Illustrating the Hotel Infinity

Description

Materials

Instructions

Ideas for discussion

Materials

• Art materials for producing the illustrations
• Extra copies of the story of the Hotel Infinity so that students can look at it some more and read it again if they want to after they have heard it.
• A large-print copy of the story to post with the illustrations.

Instructions

1. Explain to the students that you are going to read them a story, and that they should listen and try to make pictures in their minds of the things they hear about. Tell the students that they will be drawing the things that they see in their minds.

2. Read the story of the Hotel Infinity aloud.

   NOTE: Another way to present the story is for the character George, the narrator of the story to visit the class and tell the story. The advantage to doing it this way, is that George can interact with the students, invite interruptions, answer their questions, and ask them what they think as the story progresses.

3. Make the text of the story available to the students to refer to and reread.

4. Ask the students how they imagine the Hotel Infinity looks. If you visited there, what would you like to do? A few of the rooms are described in the story. What other things do you suppose the clever and imaginative Sam put in some of the other rooms? Encourage the students to conjure up details that are not in the story: what did the lobby look like? how were the hallways shaped? what did the busses look like? describe Kronecker's clothes and his truck.

5. Introduce the students to the medium you have chosen for them to use for their illustrations. Add to your presentation whatever you feel is necessary for them to make illustrations that they will enjoy and find satisfying.

6. Give the students plenty of time to work on their illustrations and complete them.

7. On a large and spacious wall, preferably one in a more public area than the classroom, arrange the illustrations interspersed with large-print copy of the text of the story.

8. Encourage the students to discuss and ask questions about the details of how the hotel works.

Ideas for Discussion

1. Once the children have heard the story, no doubt they will have ideas and questions that relate to the mathematics behind the Hotel Infinity. Students may claim that no such place could possibly really exist. The narrator in the story, George, however reminds us that although he no longer remembers how the hotel or the busses that held infinitely many people were made, they did indeed exist, because he helped to build them and he worked at the hotel for many years.

   It is not absolutely critical that the students believe George, only that they can articulate why they believe what they do in regards to the puzzling concepts behind infinity that are raised in the story.

2. Encourage the children to be creative and brainstorm about ways that the hotel and the busses could be put together. Since the Hotel Infinity is a place full of paradoxes, the following other paradoxes that involve the idea of infinity may help give students ideas for conceiving of how things worked:
   • Zeno's paradox, or the Tortoise and the Hare. The Hare challenges a tortoise to a race. Since the hare can run twice as fast as the tortoise can, the tortoise isn't very interested. So the hare agrees to give the tortoise a head start that is exactly half the distance to the finish line. The tortoise agrees and they have the race. The tortoise plods along. When he gets exactly half the distance from where he started to the finish line, the hare, who runs twice as fast, has gone twice as far, and is just at the point where the tortoise began. The tortoise is still plodding. When he again covers half the distance to the finish line, the hare has reached the point where the tortoise was. This happens again and again. And again. And again. And the hare never does catch up with the tortoise. Or does he?

   • The Great Galumphing Gnome. The Great Galumphing Gnome has the most unusual way of walking. He can start out with a step of any size. Sometimes that step is very little and sometimes that step is very big. Always, though, his second step is twice the size of the first, his third step twice the size of the second, the fourth step twice as far as the third, and so forth. (It's not known what happens when he walks backwards.)

   • The Great Galumphing Gnome's Sister, Gert. Her steps are all the same size. Sometimes she takes her first step very quickly, and sometimes her first step is very slow. When she takes her second step, it is twice as fast as the first one. Her third step is twice as fast as the second one, and so forth. Can brother and sister ever take a nice stroll down the street together side by side?

   • The Incredible Moving Sidewalk. This sidewalk is a conveyor belt of possibly infinite length. When you turn it on it goes very slowly, but every minute, its speed doubles. How long does it take to get from here to there?

3. Another idea for imagining how the Hotel and the busses could have been made is to look at prints by M.C. Escher.

4. Students may want to experiment with ways that the Hotel Infinity was laid out. Early in the story George tells us that "it was all very organized, the way that the Hotel was set up, so even though the numbers got very large, the rooms were easy to find." Later in the story we learn that Hallway Number Eight leads to all the rooms whose numbers begin with 8.

5. George tells us that it was no problem to fit in all the people on the Continue 'Em Tours busses, even when 150 busses all showed up at once. Do you think George had to make the announcement that emptied all the odd numbered rooms 150 times?

6. Do you think that George's scheme for unloading the busses was going to work? You can test it out by using rows of some kind of markers to represent the first 10 or so people on the first ten or so busses. Remove the markers according to the scheme that George describes in the story. Does it seem that all of the people will "move off of the busses in an orderly way so that no one will get skipped, and no one will have infinitely many people in front of them"? Will that be sufficient to file them all into the odd numbered rooms?

7. Georg Cantor spent a lot of time thinking about sets of numbers, particularly sets that had infinitely many numbers in them, such as the counting numbers, the odd numbers, the rational numbers (which we express as fractions), the real numbers (which we express as decimals) and so forth. He was trying to understand the notion of sizes of infinities.

   See if you can get to the bottom of this paradox that he struggled with very hard:

   • Consider the set of all sets.
   • Now consider the set of all sets who do not have themselves as members.