Function Composition:

Logic of the Lesson

The following example is of a lesson logic for teaching function composition and introducing function inverse. This lesson logic provides an outline for developing the lesson's main ideas. It does not pay attention to time, meaning that the "lesson" may transcend several class periods. It does not give the level of detail that a lesson plan gives, meaning it might not say how you will organize the classroom, how you will transition from one activity to another, etc. Instead, it focuses on the ideas you will develop, the way you develop them, and why you take the approach you take.

The following lesson logic unfolds the idea of function composition:

1)      Function composition exists in situations where you have chains of processes. By performing the actions of the first process, followed by the actions of the second, two variables are related, the input of the first process and the output of the second.  Hence the need to compose two functions often emerges in situations where you want to relate two variables but have no direct means of characterizing how they are related.

Things students must understand to complete the lesson:

1)                A function is a generalized process that accepts an argument and produces an output.

2)                Functions can be put in sequence, so that one function’s output can be another function’s input.

3)                Functions in sequence make a composite function. That is, the initial input and the final output are related by the chain of processes.

4)                Sometimes representing the chain of processes is sufficient. Sometimes one needs to represent the relationship between initial input and final output explicitly.

5)                Imagine quantities changing in a dynamically changing event.

6)                Understand how to use the finger tool to characterize how two quantities are changing in tandem.

The introductory context for this lesson logic is to coordinate the area of a growing circle with the time since the circle began to form.