Section 5: Models and Representations

Transcription

1 Section 5: Models and Representations Next comes one of the most important parts of learning to do math: building models. A model is something that makes the experience present to us. Since the experience is present again when we have the model, we say the experience is re-present. In other words, the model is a representation. We are interested in what kind of experience you had in trying to solve the puzzle. We want to give you a chance to express your ideas of what a representation of solving the puzzle could be. It doesn t matter what form you choose. If you like to draw, draw a picture that represents solving the puzzle. If you like music write a song. Or come up with your own ideas for a way to represent solving the puzzle. 28

2 Exercise 5.1. Each team should decide how they are going to represent solving the puzzle. Create your team s representation and present it to the class. Each of the models you and your classmates created has properties that reflect aspects of the experience of walking around your city attempting to get everyone to the same building. Later in this material we will discussthe representations which are used by the research mathematicians who study this puzzle. First of all, it is important to understand that this puzzle, or problem, is currently being studied at universities around the world. This is probably the first example that you have seen of a mathematics problem that is still being studied. In fact, no one knew the solution of the problem for almost 40 years. The problem was finally solved in 2007 by Professor Trahtman at Bar-Ilan University. What problem? For obvious reasons, the puzzle we have been discussing is called the Road Coloring Problem. 29

3 You may be saying to yourself, But wait, we ve also solved the problem a bunch of times already! In one sense, you are right. You and your classmates have found solutions for specific cities with three or four buildings. But to solve the Road Coloring Problem, Professor Trahtman had to prove that for every city there is a way to color its roads so that a set of directions can be found to get everyone to the same building. He had to show it works for cities with 100 buildings, or 1,000,000 buildings, or 1,000,000,000,000,000 buildings. He also had to prove that his technique works (Later in your high school careers, we will discuss what it means to prove something) You can go to the website to find out the latest news on the problem. 30

4 Let s explore this idea of a mathematical representation a little more. This idea is one of the most important concepts we will talk about in this class. A good mathematical representation finds the important features of a problem that need to be preserved and builds a model of them. For our cities, the features that are important are the buildings and the roads. This leads to the model called a directed graph. This type of model looks so much like the original city, it is usually the first representation most people build. For example, we could build the following city: The numbered circles represent the buildings and the one-way roads are shown as arrows. By representing the problem in this manner, we have concentrated on the key parts of our city. In mathematics, the circles are called vertices (one circle is a vertex) and the arrows are called directed edges. Let s start a dictionary of terms we ll use for the road coloring problem. You will find the dictionary at the back of these pages. 31

5 Whenever we use a new term that should go in the dictionary you will see this picture. Our first three dictionary terms are: vertex, directed edge, and directed graph Now that we have our directed graph representation of this city, let s begin to run some experiments. You see math, in many ways, is like other sciences. We need to perform experiments on our models to completely understand their properties. But the experiments in math are not done in laboratories. The experiments in math are thought-experiments. To do mathematics you don t need expensive equipment. Mathematics mainly requires careful observation and imagination. The second step of the puzzle involved coloring the roads of our city. Here is one possible way to color roads of the city on the last page. 32

6 Edge-colored directed graph You may have already experimented with this city and this particular way to color the roads. If someone in your class built this city during your experiments, they may have colored it differently, but let s use this as an example. We will refer to the directed graph of a city that is correctly colored as a city map. Now we ll carefully observe what happens when the command Blue is called out. We ll put names in each of the buildings to represent the people walking around the city. Candace Marcus Miguel When the command Blue is called out, here is what happens to the people in the city. a Candice leaves from building 1 and goes to building 2 a Miguel leaves from building 2 and goes to building 1 a Marcus leaves from building 3 and goes to building 2 33

7 Next, when the command Red is called here is what happens to the people in the city. a Candice leaves from building 2 and goes to building 3 a Miguel leaves from building 1 and goes to building 3 a Marcus leaves from building 2 and goes to building 3 We ve done it! The set of directions Blue - Red gets everyone to building 3. Mathematicians refer to a set of directions that solved the puzzle as synchronizing. (When your team answered exercise 4.4, did they decide on a different name?) The word synchronizing comes from the Greek language and means same time. [ Syn means same, like in synonym; chronos means time like in chronology.] For the exercises that follow choose some names of your classmates and place them in the vertices that represent the buildings of the three and four building cities your classes designed. Then record their movements around the city as the commands are called. 34

8 Exercise 5.2 Name Teacher Date Directions: (1) Draw in the colored directed edges of the city map for the 3 building city your team designed (2) Put the names of the players in their starting positions in the circles to the left. (3) In the boxes below, show how each player moved after the command was called by placing their names in the circles where they moved. Remember, all the player should move 1st command called: road 2nd command called: road 35

9 Exercise 5.2 Name Teacher Date 3rd command called: road 4th command called: road 5th command called: road 36

10 Exercise 5.3 Name Teacher Date Directions: (1) Draw in the colored directed edges of the city map of the 4 building your team designed (2) Put the names of the players in their starting positions in the circles to the left. (3) In the boxes below, show how each player moved after the command was called by placing their names in the circles where they moved. Remember, all the player should move at the same time when the command is called. 1st command called: road 2nd command called: road 37

11 Exercise 5.3 Name Teacher Date 3rd command called: road 4th command called: road 5th command called: road 38