## Common Core - Section 1: Engaging Students with Mathematical Conjecture

# Color Mapping

Lesson 6 of 7

## Objective: The students will be able to color in a map, or a shape, using the fewest number of colors possible. This will lead to discovering if the map is a Euler Circuit or Path.

*40 minutes*

Color mapping is a way to label graphs with colors to solve problems that require minimization or efficiency. It is also based in the Four Color Theory started in 1852 - any map or figure broken into segments can be colored with four or few colors. It is also part of the 15% Arizona has added to the Common Core Standards.

One history of the Four Color Theory

A portion of it states:

The Four Colour Conjecture first seems to have been made by Francis Guthrie. He was a student at University College London where he studied under De Morgan. After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan. Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them.

De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:-

A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented. ...... If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did....

Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):-

I am not likely to attempt your quaternion of colour very soon.I tell my students this story and then say we are going to work on this problem as De Morgan and Guthrie did over 160 years ago.

#### Resources

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Color mapping is the beginning of having students analyze and complete vertex edge graphs and find Euler's Circuits. I start all my lessons with the simplest lesson I can and build to complexity. I also teach this lesson after completing my geometry unit - building on the geometry vocabulary and properties of two dimensional figures.

Two vocabulary words the students need to know for this lesson are:

Vertex - the point where two rays meet at an angle

Edge or Border - the place where two sides of a shape meet

There are a few rules they will need to know prior to completing the color mapping.

1. No color can touch at an edge or border

2. Colors can touch at a vertex

3. The fewest number of colors need to be used

I walk my students through the first Map Coloring - Flags on the handout and then let them finish the rest and we discussion the options and numbers of colors of the last three flags as a class. It will help some students if you tell them they are charged $100 for each new color they need to use and they are to save money by using the fewest number of colors. I also had them get out 5 different colored pencils or crayons keeping them in range of the 4 or few colors being used.

Once the students are done color mapping and we've discussed the colors I have them write on the bottom of the page "Never more than four."

The Show Me video gives the answers to the number of colors used for each flag.

Once the students are finished with the first page I have them go on the the United States Map (blank, completed), Boxville (blank, completed) and Swirltown (blank, completed). Both Boxville and Swirltown are handouts I downloaded from an internet site years ago and it is no longer active so I've included copies of the pages in this lesson for you.

Take a look at the next section for a rule of when four colors are used.

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I teach thematically and this part of the Color Mapping lesson is also tied to my "The World Around Us - Arizona" component. It is one more connection to teaching mapping and getting my students familiar with where Arizona is in the United States. It ties perfectly with the main point I want to make with my students of how to tell when four colors are the fewest number of colors used to color map. Never more than four.

After covering the first page I tell my students they are going to tackle the problem of figuring out the number of colors they need to color the United States map west of the Mississippi and I let them get started.

Many of the discussions you will hear from your students will be about if a line is and edge/boarder or a vertex allowing opportunity for students to attend to precision of vocabulary (MP 6) and to construct viable arguments for edge vs vertex (MP3). One student connected the word vertex to being adjacent to another state, adding to the definition we had used before.

In the United States map four colors have to be used because of what I call the Wagon Wheel pattern. See the ShowMe video for and explanation.

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#### Student Reflection

*5 min*

Student reflection with collaborative work increases retention 50%. I ask my students questions from three categories. 1. Content 2. Collaborative 3. Personal

I asked my students:

*What did they learn from this lesson? *

Answers: It took a lot of thinking to color in the maps.

*What was one thing that went well with your table mates during this lesson?*

My group helped each other when they got stuck.

*Did you like doing this kind of assignment?* More thinking and coloring than working only with numbers.

I can't believe this is math, it's fun!

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