Graphs, Stories, and Games

Description

Each story can be acted out on a graph that is drawn on the floor, with a student taking the part of each one of the characters. The action and the situations in the stories explain the requirements and the constraints for how each of the characters moves about the graph. The endings of the stories remain somewhat open, because what Gertrude, Superperson and the Monster eventually do depends, not only on the choices that the students make as they act out their parts, but on the graphs that are chosen to play the games.

Play the games first with the graphs that are recommended, then try them on different graphs. As they play the games on various graphs, students can discover many different properties of the mathematical objects called graphs.

Students can also play the games on a table using a graph that is drawn on a piece of paper for a game board and moving marking pieces that show the actions of the characters.

Materials

Instructions

Ideas for discussion

Varying the Game

Materials

• A large copy of The Land of Many Ponds drawn on the classroom floor with tape. The ponds (or circles) should be large enough for a person to stand in. It should be in an area where all the students can gather round and watch what is happening.

• For each small group of students:
  • Copy of The Land of Many Ponds to use as a game board
  • 3 markers that will represent each of the characters that can be moved on the game board

• Optional: Copies of the Superperson Saves the Monster for the students to read when the first game is introduced, and copies of Gertrude Superperson and the Monster Recover From a Disaster to consider (and act out) later.

Instructions

1. Explain to the students that the figure on the floor is a game board and that they are going to play a game which is best explained by a story. The story has three characters in it: Gertrude (a goose), Superperson, and a Monster. Select a student to play the role of each of these characters.

2. Tell the students that the circles represent ponds in the land where Gertrude, Superperson and the Monster live. To begin the game, the three characters must select the ponds where they will start out. During the course of the game they will see that Gertrude has the least amount of moving power and she runs away from the Monster. So to give her as much advantage as possible, the Monster should select his place before Gertrude selects hers. Superperson chooses her place last.

3. Read the story Superperson Saves the Monster and have the students act out their parts as the story moves along. The story serves as script and rule book for the game. Encourage the students to act out their parts with as much drama as possible.

4. The game doesn't end when the story ends. The story simply provides enough information for the game to continue.

5. After Superperson has saved the Monster from Gertrude, have a different group of students play the roles of the three characters. You can continue to read the story as they play, or a student might want to read the story. Once the students know the roles of all the characters, they might want to play without reading the whole story each time.

1. Depending on the age and experience of the students, they will be able to figure out, after some number of rounds, that the Monster always catches up with Gertrude and that Superperson always saves the day.

2. Help them to explore what difference various starting points for the three characters make.

3. Help the students to articulate why it is easy to figure out what will happen. The concept of distance and diameter in a graph figure largely into the reason why. Provide them with that vocabulary when it will be useful to them in talking about their experience.

• Make the Game Last a Long Time

  Once the Monster can land in a pond that is right next to the pond that Gertrude is in, he is certain to be able to catch up with Gertrude on her next turn. This is because Gertrude can only move a distance of one. No matter where she goes she is certain to be two steps away from the Monster when it is the Monster's turn to move next.

  Notice, however, that at this point Gertrude can always assure that Superperson will be able to arrive in the pond where she and the monster are. Once Gertrude knows she is going to be "caught" she can look around and be sure that the pond she is caught in will be exactly three steps away from where Superperson is.

  When the game ends, the players have to give up their turn, sit down and let someone else play. Perhaps they don't want to do that. Let the students try to find a way to make the game last as long as possible.

• Change the Rules

  Students can decide on any rules at all to change the game to make it more challenging. Some suggestions are:

  • Let all the players move a distance of exactly two at each turn.
  • Place tokens in various ponds that the characters can pick up and "spend" to receive extra moving power when they need it.

• Change the Graph

  Use a different game board, such as any of the ones below to play the game. Because the distances in these graphs are different from the graph in The Land of Many Ponds, the students need more complex strategies to play that they had to use with that particular graph. This will help them to gain a better understanding of the concept of distance in a graph, and to see how these configurations of dots and lines have considerable structure.

  

• Change the Gender of the Characters

  In the story, Superperson is a "she", and so is Gertrude. The Monster is a "he". But girls can be big and clumsy like the Monster is, and they can scare people, too (even by mistake). It isn't unusual for boys to be afraid of something large and loud that they think might hurt them. And sometimes a boy and sometimes a girl is the person who can be a peacemaker and help everyone get along. Change the story and change the game so that boys and girls get to play whichever parts they want to.

• Change the Story, Change the Game

  Gertrude, Superperson and the Monster Recover from a Disaster is a completely new story. The three characters have a completely different problem to solve. Act out this story, then use your imagination to give new adventures and new problems to solve to our three friends. Send in your favorite ones.

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