Title
How Many Colors? (Dorothy Gieck)
Goals
1. Students will explore the concept of map coloring.
2. Students will work independently and then in small groups in order to arrive at the smallest number of colors needed and decide if that number is correct. They will also be able to explain, and justify, how they arrived at this solution.
Abstract
This activity, which is set in the context of having students find the minimum number of colors needed to color a map is related to graph theory and map coloring problems. Students are asked to come to consensus with a small group as to a method for coloring maps. Each student will come up with their own solution and share it with other students. Small groups will then synthesize a method that works for all members of the group. These methods will then be shred with the rest of the class.
Problem Statement
In reading maps, it is helpful to be able to distinguish countries or states by means of their colors. States that touch one another should not have the same color as it would be hard to find their boundaries. When coloring maps, it is optimal to use as few colors as possible. How does one determine how many different colors are needed to color any map? Students will be asked to devise methods for determining how many colors are needed to color any given map.
Instructor Suggestions
1. Set the stage by discussing the "Problem Statement" (see above) with your students.
2. Have students read through the "How many colors?" activity sheet and work through the first and second maps.
3. When students have completed their step sheets, have them exchange with another student and try to follow the other person's set of instructions.
4. Have student pairs group with another pair to discuss various solution steps. Groups should come up with a set of steps that seem the best to all members of the group.
5. Circulate between the groups and facilitate the discussions.
6. When the small groups are finished, have a spokesperson for each group share their solution and explain their method and reasoning involved in arriving at their instructions for coloring maps.
Materials
"How Many Colors?" activity sheet, colored pencils, paper for recording results and sharing with other students.
Time
Introduction of Problem Statement (5 min.), individual work (15 min.), pair work (10 min.), small group work (15 min.), presentation of small group work and large group discussion (25 min.), extension questions (15 min.).
Mathematics Concepts
Discrete Mathematics Concepts:
Representation of graphs, map colorings, four-color conjecture, chromatic number.
Related Mathematics Concepts:
Modeling with graphs, Optimizing results.
NCTM Standards Addressed
Problem solving, Communication, Reasoning, Connections
Colorado Model Content Standards Addressed
Geometric Techniques (4), Problem Solving and Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into almost any level. Discussion could include discrete mathematics topics related to graph theory, graph colorings and chromatic numbers. With a lower class level class you might use this activity to communicate methods of coming up with solutions. Writing an understandable set of steps that another can follow is a big part of this activity. This activity could be used with a traditional Advanced Algebra or Pre-Calculus class to help discuss recently solved problems and how to prove a method is true for all cases.
Further Investigation
This activity can be extended by having students look at different map colorings and discussing how the proof of the four-color theorem came to be.
Variations/Comments
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W. H. Freeman and Company.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
Colorado Model Content Standards for Mathematics (1995).