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When two people are dating, how one person feels often depends on how the other person feels. Your feelings are also informed by your own feelings, not just your partner's. In a certain well-known tragedy by Shakespeare, this feedback loop got way out of control.

In this lesson, students investigate the effect of coefficients on recursive functions. They first analyze a simpler version, where Romeo's feelings each week are simply scaled by Juliet's feelings the previous week. Then, they analyze a more complicated rule, where Romeo's feelings this week depend on both Juliet's and his own feelings the previous week. Can romantic love be modeled with mathematics? We're about to find out!

### Students will

• Evaluate a recursively-defined function at different values to see the course of a relationship
• Investigate the effects on the relationship of changing the initial values and coefficients
• Find initial values and coefficients that result in a "good" relationship
• Expand the rule so that someone's feelings are governed by her own feelings, as well as her partner's
• Explain how certain combinations of values result in certain personality types, and how these personality types play out in a relationship
• Find initial values and coefficients for the expanded model that result in both a good and bad relationship

### Before you begin

Students should have some sense of the outcomes of adding and multiplying integers. For instance, know that the product of a negative and positive value is negative without having to perform a specific calculation. They should also be familiar with function notation. If they have experience evaluating recursively-defined functions, the lesson will go faster. However, this lesson could also be used as an introduction to recursively-defined functions.