June 2006

Featured Topic:   Map Coloring

Introduce elementary students to the discrete math concept of map coloring.   The basic concept is coloring a map using the fewest colors possible with these two rules:

• Regions that only touch at a vertex or point may be the same color.
• Regions that share a side (or portion of a side) must be different colors.

Students may use a blank map of the United States or a state map of counties to experiment with map coloring skills.   It's often helpful for students to use pencil to code in colors before actually coloring so that they can easily rearrange colors, as necessary.   The challenge is to color the map using the fewest number of colors.   Can the USA map be colored using only 2 colors, or do you need 3 or 4 or 5 colors?   Allow students to work on the task and consult classmates to compare notes and discuss the process they are using to minimize colors.

• Download Blank USA map.
• Map Coloring challenges students to evaluate a student's answer to a map coloring problem and explain whether they agree or disagree and why or why not.

Four Color Theorem: In 1976, mathematicians used a super-computer to help prove that any map could be colored with four colors at most.   Knowledge of this mathematical theory helps students work to color any map in 2-4 colors.   Later, they will apply these same principles to graph theory.

• Read the history of the Four-Color Theorem on Wikipedia.
• Map Coloring: How Many Colors? also discusses the four-color theorem.

Map Coloring for Students:

• The Young Map Colorer is an excellent resource for this topic.   The site explains the topic and provides different maps for students to try as well as a collection of student-drawn maps.
• Maps Made by Kids showcases student work and provides great resources for a math center or enrichment topic.   Challenge your students to create their own maps for coloring.
• Making a 2-colorable Map is a great project for students and they will be amazed that maps created using these directions are always colorable with only 2 colors.

Featured Technology Links: Map Coloring

Students may use technology to draw and color maps.   The Paint Map Coloring directions below are easy to follow and produce good results that students can print and compare.

• Directions for Paint Map Coloring provide step-by-step directions for opening the USA map and using Paint to color the map on the computer.   These same directions will work for many of the other map coloring files found on internet sites, such as The Young Map Colorer.   NOTE: Occasionally the Paint coloring will not work if the regions of the drawing are not truly closed and this is the case with many of the student-drawn maps.
• How many colors of states are on a map of the US? provides background knowledge and a color explanation and solution.
• Students are challenged to color the map of South America using map coloring rules and the assigned costs of different colors.   NOTE: Assigning color costs is a good twist to map-coloring problems.