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Rumors - posted January 2, 2001

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The Rumors Problem

Introduction

Shanika has learned from a phone call that William has dyed his hair green. She can't wait to share this rumor with all the people at school. She stands at the door to the cafeteria, where people are entering every ten seconds, and tells every person who enters.



In real life, Shanika would probably not stand at the door of the cafeteria and tell the rumor to everyone who walks in. Here is a more realistic scenario:

Simulation

Shanika has just learned that William has dyed his hair green. Shanika is very excited and runs around telling the rumor to everybody she meets. She makes a point of telling people not to tell anyone else.

What will the shape of the graph be?

Will it be
1.   2.   3.  


To find the shape of the new graph, please click on the link below to show the applet window and run the simulation. Use what you learn from the simulation to answer the questions.

Click To Show Applet Window

Questions:

  1. Describe the shape of the new graph in terms of the x and y axes. (Hint: How is it different from the first graph?)

  2. What happened differently in how Shanika spread the rumor at the cafeteria door from how she spread it by running around during lunch that changed the way the graph looked?

  3. What else is spread in the real world, similar to the way that rumors are spread?
Bonus:

What if each person Shanika tells just can't keep the promise not to tell anyone else, and tells the rumor to other people? How will this change the shape of the graph?

Comments

Teacher Support Page

Well, this problem was harder than we thought it would be. No one got credit for it. We also didn't get a lot of submissions because of some technology problems.

The difficulty most of you had was in clearly explaining the shapes of the graphs. The answers we were looking for were these:

For question 1 (the shape of the new graph in terms of the x and y axes), saying that the curve gets steep and then levels off; a mention of the fact that it levels off or has steps would have been reasonable, too.

For question 2 (the difference between spreading the rumor at the cafeteria door and running around during lunch), the graphs look different because Shanika meets fewer and fewer people who haven't already heard the rumor. Over time, the steepness of the graph decreases until everyone in the room knows and the graph from that point on is horizontal. Or, you could have said something like meeting people at random could lead to some irregularities in the curve -- sometimes several people who don't already know the rumor will be encountered by Shanika, while at other times she may encounter no one at all.

Question 3 (what else spreads like a rumor) could have a variety of answers, for example, diseases, rabbit populations, chain letters, epidemics. We saw a few of these.

For the bonus (what if each person Shanika tells also tells the rumor to other people? How will this change the shape of the graph?), the graph would increase more steeply at first. More importantly, however, it would level off much sooner because there is a finite number of people in the room.


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