Euler’s Formula, Toothpicks and Gumdrops
Grade Level: 3-6 adjusted based on the amount of mathematics
Time: One 50 minute period of instruction
Standard: 1) Students develop number sense and use numbers and number relationships in problem-solving situations and communicate the reasoning used in solving these problems
2) Students use algebraic methods to explore, model, and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate the reasoning used in solving these problems.
3) Students use geometric concepts, properties, and relationships in problem-solving situations and communicate the reasoning used in solving these problems.
5.4.1 Construct models of two-dimensional figures and explore three- dimensional figures using a variety of materials and tools.
> Identifying a three-dimensional shape from a set of two-dimensional views (match a solid with its front, side and top views).
> Describe and draw geometric shapes.
Objectives: 1) Students will demonstrate understanding of edges, vertices and faces as they relate to a 3 dimension planar shape.
2) Students will discover and analyze Euler’s formula for the relationship between edges, vertices and faces.
3) Students will build various shapes with given materials to utilize in the comprehension of Euler’s formula.
Material: - Drawing surface, Blackboard of Dry Erase Board
- Toothpicks
- Gumdrops
Opening: Have the students discuss who uses
geometry. Allow responses and give
examples as architects, carpenters, scientists, moms, and dads, etc. Look at buildings and architecture as sources
of geometry figures. Examples of these
might be houses, churches, and all sorts of building. Explain the terms face, vertices and edge as
part of a planar solid. Ask if there a relationship between the number of
edges, vertices, and faces a shape has? Explain the terms face, vertices and
edge as part of a planar solid.
Body/Procedures:
1. Give a brief history of Euler and discuss how he might have discovered this formula.
2. Explain and discuss further the terms vertices (points), edges (lines), and faces (areas)
3. The teacher should model building a shape of have a shape ready for discussion of the building procedure. Students should then construct their own shapes.
4. Students should identify to the teacher edges, vertices, and faces of the figure build.
5. Students will build shapes that complete the attached worksheet.
6. Students will demonstrate and derive Euler’s formula, V-E+F=2
Closing: Review the relationship the students discovered. Discuss the relationship for shapes with more faces.
Assessment: Teacher will observe the students understanding of Euler’s formula through student questioning during the activity.
Extensions:
1. Look at other formulas and relationships that Euler discovered.
2. Have students demonstrate building shapes with a larger number of faces, non-Platonic Shapes, does the formula work?
3. Pose the question – Is there a limit to how many sides a shape can have? Students can research this question.
Source: Jan Swanson at www.math.twsu.edu/history/Activities/geometry-act