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