A Math Riddle: Euler‘s Formula

May 23, 2014 | by Albert J. and B. Michael

Primary Market

Education, Secondary Ed

Character Focus

Cooperation, Teamwork

Items Needed

1 Toobeez set per group, 1 package of index cards, 1 stopwatch, 1 notebook per student, 1 pen/pencil per student

The Activity Time

40 - 50 minutes

- Review the characteristics of a polyhedron
- Observe and record various qualities of different types of polyhedra
- Use deductive reasoning and observation to independently arrive at a mathematical formula
- Prove the validity of a universal mathematical formula governing polyhedra
- Work cooperatively in a deductive reasoning style with others to reach a conclusion

Character Focus

Teamwork/Cooperation

The Challenge

Through systematic observation and deductive reasoning, the group should be able to define an equation that is valid for any convex polyhedron (Euler’s formula).

Preparation

**Setup Time: **5 minutes

**Mini-Lesson Time: **10 minutes

**Materials**

- 1 Toobeez set per group
- 1 notebook per student
- pen or pencil

Activity Plan

**Time:** 40 – 50 minutes

**Instruction:** Whole class and small groups

**Space:** Medium

1. Teachers should build two Toobeez models, one of a polyhedron and one of a polygon.

1. Teachers should begin by presenting a Toobeez model of a polyhedron. Teacher Note: This should serve as a review. You can use Activity #7 “Polygons and Polyhedra” as an introduction to polyhedron theory

2. As a class, the students should be asked to recall the definition of a polyhedron and to highlight its unique properties. **Teacher Note:** It is a good idea to construct a Toobeez model of a polygon and ask the class to distinguish between the two types of figures.

3. After reviewing the definition of a polyhedron, lead the class in a review of the nomenclature of polyhedra. **Teacher Note:** Activity #7 provides an excellent introduction to this topic.

4. In addition to the number of faces, point out to students that a polyhedron can also be described by the nature of its angles.

- Convex polyhedron: All the angles are less than 180º
- Concave polyhedron: One or more of the angles is greater than 180º

**Teacher Note:** Point out that it is easy to visually distinguish the two types of polyhedra by examining the angles. If the legs of an angle appear to face outward from the figure’s center, the polyhedron is concave.

5. Inform the class that today’s lesson will focus on convex polyhedra.

6. On a blank page in their notebooks, instruct students to draw a table with four columns. The four headings for these columns are: Name of Polyhedron, Faces, Edges and Vertices.

**Helpful Hints**

- Be sure to review these tips prior to beginning the activity, and if necessary, share reminders with the group during the activity.

- Teachers should remind students to use this exercise as a review of polyhedron nomenclature
- Reinforce the important differences that distinguish polygons and polyhedra
- Remind students that the formula they are trying to deduce only governs convex polyhedra

*Here are available Training Options!*

**1. Divide the students into groups.
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**2. Read aloud the following Activity Challenge Box to the group.
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**Challenge**: Through systematic observation and deductive reasoning, the group should be able to define an equation that is valid for any convex polyhedron (Euler’s formula).

**3. **Using Toobeez, groups should be directed to build four different types of convex polyhedra. **Teacher Note:** Stress that the polyhedron must be convex and fulfill the definition of a polyhedron.

**4. After building each model, the group should observe and record the number of faces, edges and vertices on the table in their notebooks.
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**5. Using the number of faces and previous knowledge of polyhedron nomenclature, students should name the figure.
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**6. After the data are collected, each group should begin a discussion to determine the mathematical relationship between the three variables observed: faces, edges and vertices.
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**7. This discussion should continue until the students arrive at a pattern that governs each example observed. Teacher Note: Even if the pattern is quickly found by some, allow all groups adequate time to arrive at their own conclusions.
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**8. Once the pattern is observed by most groups, write Euler’s formula on the board: Faces + Vertices = Edges + 2.
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**9. After the activity, gather the class together and pose the following question: “What does Euler’s formula define as true for all convex polygons?”
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**10. Finally, move to the “Activity Discussion and Processing” section of the activity.**

- Ask students to find various convex polyhedra in the classroom and demonstrate that Euler’s formula applies
- Have students carefully graph a polyhedron that was not built in the lesson. Ask them to apply Euler’s formula

To close the lesson, end with a group discussion about what was learned during the activity. Circle up the group and work through the following questions. If possible, record the group’s responses on flip chart paper so all comments are displayed.

- What is the relationship established by Euler’s formula between the number of faces, edges and vertices found in a convex polyhedron?
- Why does Euler’s formula not apply to concave polyhedra? Use an example to illustrate
- How can the relationship established in Euler’s formula be used in everyday situations?
- How does Euler’s formula enhance mathematical communication?

Here are some additional topics for discussion:

- The definition of a polyhedron
- The difference between concave and convex polyhedra
- Deducing and defining the mathematical relationship between the number of faces, edges and vertices found in Euler’s formula
- The team effort used in analytical and deductive reasoning to communally discern Euler’s formula

**1. A different view.
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Challenge groups to build a concave polyhedron with Toobeez. After recording the required data, demonstrate that Euler’s formula does not apply. Teacher Note: If time does not permit building a model, have students carefully draw a concave polyhedron for homework on graph paper and conduct the same analysis.

**2. Extension/Follow up.
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Have the students refer to their textbooks or a provided formula list and calculate the volume for various polyhedron models.