# Mathematical Present Wrapping (Part 2)

Yesterday’s post of mathematically wrapping presents was tongue in cheek. This one is elegant and a nice application of principles from geometry.