American River Stream Stage Exploration - A Class Project in Calculus 1

This activity is a Calculus I data exploration class project that uses American River stream stage data (https://cdec.water.ca.gov/river/americanStages.html) to help students develop a conceptual and graphical understanding of derivatives and rates of change. The lesson connects core calculus concepts to a real-world environmental context by asking students to investigate how the stage, or water level, of the American River changes over time. Students work with authentic stage data presented in graphs. They begin by examining the graph of American River stream stage as a function of time and describing how the water level changes. Stream stage is an important concept for understanding how much water is flowing through a stream at a given time. “Stage” refers to the height of water above a chosen reference point in the river and is commonly measured in feet. Students use the data to calculate and interpret average rates of change over selected time intervals and use these calculations to estimate the instantaneous rate of change of the river stage at specific times. This provides a meaningful context for introducing or reinforcing the derivative as an instantaneous rate of change. The project also guides students in identifying time intervals during which the stream stage is increasing or decreasing. Students connect these intervals to the sign of the derivative, recognizing that a positive derivative corresponds to an increasing stream stage and a negative derivative corresponds to a decreasing stream stage. They also investigate times when the rate of change is zero or close to zero and relate these points to features of the original stream stage graph, such as local maxima and minima. An important component of the activity is the connection between the original stream stage graph and its derivative graph. Students analyze the shape and behavior of the stream stage graph and use that information to sketch, construct, or interpret a graph representing the rate of change. They compare the two graphs to determine how increasing, decreasing, and relatively stable river stages are reflected in derivative values. Throughout the class project, students are encouraged to interpret their mathematical results in context rather than viewing the derivative only as a computational procedure. For example, students explain what a particular derivative value means in terms of how quickly the American River’s stage is rising or falling at a given time.