GENUS NUMBER LESSON PLAN

At the Beyond Numbers exhibit, a museum employee has a cart of activities about geometric solids. This museum "explainer" shows visitors a variety of geometric solids and shows how the genus number, a topological classification, is related to the Euler characteristic. When V stands for the number of vertices, E stands for the number of edges and F stands for the number of faces:

V - E + F = 2 for genus zero,
V - E + F = 0 for genus one,
V - E + F = -2 for genus two,
V - E + F = -4 for genus four, and so on.

This is related to the exhibit's Genus Sorting Table where visitors sort familiar and unfamiliar objects by counting the number of holes through an object to determine its genus.

NCTM Standard and/or AAAS Benchmarks addressed:

Mathematics as Reasoning:
Students use patterns and relationships to analyze mathematical situations.

Number Sense and Numeration:
Students construct number meanings through real-world experiences and the use of physical materials.

Geometry and Spatial Sense:
Students relate geometric ideas to number and measurement ideas.

Mathematical Connections:
Students relate geometric ideas to number and measurement ideas.

Algebra:
Students can represent situations and number patterns with tables, graphs, verbal rules, and equations and explore the interrelationships of the representations.
Students analyze tables and graphs to identify properties and relationships.

The Nature of Mathematics, Patterns and Relationships:
Students know that mathematical ideas can be represented concretely, graphically and symbolically.

The Mathematical World, Symbolic Relationships:
Students know that table and graphs can show how values of one quantity are related to values of another.

VOCABULARY

polyhedron - a solid figure having many faces

vertex (vertices) - point or corner where edges meet

edge - a line or border

face - region within the boundaries of an edge or edges

graph - in this lesson, a network of vertices connected by edges

Leonhard Euler - (pronounced OIL-er) Swiss mathematician (1701-1783) credited with the first studies of graph theory

Euler number - the Euler number of a surface is equal to the number of vertices, minus the number of edges plus the number of faces of the surfaces.

topology - the study of certain invariant (unchanging) geometric properties. In topology, some such properties are connections, crossings, number of holes. In topology, size and shape do not matter.

INTRODUCTION

In this lesson, students explore geometric solids, record data and note patterns in order to generalize the formula for this characteristic which is named after its discoverer, Leonhard Euler (pronounced "OIL-er") .

OBJECTIVE

Students will observe attributes of polyhedra to identify their topological classification.

MATERIALS

For each student:
ruler, polyhedron cut-out (A,B,C, or D), ballpoint pen or sharp pencil, scissors , tennis balls or balloon, felt-tip markers, crayons
optional: soccer ball

PROCEDURE

Euler number
Distribute rulers and cut-out pages. Have students score the line that will not be cut, by tracing the lines with a ballpoint pen or a sharp pencil. When all the fold lines have been scored, let the students carefully cut the outlines of the shapes. Students should then crease the shapes into polyhedra. For this activity it is not necessary that students learn the names, although many students like to learn them (much the way they learn dinosaur names.) Distribute white glue in plastic bottle caps and have students use toothpicks to spread the glue on the shaded halves to complete the polyhedra. Have students use felt-tip markers to mark each corner with a dot to denote a "vertex," counting as they progress. Then have the students count and mark each edge with a line to denote an edge." Then they should mark each "face" (side of the polyhedron) with a number starting with 1 and counting up.

As they work they should fill out a table on the chalkboard.

To solve the equation in column four, have students either use a calculator or count along a number line that has negative as well as positive numbers.

Let students discover that all of the right column should have two.

Show students a prepared balloon or tennis ball. Point out that you marked points on the ball, then joined various points with lines that don't cross elsewhere, taking care that each point can be reached from every other point by going along the lines.

Count each intersection as a vertex and every line portion between two vertices as an edge. Then count the spaces as faces. (Note: don't forget to count the region 'outside' a figure as one additional face.)

Have students see if V - E + F still equals two. If you have enough balloons or tennis balls, let the students do this.

Point out that all of the objects had no holes. Topologists say these objects have genus 0 (zero). Genus could be explained as another word for "type." Genus 0 means zero holes.

EXTENSION

Have students determine the Euler number ( V - E + F ) of a soccer ball.

ASSESSMENT

Have students build genus 0 (no holes) objects with MULTiLiNK cubes. Then have them verify in two ways that the genus number is zero:
(1) The object has no holes.
(2) V - E + F = 2

MASTER

POLYHEDRON CUT-OUT A

POLYHEDRON CUT-OUT B

POLYHEDRON CUT-OUT C

POLYHEDRON CUT-OUT D

last revised 4/10/05

Link to the main Beyond Numbers Table of Contents
Link to Teacher Manual Table of Contents
Link to Maryland Science Center
Link to Eisenhower Consortium
Cathy Brady's home site
cathysfiddle@yahoo.com