Goal: Students will explore basic concepts of Graph theory through exploration. Objectives: 1. When given an order of a complete graph, students will be able to calculate the size without drawing a picture 9 out of 10 times. 2. When given a graph with vertices already labeled, students will be able to calculate edge lengths correctly 4 out of 5 times. 3. When given an even number of teams, students will be able to set up a round robin tournament so that every team plays every other team the same number of times correctly 3 out of 3 times. 4. When given experience doing math research, students will reflect upon that experience. Their reflections will be in journal format, and they will contain 3 elements of what they did today, 2 elements relating it to their life, and 2 applications that they foresee in their life. They will address the questions for the day and complete a ½ page journal each day. Students will meet 90% of this criteria. Assessment: 1. Students will set up a hypothetical tournament for an unknown, but bizarrely awesome game. There are 29 teams, 14 teams play each other at the same time. How can you schedule a tournament such that every team plays against every other team the exact same number of times. Day 1: Main Lesson Introduction to graph theory through exploration: What is a graph? Where do we see them in everyday life? o Friendship Graph example Handshake problem o There are 50 people in a room, they each want to shake everybody else s hand, but they don t want to shake anybody s hand more than once. How many handshakes will take place? o Let s start with some smaller examples. How many handshakes would occur if there s only one person in the room? How many if there are two people? How many if there are three people? Let s make a chart to see if we see any patterns. People Handshakes 1 0 2 1 3 3 4 6 5 10 o o Okay, now let s divide into groups of 6, count how many handshakes there are. How many were there? Why was there that many? Okay, so we found a pattern when we made a table. This is called a recursive pattern, because each step builds upon the previous step. Can we come up with a way to describe this pattern so that we can find the number
Day 2: of handshakes in our room full of 50 people without finding all of the number of handshakes before that? o Can we find an equation that will work for any number of people? Let s call this number n. o If each person goes in turn, shaking everyone's hand that they have not shaken, the first person will shake n-1 hands, the next n-2, the next, n-3,... the last 1. adding those in reverse order: 1 + 2 + 3 + + n 2 + n 1. o Bring up the story of Gauss and his teacher. His teacher got fed up with Gauss in class, so to keep him busy, he asked him to add up all the numbers from 1 to 100. Before she was even done asking the question, he had an answer and it was right. How did he find it so quickly? (It s the same pattern) o Lastly, draw their attention to a third way to find this equation. If each person shakes everybody else s hand, once, they shake n-1 hands, so all n people shake n-1 people s hands, however, in a handshake two people are involved, so we ll have to divide by two. Introduction to graph theory terms (vertex, edge, order, size, degree, etc) Define Complete Graph. o When we were finding the number of handshakes we were actually finding the number of edges in a complete graph of size n. Assignment: Today, we started from a real life example with the handshakes and worked to find a pattern, and then we generalized this pattern, and in doing so made it more abstract. This is what math research is, finding patterns in the world around us, generalizing them, and making them fit other real world examples, and then making them more abstract so that we can find other applications. o Journal (1/2 page written due tomorrow): What is your perception of math research? Is it useful? Are there other examples where you see patterns in the world around you? Are there any topics that you think would apply to graphs? Main Lesson- Discussion based exploration of edge length: Ask if anyone would like to share any parts of their journal? Review with them the graph terminology that they learned yesterday. Ask them: How many edges are in a complete graph with n vertices? Do you think that complete graphs have any other uses? Well obviously, since I asked you, and we re learning about them in math class. They ve got to be useful for something besides counting handshakes. Divide the class into groups of 4-6 (depending on the size of the class). If you have circle tables in the classroom, AWESOME! If you have square/rectangle tables in your classroom, those will work fine. If you have individual desks, ask the students to make a circle with their desks. This will take some time, but will be very useful for the activity. Today, we re talking about edge length. As they are getting into their groups, keep their focus on graph theory by asking them to be thinking of what they think edge length is. Ask them to come up with a group consensus on a definition, and ask each group in turn to share the definition
with the class. Possible answers are probably something like: how long an edge is. In response to this, ask what is long describing? Have each student draw on a white board a diagram of their table. With the first letter of each person as the vertex labels. Ask them to find the distance from L to C (Laura to Connie). Ask each group to pick an pair of people in the group, and then draw a line connecting that pair. From yesterday, we know that that line is a n edge. Ask the group to discuss the length of that edge. Next, ask the groups to find the person on their diagram whose name comes first alphabetically, replace their letter with a zero, and count everybody else clockwise from their (1, 2, 3, ) What is the length from 0 to 1? 0 to 2? 1 to 4? 0 to 6? Hmm, if they said six for this last one, ask if they can come with a different answer. What if they go around the circle the other way. Assign each group a section of the large white board up front, and ask them to make a complete graph of their seating arrangement with the vertices as numbers, and label all the edge lengths. We call the edges that didn t make sense at first wrap-around edges, because they wrap around the circle the other way. We denote wrap-around edges with a *. Ask the first group how many edges of length one are in their graph. Ask group two how many edges of length two are in their graph. Ask group three how many edges of length three are in their graph. Are there any of length four? Would there be any of length four in a complete graph with seven vertices? Answers will probably be yes, because obviously 7 is more than 6 so the edge lengths should be bigger. Work through an example, and show them that there aren t any of length four in a K_7. Let s take a look at a complete graph with n edges. How many of each length will it have? Let s start out like we did yesterday making a table, and see if we notice any patterns. n # Length 1 # Length 2 # Length 3 # Length 4 Total 1 0 0 0 0 0 2 1 0 0 0 1 3 3 0 0 0 3 4 4 2 0 0 6 5 5 5 0 0 10 6 6 6 3 0 15 7 7 7 7 0 21 Did you notice any patterns? (Besides the last column which is the same table that we drew yesterday if they don t notice this bring it to their attention) If n is even, then the there are n/2 of the longest edge length, and if n is odd, there are n number of all of the edge lengths. Ask them what would be the longest edge length be in a complete graph with 522 vertices. How about 1,037? How many edges are in that last graph? Observation: Bring this to their attention in case they didn t catch it, we call assigning a number to a vertex labeling and we assign them numbers to make them easier to talk about, once they have number values we can talk about things like edge length.
Day 3: Assignment: Today we discussed edge length in terms of how far away they are from you on the table, and again we made it more abstract and found edge length. This is another example of math research. A major part of doing research are the questions that are asked. We call these Gee, I wonder questions. After you journal about today s experience, I d like you to think of 2 Gee, I wonder questions that have to do about edge lengths, number of edges, or complete graphs in general. As an exit ticket, ask them to get out a sheet of paper and define: vertex, size, edge length, and complete graph. Main Lesson Round Robin Tournaments, using edge lengths in real life: Get out your Gee, I wonder questions. As a bell ringer, I would like you to find a partner, and share your questions. Then, as a class, we ll regroup, and if either you or your partner had a really good question, I would like you to share that with the class. Hmmm, those are very interesting. I ll keep those in mind, and we can address those as a have time. I wrote down the good ones as they were shared. Well, one of my Gee, I wonder questions was whether we could use complete graphs in sports, because I really like sports, and I think most of you guys either play sports or watch sports. Does anyone here know how the Olympic Committee runs their soccer, basketball, or ping pong tournaments during the Olympics? That s right! They re Round Robin tournaments, often in that level, they break them into pools and they call it pool play. The idea of a round robin tournament is that everybody plays everybody else the exact same number of times (Usually once). Let s look at setting up a round robin basketball tournament with 8 teams. o If everybody plays everybody else, how many games will each team play? (7), and how many teams are playing at a time in basketball? (2) So, we ll need a total of 8(7)/2 games. hmmm, doesn t that sound familiar. Never mind that for now. I d like you to make a table that describes each round. Let s give each team a letter so that we can distinguish between them. (A H) MATCH-UPS Game 1 Game 2 Game 3 Game 4 Round 1 A and B C and D E and F G and H Round 2 B and C D and E F and G H and A Round 3 Round 4 Round 5 Round 6 Round 7 I d like everyone to work on their own to try to set a tournament such that everybody plays everybody exactly one time. Okay, by the end there it got a little bit difficult, imagine doing something like that four 64 teams! As you could probably tell, we re going to relate this to graph theory. Okay, let s put all of these edges on a graph. So we ll have 8 vertices and they re labeled A H. (Draw this large on the white board.) Does this make it easier? If
every team has to play every other team exactly once, what type of graph are we going to end up with? (A complete graph) We ll use a different color marker to represent the different pairings. That is, round 1 we ll use orange. What colors should we use for the other rounds? I m going to pass four markers out, if you do not want to go up to the board, you can politely pass. And as you know, this is Mr. Gibson s room, so there will be plenty of opportunities for you to go up to the board, so don t feel left out. (Use name cards to hand out the markers so that it s random) Ask the first student to math up the first round, etc. Does this work? Let s look at another example of setting up an eight team round robin tournament. What if I number the teams 0 7, are you guys okay with that? That s just assigning each team a number instead of a letter. Now, instead of 7, I want to use infinity, and instead, of placing the infinity vertex on the outside with the others, I m going to place it in the middle. For my first round, (using the black marker) I m going to match up 0 with infinity, 1 with 6, 2 with five, and 3 with 4. Those are going to be my four game in round one. There s this nifty thing in graph theory called clicking. When we say we re clicking a graph, what we mean is we re adding 1 to each vertex. I m suggesting that if we click our round 1 match-ups, we ll get a unique set of round 2 match-ups. What is infinity plus one? (Still infinity). So who is going to play team infinity in round two? (1) In a different color marker, map out the round 2 match-ups, round 3 match-ups and so forth. Hey, isn t that interesting, this is the same as our normal graph of a complete graph with 8 vertices, except we stuck one in the middle. Is it really the same? We call this isomorphic. Two graphs that have the same structure are isomorphic. What do we notice about the edge lengths I used when I set up my round 1? (All the games had different length) What did you notice about clicking, and how it affected edge length? (It didn t. Our edge lengths stayed the same.) That s interesting. Using graphs, how would we set up a round robin tournament such that there are seven teams, and three teams play each other at the same time? o What kind of game can three people play at once? (Chinese Checkers) We want each team to play each other team the same number of times. (Get them started with making a graph with seven vertices, and then remind them about the fact that in the first round we used all of the edge lengths available, but exactly once) Oh look, Dani, has a good example of round 1. Let s put that up on the board. Would you like to draw it or would you like me to?
If we continue to click this around, what do you think we ll get? (A complete K_7) Isn t that interesting? So, we started out with one of each edge length that is in a K_7, and when we clicked it around 6 times, we got a K_7. Here is your challenge: You are setting up a tournament for an awesome new game, called Giblockenball. This game is awesome, because 14 teams play at once. 29 teams signed up for this tournament. You have to create a round robin tournament such that each team plays every other team the exact same number of times. You will present your tournament bracket on a poster. Contents of your poster will include: o Pictoral representation of your tournament. o A chart listing the matches of your tournament in each round. Like the one we made for the 8 team basketball tournament. o An explanation of what you did. (Tie it in with graph theory.) o Due next Wednesday. Journal: We ve spent three days discussing graph theory, and we just brushed the tip of it. There are tons of applications and lots more that you can do with it. Did anything about graph theory perk your interest? Take some extra time today to discuss your experience with math research. Write a ¾ page reflection on your thoughts and experience doing research that last few days.