## An Introduction to Calculus Thinking

All seniors take an introduction to calculus thinking at ICE. This one-semester course will provide a culmination for studentsâ€™ mathematical studies at ICE, and a bridge to possible future studies in college.

The course is structured around weekly math labs that investigate specific questions. For example, there is a lab about finding a relationship between the speed of a projectile and its acceleration. We will build on basic concepts in high school algebra and geometry (like slope and area) and look at how they apply to complex, non-linear systems. For example: How can you predict the speed of a car at any given moment in the future if the driver keeps pressing down harder and harder on the gas pedal? And how can you predict how far the car will go in a given amount of time? Ultimately, we will draw on the central concepts of calculus to answer these types of questions. The ideas of calculus will both tie together the big ideas of high school math, and push students to think about them in a more sophisticated way.

To meet the graduation requirement for math at I.C.E., students must present a roundtable, where each student provides the context for their investigation and presents their solution, demonstrating his or her conceptual understanding, procedural fluency, logical reasoning skills, and ability to solve problems using multiple solution-strategies. Students will incorporate technology into their presentation and use mathematical models and terminology appropriately. There is a question and answer period. Roundtable panelists include parents, 11th grade students, mathematicians/scientists in our community, as well as I.C.E. teachers.

Readings:
"What is Calculus About?" by W.W. Sawyer
"The Story of Zero" by Charles Seife
Media Used:
TI-84 Plus, Program Ranger, Geometer's Sketchpad, Microsoft Excel, PowerPoint
Significant Activities or Projects:

Students investigate the idea of limits and infinite sums through a cake-eating problem.

Students investigate and analyze speed by measuring time and distance of an actual moving vehicle and graph their results.

Students investigate the instantaneous speed of a quadratic function, such as a penny dropped from the George Washington Bridge.

Students investigate the instantaneous speeds of various functions by calculating the slope of a tangent line using Geometer's Sketchpad. Students also find patterns between the distance function and its corresponding velocity function (the derivative) and derive the power rule based on their conclusions.

Students prove the validity of the power rule for the derivative by creating and solving for an algebraic proof.

Students investigate the second derivative by dropping a basketball and measuring its velocity and acceleration, and analyzing the slopes of these functions.

Students derive the distance function for a given velocity function by using the anti-derivative or integral, and investigate the relationship between the area under a curve and the integral.

Sample PBATs:
Students present to a Math Roundtable in which one of the investigations above is presented by incorporating technology. Students create and solve their own problem based on their investigations.