Rumors
Posted January 2, 2001 as a Math Forum EPoW

Introduction: In this problem, students use a simulation to discover how the number of students who are told about a rumor varies with time. It also asks students to compare the difference between the rate of growth of a rumor in a controlled situation and the rate of growth in a real-life situation. Given the controlled situation graph, the students are asked to predict how the real-life situation would look as a graph.

Where’s the Math: In conjunction with the simulation, a graph is generated by the applet that illustrates the relationship between number of people and time. Students are asked to investigate how the shape of the graph changes when the parameters of the rumor-spreading are changed, i.e. how many people spread the rumor, how the rumor is spread, etc. The bonus question introduces the concept of exponential functions to students, as compared to the linear functions mentioned in the introduction. Standards: Algebra Role of Components: AgentSheets is used to display the simulation. SimCalc generates the graph that displays the data from the simulation.

Try the applet!

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Sample submitted solutions:

(No submitted solutions got credit for this problem. Here are the expected solutions)

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?)
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.

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?
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.

3. What else is spread in the real world, similar to the way that rumors are spread?
A variety of things, for example, diseases, rabbit populations, chain letters, epidemics.

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?
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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