Engaging students: Using Euler’s theorem for polyhedra

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 Megan Termini. Her topic, from Geometry: using Euler’s theorem for polyhedra.

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

It is important for us as teachers to create an activity or project that is fun and engaging for the students. One activity that I found using Euler’s Theorem for Polyhedra is a math riddle activity by Albert J. and B. Michael (Reference A). The activity requires students to work in groups and work as a team to create four different types of convex polyhedra using Toobeez. The teacher will first build two Toobeez models, one of a polyhedron and one of a polygon. Then the students will be divided into groups and will build the four different types of convex polyhedra. After building each model, they students will fill in a table for each model the number of faces, vertices and edges. Using previous knowledge, the students will look at what they have recorded in the table and name each figure. Then they will discuss as a group to try to determine the relationship between F, V, and E. Give them some time to all come to a conclusion and then discuss as a whole class their discoveries and compare to Euler’s formula. This is a great way of having the students work together in groups and having them discover Euler’s Formula on their own, instead of just giving it to them.

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

Using technology as an activity for your lesson is a good way of keeping your students engaged and wanting to learn about the topic. When learning about Euler’s Theorem for Ployhedra, I found a great website, called Annenberg Learner, for students to use to apply the equation and even find it. After the lesson, this would be a great way for students to apply what they have just learned in the lesson. The activity asks you to choose a 3D shape and it will show you the net of that shape. Then the student will fill in the table on how many faces, vertices and edges that shape has. It will have you try again if you get a number wrong on the table. After you fill in the table, you then will get asked if you see a pattern and any relationship between F, V, and E. Once they see the pattern, they then get to try to fill in the parts of the equation (Reference B). This is a fun and engaging way of students being able to see the table and discover Euler’s Formula on their own or in groups.

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

This is a very important topic for students to learn at the beginning. As they continue their mathematics or science education, they will see these 3D figures a lot and will have to find more than just the faces, sides, and vertices. This topic will help students when learning about Platonic Solids. There is a video that I found that would be a great way for them to not only remember Platonic Solids, but also Euler’s Formula! (https://www.youtube.com/watch?v=C36h00d7xGs) It is one of my favorite videos of all time and it will get stuck in your head after listening to it a few times. They also will be using Euler’s Theorem in future geometry classes and even classes in college. They will have to solve for volume and area of these 3D figures and more. So, this topic is the start to what is more to come. This is also an important topic when it comes to jobs like building and constructing buildings and bridges. They would need to know how many faces, vertices and edges the building needs.

References:

A. “Euler’s Formula | Leaning Math Through Fun Activities.” TOOBEEZ Activity Central, 2 June 2014, http://www.toobeez.com/activities/eulers-formula/.
B. “Euler’s Theorem.” Annenberg Learner, http://www.learner.org/interactives/geometry/eulers-theorem/.

Full lesson plan: Modular multiplication and encryption

Over the summer, I occasionally teach a small summer math class for my daughter and her friends around my dining room table. Mostly to preserve the memory for future years… and to provide a resource to my friends who wonder what their children are learning… I’ll write up the best of these lesson plans in full detail.

In this lesson, the students practiced their skills with multiplication and division to create modular multiplication tables. Though this is a concept ordinarily first encountered in an undergraduate class in number theory or abstract algebra, there’s absolutely no reason why elementary students who’ve mastered multiplication can’t do this exercise. This exercise strengthens the notion of dividing with a remainder and leads to a fun application with encrypting and decrypting secret messages. Indeed, this activity made be viewed as a child-appropriate version of the RSA encryption algorithm that’s used every time we use our credit cards. This was mentioned in two past posts: https://meangreenmath.com/2013/10/17/engaging-students-finding-prime-factorizations and https://meangreenmath.com/2013/07/11/cryptography-as-a-teaching-tool

This lesson plan is written in a 5E format — engage, explore, explain, elaborate, evaluate — which promotes inquiry-based learning and fosters student engagement.

Lesson Plan: Kid RSA Lesson

Full lesson plan: Platonic solids

This was the first lesson that I taught to this audience: constructing the five regular polyhedra and inductively deriving Euler’s formula. This lesson plan is written in a 5E format — engage, explore, explain, elaborate, evaluate — which promotes inquiry-based learning and fosters student engagement.

Platonic Solids Lesson