I Will if You Will

Has a situation like the following one ever happened to you?

Situation A: 
Tom and a couple of friends are discussing whether to try out for the junior class play. Tom says, "I will if one of you guys will." Dick says, "I will if you two guys will." But Harry says, "No way! I'm not going to do it." The result of the discussion is that none of the tree tries out for the play. 

Now consider a slightly different situation. 

Situation B: 
Tom, Dick, Harry, and a fourth friend, Fred, are discussing whether to try out for the junior class play. As before, Tom says, "I will if one of you guys will." Dick says, "I will if two of you guys will." Harry says, "No way! I'm not going to do it." Fred says, "I'm going to try out no matter what the rest of you do." The result of this discussion is that Fred, Tom, and Dick all try out.

Note: The information below was created with the assistance of AI.

Level of Mathematics
This module is appropriate for:

High school students (grades 9–12), especially those in Algebra I, Algebra II, or Precalculus.

Introductory college-level courses in discrete mathematics, sociology, or mathematical modeling.

While the math content is not algebraically intensive, it emphasizes logical reasoning, modeling, graphical analysis, and discrete structures, making it highly valuable in conceptual math development.

Application Areas
This lesson is rooted in real-world, interdisciplinary applications:

Sociology & Psychology:
Modeling group behavior, peer pressure, conformity, and collective action.

Political Science & Civics:
Understanding voting behavior, protest participation, or social movements.

Economics & Public Policy:
Studying consumer adoption, public compliance (e.g., water conservation).

Education & Ethics:
Decisions around academic honesty, attendance, and school initiatives.

Mathematical Modeling:
Introduction to threshold models, equilibrium points, and feedback loops.

Prerequisites
Students should be able to:

Interpret and construct tables and bar graphs.

Understand and apply logical conditions (e.g., “if X ≥ t, then…”).

Follow cumulative reasoning.

Think in terms of discrete thresholds rather than continuous variables.

No calculus or advanced algebra is required, but conceptual maturity in reading scenarios and modeling behavior mathematically is important.

Subject Matter
Core Concepts and Techniques
Threshold Analysis

Each individual has a threshold: the minimum group size required for their participation.

Individuals may or may not choose to act based on whether their threshold is met.

Equilibrium Points

A group reaches equilibrium when the number of participants equals or exceeds their thresholds.

These are points of self-sustaining participation.

Cumulative Participation Tables

Tables showing how many individuals are satisfied for each threshold level.

Enable prediction of stable group behaviors.

Graphical Representation

Threshold graphs visually indicate potential for participation and equilibrium.

Allows detection of critical mass (an individual or group that triggers further participation).

Behavior Modeling

Scenarios: school play auditions, sports teams, water conservation, haircut trends, academic honesty.

Simulates domino effects in collective behavior.

Project Component

Students conduct real-world threshold surveys and analyze their own group's behavior.

Correlation to Mathematics Standards
Common Core State Standards – High School

Algebra (Modeling)
HSA-CED.A.1–3: Create equations and inequalities that describe numbers or relationships.

HSA-REI.D.10: Understand that the graph of an equation represents all its solutions.

Functions and Graphs
HSF-IF.C.7: Graph functions expressed symbolically and show key features.

HSF-LE.A.1c: Recognize situations where one quantity grows or depends on another.

Modeling and Data Analysis
HSS-ID.B.6: Represent data with plots on a real number line.

HSS-ID.C.7: Interpret differences in shape, center, and spread in the context of the data.

Mathematical Practices

MP1: Make sense of problems and persevere in solving them.

MP2: Reason abstractly and quantitatively.

MP4: Model with mathematics.

MP5: Use appropriate tools strategically.

MP7: Look for and make use of structure.