**Mathematical Modeling**

1. In Roser Gin?©’s Midlevel and Graduate classes this year, students have used math models and mathematical reasoning to explain the impact of epidemics on the world’s population (vehicle: “Investigating Epidemics Through Mathematics”). The goal of this project was to examine how different epidemics have spread in order to make predictions using identified patterns. Students applied their knowledge of exponential functions to epidemiology. After researching a particular disease and gathering statistics, students explored appropriate mathematical models and analyzed their ‘goodness’ against the reality of the problem.

2. In a different kind of math modeling project, “Studying Conic Sections”, students in Roser Gin?©’s Midlevel and Graduate classes created parabolic dishes, whispering galleries, and three-dimensional models of our galaxy. The projects were guided by the following questions: How can we use the equations of conic sections to create real-world objects that reflect true physical laws? What may this imply about our physical surroundings? How can we use evidence from our work to explain why mathematical models are useful? Students who created a parabolic dish used the equations and graphs of parabolas and the location of their focus points to create the model, and showed how the Law of Reflection contributes its function. Students working on the 3-D models of our galaxy worked on creating appropriate scales to fit all planets in one model and used the eccentricities of the elliptical orbits of the planets to represent their paths. Students who built whispering galleries used the equation of an ellipse and algebraically located the focus points in order to show the reflective qualities of their model.

**Mathematical Proof**

1. At the Foundations level, students engage in an inductive process of proof focused on the verification of a conjecture by pattern discovery. In a vehicle designed by Stephen Cirasuolo, “Pythagorean Triplets,” students were presented with four methods that may or may not generate triplets. Their task was to explore which methods always work by conjecturing and testing each method for its validity. This problem allowed different entry points for students; those who chose could write algebraic proofs while others could delineate their inductive approach by presenting cases tested.

2. In her Midlevel class, Roser Gin?© integrated a unit on Logic and the study of trigonometry in a land-surveying project. Students worked on a project that included writing a mathematical proof of the Law of Sines, and then applied the Law of Sines and the Law of Cosines in their mapping of the school’s surroundings. They first verified the theorem through an enactive process, and followed that by constructing a valid symbolic proof.

**Problem-Solving**

1. At the inquiry level, Heather Cabrera and Abby Paske engaged students in problem solving when they investigated “Egg Safety.” Students designed and created paper cars that would prove “crashworthy” as they carried eggs and protected them against crashes into a wall. Participation in this project helped students develop skills in measuring, scaling, estimating, solving equations, graphing, and using data to justify their results. The students also explored concepts in physics as they determined the force that a car would have to survive in order to protect an egg.

2. In their yearlong study of urban ecology, students at the Foundations level work on projects in their exploration of the essential questions, “How can we obtain evidence to justify conclusions? How can we find out if our watershed is healthy?” One of the projects scheduled for implementation this year by Elizabeth Chartier and Stephen Cirasuolo challenges students to determine the amount of water that has entered the Muddy River watershed through precipitation. Students create and design their own plan for investigation while they develop their understanding of mathematical and scientific processes. As they proceed with this real-world problem, students explore three questions: “What do I know? What do I want to find out? What can I introduce?” With the guidance from their science and math teachers, they develop a real understanding of measurement and dimension while gaining skills in data collection and learning to make informed decisions.