In this unit, two undergraduate students, Zoe and Gisele, work together on a sequence of tasks, called the N-gon Tasks, aimed at the development of their reasoning about two quantities varying simultaneously (i.e., their covariational reasoning). In these tasks, an n-sided polygon is given, and students are told that a labeled point is traveling around the perimeter of the shape. They are then tasked to create a graph of the height of the point as a function of the distance traveled by the point. Through the development of their covariational reasoning, Zoe and Gisele conclude their work with the construction of the sine function.
Lesson 1: Reasoning in a Linear Context
Zoe and Gisele explore the process of creating a graph for covarying quantities grounded in an elevator context. During this lesson, they explore the relationship between height and distance traveled, wrestle with the units in this context, and generalize their graph.
Lesson 2: Covariational Reasoning on the Square Task
In this lesson, Zoe and Gisele begin working on the N-gon Task sequence. Specifically, they explore how a point’s height varies as a function of the distance traveled for a point moving counterclockwise around a square. During this lesson, they manage changes to the varying quantities, discuss starting assumptions, and create a method for coordinating segments of their graph with segments of the shape.
Lesson 3: Covariational Reasoning on the Hexagon and Octagon Tasks
Continuing their work on the N-gon Task sequence, Zoe and Gisele explore how their graphs change as the shape increases in its number of sides. Through this work, they hone their reasoning for simultaneously varying quantities. This includes a deep dive into how to describe the change in rates of change over consecutive sides of a polygon.
Lesson 4: Culmination of the N-Gon Task
Zoe and Gisele work to describe what happens to a polygon, and its related graph, when the number of sides goes to infinity.
Lesson 5: Defining a Radian
Having constructed a graph for the final task in the N-gon Task, Zoe and Gisele explore an important, and familiar, unit that will be used to hone their graph, the radian. Importantly, they work to construct the definition of a radian with quantitative meaning.
Lesson 6: Constructing the Sine Function
In the final lesson, Zoe and Gisele work to combine their insights from the N-gon Task and their definition of a radian. Through this work, they are able to identify inflection points on their graph and improve their description of their graph. This lesson concludes with a big reveal… the graph they have constructed and defined important points on is the graph of the sine function!